computer-aided design for rapid tooling: methods for mold design

Computer-Aided Design for Rapid Tooling: Methods for MoldDesign and Design-for-Manufacture

A Thesis

Presented to

the Academic Faculty

By

Yong Chen

In Partial Fulfillment

of the Requirements for the Degree of

Doctor of Philosophy in Mechanical Engineering

Georgia Institute of Technology

August 2001

Copyright © 2001 by Yong Chen

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Computer-Aided Design for Rapid Tooling: Methods for MoldDesign and Design-for-Manufacture

Approved:

David W. Rosen, ChairAssociate Professor, MechanicalEngineering

Janet AllenSenior Research Scientist, MechanicalEngineering

Charles EastmanProfessor, Architecture and ComputerScience

Farrokh MistreeProfessor, Mechanical Engineering

John MuzzyProfessor, Chemical Engineering

Thomas StarrProfessor, Chemical Engineering

Date Approved

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ACKNOWLEDGMENTS

As I am putting the finishing touches on this document that has recorded my work

for the past three years, I still clearly remember the feeling when I first read the

dissertation of Dr. Stewart Coulter in April 1998. I was anxious and tried to understand

what was required for a Ph.D. in the United States. Although getting lost at that time, I

was shocked by the work and the writing skills shown in the dissertation. To write a

dissertation like that seemed only a dream so far away. Now I have completed my

doctoral research and finished my dissertation. My years here in Atlanta have been one of

the greatest personal growth in my life. These accomplishments, however, would not

have been possible without the help of several people listed as follows. I am grateful for

the tremendous roles they have played in my life.

First sincere appreciation is expressed to my advisor Dr. David W. Rosen for his

continuous encouragement, guidance, and patience throughout my graduate studies. He

has provided generous support and has been the primary motivation for this research. His

insights have guided me to a level of understanding higher than what I thought possible.

I would like to thank the other committee members for their comments and

suggestions. In particular, I owe special thanks to Dr. Farrokh Mistree and Dr. Janet

Allen, who opened the world of decision-based design and its usage in several areas for

me. Their continued guidance and inspiration have also helped me grow both

intellectually and personally. I also owe special thanks to Professor Charles Eastman,

who guided me in the field of solid modeling. The idea of the reverse glue operation was

first thought out in my taking a course given by Prof. Eastman. I am profoundly grateful

to Dr. John Muzzy and Dr. Thomas Starr for sharing their insights on polymer injection

molding and powder injection molding.

I feel very lucky to be able to work in two distinguished laboratories, System

Realization Laboratory (SRL) and Rapid Prototyping and Manufacturing Institute

(RPMI), at the same time in the past three years. Several colleagues in SRL gave me

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great support to this research. Special thanks is given to Shiva Sambu who worked with

me together on developing the design for Rapid Tooling system and performing the case

studies of a robot arm and camera roller (Chapter 6, 7, 8). I would like to thank Yao Lin

for helping me in understanding Design of Experiments and the usage of statistic

software. Thanks also to Brian Davis for teaching me how to use the Coordinate

Measuring Machine. I cherish the support and compassion given freely by other close

colleagues in SRL – Angran Xiao, Hongqing Wang, Zahed Siddique, Sunji Jangha,

Carolyn Conner Seepersad, Marco Fernandez, Benay Sager, Scott Cowan, Scott Duncan,

and Ruhul Kulkarni. Thanks for making my study in Georgia Tech enjoyable.

I would like to thank my colleagues in RPMI for their generous help in this

research: to Reggie Ponder, for teaching me how to use SLA machines; to Giorgos

Hatzilias, for teaching me how to use Morgan Press Injection molding machine; to Giang

Pham, for teaching me how to use Sumitomo Injection molding machine; to Kent

Dawson, for sharing his experiment data and his knowledge on the properties of injection

molded parts; to Young-Bin Park, for teaching me how to use Instron universal testing

machine.

I am also a student member in Graphics, Visualization and Usability (GVU)

Center at Georgia Tech. I am very grateful to the two courses given by Dr. Jarek

Rossignac, who helped me to understand the beauty of computational geometry. The

region generation algorithm in this research was first developed as a course project in one

of these courses. The geometric modeling course given by prof. Eastman inspired me to

further develop some ideas in the field of automatic mold design. Special thanks goes to

Guoquan Zhou, a graduate student of Prof. Eastman, for sharing his code of a simple

solid modeling system.

The financial support from the National Science Foundation (NSF DMI- 96-

18039) is greatly appreciated. It makes this work possible.

Three and a half years of graduate study was a long way to go. I consider myself

blessed to know my wife, Mrs. Ying Wu. Her love and support make these years really

enjoyable. I offer deepest thanks to her for her understanding and sacrifice throughout

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this process. I am also in debt to my parents and parents-in-law for what they have done

for me. Without their continued support and belief in me, I would not have gotten to this

point. This dissertation stands as a testament to their success. I am also thankful to Sai

Zeng, Lei Wu, Cheng Zhang, Lunyu Ma, Qi Zhu, Yong Huang, and Liang Zhu, who have

given me the great gift of friendship.

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TABLE OF CONTENTS

ACKNOWLEDGMENTS iii

TABLE OF CONTENTS vi

LIST OF TABLES xi

LIST OF FIGURES xiv

NOMENCLATURE xix

SUMMARY xxi

CHAPTER 1FUNCTIONAL PROTOTYPE USING RAPID TOOLING 11.1 PRIORS – FUNCTIONAL PROTOTYPE AND RAPID TOOLING 2

1.1.1 Product Realization and Prototype 21.1.2 Rapid Prototyping 31.1.3 Injection Molding and Rapid Tooling 61.1.4 Rapid Manufacturing 8

1.2 DESIGN FOR RAPID TOOLING AND RESEARCH OPPORTUNITIES 91.2.1 Current Usage of Rapid Tooling and Related Problems 91.2.2 Motivating project – Rapid Tooling TestBed 111.2.3 Challenges of RTTB and My Research Approaches 121.2.4 Research Opportunities in Mold Design and Design-for-Manufacture 16

1.3 RESEARCH FOCUS IN THE DISSERTATION 231.3.1 The Principal Goal, Research Questions and Hypotheses in the Dissertation 241.3.2 Validation Philosophy and Strategy 261.3.3 Contributions from the Research 29

1.4 OVERVIEW OF DISSERTATION 30

CHAPTER 2 33A LITERATURE REVIEW: MOLD DESIGN AND DESIGN-FOR-MANUFACTURE2.1 TOPICS IN CHAPTER 342.2 MOLD DESIGN METHODS AND ALGORITHMS 35

2.2.1 Parting Direction 352.2.2 Parting Line 372.2.3 Parting Surface 382.2.4 External and Internal Undercut Detection 392.2.5 Synthesis Approaches of Basic Elements 392.2.6 Intersection of V-Map 412.2.7 Detection of Non-Drafted Surfaces 42

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2.3 MOLD CONSTRUCTION METHODS AND TOOLS 432.3.1 Approach Based on Extending Parting Lines 432.3.2 Approach Based on Sweeping 432.3.3 Industrial Approach 44

2.4 CAD REPRESENTATION 462.4.1 Representation of 3D Surfaces and Solids 492.4.2 Manipulation of Solid Models 512.4.3 High Level Representations 53

2.5 DESIGN FOR MANUFACTURE: STRATEGIES AND TECHNIQUES 562.6 DESIGN TECHNOLOGIES 62

2.6.1 Decision-Based Design 622.6.2 Compromise Decision Support Problem 622.6.3 Robust Concept Exploration Method (RCEM) 69

2.7 LITERATURE REVIEW SUMMARY 71

CHAPTER 3THE MULTI-PIECE MOLD DESIGN METHOD 723.1 OVERVIEW OF THE MULTI-PIECE MOLD DESIGN METHOD 733.2 PROBLEM FORMULATION FOR MULTI-PIECE MOLD DESIGN 75

3.2.1 Analysis of Existing Problem Formulations 753.2.2 Problem Formulations of Multi-Piece Mold Design 78

3.3 OVERVIEW OF THE MULTI-PIECE MOLD DESIGN PROCESS 793.4 BASIC ELEMENTS OF MPMDM 80

3.4.1 Demoldability of Mold Pieces 813.4.2 Analysis of the Basic Elements 873.4.3 The Generation Approach of Concave Regions 89

3.5 REGION COMBINATION OF MPMDM 943.5.1 Combining Criteria and Their Evaluation Approach 943.5.2 Verification of Draft Angle 983.5.3 Analysis of Region Combination Process and Related Representations 993.5.4 Region Combination Algorithm and Design Knowledge 103

3.6 MOLD PIECE CONSTRUCTION APPROACH BASED ON REVERSE GLUE 1073.6.1 Principle and Related Representations of the Approach 1083.6.2 Generation Approach of Glue Faces 1113.6.3 Reverse Glue Algorithm and Parting Surface 115

3.7 SUMMARY OF CHAPTER 3 118

CHAPTER 4RTMDS AND ITS USAGE FOR MOLD DESIGN 1224.1 OVERIEW OF RAPID TOOLING MOLD DESIGN SYSTEM 1234.2 SUPPORTING MODULES AND THEIR IMPLEMENTATIONS 124

4.2.1 ACIS Manipulation Module and Part Representation 1244.2.2 Problem PDLP and LINGO® 1284.2.3 GUI and Control Options 128

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4.3 THE IMPLEMENTATION AND LIMITATIONS OF MOLD DESIGNMODULES 1294.3.1 Region Generation 1304.3.2 Region Combination 1324.3.3 Mold Piece Construction 135

4.4 INTUITIVE EXAMPLE PARTS 1364.4.1 Test Example 1: A Box with a Rib 1374.4.2 Test Example 2: A Box with a Through Hole and Two Grooves 141

4.5 INDUSTRIAL CASES 1454.5.1 Industrial Example 1: A Housing 1464.5.2 Industrial Example 2: A Thin Wall Part 1504.5.3 Industrial Example 3: A Complex Housing 152

4.6 EVALUATION OF EXAMPLES AND CASES 1554.6.1 Region Generation Process 1554.6.2 Region Combination Process 1564.6.3 Mold Piece Construction Process 1574.6.4 The Whole Process of RTMDS 158


CHAPTER 5FORMULATING DESIGN REQUIREMENTS FOR RAPID TOOLING ASGEOMETRIC TAILORING PROBLEM 1625.1 PROPERTIES OF RAPID TOOLING 163

5.1.1 Mold Material Properties 1635.1.2 Mold Fabrication Properties 1655.1.3 Part Properties of the AIM Tooling 166

5.2 PRINCIPLES OF FUNCTIONAL TESTING AND GEOMETRIC TAILORING 1675.2.1 Principle of Functional Testing – Buckingham Π Theorem 1675.2.2 Similarity Methods 1685.2.3 Fundamentals of Geometric Tailoring 169

5.3 DESIGN DECISION TEMPLATE FOR MGT 1715.3.1 MGT Decision Template and its Methodology 1745.3.2 Formulation of the MGTDT 176

5.4 USAGE OF MGTDT 1785.4.1 Formulating Functional Properties in the MGTDT 1785.4.2 Formulating and Solving the MGT Problem 181

5.5 INITIAL CASE STUDIES 1815.5.1 Building Prototypes of a Tensile Bar 1825.5.2 Building Prototypes of a Rib Part 1885.5.3 Building Prototypes of a Ring Gear 198


CHAPTER 6A DECISION-BASED DESIGN FOR RAPID TOOLING SYSTEM 212

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6.1 OVERVIEW OF DESIGN FOR RAPID TOOLING SYSTEM 2136.2 PROCESSING PLANNING OF THE AIM TOOLING 216

6.2.1 SLA Process Planner 2166.2.2 Stereolithography Mold Life Predictor 2186.2.3 Rapid Tooling Cost Estimator 221

6.3 MPGT DECISION TEMPLATE AND MPGT PROBLEM FORMULATION 2226.3.1 MPGTDT 2236.3.2 MPGT Problem Formulation 226

6.4 SOLVING THE MPGT PROBLEM 2296.4.1 Solution Strategy 2296.4.2 Solution Process and Implementations 233

6.5 COMPARISON OF THE CURRENT USAGE AND DFRTS 2386.6 SUMMARY OF CHAPTER 6 242

CHAPTER 7FUNCTIONAL PROTOTYPES OF A ROBOT ARM 2467.1 A ROBOT ARM DESIGN – PROBLEM DESCRIPTION 2477.2 MOLD DESIGN WITH AID OF RTMDS 2487.3 GEOMETRIC TAILORING WITH AID OF DFRTS – MODELING 253

7.3.1 MPGT Decision Template for the Robot Arm 2537.3.2 Modeling Design Functions 2567.3.3 Modeling Fabrication Processes 2617.3.4 MPGT Problem Formulation 262

7.4 GEOMETRIC TAILORING WITH AID OF DFRTS – SOLVING 2657.4.1 Solving Discrete Variables 2657.4.2 Solving Other Variables 2697.4.3 Selecting A Solution 2747.4.4 Post-Solution Analysis 274

7.5 PHYSICAL VALIDATION 2787.5.1 Build Time Validation 2787.5.2 Accuracy Validation 2797.5.3 Surface Finish Validation 2807.5.4 Material Property Validation 2817.5.5 Geometry and Weight Validation 2817.5.6 Mold Life Validation 2837.5.7 Summary of Physical Validation 284

7.6 EVALUATION OF ROBOT ARM CASE – POST DFRTS 2857.7 SUMMARY OF CHAPTER 7 287

CHAPTER 8FUNCTIONAL PROTOTYPES OF A CAMERA ROLLER 2908.1 A CAMERA ROLLER DESIGN – PROBLEM DESCRIPTION 2918.2 MOLD DESIGN WITH AID OF RTMDS 2938.3 GEOMETRIC TAILORING WITH AID OF DFRTS – PROBLEM ANALYSIS 299

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8.4 GEOMETRIC TAILORING WITH AID OF DFRTS – MODELING 3038.4.1 Modeling Design Functions 3038.4.2 Modeling Fabrication Processes 3118.4.3 MPGT Problem Formulation 313

8.5 GEOMETRIC TAILORING WITH AID OF DFRTS – SOLVING 3178.5.1 Solving Discrete Variables 3178.5.2 Solving Continuous Variables 3218.5.3 Selecting A Solution 3238.5.4 Post-Solution Analysis 323

8.6 PHYSICAL VALIDATION 3268.7 EVALUATION OF CAMERA ROLLER CASE – POST DFRTS 3298.8 SUMMARY OF CHAPTER 8 331

CHAPTER 9ACHIEVEMENTS AND RECOMMENDATIONS 3349.1 ANSWERING THE RESEARCH QUESTIONS 335

9.1.1 Research Question Overview 3359.1.2 Answering Research Questions 337

9.2 ACHIEVEMENTS: REVIEW OF RESEARCH CONTRIBUTIONS 3409.3 CRITICAL ANALYSIS: LIMITATIONS OF THE RESEARCH 3439.4 FUTURE WORK 345

APPENDIX ARTMDS IMPLEMENTATION 348A.1 CODE FILES OF RTMDS 349A.2 CLASSES OF RTMDS 351A.3 IMPLEMENTATION ON ACIS 352A.4 IMPLEMENTATION ON LINGO 354

APPENDIX BCAMERA ROLLER CASE STUDY: COMPLETE SOLUTIONS OF THEMODIFIED MPGT PROBLMES 356B.1 SOLUTIONS OBTAINED FROM OPTDESX FOR SLICING SCHEMES

1~6 OF PO1 357B.2 SOLUTIONS OBTAINED FROM OPTDESX FOR SLICING SCHEMES



1~6 OF PO4 360

REFERENCES 361

VITA 375

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LIST OF TABLES

Table 1.1 – Relationship Between Hypotheses and Dissertation Sections. 26Table 1.2 - Validation of Hypotheses 1 and 2 addressed in different chapters. 28Table 4.1 – The Complexities of the Example Parts in Section 4.4 and 4.5. 136Table 4.2 – The Information for Test Example 1. 138Table 4.3 – The Information for Test Example 2. 141Table 4.4 – The Information for Industrial Example 1. 146Table 4.5 – The Information for Industrial Example 2. 150Table 4.6 – The Information for Industrial Example 3. 153Table 4.7 – Experimental Data of Region Generation Process. 155Table 4.8 – Experimental Data of Region Combination Process. 156Table 4.9 – Experimental Data of Mold Piece Construction Process. 157Table 4.10 – Experimental Data of Mold Design Process. 159Table 5.1– Tensile Properties Comparison for Atactic Polystyrene. 166Table 5.2– Flexural Properties Comparison for Atactic Polystyrene. 166Table 5.3 – Word Formulation of the MGTDT. 176Table 5.4 – Mathematical Formulation of the MGTDT. 177Table 5.5 – Experimental Plan For Testing the MGT. 182Table 5.6 – MGT Tensile Bar Problem Formulation. 184Table 5.7– Scenarios of Goal Weights and Related Results. 185Table 5.8– Material property validation results for polystyrene. 187Table 5.9 - MGT Rib Problem Formulation By the Designer. 190Table 5.10 – Design Factors and Their Ranges. 192Table 5.11 – Results of the Experiments for Rib Part. 192Table 5.12 – Complete MGT Rib Problem Formulation. 195Table 5.13– Scenarios of Goal Weights. 196Table 5.14– Rib Part MGT Results. 196Table 5.15 – Maximum Torque of The Speed Reducer. 199Table 5.16 - MGT Ring Gear Problem Formulation By the Designer. 201Table 5.17 – Design Factors and Their Ranges of Ring Gear. 203Table 5.18 – Results of the Experiments for Ring Gear. 203Table 5.19 – Complete MGT Ring Gear Problem Formulation. 205Table 5.20– Scenarios of Goal Weights. 206Table 5.21– Ring Gear MGT Results. 207Table 6.1 – SLA Process Planning Word Formulation [Sambu, 2001 #967]. 216Table 6.2 – SLA Process Planning Mathematical Formulation. 217Table 6.3 – IJM Process Planning Word Formulation [Sambu, 2001 #967]. 218Table 6.4 – Factors Selected for Mold Life Prediction [Sambu, 2001 #967]. 219Table 6.5 – IJM Process Planning Mathematical Formulation. 220

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Table 6.6 – Cost and Time Estimates for Different Steps in Rapid Tooling Process(Sambu, 2001). 221

Table 6.7– Word Formulation of the MPGTDT. 223Table 6.8 - Mathematical Formulation of the MPGTDT. 224Table 6.9 – MPGT Problem Word Formulation. 226Table 6.10 – MPGT Problem Mathematical Formulation. 227Table 7.1 – The Information for Robot Arm. 248Table 7.2 – MPGT Robot Arm Problem Formulation by the Designer. 253Table 7.3 – Design Factors and Their Ranges of Robot Arm. 256Table 7.4 - List of Experiments for Response Surface Generation. 256Table 7.5 – Results of the Experiments for Robot Arm. 258Table 7.6 - Results of Validation Experiments for the Robot Arm. 260Table 7.7 – MPGT Robot Arm Problem Formulation. 262Table 7.8 – Modified MPGT Robot Arm Problem Formulation. 269Table 7.9 - Starting Points Investigated for Each Slicing Scheme. 272Table 7.10 – Solutions of System Variables for Different Slicing Schemes. 272Table 7.11 – Solutions of Goals for Different Slicing Schemes. 273Table 7.12 – Solutions of the MPGT Problem for the Robot Arm. 274Table 7.13 - Target Modification Experiments. 276Table 7.14 – Results of Target Modification Experiments. 277Table 7.15 - Individual Functional Target Modification Experiments. 277Table 7.16 – Results of Individual Functional Target Modification Experiments. 277Table 7.17 - SLA Build Time for the Mold Pieces of Robot Arms. 278Table 7.18 – Accuracy of the Robot Arms. 279Table 7.19 - Surface Finish Experiment Results on Prototype Parts. 280Table 7.20 - Material Property Validation Results for Polystyrene. 281Table 7.21 - Injection-molded part dimensions and weight. 282Table 7.22 - Injection Molding Parameters Used for the Robot Arm. 283Table 7.23 - Summary of Physical Validation Results for MPGT of Robot Arm. 285Table 7.24 - Solutions of Sequential and Concurrent Solution Process. 286Table 7.25 - Goals Achievements for Sequential and Concurrent Solution

Processes. 286Table 8.1 – The Information for Robot Arm. 294Table 8.2 – Estimated Mold Life and Required Number of Each Mold Piece. 300Table 8.3 – Design Factors and Their Ranges of Camera Roller. 304Table 8.4 - List of Experiments for Response Surface Generation. 304Table 8.5 - Experiments to Study the Effect of Mesh Size. 305Table 8.6 – Results of the Z-Rotation Experiments for Camera Roller. 306Table 8.7 - Results of Validation Experiments of Z-rotation. 308Table 8.8 – Results of the Volume Experiments for Camera Roller. 309Table 8.9 - Results of Validation Experiments of Volume. 311Table 8.10 – MPGT Camera Roller Problem Formulation. 314Table 8.11 - Dimensions of the Different Mold Features. 317Table 8.12 - Surface Finish and Build Time Achievements for Slicing Schemes. 320

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Table 8.13 - Starting Points Investigated for Each Slicing Scheme. 321Table 8.14 - Solutions Obtained for Slicing Scheme 1 for Different Starting

Points. 322Table 8.15 - Objective Function Values Obtained for Different Slicing Schemes. 323Table 8.16 – Solutions of the MPGT Problem for Camera Roller. 323Table 8.17 - Values of Goals and Intermediate Responses for the Solution. 323Table 8.18 - Target Modification Experiments. 325Table 8.19 – Results of Target Modification Experiments. 325Table 8.20 – Percent Change of the Goals for Target Modification Experiments. 325Table 8.21 - Values of System Variables for Target Modification Experiments. 326Table 8.22 - Injection Molding Parameters Used for the Camera Roller. 328Table 8.23 - Solutions of Sequential and Concurrent Solution Process. 330Table 8.24 - Goals Achievements for Sequential and Concurrent Solution

Processes. 330

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LIST OF FIGURES

Figure 1.1 - The Cost of an Design Change Related to Development Phase 2Figure 1.2 - A Schematic Drawing of a SLA 4Figure 1.3 - A CAD Model and a SL Model of a Robot Arm 5Figure 1.4 - A Stereolithography Model for Assembly Studies and Tests with

Regard to Collision Behavior 5Figure 1.5 – Direct AIM Tooling Process 7Figure 1.6 – Processes of Using Direct AIM Tooling to Build Prototypes 9Figure 1.7 – RTTB System and My Focus 12Figure 1.8 – Different Geometries of Some Industrial Parts 13Figure 1.9 – Design for Rapid Tooling 15Figure 1.10 – An Illustrative Mold for Injection Molding 16Figure 1.11 – Mold Design Process and My Considerations 17Figure 1.12 – Standardized Master Frame Used in Morgan Press at RPMI 18Figure 1.13 – A Multi-piece Molds for a Part Given by Protoform GmbH 19Figure 1.14 – Relations of DFRT, DFM and Geometric Tailoring 23Figure 1.15 -Validation Square (Pedersen, 1999) 27Figure 1.16 - Overview of Dissertation Chapters 30Figure 1.17 - Pictorial Overview of the Dissertation 32Figure 2.1 - Literature Review Topics and Relationship with Hypotheses. 34Figure 2.2 – Two Examples of V-map. 41Figure 2.3 – A Part with Mold Pieces Generated by Two Approaches. 43Figure 2.4 – Mold Construction Process of Unigraphics/MoldWizard for an

Industrial Part. 45Figure 2.5 – An Example Mold Design Given by Magics RP. 45Figure 2.6 – An Example Mold Design Given by IMOLD. 46Figure 2.7 – An Example Mold Design Given by Moldplus. 46Figure 2.8 – An Example Mold Design Given by QuickSplit. 47Figure 2.9 – An Example Mold Design Given by I-DEAS. 47Figure 2.10 – Coedges of a Part. 50Figure 2.11 – The Distinction between Geometric and Topological Information

within the ACIS Solid Modeler (Spatial Technology, 2000). 51Figure 2.12 – An Example of Gluing Operation. 52Figure 2.13 – Boolean Operations. 53Figure 2.14 – An Example of Behavioral Modeling (PTC). 55Figure 2.15 - Word formulation of a Compromise DSP problem (Mistree,

et al., 1993). 63Figure 2.16 - Mathematical formulation of a Compromise DSP problem

(Mistree, et al., 1993). 64

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Figure 2.17 - The Compromise DSP with Modifications based on the LinearPhysical Programming Model (Hernandez and Mistree, 2001). 66

Figure 2.18 - Class Function Regions for a Generic ith Objective (Hernandezand Mistree, 2001). 68

Figure 2.19 - The Robust Concept Exploration Method (RCEM) (Chen, 1995). 69Figure 2.20 – Summary of Chapter 2 and Preview of Chapter 3. 71Figure 3.1 – The Elements of the Multi-Piece Mold Design Method. 73Figure 3.2 – Relationship between the Elements of MPMDM and Dissertation

Hypotheses. 74Figure 3.3 – An Approach Based on V-map. 76Figure 3.4 – Examples for the Problem Formulation. 76Figure 3.5 – Mold Pieces for a Pocket with Empty V-Map. 77Figure 3.6 – Steps of the Mold Design Process. 80Figure 3.7 – Demoldability of Mold Pieces. 81Figure 3.8 – Demoldability of Neighboring Mold Pieces. 83Figure 3.9 – Dividing of Concave Regions. 85Figure 3.10 – Splitting of a Combined Region. 86Figure 3.11 – Concave Regions and Combined Regions. 87Figure 3.12 – Combined Region Example. 88Figure 3.13 – Edge Classification. 89Figure 3.14 – Different Splitting Faces and Orders. 91Figure 3.15 – PD Evaluation Approach. 96Figure 3.16 – A Region with Different Mesh Size. 98Figure 3.17 – Different Draft Angles. 98Figure 3.18 – Relationships Between Regions and Minimum Draft Angle. 99Figure 3.19 – Edges and Faces of a Region in a CXF Combining Step. 100Figure 3.20 – Edges and Faces of Two CRs in a Combining Step. 102Figure 3.21 – Different Combining Results of a Part. 104Figure 3.22– Face Dividing for A Region of Cavity. 105Figure 3.23 – Two Example Parts for Combining Order. 107Figure 3.24 – Regions and Mold Pieces of a Part. 107Figure 3.25 – An Example of a Boolean Mold Base and Mold Pieces. 109Figure 3.26 – Glue Faces of an Example Part. 111Figure 3.27 – Generation of Inner Glue Faces. 113Figure 3.28 – Process of Face Generation. 114Figure 3.29 – An Example for Coedge Splitting. 116Figure 3.30 – An Example of Different Parting Surface. 118Figure 3.31 – Partial Theoretical Validation. 119Figure 3.32 - Summary of Chapter 3 and Preview of Chapter 4 121Figure 4.1 – The Organization and Data Flow of the RTMDS. 123Figure 4.2 – The Process to Generate an Input File for RTMDS. 125Figure 4.3 – Focus of the ACIS Entity Classes in RTMDS. 126Figure 4.4 – The Relationship of Topology Entity Classes in ACIS. 127Figure 4.5 – Screen Capture of the RTMDS Interface. 128

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Figure 4.6 – Setting Tolerance Values in RTMDS. 129Figure 4.7 – A Simple Example of Face Splitting. 131Figure 4.8 – Two Examples of Face Splitting. 131Figure 4.9 – A Complicated Example of Face Splitting. 132Figure 4.10 – The Combining Order of Faces in a Same Plane. 133Figure 4.11 – An Example for Setting Tolerance Value. 134Figure 4.12 – Mold Pieces for a Pocket with Empty V-Map. 136Figure 4.13 – A Test Example of a Rib Part. 138Figure 4.14 – Graphical Results of a Mold Design for Test Example 1. 139Figure 4.15 – Graphical Results of Another Mold Design for Test Example 1. 140Figure 4.16 – A Test Example of a Box with a Through Hole and Two Grooves. 141Figure 4.17 – Illustration of the Running Results of Test Example 2. 142Figure 4.18 – Graphical Results of Mold Design for Test Example 2. (Step 1~7) 143Figure 4.19 – Graphical Results of Mold Design for Test Example 2 (Step 8). 144Figure 4.20 – A Housing for A Phone Adapter. 146Figure 4.21 – Graphical Results of a Mold Design for Industrial Example 1. 148Figure 4.22 – Physical Validation of the Mold Design for Industrial Example 1. 149Figure 4.23 – A Thin Wall Part for A Phone Adapter. 150Figure 4.24 – Graphical Results of a Mold Design for Industrial Example 2. 151Figure 4.25 – Generated Mold Pieces for Industrial Example 2. 152Figure 4.26 – A Complex Housing. 153Figure 4.27 – Region Combination (2 regions). 153Figure 4.28 – Mold Pieces for Industrial Part 3. 154Figure 4.29 – Relations of Region Generation Time with Face/Edge Number. 156Figure 4.30 – Relations of Region Combination Time with Face/Edge Number. 157Figure 4.31 – Relations of Mold Piece Construction Time with Face/Edge

Number. 158Figure 4.32 – Relations of Mold Design Running Time with Face/Edge Number. 158Figure 4.33 – Empirical Structural and Performance Validation for Hypothesis 1. 160Figure 4.34 – Summary of Chapter 4 and Preview of Chapter 5. 161Figure 5.1– Maximum Tensile and Shear Strength of SL5170 Versus Temp. 164Figure 5.2 – Stair Stepping Effect of Layer Manufacturing and Reason. 165Figure 5.3 – Fundamental Terms and Concept of the ESM [Cho, 1999 #964]. 169Figure 5.4 – Steps of a Scenario based on the MGT Decision Template. 172Figure 5.5 – The Decision Template for the MGT and Design Freedom. 174Figure 5.6 - A Cantilever Beam. 179Figure 5.7 – The Illustration of a Tensile Bar Example. 183Figure 5.8 – Photo of SLA Tools and Prototype Tensile Bars. 186Figure 5.9 – Photo of Prototype Tensile Bars Used for Tensile test. 187Figure 5.10 – Quality Distribution and Geometric Tailoring. 188Figure 5.11 – The Illustration of a Rib Example. 189Figure 5.12 – Geometry Variables of the Rib by Adding Draft Angle. 191Figure 5.13 – Two Screen Dumps of Experiment Results for Rib Part. 193Figure 5.14 – Graphical Relations of The Maximum Deflection and Variables. 194

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Figure 5.15 – Parts Produced by SLA Molds. 197Figure 5.16 – SLA Molds and Pullout Failure. 198Figure 5.17 – The Ring Gear and the Speed Reducer of a Cordless Drill. 198Figure 5.18 – Some Terminology Describing a Spur Gear (Shigley and Mischke,

1989). 199Figure 5.19 – Fatigue of Ring Gear. 200Figure 5.20 – Two Screen Dumps of Experiment Results for Ring Gear. 204Figure 5.21 – Graphical Relation of the Maximum Stress with PD and W. 205Figure 5.22 – A Photo of the Prototype Ring Gears. 208Figure 5.23 – A SLA Mold Piece in a Standard Mold Plate. 208Figure 5.24– Summary of Hypothesis Validation. 210Figure 5.25 – Summary of Chapter 5 and Preview of Chapter 6. 211Figure 6.1 – Relations of DFRT, Geometric Tailoring and DFM. 213Figure 6.2 – Infrastructure of the DFRTS and Related Sections. 214Figure 6.3 – Research Scope of the DFRTS. 215Figure 6.4 – Screen Capture of the SLA Process Planner. 218Figure 6.5 – Screen Capture of SL Mold Life Predictor. 221Figure 6.6 – Relations of Designer’s Requirements and MGTDT/MPGTDT. 222Figure 6.7 – Relations of the MPGTDT and MPGT Problem. 225Figure 6.8 – Possible Values of PO and LTi for a Example Part. 230Figure 6.9 – The Solution Strategy for the MPGT Problem. 231Figure 6.10 – Two Sub-problems of the MPGT. 232Figure 6.11 – The Solution Process of the DFRTS. 234Figure 6.12 – Surface Finish Test Piece. 235Figure 6.13 – The Current Usage of RT and Related Decision Order of Variables. 239Figure 6.14 – The DFRTS and Related Decision Order of Variables. 240Figure 6.15 – A Two Variable Design Space [Karandikar, 1991 #269]. 241Figure 6.16 – Summary of Chapter 6 and Preview of Chapter 7. 245Figure 7.1 – A Robot Arm Design. 247Figure 7.2 – Generated Regions for the Robot Arm. 249Figure 7.3 – Graphical Results of a Mold Design for the Robot Arm. 250Figure 7.4 – Graphical Results of Another Mold Design for the Robot Arm. 251Figure 7.5 – A Completer Mold Design for the Robot Arm. 252Figure 7.6 – Physical Validation of the Mold Design for the Robot Arm. 252Figure 7.7 – Process-related Goals of Robot Arm Design. 253Figure 7.8 – Example ANSYS Output of Analysis for the Robot Arm. 257Figure 7.9 – Graphical Relations of Responses and Variables. 259Figure 7.10 – Validation Experiment Design for the Response Surfaces. 260Figure 7.11 – Some Process Planning Goals. 262Figure 7.12 – Two Considered Mold Designs for the Robot Arm. 266Figure 7.13 – Four Considered Part Orientations for the Mold Designs. 267Figure 7.14 – Eight Considered Slicing Schemes for the Part Orientations. 268Figure 7.15 – Inter-Relationship Diagram of Goals and Variables. 275Figure 7.16– Objective Function vs. Iteration No. for Modified MPGT Problem. 276

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Figure 7.17 – Mold Piece Layout on SLA 3500 Platform [Sambu, 2001 #967]. 279Figure 7.18 – Pieces of Robot Arm Used in Surface Finish Experiments. 280Figure 7.19 - Gap Between the Parting Surfaces due to Mold Warpage. 282Figure 7.20 - Short (left) and Complete (right) Shots obtained from IJM. 284Figure 7.21 - Chipped Mold and Parts Obtained from the Mold. 284Figure 7.22 – The Ranges of Goals for the Robot Arm Problem. 287Figure 7.23 – Empirical Structural and Performance Validation for H1 and H2. 288Figure 7.24 – Preview of Chapter 8. 289Figure 8.1 – A Production Camera Roller with a Camera. 291Figure 8.2 - Loading Conditions on the Camera Roller. 292Figure 8.3 - Surface Finish and Tolerance Requirements. 292Figure 8.4 – Dividing Cylindrical Faces for the RTMDS. 293Figure 8.5 – Graphical Results of Region Combination Process. 295Figure 8.6 – Changing Region Number of Selected Faces in the RTMDS. 296Figure 8.7 – Graphical Results of a Mold Design for the Camera Roller –Phase 1. 297Figure 8.8 – Graphical Results of a Mold Design for the Camera Roller – Phase 2. 298Figure 8.9 – A Completer Mold Design for the Camera Roller. 299Figure 8.10 – Physical Validation of the Mold Design for the Camera Roller. 299Figure 8.11 - Mold Life of Different Features in Mold Pieces. 300Figure 8.12 – Problem Identified in the Preliminary Experiment. 301Figure 8.13 - Different Features in Camera Roller. 302Figure 8.14 - Modified Camera Roller Part. 302Figure 8.15 - Geometry Variables in the Modified Camera Roller. 303Figure 8.16 - ANSYS Output of Z-Rotation for Camera Roller (Exp.1: NC=

NR = 3). 305Figure 8.17 – Graphical Relations of z-rotation and Variables. 308Figure 8.18 – Graphical Relations of Volume and Variables. 310Figure 8.19 – Process Planning Goals for the Mold Pieces. 313Figure 8.20 – Four Part Orientations for the Mold Design [Sambu, 2001 #967]. 319Figure 8.21 - Promising Slicing Schemes for PO1 [Sambu, 2001 #967]. 320Figure 8.22 - Objective Function vs. Iteration Number for MPGT Problem. 322Figure 8.23 – Mold Design for the Tailored Camera Roller Part. 327Figure 8.24 – Physical Mold Pieces and Injection Molded Camera Roller. 328Figure 8.25 – Comparison the Camera Roller before and after Geometric

Tailoring. 329Figure 8.26 – The Dimensions of the Third Mold Piece. 330Figure 8.27 – Empirical Structural and Performance Validation for H1 and H2. 332Figure 8.28 – Preview of Chapter 9. 333Figure 9.1 – Context for Research Questions and Hypotheses. 335Figure 9.2 – Validity Square. 337Figure 9.3 – An Ideal System Working Between the Designer and Manufacturer. 347

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NOMENCLATURE

ACES Accurate Clear Epoxy SolidACIS Three-dimensional Geometric ModelerAIM Tooling ACES Injection Molding ToolingANOVA Analysis Of VarianceANSYS Finite Element Analysis SoftwareB-rep Boundary RepresentationCAD Computer Aided DesignCCD Central Composite DesignCE Concurrent EngineeringCMM Coordinate Measuring MachineCPL Closed Parting LoopCR Combined RegionCSG Constructive Solid GeometryCVR Concave RegionCXF Convex Face

di+, di- positive and negative deviation variables in compromise DSP

DBD Decision-Based DesignDFM Design for ManufactureDFRTS Design for Rapid Tooling SystemDOE Design of ExperimentscDSP Compromise Decision Support ProblemDSP Decision Support ProblemFEA Finite Element AnalysisGF Glue FaceGUI Graphical User InterfaceIJM Injection MoldingLINGO Engineering Optimization SoftwareLT Layer Thickness in SLA processLP Linear ProgrammingLPP Linear Physical ProgrammingMB Mold BaseMCD Mold Configuration DesignMD Mold DesignMFC Microsoft Foundation ClassesMGT Material Geometric TailoringMGTDT Material Geometric Tailoring Decision TemplateML Mold Life

xx

MPC Mold Piece ConstructionMPD Mold Parting DirectionMPGT Material-Process Geometric TailoringMPGTDT Material-Process Geometric Tailoring Decision TemplateMPMDM Multi-Piece Mold Design MethodNF Neighboring FaceOptdesX Engineering Optimization SoftwarePC Personal ComputerPD Parting DirectionPE Parting EdgePIM Powder Injection MoldingPL Parting LinePO Part Orientation in SLA ProcessPS Parting SurfaceR2, R2

adj The ratio of the model sum of squares to the total sum of squares, andthat ratio adjusted for the number of parameters in the model

RCEM Robust Concept Exploration MethodResponse Performance parameter of the system, i.e., a system constraint or goalRM Rapid ManufacturingRMCD Region-based Mold Configuration DesignRSE Response Surface EquationRSM Response Surface MethodologyRTMDS Rapid Tooling Mold Design SystemRTTB Rapid Tooling TestBedRP Rapid PrototypingSLA Stereolithography ApparatusV-map Visibility Map

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SUMMARY

Physical models and prototypes are fundamental to superior product development

and production. Rapid Tooling techniques have the potential to dramatically reduce the

time and cost in producing limited quantities of functional prototypes in final material.

However, the current usage of Rapid Tooling has two problems: (1) mold design for parts

with a wide variety of geometries may take a long time; (2) design iterations between

designers and manufacturers may take a long time before different design requirements

are achieved in the prototypes. They overshadow the time and cost benefits of Rapid

Tooling. In this dissertation, these two problems are addressed by developing a Mold

Design Method and a Design-for-Manufacturing Method respectively.

Based on Computation Geometry, a Multi-piece Mold Design Method is

developed to automate several important mold design steps, including determining

parting directions, parting lines, and parting surfaces, and constructing mold pieces for

multi-piece molds. The method has three steps. First concave regions and convex faces

are generated from a given part. Second the generated concave regions and convex faces

are combined into several regions. Finally mold pieces are constructed for the combined

regions based on a reverse glue operation. The method is employed to develop a Rapid

Tooling Mold Design System, which has been used to design molds to fabricate

prototype parts with widely varying complexities. Two test parts and five industrial parts

are presented in the dissertation to illustrate the usage of the mold design system. The

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running time of the system is tested for each part. The mold design results are also

verified by physical experiments.

Based on Decision-Based Design, a Design for Rapid Tooling System is

developed to aid manufacturers to tailor a submitted part efficiently and effectively. A

basic idea of this work is that for Rapid Tooling the burden of design-for-manufacture

can be transferred to the manufacturer by geometric tailoring decision templates. The

designer’s requirements on functional prototypes are formulated in the decision

templates, which are in the compromise DSP format. Three testing examples illustrate

that the design freedom given by the designer to the manufacturer is important for

reducing the iterations between the designer and manufacturer. By synthesizing

decisions of part design, rapid prototyping process, and injection molding process

variables, a solution strategy and a three-stage solution process are proposed for the

system. Compared with the current usage of Rapid Tooling, better decision order of

design variables is used in the design for Rapid Tooling system for the geometric

tailoring problem. Two case studies, a robot arm and a camera roller, are used to test the

system. Physical prototypes of the tailored part designs are produced for the validation of

the DFM method.

Chapter 1 – Functional Prototype Using Rapid Tooling

1

CHAPTER 1

FUNCTIONAL PROTOTYPE USING RAPID TOOLING

The product realization process, driven by market factors, is changing dramatically.Building better physical prototypes in shorter period of time becomes more important inproduct development and production. One way to achieve this goal for injection moldedproducts is through Rapid Tooling. In this chapter we will give a brief introduction toRapid Tooling and design for Rapid Tooling. The project by which this research ismotivated is presented to highlight some of the challenging issues involved in thedevelopment of a Rapid Tooling Testbed. Research opportunities and foundations forthis research will also be presented followed by the research questions, goals andhypotheses. In the last Section an overview of the entire dissertation is given.


2

1.1 PRIORS – FUNCTIONAL PROTOTYPE AND RAPID TOOLING

The principal objective in this dissertation is to develop methods to facilitate theusage of Rapid Tooling for a part design. However, before the current usage of RapidTooling and research opportunities are presented in Section 1.2, several important notionsare introduced in this section to establish a context for the reader. These notions includethe importance of functional prototype (Section 1.1.1), Rapid Prototyping (Section 1.1.2),Rapid Tooling (Section 1.1.3) and Rapid Manufacturing (Section 1.1.4). The focus ofthis research, which includes research questions and related hypotheses, is presented inSection 1.3. Finally the overview of the entire dissertation is presented in Section 1.4).

1.1.1 Product Realization and Prototype

The product realization process, driven by market factors, is changing dramatically.There are increasing global competition, decreasing levels of demand for discreteproducts, and declining profitability. Consequently, only the companies who are able tolaunch quality products ahead of their competitors can continue to enjoy healthy markets.Smith and Reinertsen (1991) studied competitiveness of different products and stated that“A company able to launch a quality product ahead of their competition not only realizes100% of the market before rival products arrive but also tends to maintain a dominantposition for a few years even after competitive products have finally been announced.”This is also evidenced by a vice-president of Hewlett Packard who testified that “areduction of one month in the printer development time would result in additional profitsin excess of the entire product development cost” (Kazmer and Speight, 1997).

As many executives now realize how vital it is to move new products to marketrapidly, designers, who play a key role in the product realization process, face theconstant pressure to cut the product development time and costs. In the mean time,designers also face the sustained pressure on avoiding design errors. It is well known thatcost of design errors and related changes increases substantially with each step of productdevelopment. According to Wohlers Associates, the cost of an engineering changeincreases roughly by an order of magnitude (see Figure 1.1) as the design progressesfrom one significant development phase to the next.

Prototypes of the design can aid designers to find defects as early as possible andcorrect them when it is still relatively cheap and easy. As stated in a paper from Harvard

CONCEPTUAL MODELLING

DETAIL DESIGN

TEST

MANUFACTURING

PRODUCT REALSE

$10

$100

$1,000

$10,000

$100,000

Product Development Phase Cost of Design Change

Figure 1.1 - The Cost of a Design Change Related to Development Phase(Wohlers Associates).


3

Business Review (Bowen, et al., 1994), “… prototypes not only enable products to bedeveloped and launched more quickly, but also result in products that are both higher-quality and more effective in fulfilling their intended purpose in the marketplace.”

As computers become more powerful, Virtual Prototyping (VR), which involvesanalyzing Computer Aided Design (CAD) models for different end applications, isdeveloping quickly. Various analysis and simulation packages enable assessment ofproduct functionality; for example, kinematic modeling enables motion simulation, CFD(Computational Fluid Dynamics) can replicate a wind tunnel and assess fluid flow, FEA(Finite Element Analysis) can be used to determine load-carrying capacity and to predicttemperature distributions. Other analysis tools simulate various manufacturing andassembly processes; for example, software packages, like Moldflow and CMold, areavailable to model most of the usual materials-forming processes such as injectionmolding, investment casting, and so forth. Other packages simulate the various assemblyoperations, providing insight on setting up a manufacturing line (Siddique and Rosen,1997; Gadh and Ratnakar, 1998). In many cases, these VP techniques and simulationsoftware systems can help designers to evaluate their designs quickly and cheaply.

Unfortunately, despite the promise of these tools with the advances in computersimulation, not all aspects of design testing can be formulated in mathematical models.Also as products and processes become increasingly complex, the confidence andaccuracy of a Virtual Prototype system may still not be able to satisfy a designer’sexpectations. Therefore, physical prototypes are still fundamental to superior productdevelopment and production. In the words of Michael Schrage of the MIT Sloan Schoolof Management, “For companies that genuinely care about incremental and breakthroughinnovations, organizational redesign and core process re-engineering are not enough.Companies that want to build better products must learn how to build better physicalprototypes.” (Schrage, 1993) Similar themes also pervaded in an address at the IMSInternational Conference on Rapid Product Development given by Daimler-Benz’s Dr.Werner Pollmann. He examined prototyping at Daimler-Benz and wrote as follows:

“Purchase of a car depends strongly on subjective impressions. Next to technicalproperties like horsepower or security equipment, properties like noise, handling,or styling are key factors for a purchase decision. But these properties can only beevaluated by physical prototypes. For that reason, availability of high qualityfunctional prototypes will remain an important element of product developmentand cannot be substituted by digital models and analysis.” (Pollmann, 1994)

From late 1980s, a new kind of technology, which is based on additive fabrication,was developed to make physical prototypes more quickly. These techniques were namedRapid Prototyping (RP), or Solid Freeform Fabrication (SFF). They are discussed inSection 1.1.2 as the basis of Rapid Tooling, which is discussed in Section 1.1.3.

1.1.2 Rapid Prototyping

Rapid Prototyping, also called Solid Freeform Fabrication, or LayeredManufacturing (LM), refers to the fabrication of physical parts layer-by-layer directlyfrom CAD data. It involves successively adding new materials, in layers, to create a solidof some predefined shape. Therefore it is a fundamentally different method of fabrication


4

from traditional approaches like turning and milling, which are based on removingmaterials, or rolling or casting, which are based on deforming materials into desiredshapes.

Currently there are over 30 RP technologies available for model production based onthe principle of additive fabrication, including Stereolithography (SL), Selective lasersintering (SLS), Ink-Jet Printing, Fused Deposition Modeling (FDM). The majordifferences among these technologies are in two aspects: (1) material used; and (2) partbuilding techniques. Figure 1.2 is a schematic drawing to show the building technique ofStereolithography (SL), the most prevalent technology on the market, which waspioneered by 3D Systems. A short explanation on how the process operates is as follows.Initially, the elevator is located at a distance from the surface of the liquid equal to thethickness of the first, bottom-most layer. The laser beam will scan the surface followingthe contours of the slice. The interior of the contour is then hatched using a hatch pattern.The liquid is a photopolymer that solidifies when exposed to the ultra-violet laser beam.The elevator is moved downwards, and the subsequent layers are produced analogously.Finally, the part is removed from the vat, and cured in a special oven.

Rapid Prototyping has several advantages due to its unique fabrication process. It isfaster. Also the geometric complexity of the part has a significantly smaller impact onthe fabrication process. That is, when creating a part layer-by-layer, a simple cube and asculptured solid are equally easy to manufacture for RP. So some designs, which are socomplicated that it was impossible to make them before, can now be fabricated via RPtechnologies. Figure 1.3 is a robot arm design based on the principle of TensegrityStructures in Rapid Prototyping and Manufacturing Institute (RPMI) at Georgia Instituteof Technology. The new design is light yet strong, which is suitable for a high-speedrobot. Furthermore, multi-material structures and parts with embedded electronics can becreated using the layer-by-layer fabrication of RP (Merz, et al., 1994; Kumar and Dutta,1998; Jackson, 2000).

Figure 1.2 - A Schematic Drawing of a SLA.


5

Currently there are roughly 7000 RP systems installed worldwide (according toWohlers Associates at www.WohlersAssociates.com). The use of RP technologies hasresulted in a significant reduction in the prototyping time for numerous industrial cases(Jacobs, 1992; Jacobs, 1996; Wohlers, 1998). One demonstrative example shown inFigure 1.4 is Blaupunkt’s front cap. According to (Edelmann, 2000), Blaupunkt, aGerman mobile electronics company, developed three functional models of a swivel-typefront cap with the help of Stereolithography. The company was able to save 7 weeks’development time using the Stereolithography process compared with the conventionalmilling method. Also the cost for prototypes was cut tremendously. The traditionalmethod – milling from solids – would have cost the company between 30,000 and 40,000

Figure 1.3 - A CAD Model and a SL Model of a Robot Arm (RPMI at Gatech).

Figure 1.4 - A Stereolithography Model for Assembly Studies and Tests withRegard to Collision Behavior (Blaupunkt).


6

DM, the costs with Stereolithography were 12,000 DM.

Karapatis and coauthors (1998) classified the functions of prototypes in four mainclasses: the visual prototype, which is just a “solid”, real CAD model used to revealdesign defects and gain marketing clearance; the functional prototype (“form & fit”),which is mainly used to detect assembly problems; the material prototype, which has thesame geometrical and mechanical properties as the final part; the production prototype,which is made by the same process as the final part. To make material and productionprototypes of injection molding parts, a new kind of techniques, Rapid Tooling, wasdeveloped based on Rapid Prototyping technologies in 1990s. They are discussed furtherin Section 1.1.3.

1.1.3 Injection Molding and Rapid Tooling

Many consumer products involve the design and fabrication of injection moldedthermoplastic parts. Injection molding is the most widely used process for manufacturingthermoplastic products (Mitchell, 1996). It has many benefits like forming near net shapeproduct with nothing or very little secondary machining process, and producing parts thatcombine the functionality of many parts into one. The main character of injectionmolding process is that shapeless material is formed by a tool (i.e. molds) with acomplementing cavity in the shape of the desired part. Injection molded parts areproduced by closing mold pieces and injecting molten material into the formed cavity.After the molten material solidifies, the part is ejected from the mold. This process isalso applicable to metals and ceramics by a new technology known as powder injectionmolding (PIM) (German and Bose, 1997).

Tools, which define the shape of the part, play a vital role in the injection moldingprocess. Traditionally, the manufacture of tooling for both prototype parts andproduction components represents one of the longest and most costly phases in thedevelopment of most new products. It typically took a month to produce a prototypemold and another three months to complete mold design and manufacturing (Beckert,1999). The cost and time implications of the tooling process are particularly problematicfor low-volume products aimed at niche markets, or alternatively for rapidly changinghigh-volume products.

By using Rapid Prototyping technologies in the fabrication of tools or patterns, anew kind of techniques, named Rapid Tooling (RT), can reduce tooling cost and timeespecially when only small quantities of a part are needed (Jacobs, 1996). Moreimportantly the parts made by Rapid Tooling can be made of end-state materials likepolystyrene, which are required in the testing of functional prototypes in some industrialcases.

Since the first Rapid Tooling process (QuickCast) was developed by 3D Systems in1992 (Jacobs 1996), there are more than 20 Rapid Tooling methods until today (Wohlers,2000), like Epoxy Tooling, Composite Tooling, Kirksite Tooling, Spray Metal Tooling,Direct ACES Tooling, DTM RapidTool. Different tooling processes have different timeand material requirements.

Williams (1999) classifies Rapid Tooling processes into two categories: transfer anddirect. Transfer Rapid Tooling process utilizes RP model as a master pattern. An epoxy


7

is cast around the master to form a mold. Direct Rapid Tooling process uses RapidPrototyping machines to fabricate the tools directly. Tromans and Wimpenny (1998)compared direct and indirect tooling, and predicted that “the use of rapid prototypingtechnology to manufacture tooling will evolve from its indirect use as master patterns forsoft tooling, eventually providing direct methods of manufacturing tools on rapidprototyping machines.” Several reasons they gave included accuracy of the tools, timeand easiness of the process. For similar reasons, Wohlers (2000) also believed the directapproaches would gain an edge in the long term (6 to 10 years).

In addition, since only the Direct Rapid Tooling processes are investigated in our lab(Rapid Prototyping and Manufacturing Institute), in this dissertation the author will onlyconsider the direct Rapid Tooling processes. Specifically, this research of design forRapid Tooling is mainly focused on one typical direct Rapid Tooling process, direct AIM(ACES Injection Molding). The main steps of the direct AIM tooling are shown in Figure1.5. After finishing the mold design in CAD models, a SLA system is used to build themold using the ACES (Accurate Clear Epoxy Solid) build style. The mold is shelled outon the bottom side to leave a cavity which can be backfilled with various materialsincluding aluminum-filled epoxy, ceramics, and low-melting metals. After some optionalsupplement processes such as milling or polishing, the mold is ready to be used in aninjection molding machine.

Direct Rapid Tooling has been used in many industrial cases and proven useful inbuilding functional prototypes. An example given here came from Xerox Corporation(Jacobs, 2001). An internal Xerox customer required 100 polystyrene switch actuators ina very short time period. After evaluating various alternatives, Jeffery Heath and histeam decided to try Direct AIM. The team was able to injection mold the required 100polystyrene parts just 5 days after the CAD design was completed!

Compared to traditional tool making approaches like CNC and EDM, direct RapidTooling has its limitations in controlling the accuracy and material properties of theresulting tools, which is discussed further in Section 1.2. These limitations are beingaddressed in development efforts and are expected to be mitigated in the near future. AsRP and RT technologies become more mature, Rapid Manufacturing (RM), that is, usingRP and RT technologies to manufacture end use parts directly becomes more feasible.

MoldCAD

Model

Mold build by SLA Mold after Epoxy Back Fill

Direct AIM Tooling

MoldPhysicalModel

Figure 1.5 – Direct AIM Tooling Process.


8

1.1.4 Rapid Manufacturing

Although this research is based on the prototype development, the methods presentedin the dissertation also have implications on Rapid Manufacturing. Therefore the currentdevelopment of Rapid Manufacturing, especially Rapid Production Tooling, is describedbriefly in this section.

If RP and RT today can produce functional models quickly and inexpensively, alogical question is why not use them to produce actual production parts? For mostapplications today, the idea is not practical due to the limitations of speed, part size,accuracy, surface finish, and material properties. However, some RP and RT techniqueshave already been used to manufacture finished products, especially for low-volumemanufacturing and customized products.

In the development of Rapid Manufacturing, Wohlers observed as follows,

"Initially, it will happen where the unit price of a part is high and the volume is low,and also where the part is small — about the size that would fit within a 6-in.cube… In the automotive industry, at first, such activity will be limited to high-endpersonalized vehicles where volumes are lower … In the medical area, rapidmanufacturing will continue to make inroads in applications such as customimplants. Small customized parts for helicopter interiors such as structural parts forseats, cabinetry, or nameplates are other potential applications.”

Currently, several commercially available processes for producing products exist.As an example, 3D Keltool given by 3D Systems, has achieved more than 3 million shotsfor unfilled thermoplastics and over 500,000 shots with glass-fiber-filled thermoplastics.Although most of these processes are transfer Rapid Tooling, the direct Rapid Toolingtechniques are continuously improved. New technologies are also in development. Forexample, the researchers in Sandia are developing a system which uses laser to meltpowdered materials in layers to produce the final part. The part is as metallically dense asone made by conventional means (Bylinsky, 1998). Recently this technology (LaserEngineered Net Shaping – LENS) has been commercialized by OptomecTM.

The pressure of time, quality and cost, together with increasing product variety, morecustomized products and world-wide competition is driving technology development andimplementation in the area of rapid manufacturing. Hilton (2001) presented a conceptualmodel of “the disposable tool”, which uses lower-cost, lower-volume tools to produceproducts instead of high-cost, high-volume tools. It is an optional approach consideringthe trade-off between tool performance with cost and time. Currently, at least 25different groups are investigating rapid production tooling (Jacobs, 2001). In (Ashley,1994), a long list of top US companies, including Ford Motor Co. and Pitney Bowes Inc.,had joined a total of nine new research consortia devoted to the development of processesby which large-volume-capable production tooling could be produced quickly andinexpensively.

In order to compress the product development process, the technologies includingRapid Prototyping, Rapid Tooling and Rapid Manufacturing are introduced in thissection. In the next section, design for Rapid Tooling will be discussed in order tooutline the research focus for the dissertation in Section 1.3.


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1.2 DESIGN FOR RAPID TOOLING AND RESEARCH OPPORTUNITIES

The motivation and foundation for investigating the proposed research are describedin this section. First an overview of the current usage of Rapid Tooling is described inSection 1.2.1 with two related problems identified. The Rapid Tooling Testbed ispresented in Section 1.2.2 to provide a context and foster a better understanding of themotivation. Two main challenges related to the testbed are described in Section 1.2.3 tojustify the research approaches used in this research. Finally research opportunities arediscussed in Section 1.2.4 before the focus of this research is presented in Section 1.3.

1.2.1 Current Usage of Rapid Tooling and Related Problems

Rapid Tooling has the potential to dramatically shorten the time required to producefunctional prototypes or products. However, several problems may extend the lead-timeto get the functional prototypes or products. These problems should be addressed torealize this potential. In this section the current usage of Rapid Tooling is examined toshow these problems.

The whole process of using direct AIM tooling to get functional prototypes includespart design, mold design, mold fabrication by SLA, and part fabrication by injectionmolding (refer to Figure 1.6). Designers carry out part design, which results in a CADmodel of the part to be fabricated. The CAD model is then sent to manufacturers, forexample, a Rapid Tooling service bureau. According to the part design, themanufacturers will design and build molds to produce prototype parts. Finally theprototypes are sent back to the designers for functional testing. So the manufacturer, inthis dissertation, actually includes mold designer, rapid prototyping engineer, andinjection molding engineer.

As described in (Jacobs, 1996), a designer is required to do the data preparation forusing Rapid Tooling. That is, the designer should send a fabricatable CAD model(usually in STL file format) to a manufacturer. If some features of the part design arefound inappropriate for the manufacturing process, the designer is required to change thefeatures according to the feedbacks given by the manufacturer. Often, many iterations ofchanging are required before satisfactory prototypes can be fabricated.

Some tasks, like the mold design for a given part, are elaborate and difficult for themanufacturer, especially if the part is designed without considering the manufacturingrequirements. By using RP machines, the fabrication of mold pieces may only take oneor two days. However, it may take one week to generate a qualified mold design that is

PartCAD

ModelMold CAD Model

Mold build by RPMold after Epoxy Back Fill Injection Part

Phase I:Design Part

Phase II:Design Mold

Phase III:Build Mold

Phase IV:Build Part

ManufacturerDesigner

Figure 1.6 – Processes of Using Direct AIM Tooling to Build Prototypes.


10

compatible with the RP machine and the injection molding machine. Some requirementsgiven by the designer (such as tight accuracy and surface finish) may make the molddesign and process planning rather difficult, or even impossible. Whenever the designeris required to modify the part design, the manufacturer needs to go through the wholeprocesses (mold design, RP process planning, and IJM process planning) again for thenew part design.

Therefore, two kinds of problems are identified in the current usage of RapidTooling, which may significantly impede the applications of Rapid Tooling in theproduct development processes.

(1) Several steps in the processes of using Rapid Tooling may take a long time.

In the phase of part design, a designer often spends a lot of time in Design-for-Manufacture (DFM) step in order to get a fabricatable part design. The task of Design-for-Manufacture proves to be a heavy burden for designers since most designers are stillnot familiar with Rapid Tooling requirements. Even worse, as a new kind ofmanufacturing method, Rapid Tooling is developing so quickly that no accurateinformation about how to design for Rapid Tooling is available on any publishedhandbooks. All these factors may make the Design-for-Manufacture step take a longtime. In some cases, it even makes designers hesitate to use Rapid Tooling because ofthe unfamiliarity with the DFM.

The mold design for a given part also proves to be an elaborate and difficult job forthe manufacturer. For example, “designing the parting lines and surfaces for a mold isone of those thankless industrial jobs that requires intelligence and experience but isn’tthe least bit interesting or fun.” (CAD/CAM Publishing Inc., 1999) Even worse, the molddesign step may repeat several times whenever the part design is changed. Materialise(www.Materialise.com), a leading Rapid Prototyping provider, once gave a report named‘Data Preparation as a Bottleneck’, which stated that:

“The actual production time of the rapid tool insert varies from 15 to 35 hours, andthe post processing takes around 20 to 40 hours of work. Where traditional toolinguses 8 to 10 weeks to make the tool, rapid tooling does the job in one to two weeks.The cost benefit of rapid tooling compared to traditional tooling is about 25% to30%. … However, the lead-time necessary for obtaining the tool design lays ashadow over the time and cost benefits of the rapid tooling process. … The lead-time from part design to tool design can be more than three weeks, whicheliminates the time and especially the cost benefits of the rapid tooling process.”

(2) The iterations of design changes may take a long time.

Since most designers are not familiar with the Rapid Tooling technologies, somerequirements of a part design may be unreasonable, or even out of the capability of theRT technologies. Whenever these factors are identified in the mold design and processplanning phases, the designer is notified to change them according to the feedbacks givenby the manufacturer. Sometimes, negotiation between designer and manufacturer isnecessary. It is well known that the two types of engineers “do not speak the samelanguage,” a situation which has often been described by saying that design engineer donot bother to speak to manufacturing engineers and just throw their drawings over the


11

wall separating them. Currently many Rapid Tooling facilities are located at ServiceBureaus (Wohlers, 1999), which are separated from designers geographically andorganizationally. This makes communication between designer and manufacturer evenmore difficult.

Thus we believe Rapid Tooling will never be rapid if the designer still needs one ortwo weeks to modify his/her design to facilitate Rapid Tooling process after initiallysubmitting the design, and the tool design for the given part will take another week beforethe mold can be fabricated. Therefore the key question to be investigated in thisdissertation is:

How to reduce the lead-time in the usage of Rapid Tooling to produce functionalprototypes for a part design?

Being related to the two problems identified in this section which cause long lead-time in the application of Rapid Tooling, this question can be answered in two levels: (1)How to reduce the time of several time-consuming steps? and 2) How to reduce thenumber of iterations between the designer and manufacturer? To effectively answerthese two questions, research opportunities in design for Rapid Tooling are discussed inSection 1.2.4. Based on the opportunities, the research questions and related hypothesesare presented in Section 1.3.

In the next section, the motivating project (RTTB) of this research is presented toprovide a context and foster a better understanding of the DFM method to be presented inthis dissertation.

1.2.2 Motivating Project – Rapid Tooling TestBed

The research in this dissertation is motivated by the Rapid Tooling TestBed (RTTB),which is sponsored by National Science Foundation (NSF DDF DMI-9618039). TheRTTB is intended to be an experimental testbed that supports exploration of design anddesign-for-manufacture issues related to rapid prototyping and rapid tooling for injectionmolding (Allen and Rosen, 1997; Rosen, 2000). It supports the design of parts andmolds, the selection of prototyping technologies and vendors, and the fabrication of thoseparts and molds.

As depicted in Figure 1.7, the RTTB is divided into three stages: the design stage,the design for manufacture stage, and the manufacturing stage. During the design stage,requirements are entered into the testbed. These requirements consist of a partrepresentation along with the design specifications for the part. In the DFM stage,suitable part and mold materials, rapid prototype technologies, and injection moldingtechnologies, are selected for the fabrication of both the injection molding tooling and thepart. In addition, the injection molding tooling is designed and the part is tailored tofacilitate the manufacturing of both the mold and part. The final stage involves the actualmanufacturing of both the tools and the part. The work in this dissertation is related togeometric tailoring and generation and selection of mold configuration, which aremarked in black lines in Figure 1.7. Before these two modules, the appropriate materialand process have been determined for the given part (Herrmann and Allen, 1999).


12

As a result of the RTTB project, “customer” of the rapid tooling testbed will receivenearly production-representative components in a variety of polymer, ceramic, and metalmaterials within 3-4 days of submitting a product model. Related to RTTB project,challenges and the related research approaches are discussed in the next section.

1.2.3 Challenges of RTTB and My Research Approach

The Rapid Tooling TestBed is intended to assist a designer in obtaining usefulprototypes for his/her part design, where “useful” means that the prototype mimics somedesired production characteristics. A part given by the designer may have different kindsof shapes. The designer may care about different kinds of production characteristics andrequirements. Therefore, the research in this dissertation is not considering theapplication of Rapid Tooling for a part or just one kind of parts. Instead general methodsfor handling different kinds of part designs are explored. To develop these methods, twochallenges of RTTB are identified in this dissertation.

(1) A part may have a wide variety of geometries.

To implement different design functions, a part designed by the designer may havedifferent geometries. Figure 1.8 shows some industrial parts considered in thisdissertation. It is quite obvious that different designs require different shapes.

In Section 1.2.1, the first problem identified by the author in the current usage ofRapid Tooling is that several steps, especially the mold design step, take a long time.One reason for the problem is the wide variety of possible geometries that a part mayhave. Because the mold design is tightly related to the shape of the part, themanufacturer may need to spend a long time in this step.

PartRepresentation

DesignSpecification

PreliminarySelection of

Materials

PreliminarySelection of

SFFTechniques

PremilinarySelection of

IJMTechniques

CombiningSFF & IJMTechniques

GeometricTailoring

ResourceSelection

Generation &Selection of

MoldConfiguration

MoldFabrication

PartFabrication

CADTools

DesignSolutionMethods

Design for ManufactureDesign Manufacture

Event Timeline

Figure 1.7 – RTTB System and My Focus.


13

(2) A part may have a wide variety of design requirements.

A designer can have a wide variety of design requirements for the part design.Related to the different geometries in the part, there are different kinds of variables andparameters. Also the attributes or design goals are quite different according to the designfunctions. For example, the production characteristics of a part can be the maximumstress, or the deflection of a feature. In addition the designer may give requirements suchas cost, time, surface finish and accuracy of some important surfaces.

Figure 1.8 – Different Geometries of Some Industrial Parts.


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In most situations, the designers’ preferences on these design requirements are quitedifferent. So the “customer” of RTTB can be classified in different categories with themost frequently asked question they may have as shown below:

• Some care more about time: How soon I can get my parts?• Some care more about tolerance: What tolerance can be achieved for specified

design features?• Some care more about cost and tolerance: How are the tolerances correlated with

cost?• Some care more about functions: What will be the material properties, internal

stresses, etc.?

This list goes on. It is quite obvious from the list that the tradeoffs between thedifferent design requirements are an important consideration in the RTTB system.

In Section 1.2.1, the iterations between the designer and manufacturer are identifiedas a problem in the current usage of Rapid Tooling. These iterations mean the designerand manufacturer need to make tradeoffs between their requirements. This is ratherdifficult and time-consuming. The author believes one reason for this problem is the lackof tools to help the designer and manufacturer to make the tradeoffs between theirrequirements.

Therefore, the research approaches to be presented in this dissertation are based onComputational Geometry and Decision-based Design, which correspond to thechallenges of wide varieties of geometries and design requirements respectively.

(1) Mold design based on Computational Geometry.

For the wide variety of geometries considered in the mold design, the author believesthat the computer can aid the manufacturer to dramatically reduce the time needed in thisstep. Whenever the designer changes the part design slightly, the manufacturer should beable to repeat the mold design in even less time. Solving geometric problems with theaid of computer requires carefully designed geometric algorithms and data structures,which are exactly the research areas of computational geometry.

Computational geometry emerged from the field of algorithms design and analysis inthe late 1970s. It has grown into a recognized discipline partly due to the advent ofpowerful computers, and the explosion of application domains – computer graphics,geographic information system, robotics, computer aided design and manufacture – inwhich geometric algorithms play a fundamental role. According to (Goodman andO'Rourke, 1997), computational geometry means “the study of geometric problems froma computational point of view, including computational complexity, computationaltopology, and questions involving the combinatorial complexity of arrangements andpolyhedra.”

Computational geometry and the research opportunities in mold design are furtheranalyzed in Section 1.2.4. Based on the analysis, the research questions and relatedhypotheses on mold design are presented in Section 1.3.2.


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(2) Design-for-Manufacture based on Decision-Based Design.

The cost, lead-time and quality of injection molded parts are tightly related to thepart design, mold design, and fabrication process planning (Pye, 1989; Rosato andRosato, 1995). These tight interrelations make the tradeoffs between the wide variety ofdesign and manufacturing requirements rather difficult to make.

In this dissertation, the fabrication process planning, in which the manufacturingparameters and variables are determined, is considered as a design process to distinguishit from the actual fabrication process itself. So the main design processes of using RT toget the prototypes include part design, mold design, RP process planning, and IJMprocess planning. Currently these design processes are sequential. That is, mold designis started after the designer fixes all the part design variables and sends the CAD modelof the part to the manufacturer. Similarly, the RP process planning needs the molddesign before it starts, and the injection molding process planning needs the informationof part design and mold design. These sequential processes can be shown in Figure1.9.(a) as one of the product development spirals.

In the above sequential design processes, the designer and manufacturer makedecisions about part design variables, mold design variables, RP process variables, andIJM process variables. These decisions will affect the attributes such as cost, time andfunction of the prototypes, as shown in Figure 1.9.(b). So from the viewpoint ofdecision-making, the author believes the decisions in each design process are notnecessarily sequential or in the same order as that of the information flow. To someextent, they should be integrated or concurrent because of the interrelations between thevariables. The work presented in Chapter 5 and 6 is toward this direction.

In this dissertation, the framework of Decision-Based Design, DBD, is used fordescribing the part realization process. In Decision-Based Design, decisions serve asmarkers to identify the progression of a design from initiation to implementation totermination. In DBD, decisions represent a unit of communication; one that has bothdomain-dependent and domain-independent features (Shupe, 1988; Mistree et al., 1989).

Mold DesignParameters &

Variables

Part DesignParameters &

Variables

IJM ProcessParameters &

Variables

FunctionCostTime

Quality, etc

Attributes

Part Design

Mold Design

RapidPrototyping

ProcessPlanning

InjectionMoldingProcessPlanning

Testing

Better PartDesign

ManufacturerRP Process

Parameters &VariablesFunctional

Prototypes

Designer

(a) Product Development Spiral (b) Relations of Parameters and Attributes

Figure 1.9 – Design for Rapid Tooling.


16

By approaching design from the perspective of making decisions, it becomes possible toenvision a unified approach for design-for-manufacture problem, which is furtheranalyzed in Section 1.2.4.

Related to the research approaches used in this dissertation, research opportunities inmold design and design-for-manufacture are identified in the next section.

1.2.4 Research Opportunities in Mold Design and Design-for-Manufacture

Considering the two challenges of the Rapid Tooling TestBed, the researchapproaches used in this research are presented in the last section. In this section, theopportunities for making contributions in mold design and Design-for-Manufacture arefurther discussed before the research focus is presented in Section 1.3.

• Mold Design for Different GeometriesA mold is a highly sophisticated design that consists of several components for even

a very simple part. A typical layout and descriptions of parts in the mold is shown inFigure 1.10. The function of the mold is threefold: injecting the plasticized plastics in thedesired shape, solidifying the injecting molding part (cooling for thermoplastics andheating for thermosets), and ejecting the solidified parts out of the mold.Correspondingly, the mold has five sets of components: (1) the components for injection(e.g. sprue bushing); (2) the cavity and core; (3) the components for cooling (e.g. coolingchannels in core and cavity); (4) the components for ejection (e.g. ejector pins and ejector

Figure 1.10 – An Illustrative Mold for Injection Molding.


17

plate); and (5) the components for assembling other parts in the machine (e.g. leaderpins). The components of core and cavity are to form the shape of the part. Thereforethey are the most important component in the mold design.

To design such a mold for a part with varied geometries is a rather complicatedprocess involving several steps. The author classified the main steps as shown in Figure1.11 which are based on the handbooks of (Rees, 1995) and (Rosato and Rosato, 1995).They are simplified as sequential processes although some steps may actually be paralleland some loops may exist.

The mold design process is divided into two phases: configuration design and detaildesign. The mold configuration design includes the selection of parting direction, partingsurface, ejection method, feeding method, and cooling method. These selectionssignificantly affect the mold cost and DFM features. The mold detail design is mainly toconstruct the mold pieces according to the mold configurations. CAD systems likePro/Mold and SolidWorks can help mold designers to reduce the time in this phase.

Ejection typesand position

Splitting moldbase into mold

pieces

Add ancillary items (guidebushes, guide pillars,

spure bush, etc.)

Feed systems (sprue,runner, gate) types

and layout.

DetermineParting Lines

Creating Coolingsystem

Determine external andinternal side actions andmovement requirements,

if any

Determine Coreand Cavity

Determine PartOrientation andPosition in Mold

Base

DetermineParting

directions

DeterminePartingSurface

(a)

MoldConfiguration

Design(choose layout)

Add holes forejection pins in mold

Cooling systemconfiguration

Add feed systems(sprue, runner, gate)

in mold.

Mold Base Selection

Determine moldtype (integral/

insert)

Determineimpression

number in a mold

(b)

Mold DetailDesign

(construction)

Determine partshrinkage and

apply shrinkagecompensation

Check Part forDraft Angle,

Thickness, etc.

Check Interferenceby Simulation

CompleteAssembly

Design

Figure 1.11 – Mold Design Process and My Considerations.


18

In prototype tooling, many components of the mold are standardized. Specifically,the usage of interchangeable mold inserts into a standardized mold base, or master frame,significantly simplifies the mold design. The inserts encompass the mold core andcavity, from which parts are molded. The master frame contains standardizedcomponents, such as the ejector system, gating, nozzles, and built-in water jackets forcooling. Standardizing these components in the master frame eliminated much of thework associated with design and manufacturing the components in a new mold. Figure1.12 is a picture showing the master frame used in Morgan Press at RPMI with a SLAmold insert.

The usage of the standardized master frame saves time in several steps including thedesign of sprue, runner, ejection pins, and cooling lines. However, the design andconstruction of the mold inserts (core and cavity for two-piece mold) are still time-consuming due to the varied geometries of a part.

Several commercial software systems have been developed to aid mold designers indesigning mold inserts. However, these software systems are mainly on moldconstruction instead of mold configuration design. And lots of improvements areexpected from mold designers. In (Beckert, 1999), Steve Sivitter, vice president ofDelcam International Inc., gives suggestions for mold design software according to theirsurvey of mold designers as below:

Moldmakers look for design software that allows core and cavity surfaces to beseparated automatically and that can automatically determine the natural parting linefor shapes and use that curve to create the split surfaces. Because plastic parts musthave sloping surfaces to be removed from the mold, software must also be able tocheck, create or modify the draft on surface models. What’s more, software shouldalso allow for even multi-surface, variable radius fillets to be created quickly andeasily.

Cavity

EjectorPlate

Sprue

LeaderPin

Runner

B-Plate

MasterFrame

Figure 1.12 – Standardized Master Frame Used in Morgan Press at RPMI.


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Therefore, the configuration design of parting direction, parting lines, and partingsurface, and the construction of mold pieces are investigated in this dissertation. Theyare shown in Figure 1.11 enclosed with thick boundary lines.

Another technique in prototype tooling that is also considered in this research ismulti-piece molding. Being different from two-piece molds, multi-piece molds are themolds that contain more than two mold pieces. Each mold piece can be hand-loaded intoa mold base mounted on the injection molding machine platens. During mold injectionand part cooling process, the molds are accurately and securely clamped into the holdingdevice. Finally each mold can be hand-removed from the mold base to release the part.Although the technique is only suitable for producing a small number of parts, it canfabricate the prototypes of parts with more complicated geometries. An example ofmulti-piece molding, which is provided by Protoform GmbH is shown in Figure 1.13.The injection molded part has several undercuts (features that would prevent the partfrom ejection out of the mold), therefore it cannot be formed by only two mold pieces.

Since multi-piece molds have more than a pair of opposite parting directions, it ismore difficult and time-consuming to generate a good mold design. The author believesthe design for multi-piece molding also provides plenty of research opportunities.Therefore,

(a) A part to be built (b) CAD models of disassembled mold pieces

(c) Photo of mold pieces (d) Injection molded parts

Figure 1.13 – A Multi-piece Molds for a Part Given by Protoform GmbH.


20

it is proposed to develop a systematic approach from geometrical perspective toautomate several important mold design steps, including selecting partingdirections, parting lines, and parting surfaces, and constructing mold pieces formulti-piece mold design.

The Multi-Piece Mold Design Method (MPMDM) is developed in this dissertation toaddress this question. The MPMDM and its associated steps are introduced in Section3.1 and 3.3. The associated system (Rapid Tooling Mold Design System) is introduced inSection 4.1.

• Design-for-Manufacture for Different RequirementsDesign-for-manufacture (DFM) “is concerned with understanding how product

design interacts with the other components of the manufacturing system and in definingproduct design alternatives which help facilitate ‘global’ optimization of themanufacturing system as a whole” (Stoll, 1991). In the past 10 years the area of DFM isunder intense investigation. A large number of DFM methods and tools have beendeveloped in various manufacturing domains for a part design.

Although there are a number of ways to categorize these approaches, the author willuse four categorization methods (analyzing approach, design modification scope,measure of manufacturability, and responsibility of DFM) to identify researchopportunities for the usage of Rapid Tooling. More detailed descriptions on some ofthese approaches are presented in Section 2.5.

(1) Analyzing approach.

Based on what analyzing approach may be taken, the DFM approaches are classifiedroughly as follows:

• Rule-based approaches. In these approaches, rules are used to identify infeasibledesign attributes from direct inspection of the design description. A typicalexample of the rule-based approaches is the guidelines in a handbook. Theseguidelines enumerated design configurations that posed manufacturabilityproblems. The designer had to carefully study these guidelines and avoid theseconfigurations that resulted in poor manufacturability.

• Plan-based approaches. In these approaches, manufacturing plans for the designare generated first. If there is more than one possible plan, then the mostpromising plan should be used for analyzing manufacturability.

Rapid Tooling has several manufacturing operations. Therefore it is rather difficultto formulate design guidelines and to determine the manufacturability of a design directlyfrom the design description. Thus the plan-based approach is more suitable for RapidTooling.

(2) Design modification scope.

A part design usually consists of three kinds of requirements. First is the basic shapeof the part, or topology. Most parts are shaped the way they are for a specific reason.Second is the geometry, or parametric values of the part. Third are other requirements,including fit (e.g. tolerance, accuracy), function (e.g. materials, physical and mechanical


21

properties, thermal properties, electrical properties), appearance (e.g. color, texture, andcontour), time, cost, safety and environmental issues.

Different DFM approaches focus on different design requirements. So they can beclassified according to their design modification scopes. For example, if the guidelines ina handbook focus on the design configurations, then the design modification scope is thetopology of the part design.

The design requirements that need to be tested in the functional prototypes producedby Rapid Tooling varied, including the form, fit, function, and appearance. In thisdissertation, the design modification scope of the DFM approach includes geometry,tolerance, accuracy, physical and mechanical properties, time, and cost. The changing ofthe part shape or topology is not considered in this research.

(3) Measure of Manufacturability.

Based on the scales on which manufacturability is measured, the DFM approachesare classified as follows (Gupta, et al., 1995):

• Binary measures. The approaches using this manufacturability rating simplyreport whether or not a given set of design attributes is manufacturable.

• Qualitative measures. Designs are given qualitative grades (such as “poor”,“average”, “good”, or “excellent”) based on their manufacturability by a certainproduction process.

• Abstract quantitative. This type of approaches involves rating a design byassigning numerical ratings along some abstract scale, for example amanufacturability index between 1 and 10.

• Time and cost. In general, a design’s manufacturability is a measure of the effortrequired to manufacture the part according to the design specifications. Since allmanufacturing operations have measurable time and cost, this type of approachesuse time and cost as an underlying basis to form a suitable manufacturabilityrating.

For the approaches based on the measure of manufacturability by binary measures,qualitative measures, and abstract quantitative, it can be difficult to interpret the measuresor to compare and combine them. However, the ratings based on time and cost can easilybe combined into an overall rating. Since several requirements are considered in thisdissertation and the tradeoffs between them should be made, the measure ofmanufacturability by time and cost is more suitable for Rapid Tooling.

(4) Responsibility of DFM.

According to the people who are responsible for DFM, the DFM approaches areclassified as follows.

• Designer-based approaches. In several DFM approaches, it is taken for grantedthat the designer takes the full responsibility of DFM. In these approaches, “thepurpose of DFM is to ensure that the designer considers manufacturing issuesduring the design.” (Dissinger and Magrab, 1996) So this type of approach tries todeveloped tools from design guidelines to automated manufacturability analysissystems, which are to be used by the designer. However, automated analysis


22

meets difficulties if a part or a manufacturing process is complex. It is ratherdifficult, if not impossible, to formulate all manufacturer’s experience anddecisions into rules and algorithms.

• Approaches based on an integrated team of designer and manufacturer. Ideallydesigner and manufacturer should cooperate in DFM problem. ConcurrentEngineering (CE) advocates designers to work with manufacturing engineersduring the design process. However, these approaches are more on organizationalstructures and information sharing than on integration of decisions. The authorbelieves the collaborations between the designer and manufacturer are notguaranteed even if they can work as a team. More importantly, a coordinationmethod and a uniform decision framework need to be developed for thecollaborative decision-making.

• Manufacturer-based approaches. Recently much of the research on DFM haslooked to the VLSI (very large scale integrated circuits) community because ofthe tremendous success of the VLSI industry. A group of researchers in a NSFworkshop on structured design methods observed that a clean interface, whichseparate design efforts at increasingly high levels of abstraction from the growingcomplexities of the fabrication processes, is the key to the rapid success of theVLSI development (Antonsson, 1996). That is, a VLSI designer does not have tofully detail his/her design; a process compiler will fill in the process-dependentdetails. So in this kind of approach, the designer actually transfers part of theburden of DFM for a fabrication process to the manufacturer.

The Rapid Tooling TestBed is partly motivated by developing approaches to achievea clean interface between the designer and manufacturer for Rapid Prototyping and RapidTooling processes (Allen and Rosen, 1997; Rosen, 1998). The research presented in thisdissertation also proceeds in this direction. Since the objective is to develop production-representative prototypes, part of the DFM for Rapid Tooling is actually design-for-prototyping. The author believes these DFM tasks (Geometric Tailoring which is definedlater) can be transferred to the manufacturer. In doing so, it is important to develop anapproach and a system to aid the manufacturer to integrate design and manufacturingrequirements, and to gain a better understanding of the tradeoffs between theserequirements.

As a summary, a DFM approach, which is based on process planning, consideringthe tradeoffs of many design requirements, and executed by the manufacturer, is exploredin this dissertation for Rapid Tooling process. But the current state of research does notaddress the issues of how to transform DFM to the manufacturer and how to integrateDFM with the process planning in the manufacturer side. Therefore,

it is proposed to develop a formal, technical approach founded in Decision-BasedDesign to tailor a submitted part design according to both product and processconsiderations for Rapid Tooling process.

In light of the DFM approach explored in this dissertation, the following definitionsfor geometric tailoring and design for Rapid Tooling are offered to provide a context forthe remainder of the dissertation.


23

Geometric Tailoring, in this dissertation, is to change the geometry of a part tolower fabrication cost and time, and to produce functional prototypes to mimic theproduction functions, because the material and fabrication process to get molds and partsare different from those in producing products. Geometric Tailoring is actually part ofdesign-for-manufacture.

Design for Rapid Tooling, in this dissertation, is the integration of the requirementsof geometric tailoring, mold design, and fabrication process planning to make decisionson their variables for producing the prototypes of a part design.

The relations of Design for Rapid Tooling (DFRT), Design-for-Manufacture andGeometric Tailoring are shown in Figure 1.14. The research work related to GeometricTailoring is elaborated in Chapter 5, and the research work related to DFRT is presentedin Chapter 6.

With the research opportunities identified and two definitions presented, the focus ofthis dissertation will be described in the next section.

1.3 RESEARCH FOCUS IN THE DISSERTATION

The research focus in this dissertation is embodied as follows:

! a set of research questions that captures motivation and specific issues to beaddressed,

! a set of corresponding research hypotheses that offers a context by which theresearch proceeds,

! a set of verification studies performed in this work, and! a set of resulting research contributions that embody the deliverables from the

research in terms of intellectual value, a repeatable method of solution,limitations, and avenues of further investigation.

The research questions are presented in Section 1.3.1 along with the correspondingresearch hypotheses and related tasks. The validation strategy for this work is presentedin Section 1.3.2. The resulting research contributions are introduced in Section 1.3.3.

Mold Design

RP ProcessPlanning

IJM ProcessPlanning

GeometricTailoring(Chp 5)

Design forRapid Tooling

(Chp 6)

Design forManufacture

Considerationson Topology,

Assembly,etc.

Figure 1.14 – Relations of DFRT, DFM and Geometric Tailoring.


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1.3.1 The Principal Goal, Research Questions and Hypotheses in the Dissertation

The principal goal in this dissertation is the development of methods to reduce thelead-time in the usage of Rapid Tooling to produce functional prototypes for a partdesign. The principal research goal is to be achieved by two sub-goals: 1) To reduce thetime of several time-consuming steps in mold design, and 2) To reduce the number ofiterations between the designer and manufacturer. As discussed in the previous section,Computational Geometry and Decision-Based Design provide the foundation on whichthis work is built. Given these foundations and goals, the motivation for this research isembodied in two primary research questions which are stated as follows.

Primary Research Questions:

Q1. How to aid the mold designer to reduce the mold design time for a widevariety of part geometries in the design for Rapid Tooling process?

Q2. How to reduce the time of iteration between the designer and manufacturerin the usage of Rapid Tooling for a wide variety of design requirements?

These research questions are related directly to the sub-goals in this research. Thefollowing hypotheses are investigated in this dissertation in response to the primaryresearch questions. There is a one-to-one correspondence between the hypotheses andprimary research questions.

Hypothesis 1: Multi-Piece Mold Design Method provides a method to automateseveral key steps of the mold design process, which can greatly reduce themold design time for a wide variety of part geometries.

Hypothesis 2: Geometric tailoring for Rapid Tooling can be integrated withprocess planning based on decision templates and solved by the manufacturer,which can reduce the time of iteration between the designer and manufacturer.

Since Questions 1 and 2 are quite broad, three supporting research questions andsub-hypotheses are proposed to facilitate the verification of Hypothesis 1. Threesupporting research questions and sub-hypotheses are proposed to facilitate theverification of Hypothesis 2. The supporting questions and sub-hypotheses forHypothesis 1 are stated as follows.

Q1.1. What are appropriate basic elements to automate several mold design stepsfor a wide variety of part geometries?

Q1.2. How to generate mold configurations by a systematic design process basedon the basic elements?

Q1.3. How to generate mold pieces from a given mold base effectively andefficiently according to a mold configuration design?

Sub-Hypothesis 1.1: Concave region and convex face are two kinds of basicelements that provide an efficient and effective approach for exploring anddeveloping molds design method for Rapid Tooling.


25

Sub-Hypothesis 1.2: The Region based combining process and related algorithmsprovide a systematic method to generate mold configurations of multi-piecemolds design.

Sub-Hypothesis 1.3: The Reverse Glue Mold Construction Method and relatedalgorithms provide an efficient and effective method for constructing multi-piece mold designs for Rapid Tooling.

There is a one-to-one correspondence between the supporting questions and sub-hypotheses. Although developed for Rapid Tooling, sub-hypotheses 1.1, 1.2 and 1.3 alsohave implications on the mold design for the production injection molding (refer toSection 3.4, 3.5 and 3.6).

The second hypothesis is related to the Design-for-Manufacture problem for RapidTooling. The supporting questions and sub-hypotheses for Hypothesis 2 are stated asfollows.

Q2.1. How to reduce the iterations between the designer and manufacturer inproducing functional prototypes that have different material propertiesfrom products?

Q2.2. How to formulate the design for Rapid Tooling problem which integratesdecisions on design and manufacturing variables and other design andmanufacturing requirements including goals, constraints, and preferences?

Q2.3. How to solve the design for Rapid Tooling problem effectively andefficiently?

Sub-Hypothesis 2.1: The designer can initiate a material geometric tailoring(MGT) formulation based on a MGT decision template; therefore themanufacturer, who completes and solves the MGT problem, can produceproduction-representative prototypes more quickly.

Sub-Hypothesis 2.2: The design for Rapid Tooling problem can be formulated byseveral compromise DSPs and tasks, which can then be integrated into adesign for Rapid Tooling system (DFRTS).

Sub-Hypothesis 2.3: A three-stage solution process can be utilized to get asatisficing solution effectively and efficiently based on design andmanufacturing models and continuous/discrete variables.

The research questions and hypotheses provide a frame upon which the differentdevelopments and contributions from this research will be achieved. The relationshipbetween the hypotheses and the various sections of the dissertation are summarized inTable 1.1. Verification and validation of the issues are discussed in the next section, andtesting of the individual hypotheses commences in Chapter 3, lasting until Chapter 6.Further verification of the hypotheses comes from application of the Rapid Tooling MoldDesign System and Design for Rapid Tooling System on producing prototypes for twoindustrial parts. The application of the systems is presented in Chapter 7 and 8. Chapter 9contains a review of verification of hypotheses. In the next section the strategy forhypotheses validation in this dissertation is presented.


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1.3.2 Validation Philosophy and Strategy

According to Simpson (1998) the validation process for engineering design "is toshow the research and its products to be sound, well grounded on principles of evidence,able to withstand criticism or objection, powerful, convincing and conclusive, andprovable." The validation of a design method can be difficult and therefore a validationstrategy needs to be developed carefully. Referring to a validation strategy developed by(Pedersen, 1999), Siddique presented a validation strategy in his dissertation (Siddique,2000) to test a “Product Family Reasoning System” (PFRS). The validation strategy issound, and therefore is adapted in this dissertation to test RTMDS and DFRTS.

The argument of this dissertation is embodied in a structure characterized byresearch questions, hypotheses, and hypothesis-testing (the Scientific Method). In thecontext of Natural Science, research questions refer to observations (articulating the‘truth’), hypotheses refer to explaining the observations (understanding the ‘truth’), andhypothesis-testing refers to validating the explanation (accepting knowledge about the‘truth’). Hence, in the context of Engineering Design, hypothesis-testing becomes thevehicle by which new scientific knowledge is accepted and added to the current pool ofknowledge. This ties the research validity discussion strongly to a fundamental problemaddressed early in epistemology and later in the philosophy of science: what is scientificknowledge, and what constitutes confirmation of a knowledge claim? Pedersen (1999)discusses the answer to this question from two major philosophical schools: theReductionist / Formalist / Foundationalist school and the Holistic / Social / Relativistschool.

Table 1.1 – Relationship Between Hypotheses and Dissertation Sections

Hypothesis and Tasks SectionDiscussed

Section Tested

H1. Geometric reasoning algorithms formold design

Chap 3 Chap 4Chap 7Chap 8

H1.1 Basic elements and related algorithms §3.4 §4.4, §4.5,§7.2, §8.2

H1.2 Region based combining process andrelated algorithms

§3.5 §4.4, §4.5,§7.2, §8.2

H1.3 Reverse Glue Mold Constructionmethod and related algorithms

§3.6 §4.4, §4.5,§7.2, §8.2

H2. Design-for-manufacture method forRapid Tooling

Chap 5Chap 6

Chap 5Chap 7Chap 8

H2.1 Material geometric tailoring formulationand solution

Chap 5 §5.5, §7.3, §8.3

H2.2 Design for Rapid Tooling problemformulation

§6.1, §6.2,§6.3

§7.3, §8.4

H2.3 Design for Rapid Tooling solutionProcess

§6.4, §6.5 §7.4, §8.5


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Research validation is asserted to be a process of building confidence in itsusefulness. As stated the purpose of this research is to develop the “Rapid Tooling MoldDesign System” (RTMDS) and “Design for Rapid Tooling System” (DFRTS) to facilitateproducing functional prototypes for a set of part designs, which may have wide varietiesof geometries and requirements. The usefulness of the RTMDS and DFRTS is associatedwith whether the method is effective and efficient. Pedersen (1999) partitions thevalidation process into (1) Qualitative Process and (2) Quantitative Process.

Structural Validation – A Qualitative Process (Adapted from Pedersen, 1999)

Effective is referred to as being structurally valid. In context of design methodvalidation, this implies three things. It implies (1) accepting the ‘correctness’ of theindividual constructs constituting the method; (2) accepting the internal consistency ofthe way the constructs are put together in the method; and (3) accepting theappropriateness of the examples / case studies that will be used to verify the performanceof the method. The validity of the method constructs – individually and integrated –deals with the structural ‘soundness’ of the method in a more general sense, and istherefore denoted as Theoretical Structural Validity. The validity of demonstrating themethod with the chosen case studies deals with the structural ‘soundness’ of the methodfor some particular instances, and is therefore denoted Empirical Structural Validity, andboth ‘validity’ are evaluated qualitatively.

Performance Validation – A Quantitative Process (Adapted from Pedersen, 1999)

Efficient is referred to as being valid from a performance perspective. In the contextof design method validation, this implies three things. It implies (1) accepting that theoutcome of the method is useful with respect to the initial purpose; (2) accepting that thehypothesized constructs are contributing to making the outcome useful; and (3) acceptingthat the usefulness of the method is beyond the case studies. The validity of the methodperformance and the outcome performance which is based on documenting the usefulnessfor some particular instances, is denoted Empirical Performance Validity. Similarly, thevalidity of the method performance and the outcome performance which is assertedbeyond some particular instances, deals with generality and is therefore denotedTheoretical Performance Validity.

TheoreticalStructuralValidity

EmpiricalStructuralValidity

EmpiricalPerformance

Validity

TheoreticalPerformance

Validity

Figure 1.15 -Validation Square (Pedersen, et al.,, 2000).


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To perform the validation in a systematic manner Pedersen (1999) introduced a"validation square" (Figure 1.15). The validation square is applied in this dissertation totest the hypotheses. More specifically:

(1) Mathematical proofs and constructions of tools used in different steps of theRTMDS and DFRTS will be used to provide structural validation.

(2) Partial performance validation of the hypotheses will be performed bydetermining the usefulness of the different mathematical tools on simpleexamples.

(3) Further performance verification of the RTMDS and DFRTS will be providedusing integrated case studies.

In the mold design problem, math models of basic elements and their combiningprocess will be developed. Some of the properties of these math models will be formallyproven to provide theoretical structural validation of RTMDS. Mathematicalconstruction of the tools along with some of the well-matured mathematical concepts,used in both mold design and DFM problems, will provide partial structural validation ofthe hypotheses.

Simple examples are used to test the usefulness of the mathematical tools and thealgorithms developed for mold design and DFM problem of Rapid Tooling. Althoughthese examples will be simple, results obtained from applying developed tools andalgorithms will be used to validate the performance of RTMDS and DFRTS.

Two case studies are implemented to further validate the performance of RTMDSand DFRTS, and to validate and verify this research. The two case studies are:

• A robot arm design• A camera roller design

Validity of hypotheses 1 and 2 is addressed in different chapters of the dissertation,which are summarized in Table 1.2. With the validation strategy for this dissertationpresented in this section, the contributions from this research are presented.

Table 1.2 - Validation of Hypotheses 1 and 2 addressed in different chapters

Chap 3 Chap 4 Chap 5 Chap 6 Chap 7 Chap 8 Chap 9

Hypotheses 1Theoretical

Performance ValidityX X

Theoretical StructuralValidity

X XEmpirical Structural

ValidityX X X

EmpiricalPerformance Validity

X X X

Hypotheses 2Theoretical

Performance ValidityX X

Theoretical StructuralValidity

X XEmpirical Structural

ValidityX X X

EmpiricalPerformance Validity

X X X


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1.3.3 Contributions from the Research

Rapid Tooling technologies enable the injection molding of near production-representative prototypes (accuracy and material properties, etc.) to be made within lesstime and in a lower cost. But the current state of research does not address the issue ofhow to reduce the lead-time in the usage of Rapid Tooling. The basic idea of this work isthat by using mold design methods, mold inserts for a given part can be designed quicker;and by using design-for-manufacture methods, the time of iteration between designer andmanufacturer can be reduced. The hypotheses and sub-hypotheses, taken together, definethe research presented in this dissertation and hence the contributions from the research.

Contributions related to Computer-aided Mold Design:

• A problem definition for multi-piece mold design and a solution process for theformulated problem.

• The notions of concave region and convex face in mold configuration design anda means of generating them for a given part.

• A combining approach to generate regions from the basic elements based onlinear programming and mold design knowledge.

• A mold construction method based on the reverse glue operation and a geometricreconstruction problem for any mold base.

Contributions related to Design-for-Manufacture:

• A demonstration of transferring part of DFM responsibility (geometric tailoring)to the manufacturer is feasible for Rapid Tooling.

• The development of design decision templates which are used as the digitalinterface between the designer and manufacturer for Geometric Tailoringproblem.

• The development of an integrated system for Design for Rapid Tooling problemand a solution process with three stages.

Contributions related to Rapid Tooling:

• A tool to aid the mold designer in designing molds for a part design.

• An integrated system to aid the manufacturer to change part design and processplanning according to design and manufacturing requirements.

• Several industrial cases are investigated in the system which is importantcomponents of the Rapid Tooling TestBed.

This being the first chapter of the dissertation, these contributions cannot besubstantiated; therefore, they are revisited in Section 9.2 after all of the research findingshave been documented and discussed. An overview of the dissertation is presented next.


30

1.4 OVERVIEW OF DISSERTATION

To facilitate this discussion, an overview of the chapters in the dissertation is shownin Figure 1.16. Having been lain the foundation by introducing the research questionsand hypotheses for the work in this chapter, the next chapter contains a literature reviewof research related to different product family avenues. Research areas that are reviewedinclude: (1) mold configuration design methods (Section 2.2); (2) mold constructionmethods and tools (Section 2.3); (3) CAD representation (Section 2.4); (4) Design-for-Manufacture strategies and approaches (Section 2.5); and (5) design technologies(Section 2.6). A discussion on how these concepts relate to the overall objective of the

Chapter 1Functional Prototype Using RapidTooling

Chapter 2A Literature Review: Mold Designand Design-for-Manufacture

Chapter 3Automatic Design andConstruction of Multi-piece Molds: Methods

Chapter 4RTMDS and its Usages

Chapter 5Formulating Design Requirementsas Geometric Tailoring

Chapter 6Design for Rapid Tooling: ADecision-Based Method

Chapter 7Functional Prototypes of a Robot Arm

Chapter 8Functional Prototypes of a Camera Roller

Chapter 9Closing Remarks

Pro

blem

Iden

tification

Develo

pm

ent

of

Meth

od

Testin

gan

dV

erification

Clo

sure

# Summarize research findings,contributions, and limitations

# Identify future work

# Intorduction, Motivation, Foundation# Research question, hypotheses

# Review different mold design methodsfrom existing literature

# Review different DFM methods andsolving techniques

# Present technological foundations

# Demonstrate usage of MDRTS andDFRTS

# Provide verification of methods

# Demonstrate usage of MDRTS andDFRTS

# Provide further verification of methods

# Problem definitions and steps of MPMDM# Basic elements and generation method# Combining critera, process and algorithms# Mold construction method and algorithms

# Implementation, limitation and testexamples

# Provide proof of concept and initialverification of method

# Modeling of Geometric Tailoring# Design decision templates with design

representation# Proof of concept and initial verification of

method

# Modeling of DFRTS# MPGT decision template and problem# A solution strategy and process for DFRTS

Figure 1.16 - Overview of Dissertation Chapters


31

dissertation is presented in Section 2.1.

The general objective and steps of Multi-Piece Mold Design method are presented inChapter 3. Section 3.1 gives an overview of the problem definition, highlighting theissues and design steps associated with the method. The basic elements of MPMDM andan approach to generate them are presented in Section 3.2. The criteria and algorithm forcombining the basic elements into regions are discussed in Section 3.3. Finally, a moldconstruction method based on the reverse glue operation and glue faces reconstruction ispresented in Section 3.4.

In Chapter 4 the implementation and initial testing of the mold design methods andalgorithms are discussed. The overview of RTMDS including software tools, modulesand limitations are presented in Section 4.1. Two simple test examples are discussed inSection 4.4 for the partial testing of the system. Three more complicated industrial casesare presented as the further verification of the RTMDS in Section 4.5 followed by asummary of the system presented in the chapter (Section 4.6).

Material geometric tailoring problem for Rapid Tooling is presented in Chapter 5.The properties of Rapid Tooling are discussed in Section 5.1. Related to the principle offunctional testing and similarity methods, the fundamentals of geometric tailoring arepresented in Section 5.2. Material geometric tailoring and MGT decision template arediscussed in Section 5.3. The usage of the MGT decision template, including formulatingfunction properties and solution approaches, is presented in Section 5.4. Three testexamples are discussed in Section 5.5 for the partial testing of the method.

After the material geometric tailoring study is completed in Chapter 5, the design forRapid Tooling problem, which consists of geometric tailoring, mold design and processplanning, is discussed in Chapter 6. An overview of the DFRTS is presented in Section6.1. Software tools related to the modules of the system are introduced in Section 6.2.The problem formulation of design for Rapid Tooling is presented in Section 6.3. Basedon the solution process for the DFRTS described in Section 6.4, a comparison of thecurrent usage of RT and DFRTS is provided in Section 6.5.

Application of RTMDS and DFRTS to a robot arm and a camera roller to illustratethe use of the method in mold design and DFM is presented in Chapter 7 and 8respectively. For each case study, an overview of the problem is given along withpertinent analysis information, the steps of the methods are performed, and theramifications of the results are discussed.

Chapter 9 is the final chapter in the dissertation and contains a summary of thedissertation, emphasizing answers to the research questions and resulting researchcontributions in Sections 9.1 and 9.2, respectively. Limitations of the methods andpossible avenues of future work are discussed in Section 9.3 and Section 9.4. Finally,some closing remarks are given in Section 9.5.

A pictorial overview of the dissertation is illustrated in Figure 1.17. It proceeds frombottom to top, beginning with the foundation provided in this chapter: ComputationalGeometry and Decision-Based Design. This figure provides a road map for thedissertation, and it is referred to at the end of each chapter to help guide the readerthrough the work as the research progresses from chapter to chapter.


32

Chapter 9: Achievements and Recommendations

Chp 7: Prototypes of aRobot Arm

Chp 8: Prototypes of aCamera Roller

Chp 3: Rapid Tooling MoldDesign method

R1

R2

Rk

R3

F1

F2

F3

F4F5

F6

Fn

PL1

PL2

PD1

PD2

Part P

F1

F2

F3

F4

F5

F7

F8

Fn

Part P

(1) (2)

(3)

F1

F2

F3

F4

F5

F7

Fn

Part P

F8F8 F6

F6

F7

F9 F9

F9 M1

M2 Mk

Mold Base

PD1

F1

F2

F3

PD2

F4F5

F7 Fn

Chp 4: RTMDS and its Usage

Chp 5: Geometric Tailoring

GivenAnalternative tobe improvedthroughmodification; Assumptionsusedtomodelthedomainof interest.

Thesystemparameters:n number of systemvariables p+q number of systemconstraintsp equalityconstraints q inequalityconstraintsm number of systemgoals Gi(X) systemconstraint functionfk(di ) functionof deviationvariables tobeminimizedatpriority level kfor thepreemptivecase.

FindValues for thesystemvariables Xi i = 1, ... , nValues for thedeviationvariables di

-, di+ i = 1, ... , m

SatisfySystemconstraints (linear, nonlinear)

gi(X) = 0; i = 1, ..., p gi(X)≥ 0; i = p+1, ..., p+qSystemgoals (linear, nonlinear)

Ai(X) + di- - di

+= Gi ; i = 1, ..., mBounds

Ximin≤ Xi ≤ Xi

max; i = 1, ..., nDeviationvariables

di-, di

+ ≥ 0; di-. di

+= 0; i = 1, ..., mMinimize

Preemptivedeviationfunction(lexicographicminimum)Z=[ f

1(d

i-,d

i+),..., f

k(d

i-, d

i+)]

ArchimedaindeviationfunctionZ= W

i(d

i

− +di

+) where Wi=1, W

i≥0∑∑

§2.2 §2.3 §2.4 §2.5 §2.6

MoldConfiguration

Design Methods

Mold ConstructionMethods and

Tools

CADRepresentation

DFM StrategiesDesign

Techniques

Foundations: Computational Geometry & Decision-Based Design

Chp 6: Design for Rapid Tooling

ParametricCAD Model

of Part

The Designer's MPGTProblem FormulationGiven

Part DesignFind

Design ParametersSatisfy

ConstraintsGoals

MinimizeDeviation

Part DesignRequirements

A. Rapid Tooling MoldDesign System

Parting DirectionsParting LinesParting SurfacesMold Piece Number

Parametric CADModels of Mold Pieces

C. Injection MoldingProcess Analyzer

Draft AngleRib Height/width RatioPart ThicknessMold Life

B. RP Process Planner

Surface FinishAccuracyCostTime

E. Rapid ToolingCost PredictorTimeCost

Part Design

(Chp3 & 4)

The RP ProcessCompromise DSP

GivenMold Design

FindRP Process Parameters

SatisfyConstraintsGoals

MinimizeDeviation

The IJM ProcessCompromise DSP

GivenMold Design

FindIJM Process Parameters


MinimizeDeviation

D. CoompromiseDSP Solver

Tailored Part Design andrelated Mold Design, RP and

IJM Process Parameters

Input and Output

Processor

C-DSP Template

(Section 6.2.1) (Section 6.2.2)

(Section 6.2.3)

(Section 6.3)

(Section 6.4& 6.5)

Figure 1.17 - Pictorial Overview of the Dissertation

Chapter 2 – A Literature Review: Mold Design and Design-for-Manufacture

33

CHAPTER 2

A LITERATURE REVIEW: MOLD DESIGN AND DESIGN-FOR-MANUFACTURE

In this chapter a survey of relevant work in different aspects of mold design anddesign-for-manufacture will be presented. In Section 2.1 a more thorough description ofhow the research topics covered in this chapter are relevant to this dissertation isprovided. Mold configuration design methods proposed by different researchers fordetermining parting directions, parting lines, parting surfaces, and undercut features arepresented in Section 2.2. Mold construction methods and commercial mold designsoftware systems are investigated in Section 2.3 for constructing mold pieces. A reviewof representation and manipulation methods of solid models is presented in Section 2.4.High-level CAD representation methods are also discussed in this section to provide abackground for the decision templates in Chapter 5. Different DFM metrics and methodsare presented in Section 2.5. The difficulties associated with these methods are alsohighlighted. Three design technologies, the Decision-Based Design, Compromise DSP,and Robust Concept Exploration Method, are reviewed in Section 2.6. They providefoundation for the design for Rapid Tooling system in Chapter 6. In the final section ofthis chapter (Section 2.7), a summary of the material presented and a preview of what isnext are provided.


34

2.1 TOPICS IN CHAPTER

In Section 1.2 motivation to mold design and design-for-manufacture (DFM) wasgiven along with the foundations of the research approaches used in this dissertation. Inthis chapter a literature review on different topics related to mold design and design-for-manufacture will be presented (Figure 2.1). Topics covered are:

(1) Mold configuration design methods: Methods used to determine partingdirections, parting lines, parting surfaces, and undercut features, have beenpresented by different researchers. They are reviewed in this section as a startingpoint for developing a multi-piece mold design method. This topic is related toHypotheses 1.1 and 1.2.

(2) Mold construction methods and tools: Methods used to construct mold pieceshave been identified and reviewed. Several commercial mold design softwaresystems are also investigated on their splitting approaches. This topic is related toHypothesis 1.3.

(3) CAD representation: Both the representation and manipulation of solid modelsare reviewed. They provide a base for developing a computer-aided mold designmethod and related algorithms. The solid modeling approaches are related toHypotheses 1.1 to 1.3. In this section high-level CAD representation methods arealso investigated to provide a background for decision template, which is relatedto Hypothesis 2.1.

(4) DFM strategies and techniques: Both process-oriented methods and information-orientation methods are reviewed. The DFM approaches in VLSI (Very LargeScale Integrated Circuits) design are also investigated to provide a backgroundfor the DFM system developed in this research. The DFM approaches provide abackground for Hypothesis 2.1 and 2.2.

(5) Design technologies: Several design technologies, including the Decision-BasedDesign, Compromise DSP, and Robust Concept Exploration Method, arereviewed. They provide foundation for the design for Rapid Tooling system,which is related to Hypotheses 2.2 and 2.3.

§2.2 §2.3 §2.4 §2.5 §2.6

MoldConfiguration

Design Methods


Tools

CADRepresentation


Technologies


H1 (H1.1, H1.2, H1.3) H2 (H2.1, H2.2, H2.3)

Figure 2.1 - Literature review topics and relationship with Hypotheses


35

With the importance and relationships of the topics covered in the literature reviewhighlighted, Mold configuration design methods are discussed in the next section.

2.2 MOLD DESIGN METHODS AND ALGORITHMS

The methods by which information is extracted from the model for use by the molddesign software can be collectively termed geometric reasoning. There appears to be nodefinitive accepted definition of geometric reasoning, although the term has been used forthe last fifteen years or so. Woodwark (1989) explained it in the following manner.

While computer graphics and image processing have been with us for thirty years orso, and look like mature disciplines, it is really only the algorithmic processes thathave been computerized. There are many more complex, less easily pinned-downthings that are still want to do with shape information and cannot. In the organizationof this conference, we called these activities Geometric Reasoning.

Jared, et al., (1994) give a refined definition of Geometric Reasoning as “the processby which application requests related to geometric properties and attributes are satisfiedfrom a geometric properties and attributes are satisfied from a geometric model usingboth inferential and algorithmic method”.

Geometric reasoning is one method of providing answers for solving variety ofgeometries in design for Rapid Tooling. In this and the next section some geometricreasoning methods on automatic mold design are presented based on literature survey.

The automation of mold design for injection molding process is studied in manypublications. However, works scatter in the determination of parting direction, partingline, parting surface, and undercut detecting individually. In this section, the reviewedworks are arranged according to their focused area in the mold design. The definitions ofparting direction, parting line, paring surface and undercut are given in mold designhandbooks (Rosato and Rosato, 1995), and hence not repeated here.

2.2.1 Parting Direction

In the determination of parting direction, there are mainly three kinds of approaches.

(1) Compute the exact global accessibility cones based on V-map.

A representative problem formulation of this approach is given by Chen, et al.,(1993). It formulates demoldability as a visibility problem. A notion of visibility map(or V-map) of surfaces is presented to define the possible parting direction for pockets ofthe given part, which are some spherically convex polygons. By computing theintersection of V-maps, the problem of finding a pair of parting directions that minimizesthe number of cores is transformed to find a pair of antipodal points p and –p thatmaximize the number of V-maps which contain either p or –p.

Based on this formulation, several papers presented other approaches for theselection of parting directions of mold (Weinstein and Manoochehri 1996; Vijay, et al.,1998). Visibility map is also used in other areas like NC machining (Woo, 1994; Gupta,et al., 1996). The basic element of the approach is a “pocket”, which can be generated bysubtracting a part from its convex hull, or by testing adjoining surfaces as shown in(Weinstein and Manoochehri, 1996).


36

(2) Compute the exact global accessibility cones based on automatic molding featurerecognition.

Gu, et al. (1999) use a universal hint-based feature recognition algorithm torecognize all features of a molded part. Their features included holes, steps, pockets,protrusions, etc. Different feature types have their candidate parting directions (CPD)stored in the system. Finally each CPD is evaluated using an object evaluation functionand the CPD with maximum evaluation value is selected as the optimal parting directionof the part.

Fu, et al. (1999) classify undercut features (features that prevent the removal of apart from the molds along the parting directions) as Inside Internal Undercut, OutsideInternal Undercut, Inside External Undercut, and Outside External Undercut. Based onthe undercut feature characteristics and geometric entities (three-edge, four-edge andmore than four-edge), algorithms to recognize them are presented in the paper.

More recently, Yin, et al. (2001) present their approach to construct core and cavitybased on the recognized undercut features. For a given part, a volume-based featurerecognition method using non-directional blocking graph is developed to recognizeundercut features. Then the optimal parting direction is determined by minimizing thenumber of undercuts for different candidate parting directions.

Krishnan (1997) describes automated two-piece and multi-piece mold design forinjection molding. The part is constructed by stacking 2.5D primitives called C-entitiesalong the Z direction through either a Constructive Solid Geometry (CSG) or DestructiveSolid Geometry (DSG) operation. Since the primitives considered are only 2.5D solidthat are stacked along the Z direction, the complexity of the part is limited. The partingsurface directions are also constrained to be along the X-axis or Y-axis direction.

Dhaliwal, et al. (2000) described a feature-based approach to automated design ofmulti-piece sacrificial molds which are to be manufactured by CNC machining.However, how to separate mold from the part is not considered since the authorconsidered only sacrificial molds, which can be destroyed after the part has beenproduced.

(3) Compute approximate global accessibility cones by sampling a set of discretedirections.

Hui and Tan (1992) heuristically generated candidate parting directions from normalvectors of planar faces and from center-lines of holes and bosses. Two criteria, theblockage factor and the performance value, are used to determine the main partingdirection and subsidiary parting direction. To evaluate the geometry of an undercut, Hui(1996) developed a partitioning scheme to subdivide the cavity solid of a componentalong a given direction. In the search for main and side core directions, the search spaceis the set of all normals to individual faces of the object and the opening of cavity solid.A search tree is built for side core selection.

Urabe and Wright (1997) selected three principal coordinate directions as thecandidate parting directions, then calculate the number of undercuts, the projected area,the number of cone surface for each candidate direction and used them as major mold


37

factors to determine the main parting direction. This approach was limited to simple 3Dparts.

Lu and Lee (2000) proposed an approach for analyzing the interference element andrelease direction in die cast or injection molded components. First a three-dimensionalray-detection method is used to recognize and extract the interference elements. Thendistribution of the release directions can be computed, and the candidate releasedirections can be prioritized based on the minimization of the number of side cores.

Review of Methods:

Three kinds of approaches are reviewed in this section. It is well know that featurerecognition of a part is rather difficult, especially as feature interactions are ratherimportant in the multi-piece mold design. Sampling a set of discrete directions cannotguarantee that a suitable mold design will be find. Therefore this research will utilize theapproach of computing the exact global accessibility cones based on V-map. However,the basic elements used in this research are different from those of the reviewedapproaches.

2.2.2 Parting Line

There are few published works on the determination of parting lines. Pye (1989)suggested that the parting lines should be around the position of maximum dimension ofa product when viewed in the draw direction of an injection molding process. The basicelement in judging parting lines is part faces in all the approaches. Also in theapproaches, either the parting direction is already set, or the parting direction will notcause any undercuts (because of convex polyhedra).

Tan, et al. (1988) proposed a parting line generation method for a triangular sub-division of the product model’s surfaces. In the method, a draw direction is selected first.Then, the algorithm triangulates and classifies the surfaces into visible faces and invisiblefaces. Since the parting lines are located in the outermost and the innermost boundariesof the product model when viewed in the draw direction, they are those edges on theproduct model separating the visible faces from the invisible faces. The limitation is thatit does not apply to models with non-drafted faces (faces parallel to the mold openingdirection).

Ravi and Srinivasan (1990) proposed the sectioning and silhouette methods forparting line generation. The section method locates the parting line by computing theintersection of the product model with a plane normal to the draw direction. Thesilhouette method is capable of dealing with non-planar parting surfaces. It projects theproduct model onto a plane normal to the draw directions to obtain the projectedboundary of the product. The projected boundary is then swept back along the drawdirection. The intersection between rays from the projected boundary and the surfaces ofthe product model form portions of the parting line.

Serrar (1995) developed a semi-automatic method for selecting parting loops. In thismethod, the user first selects an edge from the parting loop. Then the system finds thebranches (adjacent edges), and successively highlights these branches until the userconfirms which branch belongs to the parting loop. This process is repeated until theloop is closed.


38

Wong, et al. (1996) used a slicing strategy to locate the parting lines of a productmodel along a draw direction. A recursive uneven slicing method is developed to locateseveral parting surface for further evaluation. The approach is primarily proposed to dealwith free-form surfaces in product design.

Majhi, et al. (1999) discussed the problem of computing an undercut-free parting linethat is as flat as possible in mold design for a convex polyhedron. Two flatness criteriafor parting lines are given and algorithms are presented to compute a parting line basedon the criteria.

Review of Methods:

In the reviewed approaches, the parting directions have already been determined.Therefore the parting lines are determined only for the given parting direction. Also onlytwo mold pieces were considered in these approaches. In this research, a mold designmethod is developed in which parting lines and parting directions are determined at thesame time. The method is also suitable for multi-piece mold design.

2.2.3 Parting Surface

Parting surface is the surface separating the mold halves. Ganter and Truss (1990)introduced a method for locating parting surface based on a set of evenly distributedpoints lying on the surface of a sphere enclosing the object.

Ravi and Srinivasan (1990) presented a comprehensive decision model for selectionparting surfaces. A total of nine criteria were used, including projected area, flatness,draw distance, draft, number of undercuts, volume of flash, and dimensional stability.They presented algorithms for most of these criteria, but did not present deterministicalgorithms for generating a variety of candidate parting surfaces or parting directions.Although it was not automated, the comprehensiveness of their approach aids designersin decision making.

Tan, et al. (1988) presented a parting surface generation method by projection. Aftergetting the parting lines for a given parting direction, the algorithm creates planar partingsurface elements for each edge of the outer loop generated by a convex hull algorithm.Therefore the whole parting surface of the mold is segmented and not in one planeanymore. This approach is simple, however, the results do not fit with the moldingheuristics, that is parting surface is better to be planar to increase the shut-off force.

Nee and coauthors present the core and cavity creating approach of IMOLD@ (Nee,et al., 1998; Nee, et al., 1999). After the parting lines are determined, all the partingedges are classified into different groups and their extruding directions are determined.The parting surfaces are the unions of the surfaces by extruding the parting edges to theboundary of the core and cavity in directions perpendicularly outwards to the partingdirection.

Review of Methods:

Although several methods have been proposed for the determination of partingsurface from parting lines, none of them explored theoretic foundations of generatingparting surfaces. In this research, the generation of parting surface (in this dissertationthey are called glue faces because of the reverse glue operation) is transformed into a


39

geometric reconstruction problem. A formal problem formulation and several algorithmsare also presented.

2.2.4 External and Internal Undercut Detection

Undercut is one of the most important considerations in the mold design. Severalmethods are developed for their recognition from given CAD models.

Hui and Tan (1992) choose points on the edge of a solid as a set of test points. Thesetest points are then checked for obstruction in the direction considered. A technique thatis similar to that of hidden-line removal is used. A semi-infinite ray that originates fromthe point is cast in the parting direction under evaluation. A series of test points are thengenerated on the ray to decide if the point is obstructed.

Rosen (1994) focused on finding undercuts and determining whether or not they areexternal or internal. The shadow casting approach is used to find undercuts. However,no parting lines are considered. So the results are actually only potential undercuts.

Shin and Lee (1993) developed a procedure to identify the interference facesbetween the product and the mold from the core plate (or cavity plate) alone withoutconsidering the product. Based on the results, an algorithm based on Euler operation isdeveloped to generate the side cores and the corresponding core and cavity plates of amold. In the approach, core plate need to be specified by a designer and glued to thegenerated cores manually.

Fu, et al. (1999) presented undercut feature definition, classification, parameters andthe recognition criteria. It classified undercut feature as Inside Internal Undercut, OutsideInternal Undercut, Inside External Undercut, and Outside External Undercut. Based onthe undercut feature characteristics and geometric entities, algorithm for classificationand recognition are presented.

Stefano (1997) propose an approach based on the analysis of the topology of concaveobject parts to identify and extract features from a solid model representation for castingprocess. Two topological invariants are defined for feature classification. Extractedfeatures can then be used for automatic core pattern development and mold design.

Review of Methods:

Undercuts are main considerations in the determination of mold design. In thisresearch they are considered in the combination criteria in the mold design process.Therefore no feature recognition algorithms are considered, which saves a lot of work.

2.2.5 Synthesis Approaches of Basic Elements

As stated in Section 2.2.1, the approaches for determining parting directions useddifferent basic elements, which include pockets (Chen, et al., 1993; Weinstein andManoochehri, 1996; Vijay, et al., 1998), features (Gu, et al., 1999; Lu and Lee, 2000;Yin, et al., 2001), and faces (Hui and Tan, 1992; Urabe and Wright, 1997). All theapproaches use a similar synthesis approach to evaluate a parting direction. That is,

(1) For each candidate direction dk, V(dk) =i∑ weight_factori × mold_factori;


40

(2) Parting direction d: V(d) = Min [V(dk)].

In Step (1), mold factors are some rules used to determine the parting directions,which may include number of undercuts, projected area, draft angles, etc. Weight factorsdistinguish the importance of one mold factor relative to another. The assignment ofthese weight factors is based on how the corresponding mold factor affects the cost,quality, and productivity of the mold.

Depending on how to evaluate mold factors, there are two kinds of approaches.

(i) Evaluate mold factor for each pocket/features.

In (Chen, et al., 1993), candidate directions are the directions computed from the V-map of pockets. Only one mold factor, number of undercuts, is considered. So the moldfactor for each candidate direction is actually the number of V-maps that do not containthe candidate direction. In (Weinstein and Manoochehri, 1996; Lu and Lee, 2000; andYin, et al., 2001), a similar approach is used in evaluating parting directions. Besidesnumber of undercuts, Gu et al. (1999) also considers the projected area and thickness ofthe molded part, which are evaluated based on the bounding box of a part.

Since face connectivity and faces that do not belong to pockets or features are notconsidered in these approaches, they can explore the combinations of pocket/featurerather quickly, usually with the aid of visibility map. However, this may cause problemsin constructing mold pieces for non-connected faces. Also if we want to consider moremold design knowledge, we need to add more mold factors. Since several mold factorslike minimal draft angle are also related to faces that do not belong to pockets or features,it is not clear how they should be added.

(ii) Evaluate mold factor for all faces.

In this approach, mold factors are evaluated for all faces. So more mold factors canbe considered. Urabe and Wright (1997) consider boxed area, projected area, number ofno-hidden faces, number of undercuts and cone surfaces. Hui and Tan (1992) considernumber of undercuts and projected area. A more comprehensive decision model ispresented in (Ravi and Srinivasan, 1990) for the selection of parting surfaces for castingparts. A total of nine factors were used, including projected area, flatness, draw distance,draft, number of undercuts, volume of flash, and dimensional stability. Relatedapproaches and expressions for most of these criteria are also presented.

Since it is quite time-consuming to test these mold factors for all faces, only a smallnumber of candidate parting directions or parting surfaces are considered in theseapproaches. Urabe and Wright (1997) consider the principal axis directions as thecandidate directions. Hui and Tan (1992) consider the normals of planar surfaces and theaxis of a cylindrical or helical surface as the candidate directions.

Review of Methods:

In this research the first synthesis method is used. However, the basic elements usedin the mold design process are different from pockets or features, as discussed in Section3.4.2. In the synthesis process mold design knowledge is also considered in this research.


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2.2.6 Intersection of V-Map

Woo (1994) defines “Visibility Map (V-map)” from Gauss Map to formulate thevisibility problem. For a planar surface F, the V-map of F is a hemisphere centered onthe unit outward normal. So the V-map of a region is the range formed by theintersection of the allowable draw ranges for each individual surface (Figure 2.2.a). Bycalculating the intersection of V-maps, the range of feasible solutions can be representedas a subset of a spherical surface.

Since all V-maps and their intersections are calculated on spherical surfaces, severalapproaches and algorithms based on spherical polygons have been presented for differentapplications (Chen and Woo, 1992; Woo, 1994; Gupta, et al., 1996; Kweon andMedeiros, 1998). Using the central projection to convert a spherical problem to a planarone, Chen and Woo (1992) present four spherical algorithms, which include detection ofconvexity on the sphere, computation for spherical convex hull, determination of thespherical convexity of a union, and the intersection of hemispheres. They are used innumerical control (NC) machining planning. Kweon and Medeiros (1998) utilize V-mapto represent accessible directions for measurements of tolerance. The concept of V-mapdimensionality is proposed to provide a method of clustering V-maps. Related algorithmsand data structures are presented and tested for CMM inspection.

Feasible withdrawaldirections Draw-range

CR

(a) V-map and draw-range

(b) A Case given in (Dhaliwal et al., 2000)

Figure 2.2 – Two Examples of V-map.


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The algorithms and related data structures to calculate the intersection of V-maps onspherical surfaces are rather complicated. Typically, computations take long timeespecially for a complex part. As an example, most recently Dhaliwal, et al. (2000)present an algorithm for computing exact global accessibility cones for various faces of apolyhedral object. One example given in the paper, which is shown in Figure 2.2.b,would take 77 minutes on a Sun Ultra10 workstation.

Balasubramaniam, et al. (2000) developed a method by taking advantage ofcomputer graphics hardware to generate 5-axis roughing tool paths directly from atessellated representation of a body. Graphics cards make use of the depth-bufferimplemented using hardware to perform fast hidden surface removal and render theobject in a given scene. If all the individual faces on the object have been assigneddifferent colors, then the accessibility of each face in a given direction can be detected byrendering the object using the given direction as the viewing direction, and querying thecolors that appear on the pixel map after rendering.

Review of Methods:

The idea of V-map is also used in this research. However, the calculation of drawranges is omitted. Instead a parting direction is calculated directly from the given facesby solving an optimization problem. The optimization problem is further simplified byapproximating the spherical surface by a set of triangles. Therefore the intersection of V-maps of a set of faces can be determined by solving a linear problem. The approachpresent in this research is much easier and quicker than the spherical algorithmspresented in the reviewed approaches.

2.2.7 Detection of Non-Drafted Surfaces

Surfaces parallel to the parting direction must be drafted at least a minimum draftangle for injection molding process (Rosato and Rosato, 1995). In order to guarantee afully drafted model, it is critical to automate the detection of non-drafted surfaces.Relatively few published works cover this topic. Serrar (1995) presents a non-draftedsurface detection algorithm for a given part. In his approach, the parting direction isgiven as the input. So the only task is to find faces with outward normals forming anangle between 90o-min_draft_angle and 90o+min_draft_angle with the parting direction.However, since draft angle of a face is tightly related to a given parting direction andparting lines, it is better to detect non-drafted surfaces in the determination process ofparting direction and parting lines.

Review of Methods:

In the reviewed approaches, the parting direction is given as the input, which greatlysimplifies the problem. In this research, the detection of non-drafted surfaces isintegrated with the determination of parting direction and parting lines, which is moregeneral and useful.

After the methods on computer-aided mold configuration design are presented, themethods and commercial tools on mold piece construction are described in the nextsection.


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2.3 MOLD CONSTRUCTION METHODS AND TOOLS

After parting direction and parting lines are selected (Section 2.2), mold designer canuse general CAD software such as Pro/Engineer, SolidWorks, and CATIA to constructmold halves manually. However it is a tedious process that takes a long time. Currentlytwo approaches are proposed for automatically splitting the core and cavity inserts fortwo-pieces mold design.

2.3.1 Approach Based on Extending Parting Lines

A mold base can be cut into two pieces by a parting surface. For parting lines not ina plane, Tan, et al. (1988) presents a parting surface generation method. After getting theparting lines for a given parting direction, an outer loop and inner loops are generated. Aconvex hull algorithm is applied to the outer loop. Each edge of the hull is projected toan adjacent side face of the mould block. The projection direction is perpendicular to theparting direction but parallel to the surface normal of the side face of the mould block.All planar faces generated by projecting hull edges can form a parting surface for thepart. Shin and Lee (1993), Serrar (1995) and Nee, et al. (1998) use a similar approach toform the parting surface to split mold base into two halves.

Review of Methods:

This approach is quite straightforward. Extending the given parting lines outwardinto faces can split mold base into two mold pieces. However, for non-flat parting lines,the parting surface generated by this approach is also not flat. This is not accordant withthe best practice of mold design, that is parting surface is better to be planar to decreasethe manufacturing complexity and to increase the shut-off force to reduce material flash(Ravi and Srinivasan, 1990). For example, for a part as shown in Figure 2.3.a, one moldpiece generated by the approach of extending parting lines is shown in Figure 2.3.b, andthe mold piece generated by algorithm Two_Mold_Piece_Generation (Section 3.6.3) isshown in Figure 2.3.c. Compared with the mold piece in Figure 2.3.b, the mold piece inFigure 2.3.c is cheaper to fabricate because of less benchwork, and its parting surface hashigher accuracy and surface finish. Consequently we can expect less material flash in itsinjected parts.

2.3.2 Approach Based on Sweeping

Conceptually removing an injected part from a mold in a parting direction is similarto sweeping part faces in the same parting direction. Hui and Tan (1992) described an

(a) A part (b) Mold piece 1 (c) Mold Piece 2

Figure 2.3 – A Part with Mold Pieces Generated by Two Approaches.


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algorithm using sweep operations and Boolean operations to generate mold core andcavity. First sweeping the mold part in the parting direction to generate a solid. Thenusing two mold plates to subtract each end of the solid can generate two mold pieces.The algorithm does not consider the internal parting lines. So for a shape with a throughhole, the algorithm will not generate the desired mold pieces. Urabe and Wright (1997)also presented a mold construction method based on the sweeping of FACE_STRUCTinto BODYs. Then they are united with plates and mold walls to form core and cavity.

Review of Methods:

The sweeping approach has two problems:

(1) The mold construction process takes a long time by sweeping each face anddoing Boolean operations with mold plates, especially for a complex part. Forexample, for an industrial part with normal mesh size as shown in Figure 4.26,there are more than 5493 faces. Among them if only half faces need to beswept, it is quite easily to take several hours to generate the mold pieces by a PCmachine.

(2) It is well known that Boolean operation for coincident geometry is a verydifficult problem. Most CAD packages still cannot handle it properly,especially for the subtraction of two bodies with many vertices, edges or facesin exactly same positions. So although the sweeping approach is theoreticallyfeasible, one may meet problems related to Boolean operations in theimplementation of the approach.

2.3.3 Industrial Approach

Currently there are several commercial mold design software systems that canautomatically split core and cavity for a part. The author investigated six leading systemsas listed below. For each system, an example is given to show the splitting approach ituses.

(1) MoldWizard

Unigraphics/MoldWizard is a highly automated product for designing plastic molds.It incorporates industry best practices to guide users through the steps required toconstruct a mold. Some brief steps for constructing core and cavity for an industrial partare shown in Figure 2.4. First parting lines for the given part are identified by theinteractive input from designers (Figure 2.4.b). Then for each parting edges a sweepingdirection is specified and parting surfaces are generated (Figure 2.4.c). These partingsurfaces together with others generated from inner loops can cut mould box into twomold pieces. Figure 2.4.d shows one of the mold pieces.


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(a) An industrial part (b) Parting lines of the part

(c) Parting surface to do Boolean operation (d) One of mold pieces

Figure 2.4 – Mold Construction Process of Unigraphics/MoldWizard for an Industrial Part.

Figure 2.5 – An Example Mold Design Given by Magics RP.


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(2) Magics RP

Materialise (www.materialise.com) develops a Rapid Tooling module in its MagicsRP system, which automates the design of the insert tool. Figure 2.5 shows an examplegenerated by the system.

(3) IMOLD

IMOLDTM is a supplementary program for Unigraphics and SolidWorks. It has amodule, Core/Cavity Builder, to handle the paring of cores and cavities for both solid andsurface product models. An example generated by the system is shown in Figure 2.6.

Figure 2.6 – An Example Mold Design Given by IMOLD.

Figure 2.7 – An Example Mold Design Given by Moldplus.


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(4) Moldplus

Moldplus (www.moldplus.com) is a supplementary program for Mastercam®. It cangenerate parting surfaces for given parting lines. Figure 2.7 shows an example of partingsurface.

(5) QuickSplit

QuickSplit is a splitting tool developed by Cimatron Ltd. (www.cimatron.com) thatcan separate core and cavity with sliders and inserts. Figure 2.8 shows an examplegenerated by the system.

Figure 2.8 – An Example Mold Design Given by QuickSplit.

(a) Part (b) Mold Design

Figure 2.9 – An Example Mold Design Given by I-DEAS.


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(6) VGXTM Core/Cavity Design

VGXTM Core/Cavity Design is the mold design model of I-DEAS. An example partand its mold design are shown in Figure 2.9.

Nee (1999) presents the generation approach of parting surface used in IMOLD,which is based on extending parting lines. For other systems, the author did not find anypublished works related to their splitting approach. However, according to the moldresults generated by the systems, it seems that all the systems use the approach based onextending parting lines.

Test reports for each system are not given by its software company nor found in anypublished works. Hogarth (1999) describes a test done by Minco Tool & Mold (Dayton,OH) to split a part with more than 5000 faces. The splitting process byUnigraphics/MoldWizard took four hours. In the same article, Cimatron claims a 90%saving of core/cavity splitting time by using Quick-Split in projects that used to take20~40 hours. Although not the same part nor computer, an industrial part with 5493faces, which is shown in Figure 4.27, was tested with a splitting approach to be presentedin Section 3.6. The splitting process took less than 1 minute.

Review of Methods:

According to the mold design results given by the companies, the approach based onextending parting lines is used in all the reviewed software systems. Consequently thedisadvantages of the approach of extending parting lines (Section 2.3.1) can be found inthese systems.

2.4 CAD REPRESENTATION

The CAD representation of parts is a critical element in this research. In a computeraided mold design method, a part design need to be represented by some data structuresand manipulated by some algorithms. Also the generated mold pieces for the part shouldbe represented in CAD models. Besides the shape of a part, the design features anddesign intents are also critical in a design-for-manufacture system. The decision-templateproposed in this dissertation (Section 5.3) is actually a high-level CAD representation ofa part design. Therefore, the CAD representation of a part design is reviewed in thissection.

The representation and manipulation of 3-dimensional surfaces and volumes arestudied in the field of solid modeling, which is a critical element in a growing number ofapplication areas, including CAD/CAM, robotics, simulation, analysis and graphics.Conceptually, work in solid modeling has proceeded in three levels of abstraction(Hoffman, 1989):

1. Symbolic and arithmetic foundations represent the lowest level of abstraction andare concerned with the computer hardware support of integer and floating pointarithmetic as well as with the abilities of a programming language to expresscomputations and manipulate memory. Examples include the development ofdata structures and handling of issues of mathematical robustness.

2. Mathematical and algorithmic infrastructure describes the fundamental operationsas implemented in the foundation above. Examples include operations for


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creating solid, for performing interference tests among solids, and for developingefficient algorithms for basic manipulations.

3. Application and user interface is the highest level of abstraction, focusing on thedevelopment of applications in terms of the infrastructure created by the above.

Mold design is primarily concerned with problems that lie in the second category.Decision template is concerned with problems that lie in the third category. There arenumerous important concepts from the core solid modeling literature that are directlyrelevant to the work described in this dissertation. The remainder of this section verybriefly reviews some solid modeling concepts and terminology which will be usedthroughout this dissertation – in particular, concepts concerned with manipulation andrepresentation of solid models. Some related work on high-level CAD representation isalso provided.

For more in-depth coverage of the field of solid modeling, readers can refer to thetexts by (Mantÿlä, 1988; Hoffman, 1989; Mortenson, 1997; and Woodwark, 1989), aswell as papers of (Requicha and Rossignac, 1992; and Miller, 1989). The classic text oncomputer graphics of (Foley, et al., 1990) also covers solid modeling and its relationshipto graphics and rendering. For information regarding commercial solid modelingsystems, readers can refer to the product and reference information for SpatialTechnologies ACIS modeler (www.spatial.com) as well as the EDS/UNIGRAPHICSParasolid solid modeling kernel (www.ugs.com).

2.4.1 Representation of 3D Surfaces and Solids

Solid modeling systems are able to distinguish between the inside and outside of theobject. This capability distinguishes solid modelers from wire frame models, which storeonly enough information to describe the edges bounding an object. There are three broadclasses of schemes for representation of solid models (Mantÿlä, 1988):

1. Decomposition approaches that model a solid as collections of primitive objectsconnected in some way. Examples of these data structures include Quadtree fortwo-dimensional objects and octree for solid objects.

2. Constructive approaches that model a solid as a combination of primitive solidtemplates. The most famous constructive approach is Constructive SolidGeometry (CSG). CSG adopts the “building block” approach to solid modeling inits pure form. The user of a CSG modeler operates only on parameterizedinstances of solid primitives and Boolean set operations on them. The part isstored in the database as a tree in which the leaves of the tree correspond toprimitives and the nodes correspond to the Boolean operations.

3. Boundary-based approaches that model a solid using a data structure thatrepresents the geometry and topology of its bounding faces. A face is bounded byedges, and edges are bounded by two vertices. Parts are stored in the database asa linked-list of vertices, edges, and faces. The database must store informationabout the connectivity of the faces and the equations defining the geometry of thefaces.


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Currently the boundary-representation (b-rep) approach has emerged as the dominantrepresentation scheme in solid modeling and CAD systems. This is due in large part tothe representational power and flexibility of boundary models, as well as to recentadvances in numeric computation that overcame earlier problems with models becomingunstable and inconsistent.

The three object types (face, edge, and vertex) form the basic constituents ofboundary models. Correspondingly there are polygon-based, vertex-based and edge-based boundary models. Among them, one of the most popular b-rep structures is theWinged-Edge representation and its variations. A basic idea in this representation is theintroduction of coedges. A coedge records the occurrence of an edge in a loop of a face.Coedges have partner pointers that point to the other coedge list associated with the edge.An example of a simple part is shown in Figure 2.10. Normally an edge is adjacent totwo faces; therefore, the edge has two coedges, each associated with a loop in one of thefaces. The two coedges always go in opposite directions along the edges.

The work described in this dissertation assumes a boundary-representation solidmodel represented in winged-edge data structure. A b-rep usually consists of a graphicalstructure that models an entity’s topology. The connections between the nodes in the b-rep graph represent the connections between the topological components of the entity’sboundary. These topology nodes then contain pointers to their underlying geometricentities; for example, a face of a solid is a topological entity (represented as a collectionof bounding edges) and it has associated with its surface (represented as an equation). Anillustration of the distinction between geometric and topological information from theACIS Solid Modeling Kernel is given in Figure 2.11.

Figure 2.10 – Coedges of a Part.


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In this research, only manifold solids are considered. Intuitively, a manifold solid isone in which each point on the boundary of the object has a neighborhood that isequivalent to a two-dimensional disk. Practically, this means: (1) each edge belongs toexactly two faces; (2) each vertex is surrounded by one sequence of edges and faces; (3)faces intersect at common edges and vertices only; and (4) there is volume on only oneside of a face. The manifold conditions also exclude solid whose bounding surfaces areself-intersecting. Non-manifold objects can be of mixed dimension and may havevertices, edges, and faces that do not meet these requirements.

Review of Methods:

The representation method of solid models is considered in developing datastructures for a mold design method. They also determine the capability of a mold designsystem in manipulating solid models.

After the representation of a part is introduced, the manipulation of parts is presentedin the next section.

2.4.2 Manipulation of Solid Models

This section briefly describes some of the common operators used to manipulategeometric and solid models. In addition to operators such as transformations, rotations,

Figure 2.11 – The Distinction between Geometric and Topological Information within theACIS Solid Modeler (Spatial Technology, 2000).


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and scaling, there are additional functions specific to the nature of geometric modeling.These operations include Euler operations and Boolean operations, which are used inChapter 3 and 4 in this dissertation.

• Euler OperatorsBased on the theory of plane models, Euler operators act on the topology of a

boundary representation data structure. Starting from an idealized primitive “solid”consisting of one face and one vertex, one can create a solid through a series of local andglobal manipulations. Local manipulations can create (or delete) vertices, edges, andloops; global manipulations can be used to create (or delete) holes or to divide a bodyinto multiple bodies.

Gluing is a high-level Euler operator, which can combine simpler solids to morecomplicated ones. For two 6-sided cylinders (Figure 2.12), they can be glued into onealong a pair of entirely coincident faces. The gluing operation consists three steps(Mantÿlä, 1988): (1) the merging of two half-edge data structures into one data structurethat has two shells; (2) the joining of the shells; (3) the merging of coincident edges andvertices of the face.

• Boolean Operations.Solid models can be considered as bounded point sets. Boolean operations union

(∪ ), intersection (∩) and difference (-) can be defined on solid models based on theiraction on point set. For non-manifold models we are interested in, regularized Booleanoperations correspond to the counterparts of the ordinary Boolean operations. If A is asolid, i(A) is the interior of A (point set A minus its boundary) and c(A) is the closure of A(point set A plus its boundary); the regularized Boolean operations are defined as follows(Mantÿlä, 1988):

∪ *, Union: A∪ *B = c(i(A∪ * B));

∩*, Intersection: A∩*B = c(i(A∩*B));

-*, Difference: A-*B = c(i(A-*B));

Examples of regularized Boolean operations are shown in Figure 2.13.

Figure 2.12 – An Example of Gluing Operation.


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The Boolean operations can also be represented as gluing operation. From theboundary classification of A and B, all their Boolean combinations are:

∪ , Union: A∪ B = A out B ⊕ B out A;

∩, Intersection: A∩B = A in B ⊕ B in A;

-, Difference: A-B = A out B ⊕ (B in A)-1;

where ⊕ denotes the gluing operation, and (B in A)-1Examples denotes the “complement”of B in A, i.e., B in A with the orientation of all faces reversed.

Review of Methods:

Euler operators and Boolean operation are used in the mold design method to bepresented in Chapter 3. The notations reviewed in this section will be used in thepresentation of the mold design method and revisited in Chapter 3.

2.4.3 High Level Representations

As solid modeling has become better understood, increasingly sophisticated androbust solid modeling systems have emerged. Given that adequate solid modeling toolsare available at one’s disposal, one begins to ask questions such as: How shouldfunctionality be represented? In this section several high-level representation methodsare reviewed, which provide a context for the decision template to be presented inChapter 5.

Features and feature-based approaches have proven popular in a variety ofCAD/CAM application domains. The concept of a feature attempts to reason about

Figure 2.13 – Boolean Operations.


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design and manufacturing activities by modeling the relationship between the localgeometric and topological configurations of a design and the higher-level abstractions. Inthis way, semantic information can be conveyed along with the shape. Significant workhas been directed toward feature-based design and manufacturability evaluation of partdesigns (Shah, 1991; Rosen, 1992; Regli, 1995; Shah and Mantyla, 1995).

Currently most researchers are convinced that no single set of features can satisfy therequirements of every possible design and manufacturing domain (Regli, 1995).Generally a feature is regarded as a functional entity that is meaningful in certaindomains. Different feature types exist. Some typical feature types include (1) formfeatures, which are portions of the geometry; (2) precision features, which are deviationsfrom nominal form, size or location; (3) technological features, which are non-geometricparameters related to the function or performance; (4) material features, which related tomaterial properties; and (5) assembly features, which include part relative orientations,interaction surfaces, and fits. However most feature-based models address form featuresonly (e.g. hole and rib).

Rosen and coauthors (1994) proposed a computational framework, which was calledGoal-Directed Geometry, for early design stages. Tools for parametric geometry,variational modeling, and feature-based design were combined with a multi-objectiveoptimization code to provide robust support for parametric design problems, whereparameter values are desired that best meet a set of goals and constraints. Geometric andengineering models of a design are combined into a multi-objective optimizationformulation in Compromise Decision Support Problem (DSP). This idea is extended inthis research into the field of design-for-manufacturing.

Recently integrating part design knowledge into a CAD model is a developingdirection of mechanical design automation modeling. Parametric Technology Corp.(PTC), one of the world’s leading CAD companies, recently divided the mechanicaldesign automation modeling into five generations. Initially, two-dimensional drafting,then three-dimensional wireframe modeling, and finally three-dimensional solidmodeling. The current fourth generation is the parametric feature-based modeling. Thefifth generation is proposed and named by PTC as behavioral modeling(http://www.ptc.com/products/proe/bmx/index.htm). The basic idea of the behavioral modeling isto capture intelligence within features in the design side. Capturing product-intent istaken as a natural part of the engineering process, and then automatically builds virtualprototypes that satisfy multiple objectives. It advances feature-based modeling toaccommodate a set of adaptive process features that go beyond the traditional coregeometric features. The intent and performance of the design are also modeled.

To illustrate the behavior modeling, an example of a golf club design is shown inFigure 2.14 (www.ptc.com/products/proe/bmx/examples/golf_club.htm). "Forgiveness"and "feel" are two important characteristics in golf club design. Forgiveness refers to thesensitivity of a club to off center hits and is influenced by location of the center of gravityin the club head. Feel refers to the ease with which a golfer can comfortably swing theclub and is influenced by "swingweight", the measurement of a club's weight distributionabout a fulcrum point which is 14 inches from the grip end of the club. To solve the golfclub design problem, analysis features are created to capture mass and the center ofgravity of the head. To gain design insight to improving the forgiveness characteristics of


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the club, a User Defined Analysis with a field point is created in this example to measurethe twisting force, or torque, applied to the club face when a ball is hit off center.Sensitivity studies are then performed to determine the effect of toe thickness of the clubhead on the location of the center of gravity and twisting force. To gain design insight toimproving the feel characteristics of the club a sensitivity study is performed to determinethe effect of the flange thickness of the toe of the club head on the weight of the head. Afinal design is then identified through optimization studies.

Zhu and Kazmer (1999; 2000) presented a performance-based representationapproach, which is called Performance Orientation Chart (POC). POC is in the sameformat as House of Quality (HOQ) but has different contents. Performances of thedesign are visualized by relation figures. The designer can interactively develop thedesign solution to satisfy multiple specifications guided by these graphical matrices.

Another interesting research work on transmitting CAD models from designers tomanufacturers is presented in (Storti, et al., 1999). Motivated by recent advances insoftware development associated with object oriented programming style and dataencapsulation, Storti and coauthors developed an approach to the transmission of partspecifications for distributed solid freeform fabrication. Rather than translating astandard format of CAD model, the authors specified a set of public methods necessaryfor solid freeform fabrication. By specifying public members and methods that providefabrication systems with all the information needed to build parts, SFF systems can buildparts based on models constructed in any modeling environment for which the methodsare available.

Figure 2.14 – An Example of Behavioral Modeling (PTC).


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Future design technologies will shift from the “geometry-centric” view to the“knowledge-centric” view. Knowledge representations will play a larger role in thedesign process for creation of product and assembly models. In this research, the authoruses a compromise-DSP (C-DSP) formulation to formulate design knowledge, which iscalled decision template (Chapter 5) in this dissertation. A decision template can belinked with a parametric feature-based CAD model, and transferred from designers tomanufacturers.

Review of Methods:

The high-level representation methods are briefly reviewed in this section to providea context for the decision templates to be discussed in Chapter 5.

In Section 1.2.4, design-for-manufacture methods were briefly evaluated in order toidentify research opportunities in design for Rapid Tooling. More detailed literaturereview on design-for-manufacture is presented in the next section.

2.5 DESIGN FOR MANUFACTURE: STRATEGIES AND TECHNIQUES

The term design for manufacture, or DFM, is used to characterize efforts by designand manufacturing to improve the product-process fit or to increase the degree to whichthe product and process are designed simultaneously (Susman, 1992). Accordingy tothese two goals, the literature review on DFM is divided into two groups in this section.

• Improve Product-Process FitToday most information exchange between design and manufacturing occurs through

informal human communication, often requiring several iterations to get a part right.Because of the drawbacks associated with this serial approach, design for manufacture(DFM) has received considerable attention.

In order to improve product-process fit, a dominant approach in field of DFM is toincorporate manufacturing concerns into the design process, with the goal of improvingproduct quality, decreasing product cost, and reducing product development time. Inessence, the purpose of this approach is to ensure that the designer considersmanufacturing issues during the design stage.

A DFM system related to this approach should guide the user through the design sothat a part is compatible with a process or it should provide the user with feedback so thatthe user can decide if the part needs to be modified. The DFM system requires twoprimary components: (1) a means to evaluate the part for manufacturability and (2) theinformation needed to support the evaluation. Currently there are a number ofmechanisms that can be used for manufacturability evaluation.

One is to estimate the cost of a part. Cost estimation systems are available for anumber of processes. Most of these systems are based on empirical cost models for theprocess and do not require a full three-dimension model of the part. Additionally, thesesystems often require knowledge of the process not ordinarily possessed by the designer.For example, one cost estimating system for injection molding requires the user to knowthe injection temperature and pressure, the coolant temperature, and whether two-plate orthree-plate molds will be used for the part (Poli, et al., 1988).


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A second approach for evaluating manufacturability is to apply manufacturingguidelines to the geometry of the part to identify attributes of the geometry that aredifficult to manufacture. Often the guidelines are stated in terms of good and badpractices which are based on years of experience. Unless some measure ofmanufacturability is used, these systems do not enable two different part designs to becompared. Most leading handbooks on DFM, such as (Pye, 1989; Broothroyd andDewhurst, 1991; Bralla, 1998), use this approach.

A third approach for evaluating manufacturability is to perform a manufacturingsimulating of the part and then to use the simulating to make redesign suggestions. Forexample, plastic part designers already practice this approach to some extent bymodifying a part based on the results of flow simulations. Several commercial softwaresystems are available such as Part Advisor and QuickFill, which are developed byMoldflow Corp. and C-Mold respectively.

Each approach has a set of benefits and drawbacks associated with it. Costestimation enables different parts to be compared and permits the comparison of partsproduced by different methods. The application of guidelines to the part geometry can beaccomplished at the design stage when the details are being generated. Simulation maybest be applied after the part has been designed, when the detail design is completelyknown. Therefore many systems use a combination of the approaches. For example,Poli and his coworkers have developed DFM systems for forging (Knight and Poli,1985), die casting (Poli and Shanmugasundaram, 1991), sheet metal stamping (Poli, etal., 1993), and injection molding (Poli, et al., 1988) based on a group technologyapproach. In this approach, cost drivers were identified and then associated withprocessing and tooling costs. Classification codes are associated with these cost drivers.Parts are then classified according to a multi-digit code in order to provide their relativecosts so that candidate configurations can be compared. These systems are not linked toa geometric model of the part. Instead, the users must fill out spreadsheets to define thecost drivers associated with the part.

Besides the DFM metrics, a DFM method is any systematic procedure or strategythat search for, analyzes, evaluates, or improves the manufacturability of a design (Shahand Wright, 2000). DFM methods can be divided as the following:

- Technological Feasibility: Can a given process manufacture a given design? Canwe find a process from a finite set of processes to produce the desired design?

- Economic Feasibility: Is it cost effective to produce a design in desired batch sizeby a given process plan? Can the desired quality be produced/maintainedconsistently by the process plan without a high rejection rate? What is the time tomarket if a given process plan is used?

- Trade-off Study/Optimization: Consideration of changes to design parameters toreduce manufacturing cost by a greater amount than the loss of design quality.

The above DFM methods can be found in numerous literatures on DFM research fordifferent manufacturing processes. For example, (Boothroyd and Dewhurst, 1989)studied the assembly process and proposes a Design for Assembly (DFA) method;(Sarma, et al., 1996) presented an integrated approach for a CNC milling process; (Lee, et


58

al., 1998) presented a framework for concurrent process planning for injection moldingprocess; and (Dissinger and Magrab, 1996) discussed design for powder metallurgyprocess. All these approaches largely depend on the characteristic of the manufacturingprocess.

Similarly (van Vliet, et al., 1999) divided the DFM approaches into three phases:Verifying, qualifying and optimizing the product manufacturability. The approachesdiscussed before are in the categories of verifying and qualifying. In optimizing categoryGrace and Billatos (in Van Vliet, et al., 1999) proposed a re-design approach foroptimization. In order to generate re-design suggestions, it is necessary to know thefunctionality of the part. For this purpose, El Maraghy, et al. (in Gupta, et al., 1995) usedpre-defined functional features. Henderson, et al. (in Gupta, et al. 1995) developed amethod for representing functionality semantically within a solid modeling system.

Technologies from other research fields, such as neural networks and visualization,can also be utilized in DFM research. He, et al. (1998) presented an intelligent systememploying fuzzy sets and neural networks, which is able to predict the process parameterresetting automatically to achieve better product quality. Seven commonly encounteredinjection molded product defects (short shot, flash, sink-mark, flow-mark, weld line,cracking, and warpage) and two key injection mold parameters (part flow length and flowthickness) are used as system input which are described using fuzzy terms. On the otherhand, nine process parameter adjusters (pressure, speed, resin temperature, clampingforce, holding time, mold temperature, injection holding pressure, back pressure, andcooling time) are the system output. A back-propagation neural network has beenconstructed and trained using a large number of defects -> parameter adjustersexpert rules. They system is able to predict the exact amount to be adjusted for eachparameter towards reducing or eliminating the observed defects.

Lu, et al. (1997) presented a volume-based geometric reasoning and visualizationapproach to support design evaluation during preliminary design. The system extractedthe underlying geometric characteristics which usually affect part quality or increasemanufacturing difficulties from the part model. All the information is then presented todesigners in a form which can be easily interpreted via visualization techniques. Thepresented system focused on thermal- and flow-related problems in die casting.

With the development of Internet, several Internet-based DFM system have beendeveloped. For example, an Internet-based design for Rapid Prototyping system isdeveloped at Stanford University to use Rapid Prototyping effectively (Frost andCutkosky, 1996; Rajagopalan and Pinilla, 1998). By formalizing RP process constraintsas design rules, the system creates a clean interface to decouple design frommanufacturing processes. Similarly, another web-based design-for-manufacture systemnamed CyberCut is developed for 3-axis CNC machining at the University of California,Berkeley (Sarma, et al., 1996; Wang and Wright, 1998). The system combines facilitiesfor CAD model creation, computer aided process planning and NC code generation.

However, in these methods, researchers have to formulate requirements of theinvestigated manufacturing process into knowledge (rule or algorithm), then developsoftware systems to allow designers to analyze manufacturability during the design stage.The approach seems to work well for processes with little requirements like rapid


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prototyping and for conceptual design stage by using approximation codes. However,automated analysis meets difficulties if a part or a fabrication process (like RapidTooling) is complex. For a complex part, complex interactions of three-dimensionalgeometries have largely prevented formalization of this knowledge. For a complexfabrication process, it is rather difficult, if not impossible, to formulate all fabricator’sexperience and decision into rules and algorithms in the designer side.

In the usage of Rapid Tooling in producing functional prototypes, designers haveknown functional requirements of a part before the prototypes are to be made. Thereforeif the part design requirements are formulated understandable for the manufacturer, thenthe manufacturer may be in a better position to adjust the design to facilitatemanufacturing without compromising its functionality.

• Design Product and Process SimultaneouslyIdeally designer and manufacturer should cooperate in DFM problem. Concurrent

Engineering (CE), which is “a systematic approach to the integrated, concurrent design ofproducts and their related processes, including manufacture and support” (Winner, et al.,1988), was proposed to solve DFM problem. Over the years, numerous research effortshave been focused on different methods of integrating product and process planning(Prasad, 1996). While there are a number of ways to categorize these approaches, thecategorization developed by Eversheim (1997) is: the first category is organizationalstructure-oriented integration, which is based on the formation of multidisciplinarydesign teams, and enforced coordination. The second category is process-orientedintegration through parallel (as opposed to integrated) execution of design and processplanning activities to reduce development time. The final category is information-oriented integration, which deals with the integration of computer based information torealize information flow between design and process planning through improving dataexchange.

In the category of organizational structure-oriented integration, research is performedmainly from the management (Adler, 1992), social (Susman and Dean, 1992) and culturalcontext (Liker and Fleischer, 1992). Within industry, concurrent design often takes theform of “Integrated Product Development Teams”, compromise of representatives fromall aspects of the product development process, who meet together to better design theproduct. However, it is well known that the two types of engineer “don’t speak the samelanguage”. Rosenthal (1990) observed “one prerequisite for collaborative efforts tosucceed is that there be effective technological capabilities for the focused assembly ofinformation”. Thus, even if designer and manufacturer can work as a team, acoordination method and a uniform decision framework are important for thecollaborative decision-making.

The approaches on the category of process-oriented integration are discussed beforeand are not repeated here.

In the category of information-oriented integration, Collaborative Engineering isproposed to cause product team members to consider all elements of the product lifecycle. Jin, et al. (1997) point out that there are three basic issues involved in providingcomputer support for collaborative engineering design. The first issue is taskdecomposition and representation. Task decomposition is concerned with identifying the


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sub-tasks that can be divided in the way that minimizes interactions among the sub-tasks.Task representation is related to defining a design task and its sub-tasks in the form thatcan be easily handled by designers and computers to identify interactions among the sub-tasks and cross the life-cycle of product development. The second issue is the need for acommunication infrastructure to facilitate communications among designers. Since inmost cases complete task decomposition, i.e., no interactions exist among sub-tasks, isimpossible, a sophisticated communication infrastructure is needed to facilitate flow ofinformation among designers. The third issue is coordination support. Coordination isgenerally considered as the activity to resolve dependencies among sub-tasks.

Pahng, et al. (1998) presented an integrated and open product developmentenvironment for distributed and collaborative design. The web-based framework, calledDOME (Distributed Object-based Modeling and Evaluation), allows designers to buildintegrated models using both local and distributed resources and to collaborate byexchanging services.

A computer-based design system developed by Sriram et al. provides a sharedworkspace where multiple designers work in separate engineering disciplines (Sriram andLogcher, 1993). In their DICE (Distributed and Integrated Environment for Computer-aided Engineering) program, an object-oriented database management system with aglobal control mechanism is utilized to resolve coordination and communicationproblems. Design rationale provided during the product design process is also used forresolving design conflicts.

Currently collaboration between team members is supported at a number of levels,from simple data sharing, through single user view and mark-up, to co-viewing, to co-modeling. It is believed that eventually team will be able to work in a productdevelopment process that will support cooperative innovation through a shared workenvironment.

With advances in computing and increased Internet usage, many companies take astrong interest in the collaborative approach. Several commercial software systems weredeveloped and are available. For example, OneSapce by CoCreate Software Inc.(www.cocreate.com) allows a number of people to participate in an on-line work session,viewing a design (assembly) while cooperatively making comments (annotations)directly on the 3D model and changing the model as desired. The basic principle of thesoftware is to operate it in a client-server environment. The server providescommunication coordination and prepares the data for the session. Others similarproducts include Windchill by PTC (www.ptc.com) and e!Vista from SDRC(www.sdrc.com). They support managing and communicating information about productstructures and changes throughout product life cycles. Both Windchill and e!Vista arewritten in Java on both the client and server side. They appear as web pages within acommercial web browser such as MS-Internet Explorer and Netscape.

• Design for Manufacture in other FieldsAs to design and manufacturing processes, it is widely agreed that the computer

support of VLSI (Very Large Scale Integrated Circuits) design is generally more maturethan that of mechanical items. A group of researchers in a NSF workshop on structureddesign methods observed that a clean interface, which separate design efforts at


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increasingly high levels of abstraction from the growing complexities of the fabricationprocesses, is the key to the rapid success of the VLSI development (Antonsson, 1996).That is, VLSI designer can design circuits at the block-diagram level, a higher level thandetails. This capability allows designers to quickly construct complex circuits, andfacilitates re-using parts of one design in another product. For the block design given bythe designer, a process compiler will fill in the process-dependent details. Mead (1994)believed that a major factor in the success of the VLSI design is the ability to submitdesigns with confidence that products would come back meeting specifications.

Inspired by the NSF workshop, some researchers (Rajagopalan and Pinilla 1998;Frost and Cutkosky 1996) claimed a clean interface is achieved for RP process byformalizing process constraints as design rules for designer. Jerard and coauthors (1998;2000) also proposed a system, FACILE (Fast Associative Clean Interface Language andEnvironment), to achieve a clean interface for NC machining.

However, argument that mechanical design cannot be like VLSI design can also befound (Whitney, 1996). Whitney observed that the final designs are not optimized forsize or power consumption in VLSI design. He believed that this is the trade-off thatmust be made to design at a function level. However in mechanical systems, minimizingsize, weight, and power consumption are often primary concerns. Therefore when it isessential to minimize manufacturing costs or to meet stringent demands on tolerances ormaterials properties, the designer will have to know detailed process characteristics,constraints and costs.

As mentioned before, the research project of this dissertation, Rapid ToolingTestBed, is also motivated by the NSF workshop and try to develop approaches toachieve a clean interface between the designer and manufacturer for Rapid Prototypingand Rapid Tooling processes (Allen and Rosen, 1997; Rosen, 1998). However, insteadof requiring the designer to know detailed process characteristics, constraints and costs, adifferent approach is proposed in this research to achieve the clean interface. Ourapproach is to develop a method to aid the manufacturer, who is familiar with the processcharacteristics, constraints and costs, to integrate design and manufacturing requirements.The author believes this may be a better approach to achieve the clean interface betweendesigners and manufacturers for a complex fabrication process such as Rapid Tooling.

Review of Methods:

Although Concurrent and Collaborative Engineering can offer substantial benefits onsharing information and knowledge, one difficulty associated with them is an effectivecoordination mechanism to handle conflicts between different designers andmanufacturers. As mentioned by Jin, et al. (1997), “coordination means to take intoconsideration decisions made by others in making local decisions … The research isconcerned with exploring coordination strategies, and developing effective tools to carryout coordination activities of designers”.

In this research, the author addresses the problem by proposing a decision supportsystem (Chapter 6), in which the benefits and drawbacks of designs and manufacturingprocesses are quantified. Therefore tradeoffs between design and manufacturingvariables can be determined by solving requirements formulated in Compromise DSPwith the aid of optimization software systems.


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After a DFM strategy is identified, several design technologies used in developing aDFM system are presented in the next section.

2.6 DESIGN TECHNOLOGIES

Several design technologies, methods, and tools are utilized to support the design formanufacture in this research. The design technologies of interest include the Decision-Based Design (DBD), Compromise DSP (cDSP), and Robust Concept ExplorationMethod (RCEM). They are described in more details as follows.

2.6.1 Decision-Based Design

Decision-Based Design provides the foundational base in this research fordeveloping an approach to integrate design and manufacturing requirements. Decision-Based Design (DBD) is rooted in the notion that the principal role of a designer is tomake decisions (Shupe 1988; Mistree et al. 1989). In design, decisions are invariablymultileveled and multidimensional in nature. By approaching design from theperspective of making decisions, it becomes possible to envision a unified approach fordesign-for-manufacture problem.

An implementation of Decision-Based Design is the Decision Support Problem(DSP) Technique (see, e.g., Bras and Mistree, 1991). In the DSP Technique, designing isdefined as the process of converting information that characterizes the needs andrequirements for a product into knowledge about a product (Mistree, et al., 1990). Acomplete description of the DSP Technique can be found in, e.g., (e.g., Mistree, et al.,1990). Among the tools available within the DSP Technique, the compromise DSP(Mistree, et al., 1993) is a general framework for solving multiobjective, non-linear,satisficing problems. In this dissertation, the compromise DSP is central to modelingmultiple design objectives and assessing the tradeoffs pertinent to design formanufacture. Examples of these tradeoffs are discussed in the context of the two exampleproblems in Chapters 7 and 8. Decision Support Problems have been used in a variety ofdomains, including design of ships, aircraft, mechanical systems etc (Mistree, et al.,1990; Koch, et al., 1996). There are two basic types of DSP, selection and compromise.These decisions can also be coupled and solved simultaneously. The procedures and toolswithin the DSP Technique are detailed by Mistree, et al., (1993). The preliminaryselection and selection DSPs are discussed by Kuppuraju, et al., (1985) and Mistree, etal., (1994); the compromise DSP is detailed within (Mistree, et al., 1993). Thecompromise DSP will be used in this research, hence it is further described in the nextsection.

2.6.2 Compromise Decision Support Problem

Solving design-for-manufacture problem will require decisions between conflictinggoals – determining the best set of design variables that will achieve both design andmanufacturing goals. Compromise DSP (cDSP) is a multiobjective decision modelwhich is a hybrid formulation based on Mathematical Programming and GoalProgramming (Mistree, et al., 1993) to satisfy a set of constraints while achieving a set ofconflicting goals as well as possible. The cDSP has been successfully used in severaldesign applications (Chen, et al., 1996; Koch, et al., 1996). In this research, cDSP is used


63

as a template to transfer the information and knowledge between designer andmanufacturer in DFM phase.

Formulation of a compromise DSP begins with a word formulation and proceeds to amathematical formulation. The word formulation consists of the keywords given, find,satisfy, minimize and their associated descriptors, as shown in Figure 2.15. Given analternative and domain information for a problem at hand, the objective in thecompromise DSP is to find the values of system design variables which satisfy a set ofconstraints and bounds and achieve as closely as possible a set of conflicting goals whileminimizing a deviation function.

The generic mathematical formulation of the compromise DSP is presented in Figure2.16. In general, compromise DSP’s are written in terms of n system variables, whichare represented as a vector X. They define the physical attributes of an artifact that canbe altered. A set of p+q system constraints are used to model the limits placed on asystem design, and must be satisfied for feasibility. Mathematically, system constraintsare functions of system variables only, and may be a mix of linear and nonlinearfunctions. Bounds are specific limits placed on the magnitude of each of the systemvariables. A set of m system goals is used to model the aspirations for the design. Itrelates the goal target, Gi, to the actual performance, Ai(X), of the system with respect tothe goal. The deviation variables, di

- and di+, are introduced as a measure of

achievement, the difference between Ai(X) and Gi. The deviation variables, di+ and di

-,are always non-negative, and the product constraint, d i

+ • di- = 0 , ensures that at least one

of the deviation variables for a particular goal is always zero. In the compromise DSPthe objective is to minimize a deviation function, Z(d-, d+), a function of the deviationvariables. Deviation function (objective function) formulations are classified asArchimedean or Preemptive -- based on the manner in which importance is assigned tosatisfying the goals.

Keywords Descriptors

Given An alternative to be improved through modification;assumptions, system parameters, constraints,bounds, goals, and the deviation function.

Find Values of system variables and deviation variables.

Satisfy System constraints and bounds (feasibility), and goals(desired target values or objectives).

Minimize A deviation function.

COMPROMISE DSP

Figure 2.15 - Word formulation of a Compromise DSP problem (Mistree, et al., 1993).


64

Objective function captures all the goals in a single function and takes differentforms depending on the way different goals are synthesized. The solutions obtained fordifferent formulations of objective function could be significantly different. Hence theobjective function formulation is an important step that should be formulated to representthe original problem requirements. The two types of objective function formulation incDSP are explained here.

In pre-emptive formulation, objective function is a list of rank ordered goals. Whileminimizing this objective function, the goals with higher rank are minimized beforeminimizing the goals with lower rank. Each time a goal is minimized, it is ensured thatits value is not changed while minimizing the subsequent goals (lower ranked goals)

GivenAn alternative that is to be improved through modification.Assumptions used to model the domain of interest.The system parameters:

n number of system variablesl number of discrete/integer system variablesp+q number of system constraintsp equality constraintsq inequality constraintsm number of system goals

gi(X) system constraint functionsgi(X) = Ci(X) - Di(X)

fk(di) function of deviation variables to be minimized at prioritylevel k for the preemptive case

Wi weight for the Archimedean caseFind

The values of the independent system variables (they describe the physicalattributes of an artifact).

Xi i = 1,..., nThe values of the deviation variables (they indicate the extent to whichthe goals are achieved).

di-, di

+ i = 1,..., mSatisfy

The system constraints (linear, nonlinear) that must be satisfied for thesolution to be feasible. There is no restriction placed on linearity or convexity.

gi(X) = 0; i = 1,..., pgi(X) 0; i = p+1,...,p+q

The system goals that must achieve a specified target value as far as possible.There is no restriction placed on linearity or convexity.

Ai(X) + di- - di+ = Gi ; i = 1,..., mThe lower and upper bounds on the system.

Ximin Š Xi Š Ximax; i = 1,..., ndi

- , di+ 0 and di

- • di+ = 0

MinimizeThe deviation function which is a measure of the deviation of the systemperformance from that implied by the set of goals and their associated prioritylevels or relative weights:

Case a: Preemptive (lexicographic minimum)Z = [ fl( di- , di+), . . ,fm( di- , di+) ]

Case b: ArchimedeanZ = Σ Wi(di

- + di+) ; ΣWi = 1; Wi 0; i = 1,...,m

Figure 2.16 - Mathematical formulation of a Compromise DSP problem(Mistree, et al., 1993).


65

(Mistree, et al., 1993). In this approach, all the available design freedom is used tominimize the highest ranked goal. Then the design space is narrowed to the region thatyields the minimum value for the first goal. This design space is used to minimize thesecond ranked goal. This is continued either till all the goals are minimized or till thedesign space reduces to a single point.

In Archimedean formulation, the objective function is formulated as a weighted sumof the appropriate deviation variables. Deviation variables are a measure of the goalachievement. If the goal is a target-matching goal then both positive and negativedeviations are undesirable and hence both are considered in objective functionformulation. If the goal is a minimization goal then negative deviation indicatesoverachievement and is desirable but positive deviation indicates underachievement andis undesirable. Hence, only positive deviation is considered in objective functionformulation. For a maximization goal, negative deviations are undesirable and only theseare considered in objective function formulation. In archimedean formulation, mostimportant goal is given highest weight. Also, it is a common practice to normalizeweights so that their sum equals one.

These Archimedean and preemptive formulations that are typically used in theCompromise DSP formulation suffer two major drawbacks associated to the arbitrarydefinition of priority levels and targets for multiple objectives. The Archimedean one isvery difficult to implement with meaningful results because there is no consistent way todetermine a priori the right set of weights. Thus, choosing weights is either donearbitrarily or through cumbersome iterations. The preemptive approach has the problemthat one objective is assumed infinitely more important than other. Also, in theCompromise DSP is needed to define targets for each goal, and usually those targets areselected based on "educated guesses" or through an inefficient process of iteration. Theseshortcomings undermine the effectiveness of the Compromise DSP as a design tool.These problems can be amended by modifying the objective function formulation inCompromise DSP according to the Linear Physical Programming (LPP) formulationproposed by Messac and coauthors (1996). Hernandez and Mistree (2001) modified theobjective function formulation of a cDSP to incorporate the features of LPP. This newformulation of Compromise DSP is presented in Figure 2.17. The cDSP’s in the robotarm and camera roller cases (Chapter 7 and 8) are formulated and solved using thisapproach. The approach judiciously exploits the designer knowledge of the problem byallowing to express preferences of each objective through various degrees of desirability:unacceptable, highly undesirable, undesirable, tolerable, desirable, and ideal. It alsoeliminates the need for iterative (or arbitrary) weights setting, making the use ofoptimization technology more appealing to design engineer in an industrial setting.


66

In physical programming a designer provides targets for different levels ofsatisfaction of each of the goals. Initially, all the target-matching goals are divided intotwo goals: one minimization goal and one maximization goal. Each of these goals isfurther divided into 6 regions: ideal, desirable, tolerable, undesirable, highly undesirableand unacceptable based on target values. If the goal achievement is in ideal region, thenit is completely satisfied and its value need not be considered in the objective functionformulation. If the goal achievement is in unacceptable region, then it is consideredinfeasible. Hence the six regions of a goal result in 4 sub-goals (corresponding todesirable, tolerable, undesirable and highly undesirable regions) and a constraint

GivenAn alternative to be improved through modification.Assumptions used to model the domain of interest.The system parameters:

n number of system variablesp+q number of system constraintsp equality constraintsq inequality constraintsh number of system goals of class 1-Sm number of system goals of class 2-S

gi(x) system constraint function: )x(D)x(C)x(g iii −=zk(di) function of deviation variables to be minimized at priority level k

Findxi design variables: i = 1, …, n

d di k l k, ,,+ − deviation variables i = 1, …, h; l = 1,…,m; k = 1,…,4

SatisfySystem constraints (linear, nonlinear)

0=)x(g i ; i = 1, …, p

0≥)x(g i ; i = p+1, …, p+q++ ≤ 5,ii t)x(A i = 1, …, h−− ≥ 5,ii t)x(A i = 1, …, m

System goals (linear, nonlinear)++

++−+ =−−+ k,ik,ik,ik,ii tddd)x(A 1 i = 1, …, h; k = 1,…,4; 05 =+

,id−−

++−− =+−+ k,ik,ik,ik,ii tddd)x(A 1 i = 1, …, m; k = 1,…,4; 05 =−

,id

Bounds

ximin ≤ xi ≤ xi

max ; i = 1, …, n−+k,lk,i d,d ≥ 0 ; i = 1, …, h; l = 1,…,m

+k,id . 0=−

k,id ; i = 1, … h,…m

MinimizeLexicographic FormulationDeviation function with four preemptive levels: ]z,...,z[Z 41=

where ( )∑+

=

−−

+−− +=

mh

ij,ij,ij,ij ddwz

1555

Archimedean Formulation ( )∑ ∑=

+

=

−+ +=4

1 1k

mh

ik,ik,ik,i ddwZ

Figure 2.17 - The Compromise DSP with Modifications based on the Linear PhysicalProgramming Model (Hernandez and Mistree, 2001).


67

(corresponding to unacceptable region). Each of the 4 sub-goals has different weightsand is calculated from their target values. The values of weights increase from desirableto highly undesirable sub-goals. The contribution of a goal achievement (in differentregions) to the objective function is shown in Figure 2.18. In the figure, goals of 4different classes are presented. Class 1S corresponds to minimization goals, class 2Scorresponds to maximization goals, class 3S corresponds to target-matching goals andclass 4S corresponds to range-matching goals. Goals in class 3S and class 4S are dividedinto two goals: one of class 1S and one of class 2S. From Figure 2.18, it can also be seenthat the rate of increase of objective function in farther regions (from ideal) is higher thanthe closer regions.

The compromise DSP may be solved using the ALP algorithm (Mistree, et al., 1993).A solution to the compromise DSP is called a "satisficing" solution since it is a feasiblepoint that achieves the system goals to the extent that is possible (Simon, 1996). Byspecifying ranged sets of design parameters rather than point solutions, design flexibilitycan be maintained, and part design can be more easily adapted to meet manufacturingrequirements. In this research, another term ‘Compromise DSP template’ indicates thatthe cDSP formulation is a template and does not correspond to a specific problem. Basedon the requirements of the problem, the goals and constraints are formulated. Also, someof the system variables in the template could become parameters (that are fixed) in thecDSP formulation of a specific problem. Compromise DSP template formulations can beused to represent the problem formulation for a group of similar problems and can beformulated even before the specific problem requirements are defined.


68

UnacceptableHighly

UndesirableUndesirableTolerableDesirableIdeal

iz

)x(gi+5it

+3it

+2it

+1it

+4it

2z~

3z~

4z~

5z~

Class-1S

UnacceptableHighly

UndesirableUndesirable Tolerable Desirable Ideal

iz

)x(gi−5it

−3it

−2it −

1it−4it

2z~

3z~

4z~

5z~

Class-2S

HighlyUndesirable

)x(gi

Unacceptable Undesirable Tolerable Desirable

iz

+5it+

3it+2it

1it+4it

Class-3S

Desirable Tolerable UndesirableHighly

UndesirableUnacceptable

−5it −

3it−2it

−4it

HighlyUndesirable

)x(gi

Unacceptable Undesirable Tolerable Desirable

iz

+5it+

3it+2it−

1it+4it

Class-4S

Desirable Tolerable UndesirableHighly

UndesirableUnacceptable

−5it

−3it

−2it

−4it +

1it

Ideal

Figure 2.18 - Class Function Regions for a Generic ith Objective (Hernandez and Mistree,2001).


69

2.6.3 Robust Concept Exploration Method (RCEM)

Robust Concept Exploration Method (RCEM) (Chen, 1995; Chen, et al., 1995) is amethodology resulted from the integration of robust design techniques, design ofexperiment techniques, and response surface methodology within the framework of thecompromise DSP. The RCEM facilitates an efficient and effective concept explorationprocess as robust top-level design specifications are identified for the design of complexsystems. In this context, robustness of specifications is measured in terms of sensitivityto changes in requirements - thus the focus is to minimize the effects on the conceptualdesign of downstream design changes.

The computer infrastructure for implementing the RCEM is composed of fourgeneric processors surrounding a central ‘slot’ for inserting existing, domain-dependentanalysis tools as simulation programs, as shown in Figure 2.19. The simulation programs(existing analysis programs) are used to evaluate the performance of a minimum numberof conceptual designs. The RCEM processors increase computational efficiency andfacilitate the generation of top-level design specifications. The point generator (processorB) is used to design the necessary screening experiments. The experiments analyzer(processor D) is used to evaluate the results of the screening and to plan additionalexperiments. The response surface model processor (E) is used to create response surfacemodels, and the compromise DSP processor (F) is used to develop robust top-level designspecifications.

A. Factors and Ranges: Design variables are classified following the terminology andprinciples used in Taguchi’s robust design to define the initial concept exploration space.

C. SimulationPrograms

(Rigorous AnalysisTools)

Overall DesignRequirements

D. Experiments Analyzer

Eliminate unimportant factorsReduce the design space to the region

of interestPlan additional experiments

Robust, Top-LevelDesign Specifications

A. Factors and Ranges

Product/Process

Noise zFactors

yResponse

xControlFactors

F. The Compromise DSP

FindControl Variables

SatisfyConstraintsGoals"Mean on Target""Minimize Deviation"“Maximize the independence”

BoundsMinimize

Deviation Function

Input and Output

Processor

Simulation Program

E. Response Surface Model

y=f(x, z)

µˆy = f(x,µz)

σ2ˆy=Σi=1

k ŽfŽzi( )

2σ2ˆz i

i=1

l ŽfŽxi( )

2σ2ˆx

i+Σ

x1

x2

y

B. Point Generator

Design of ExperimentsPlackett-Burman

Full Factorial DesignFractional Factorial DesignTaguchi Orthogonal ArrayCentral Composite Design

etc.

Figure 2.19 - The Robust Concept Exploration Method (RCEM) (Chen, 1995).


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Design variables are defined as either control factors (under designer’s control) or noisefactors (not under designer's control), and the appropriate range of values for each isspecified. The responses (performance measures) are also identified, along with theperformance goals (signals). The range of interest for each response is also determinedfor use in reducing the problem. The means for predicting the performance must also beidentified. The focus in robust design is to reduce both the effect on performance of thenoise factors, and the effect of variations in control factor values on performance.

B, C, D. Sequential Experimentation: A low order experiment is designed, theexperiments simulated (conceptual designs generated), and the results analyzed.Significant design variables are identified (design drivers), and insignificant parametersare fixed. Higher order experiments are designed and conducted as necessary and theresults analyzed. Thus, the number of experiments and order of the experiments isgradually increased while the size of the problem is gradually reduced.

E. Elaborate Response Surface Models: Response surface models are created toreplace the original analysis tools when exploring concepts to generate top-levelspecifications. The response surface equations map the factor-response relationship.When the order of experimentation is satisfactory, the results are analyzed usingregression analysis and analysis of variance to determine the significance of the fit.When the fit is significant, the final response surface models are defined.

F. Determine the Top-Level Specifications: The response surface models and overalldesign requirements are formulated within the compromise DSP to generate the top-leveldesign specifications. The values of control factors identified in this step become the top-level design specifications. Different design scenarios can be rapidly explored bychanging the priority levels of the goals.

Using the RCEM, a design space can be quickly and efficiently populated in theearly stages of design. Simulations are run at a set of design points, and responsesurfaces are generated that relate product performance to design variable values. Thesefast analysis modules are then integrated into the compromise DSP and the best regionsof design solutions are determined based on multiple measures of merit.

Response surface methodology (blocks A – E in Figure 2.19) is used for generatingrelationship between several responses and design /manufacturing variables in thisresearch. A variation of RCEM to solve geometric tailoring problems is also explored.

Review of Methods:

Three design technologies that are related to the development of the design for RapidTooling system (DFRTS) are reviewed in this section. As discussed in Section 1.2.3, thedecisions of design and fabrication variables provide a unified approach for DFMproblem, which is also shown in the review of Decision-Based Design. In this researchcompromise DSP is used to formulate the DFM problem in the DFRTS. A similarapproach to RCEM is used in the first phase of the solution process of the DFRTS(Section 6.4.2). Therefore they are reviewed in this section to provide a context for theDFRTS to be presented in Chapter 6.


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With some of the related and important topics reviewed in Section 2.2 ~ 2.6, the nextsection will give a summary of the chapter followed by a preview of topics that will becovered in the next chapter.

2.7 LITERATURE REVIEW SUMMARY

The literature review of the topics provides a foundation that can be used indeveloping the mold design and design-for-manufacture methods. Research areasreviewed in this chapter are:

! Mold configuration design methods

! Mold construction methods and tools

! CAD representation

! DFM strategies and techniques

! Design technologies

In the next chapter a Multiple-Piece Mold Design Method (MPMDM) will bepresented (Figure 2.20). The chapter (Section 3.1) starts with an overview of theMPMDM, followed by problem formulations considered in the method (Section 3.2) andsteps involved (Section 3.3). Approaches associated with each step are described inSection 3.4 ~3.6. The chapter closes with a summary, including hypotheses teasing.


R1

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PD2

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§2.2 §2.3 §2.4 §2.5 §2.6

MoldConfiguration

Design Methods


Tools

CADRepresentation


Technologies


Figure 2.20 – Summary of Chapter 2 and Preview of Chapter 3.

Chapter 3 – The Multi-Piece Mold Design Method

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CHAPTER 3

THE MULTI-PIECE MOLD DESIGN METHOD

Mold design can be a difficult, time-consuming process, particularly for multi-piecemolds. The automation of mold design for injection molding process is studied in manypublications. However, they usually address the determination of parting direction,parting line, parting surface, and undercut detection individually. In this chapter, asystematic approach is presented to automate several important mold design steps,including selection of parting directions, parting lines, parting surfaces, and constructionof mold pieces. Additionally, this multi-piece mold design method (MPMDM) is suitablefor simple two-piece molds (consisting of core and cavity), as well as for molds withmany additional moving sections. The method is mainly based on regions, which arecentral to the mold configuration design and mold piece construction processes. Thechapter starts with an overview of the MPMDM (Section 3.1), followed by problemformulations considered in the method (Section 3.2) and steps involved in the MPMDM(Section 3.3). With the general steps presented, approaches associated with each step aredescribed individually (Section 3.4~ 3.6). Finally the relationship between MPMDM andthe validation of the hypotheses are discussed (Section 3.7).

Chp 3: Multi-Piece MoldDesign method

R1

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3.1 OVERVIEW OF THE MULTI-PIECE MOLD DESIGN METHOD

Combining multi-piece molding and Rapid Tooling techniques, it is possible to buildinjection molding tools for complex parts in a very short period of time. However, sincemulti-piece molds have more than one pair of opposite parting directions, it is moredifficult and time-consuming to generate a good mold design. Particularly for rapidtooling applications, delivering prototype parts with turn-around times of less than twoweeks requires fast, proven mold design methods.

Given the geometry of a part, depending on the selection of mold design variables, adifferent number of mold pieces may be required to form the part. It is desired tominimize the number of required mold pieces because less mold pieces reduce the toolingcost and simplify the operation of the mold. Therefore the problem considered in thisdissertation for multi-piece mold design is described as follows.

Problem MD: Mold Design. Given a solid part and a mold base, design minimumnumber of mold pieces that can form the cavity of the part in the material injectionprocess, and can disassemble properly in the part ejection process.

Mainly from the geometric perspective, a systematical method, Multi-piece MoldDesign Method (MPMDM), is developed to automate several important mold designsteps, including selection of parting directions, parting lines, parting surfaces, andconstruction of mold pieces. The elements of the MPMDM and their relations are shownin Figure 3.1. The CAD models of part and mold base are the input. Correspondinglythe CAD models of mold pieces are generated by the MPMDM as the output. Asillustrated in the figure by a bounding box, the MPMDM actually consists of twoproblem formulations, a three-stage design process, and three approaches for each stage.

Problem MCD(Section 3.2.2)

A Mold Configuration Design Process

An approachto generate

basic elements(Section 3.4.3)

An approach tocombine basic

elements(Section 3.5.3)

An approach toconstruct mold

pieces(Section 3.6.1)

Part Basicelements

Moldpieces

Multi-PieceMold Design

Method

A MoldConstruction

Process

Problem MPC(Section 3.2.2)

Problem MD

MoldBase

Mold PieceRegions

Figure 3.1 – The Elements of the Multi-Piece Mold Design Method.


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In the MPMDM, the given Problem MD is divided into two sub-problems (ProblemMCD and MPC). Related to the sub-problems, a design process with three stages isproposed and approaches for each stage are developed (Section 3.4 ~ 3.6).

In this chapter, the Multi-piece Mold Design Method (MPMDM) is presented beforethe associated system (Rapid Tooling Mold Design System) is introduced in the nextchapter. Related to the elements given in Figure 3.1, the problem formulations ofMPMDM are presented in Section 3.2 to provide the context of the method. Based on theproblem formulations, the mold design processes, which consists of three steps togenerate mold pieces for a part, are described in Section 3.3. The basic elements used inMPMDM and an alternative approach to generate them from a CAD model of a part arepresented in Section 3.4. An approach to combine these basic elements into regionsrelated to a mold design is discussed in Section 3.5. Based on the generated regions, amold piece construction approach is presented in Section 3.6 for a given mold base.

In Section 1.3.1, research questions and hypotheses for this dissertation werepresented. These research questions and hypotheses provide the basis for this work. Toprovide a better understanding of how the MPMDM in this chapter is related to thehypotheses, the relationship between the elements of MPMDM and the dissertationhypotheses is shown in Figure 3.2. Hypothesis 1 and sub-hypotheses 1.1 ~ 1.3 werepresented in Section 1.3.1. In Section 3.7, how these hypotheses are validated will beelaborated.

In light of multi-piece molding described in Section 1.2.4, four terms extensivelyused in this chapter are defined first to provide a context for the reader. They may haveslightly different meanings from the definitions given in the handbooks like (Pye, 1989)and (Rosato & Rosato, 1995), which mainly considered two-pieces molding.

Problem MCD

A Mold Configuration Design Process

An approachto generate

basic elements

An approach tocombine basic

elements

An approachto constructmold pieces

Part Basicelements

Moldpieces

Multi-PieceMold Design

Method

A MoldConstruction

Process

Problem MPC

Problem MD

MoldBase

H1

H1.1 H1.2 H1.3

Mold PieceRegions

Figure 3.2 – Relationship between the Elements of MPMDM and DissertationHypotheses.


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A Parting Direction (PD), in this dissertation, is a direction along which a mold piece isseparated from the injection molded part.

A Parting Line (PL), in this dissertation, is the continuous closed curves on the surfaceof a part which define faces to be split into mold pieces.

A Parting surface (PS), in this dissertation, is the contacting plane of the mold piecesthat forms a seal to prevent the thermoplastic material from escaping.

A Mold base (MB), in this dissertation, is the mold plate that forms the part cavity.

In light of these definitions, the problem formulation for multi-piece mold design isdiscussed in the next section.

3.2 PROBLEM FORMULATION FOR MULTI-PIECE MOLD DESIGN

An accurate problem formulation is significant since it will affect the researchapproach and the problem solving process to be used. The current state of research onthe automation of mold design mainly focuses on the two-piece molds because they aremore commonly used and relatively easier to design and manufacture than the multi-piece molds. However, the problem formulations presented for the two-piece molddesign may not be proper for the multi-piece mold design. This is illustrated in Section3.2.1 by analyzing a representative problem formulation given by Chen, et al., (1993).Based on the analysis, the problem formulations of the MPMDM are presented in Section3.2.2.

3.2.1 Analysis of Existing Problem Formulations

The automation of mold design for injection molding process is studied in manypublications (refer to Section 2.2 and 2.3). However, each paper usually addresses thedetermination of parting direction, parting line, parting surface, and undercut detectionindividually. This may bring problems in the efforts of combining them into one systemsince different approaches may use different criteria and different basic elements.Therefore problem formulations that consider the whole mold design process are highlydesirable.

Since a systematic approach to consider all the above important considerations is notfound, the remainder of this section will mainly discuss the existing problemformulations for the determination of parting direction (PD). The determination of PD isthe first step in the automation of mold design. It is also the most important step since aPD will affect all the subsequent steps in the design of a mold. A general description ofthe existing problem formulations is: Find the best parting direction according tosome criteria among candidate parting directions, which are calculated fromindividual pockets or features. According to this formulation, other faces of the partthat do not belong to the pockets or features will not be considered in the determination.

A representative approach (Chen, et al., 1993) with its problem formulation isprovided here to illustrate this general description. The approach was developed byLinlin Chen, who was supervised by Tony Woo when they were at the University ofMichigan. For an object Ω (Figure 3.3.a), pockets of the object are the regularizeddifference between its convex hull CH(Ω) and Ω, denoted by CH(Ω) -* Ω (Figure 3.3.b).


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Based on the Visibility Map (V-map) of pocket surfaces (refer to Section 2.2), sphericalpolygons of the pockets can be generated (Figure 3.3.c).

So the problem PPD (pair of parting directions), which is described as “given anobject, find a pair of opposite parting directions that minimizes the number of moldpieces”, is transformed to a new problem formulation as follows:

Problem SPCA (spherical polygon covering by antipodes): Given a set of sphericallyconvex polygon V1, V2, …, Vm, find a pair of antipodal points p and –p that minimizethe number of Vi containing either p or –p.

This problem formulation is widely referenced. Several other approaches (Weinsteinand Manoochehri, 1996; Vijay, et al., 1998; Yin, et al., 2001) for the selection of PDwere also developed based on it. However, three problems exist especially when it isapplied to the multi-pieces mold design.

First, the position relationships among pockets and features are not considered in theformulation. So all related approaches look for a pair of parting directions (PD)according to the intersection of V-maps of pockets or features. It is assumed that ifseveral pockets share the same PD, they can be formed by a single mold piece without

(a) A part (b) Pockets of the part (c) Spherical polygon of the pockets

Figure 3.3 – An Approach Based on V-map.

PDPD1

-

PD1+

PD2+

PD2-

(a) Pockets with same PD (b) Different Criteria

Figure 3.4 – Examples for the Problem Formulation.


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considering their actual positions. However, it is not always true. For two pockets withthe same PD, if another pocket with a different parting direction lies between them, it isdifficult or even infeasible to construct a single mold piece for both of them. Forexample, in Figure 3.4.a, the PD can satisfy the V-maps of P1 and P3. However, since P2,which lies between P1 and P3, cannot utilize the same PD, a different PD needs to beassigned to it. Even if the V-maps of P1 and P3 intersect, a single mold piece for both ofthem is hard to generate. To avoid constructing a mold piece to form separated faces, itis assumed in this dissertation that all the faces of a mold piece are connected.

Second, the problem of finding a PD with the minimum number of cores isformulated so as to find the minimum number of pockets or minimum number of faces,which are not covered in PD+ and PD-. However, the criterion of minimum non-coveredpocket number is not always the same as the criterion of minimum mold piece number.For example, for a simple part as shown in Figure 3.4.b, by using the criterion ofminimum pocket number, PD1 will be chosen as the parting direction. Therefore twoadditional cores are needed to form pocket P1 and P5. However if we choose PD2, onlyone core is needed to form P2, P3 and P4. So actually the best solution to minimize themold pieces (PD2) is different from the solution to minimize non-covered pockets (PD1).Again, since the geometric relations are not considered in the problem formulation, it isdifficult to know the relationship between the number of mold pieces and the number ofnon-covered pockets.

Third, the parting direction for a face F is usually governed by the notion ofcomplete visibility. That is, for every point p on F, if the ray from p to infinity in thedirection d does not intersect the part, d is a good parting direction for face F. However,for multiple piece mold design or form pin design, the requirement is less strict. If moldpieces can form the cavity and they can be disassembled in their parting directions insome order, then it is a feasible design. As an illustrative example, the shape P in Figure3.5 is a pocket with empty V-map according to (Chen, et al., 1993) and is therefore notconsidered in the formulation. But by using multiple piece mold design, the mold piecesM1 and M2 can form the shape. In the disassembly process, M1 is first translated indirection PD1. Then M2 can be moved out first in direction PD2, then direction PD1. SoPD2 is a feasible solution for face F even if F cannot be swept to infinity in direction PD2

without interference with the part.

Fd

M1

M2

PD2

PD1

P

Figure 3.5 – Mold Pieces for a Pocket with Empty V-Map.


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Based on the example given in Figure 3.5, it is clear that multi-piece mold design ismuch more complicated than two-piece mold design. In the mold disassembly process,mold pieces can be translated in different orders, and a mold piece can also be translatedin more than one direction. Since face sweeping operations and body interference testsare rather time consuming, these tests will not be considered in this dissertation. Butbased on the mold design result, an individual simulation module can be developed tofind a suitable disassembly order to translate the generated mold pieces in related partingdirections. If interference between the part and the mold pieces is found, we know thateither a new mold design needs to be regenerated, or the given part is not moldable andtherefore some modifications are necessary. Therefore in this dissertation it is assumedthat checking a mold piece and its neighboring part faces is sufficient to determine if themold piece can be disassembled.

In light of the problems of the existing formulations, the problem formulations ofMPMDM for the mold configuration design and the mold piece construction arepresented in the next section.

3.2.2 Problem Formulations of Multi-Piece Mold Design

The generation of mold pieces for a part can be divided into two phases (Figure 3.1):

(1) Mold configuration design. The mold design variables of parting directions andparting lines are determined according to the geometry of the part.

(2) Mold piece construction. The mold pieces are generated according to a moldbase and the results given by the mold configuration design.

Accordingly the problem MD described in Section 3.1 actually consists of two sub-problems, problem MCD and problem MPC. The formal formulations of the twoproblems are presented as follows.

Problem MCD: Mold Configuration Design. Given a solid part in theBoundary Representation (Mäntylä, 1988), it can be transformed as a graph G(N, A, R, E)where N is the set of nodes, A is the set of arcs, R is the set of attributes for nodes, and Eis the set of attributes for arcs, such that:

a. for each face si ∈ Sface, there exists only one node in N;b. for each common edge between faces si, sj ∈ Sface, there exists a unique arc aij-k

connecting the nodes ni and nj (there may be more than one arc between ni and nj);c. for each node corresponding to a face si, an attribute ri (an integer number) is

assigned to represent the region number;d. for each arc aij, an attribute ek is assigned to represent the edge between two faces.

If ri ≠ rj, ek = 1. The related edge is defined as a boundary edge; otherwise ek = 0,and the related edge is defined as an internal edge.

Among all the combinations of G(N, A, R, E), find a graph G(N, A, Ri, Ej) such that:

(1) for each face pair si and sj, if rsi = rsj = r, we can find a path to link si and sj withall nodes that have the same r. In other words, all faces with r are connected;

(2) A direction PD exists for all faces with same value ri so that the related moldpiece Mi can be disassembled (demoldability);


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(3) The total number of different r values is minimized.

According to the result G(N, A, Ri, Ej) of the problem MCD, several sets Ri(F, E, D)can be generated easily for the part, where F is the nodes N with the same value Ri, E isthe set of arcs A with Ej=1, and D is the direction PD generated for each set ri. Thereforea definition of mold piece region is given as follows:

Definition 3.1. A Mold Piece Region (MP region) is a set Ri(F, E, D) where F is thenodes N with the same value Ri, E is the arc set A with Ej=1, and D is the directionPD generated for each set ri based on the result G(N, A, Ri, Ej) given by the problemMCD.

Therefore the phase of mold piece construction can be formulated as:

Problem MPC: Mold Piece Construction. A solid part P and a mold base MBare given in the Boundary Representation. Suppose set Ri(F, E, D) (1≤ i≤ k) are given,where

(1) each node of F is a face of P, and each face of P is a node of F in Ri;(2) each node of E is a boundary edge of the faces in F;(3) D is a direction which makes an angle of at most 90o with the outward normals of

all faces in F;(4) All faces in F are connected, that is, for any two nodes of F, we can find a path to

link them with all nodes in the same set.Generate bodies M1~Mk such that:

(1) Faces of Ri are formed and only formed by Mi.(2) After M1 ~ Mk are assembled together, they form MB with a cavity P inside. That

is, MB – P = M1 ∪ M2 ∪ …∪ Mk, where – and ∪ are two Boolean operators,subtraction and union respectively.

After the mold design problems considered in this dissertation are formerly defined,the mold design process related to the problems is introduced in the next section.

3.3 OVERVIEW OF THE MULTI-PIECE MOLD DESIGN PROCESS

The mold design process has been divided into two phases, mold configurationdesign and mold piece construction. However, it is still quite difficult to generate moldpiece regions from the given faces of a part. Therefore the first phase is further dividedinto two steps, generating basic elements from the faces and generating mold pieceregions from the basic elements (refer to Figure 3.1). In Section 3.4, the author willexplore what kinds of basic elements are appropriate for problem MD based on thedemoldability of a mold piece.

So the Multi-Piece Mold Design Method (MPMDM) has three steps as illustrated inFigure 3.6 with relevant section given for each step. The inputs to MPMDM are theboundary faces of a part P. Suppose the part has only one lump (a bounded, connectedregion in space). Its boundary faces can be represented as shown in Figure 3.6.a.Furthermore suppose that the part considered in this dissertation is a polyhedron. So allfaces are planar surfaces and all edges are straight lines. For other kinds of surfaces, theycan be approximated by planar surfaces in different mesh sizes. This limitation will berevisited in Section 4.2.


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From the faces of P, basic elements are generated first as the starting point for thedetermination of regions (Figure 3.6.b). An approach with related algorithms for this stepis presented in Section 3.4. The basic elements are then combined into several moldpiece regions (Figure 3.6.c). All the part faces are combined in the way such that all thegenerated regions have at least one feasible parting direction. Therefore it is guaranteedthat the mold pieces constructed in the next step can be disassembled properly. Anapproach with related algorithms for this step is presented in Section 3.5.

After the mold configuration design, mold pieces M1, M2, …, Mk are to beconstructed for the mold piece regions R1, R2, …, Rk according to their PDs and PLs(Figure 3.6.d). There is a one to one correspondence between the mold pieces and theregions, and the faces of Ri should be formed by the mold piece Mi. By assembling allthe mold pieces together, they should form a mold base with a hollow cavity space in theshape of P. An approach with related algorithms for this step is presented in Section 3.6.

As stated the generation of basic elements is the first step of MPMDM. In the nextsection, the basic elements developed for multi-piece mold design are introduced withalgorithms to generate them from the faces of a part.

3.4 BASIC ELEMENTS OF MPMDM

In this section the demoldability of a mold piece Mi is analyzed first. Based on theanalysis, the basic elements of MPMDM are presented in Section 3.4.2 with a

R 1

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Section3.4

Section3.5

Section3.6

(a) Part faces (b) Basic elements

(c) Mold Piece regions (d) Mold pieces

Figure 3.6 – Steps of the Mold Design Process.


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comparison with the basic elements of other approaches. Finally an approach and twoalgorithms to generate them are described in Section 3.4.3.

3.4.1 Demoldability of Mold Pieces

The condition of demoldability is the basis to problem MCD (Section 3.2.2).Several lemmas on the demoldability of a mold piece are derived as follows.

Suppose region Ri is related to mold piece Mi. All the faces of Ri and all the faces inother regions, which share at least one boundary edge with Ri, will affect Mi’sdemoldability. Although the faces that are not neighboring may also cause interferenceafter translating Mi for some distance, they are not considered in this dissertation (refer toSection 3.2.1). Therefore two lemmas, lemma 3.1 and 3.2, are sufficient for determiningthe demoldability of Mi. Lemma 3.1 considers the possible interference between a moldpiece and the faces of the related region.

Lemma 3.1. A mold piece Mi can be removed from the faces it forms (FRi) by atranslation in direction PD if and only if PD makes an angle from 0o to 90o with theoutward normals of all faces FRi.

Proof. The proof for the lemma is similar to a lemma presented for casting in (de Berg etal., 1997). For the “only if” part: if PD would make an angle greater than 90o withsome outward normal of a face f, then any point q in the interior of f collides with themold when translated in direction PD.

For the “if” part, suppose at some moment FRi collides with Mi when translated indirection PD. Let p be a point of Mi that collides with a face f of Fi. This means thatp is about to move into the interior of f, so the outward normal of f must make anangle greater than 90o with PD. !

Let ηηηη(f) = (ηx, ηy, ηz ) be the outward normal of a region face f. The direction PD =(dx, dy, dz) makes an angle of at most 90o with ηηηη(f) if and only if the dot product of PDand ηηηη is non-negative. Hence, a face of the region induces a constraint of the form

ηx dx + ηy dy + ηz dz ≥ 0. (3.1)

Lemma 3.2 considers the possible interference between a mold piece and its

Fj

Fi PEi

PEi

Fi Fj

+-

ηηηη(Fj)

FjFi

+-

FjFi

PDi PDj

PEi PEi

MiMj

(1) (2) (3)

Ri

Rj

PDi

PDi

PDi

(a) A neighboring face (b) Relation of PD with neighboring face

Figure 3.7 – Demoldability of Mold Pieces.


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neighboring faces. As shown in Figure 3.7.a, suppose Fi is a face of region Ri and Fj is aface of another region. If Fi and Fj share an edge PEi, Fj is called a neighboring face ofRi, and PEi is called a neighboring edge of Ri. For any two neighboring faces, if thedihedral angle of the faces is less than 180o, the edge between them is a concave edge.Otherwise the edge is a convex edge. Therefore the translation direction PDi of the moldpiece Mi needs to satisfy some additional constraints.

Lemma 3.2. The mold piece Mi can be removed from a neighboring face Fj by atranslation in direction PD without interference if and only if (1) PEi is a convexedge; or (2) PEi is a concave edge and PD • ηηηη(Fj) ≥ 0.

Proof. For the “only if” part, if PEi is a concave edge and PD •ηηηη(Fj) < 0, then any point pin PEi of the mold piece will collide with the interior of Fj when translated indirection PD, as shown in Figure 3.7.b2.

For the “if” part, suppose at some moment Fj collides with Mi when translated indirection PD. Let p be a point of Mi that contacts Fi. This means that p is about tomove into the interior of Fi from the outside, so (1) if PEi is a convex edge, since Fj isbelow Fi, Mi is translated in a direction that makes an angle greater than 90o with theoutward normal of Fi. So PD is not a valid direction for Mi according to Lemma 3.1(refer to Figure 3.7.b1). (2) If PEi is a concave edge, the outward normal of Fi mustmake an angle greater than 90o with PD, so PD • ηηηη(Fj) < 0. !

It is noticeable that following Lemma 3.2, even if a mold piece can be generated thatdoes not interfere with its neighboring faces, two neighboring mold pieces generated inthis way may not be able to be taken out at the same time. For example, as shown inFigure 3.7.b3, face Fi is related to Mi which has parting direction PDi, while face Fj isrelated to Mj with parting direction PDj. In the disassembly process, it is quite obviousthat we need to translate Mi in PDi first before we can translate Mj in PDj.

To design a suitable mold construction order, Lemma 3.3 provides guidance inidentifying the neighboring regions that cannot be removed at the same time. Therefore adisassembly order needs to be considered further for them. For two neighboring regionsRi and Rj, two of their faces Fi and Fj share an edge PEi as shown in Figure 3.8.a. Twovectors (called Coedge in the Boundary Representation) CEi and CEj share the same edgeand have reverse directions. They define the interior side of the related faces.

Lemma 3.3. Two neighboring mold pieces Mi and Mj can be removed by translations indirections PDi and PDj individually without interference with each other at edge PEi

if and only if CEi • (PDi x PDj) ≥ 0 , or CEj • (PDj x PDi) ≥ 0.

Proof. Since CEi = - CEj, the two inequalities are actually the same. So we only need toprove one. For the “only if” part, if CEi • (PDi x PDj) < 0, the vector (PDi x PDj ) isin the reverse direction as CEi, so PDi and Mi are in the different sides of PDj.Therefore the sweeping of PEi in PDi will intersect with the sweeping of PEi in PDj,as shown in Figure 3.8.b2.

For the “if” part, if two mold pieces can not be removed in directions PDi and PDj

individually, that is, the sweeping of edges PEi in PDi will intersect with thesweeping of PEi in PDj. So PDi and Mi are in the different sides of PDj. ThereforeCEi • (PDi x PDj) < 0.


83

If CEi • (PDi x PDj) = 0, that is PDi and PDj are in the same direction, so thesweeping of edges PEi in PDi will not intersect with the sweeping of the edge in PDj,and vice versa (refer to Figure 3.8.b3). !

For Problem MCD, conceptually if we try all different combinations of attribute Rfor a part represented by graph G(N, A, R, E), we can always get the best designaccording to our requirements and criteria. However, it is quite obvious that this problemis strongly NP-hard since even one of its sub-problems (pocket combination to get a pairof directions) is strongly NP-hard (Chen, et al., 1993). To develop an approach to solvethis NP-hard problem, we need to explore the properties of convex and concave edgesfurther.

Lemma 3.4. Suppose all the edges of a face Fi are convex. Fi can be added to anyneighboring region Rj without increasing the mold piece number or changing thedemoldability of mold pieces, if a parting direction PD exists which makes an angleof at least 90o with the outward normals of all the faces of the region and Fi.

Proof. First according to Lemma 3.1, if a PD exists for all the faces of the region (Fj) andFi, a new mold piece can form Fi and Fj, and be removed from the faces by atranslation in direction PD.

Suppose F1 is the face to be combined with R1. It has an edge E1 sharing with face F2,which belongs to another region R2. Since E1 is a convex edge, PD1 for R1 and PD2

for R2 will all satisfy Lemma 3.2. So the disassemblability of related mold pieces M1

and M2 is not changed. It is quite obvious that the mold piece number will not changealso. For other neighboring edges we can follow the same proof. !

Lemma 3.4 can also be extended to two regions. Suppose faces F1, F2, …, Fk

compose a region R1 and Fk+1, Fk+2, …, Fn compose a region R2. The bounding edgesbetween F1, …, Fn and all other faces are E1, E2, …, Ek.

Lemma 3.5. Suppose all bounding edges E1, E2, …, Ek are convex. Two neighboringregions R1 and R2 can be combined as one region without increasing the mold piecenumber or changing the demoldability of mold pieces, if a parting direction PD existswhich makes an angle of at least 90o with the outward normals of all the faces F1, F2,…, Fn.

CEiCEj

RiRj

Fj

Fi

PEi

PDiPDj

PDjPDi PDiPDj

CEi CEiMi MiMj Mj

PDiPDj

CEiMi Mj

(1) (2) (3)

(a) Two neighboring regions (b) Relation of Parting Direction of the regions

Figure 3.8 – Demoldability of Neighboring Mold Pieces.


84

Proof. First according to Lemma 1, if a PD exists for all the faces of the regions R1 andR2, a new mold piece can form all the faces, and be removed from the faces by atranslation in direction PD.

For a neighboring face Fi, suppose Ei is the neighboring edge. Since Ei is a convexedge, the demoldability of related mold pieces is not changed according to Lemma3.4. Also it is quite obvious that the mold piece number will not increase. Instead itwill decrease by 1. !

The mold design process is actually a process to find a good combination of R in thegraph G(N, A, R, E) according to our requirements. From the above analysis, severalproperties of the combining process become evident:

Property 3.1. To get the minimum number of mold pieces, different regions and facesare to be combined into larger regions to the extent possible.

Property 3.2. To maintain the connectivity of a region, only the region’s neighboringfaces need to be evaluated.

Property 3.3. All faces in one region satisfy Equation 3.1. According to Lemma 3.2, ifa face shares a concave edge with a region, the face normal must satisfy Equation 3.1also. Therefore after combining faces into regions, two faces with a concave edge aremore likely to be in the same region.

Property 3.4. The combinability of a face with other neighboring faces is affected bywhether the neighboring edges are concave or convex. Since all the faces that shareconcave edges need to be considered, a face with only convex edges is more easilycombined than a face with concave edges.

For further discussion, three definitions are given as follows.

Definition 3.2. A Concave Region, CVR(r), is a subgraph of the graph G(N, A, R, E)given in Problem MCD, such that: (1) for every node Ni that belongs to the subgraph,related attribute ri = r; (2) for each arc Aij related to Ni and Nj, if attribute ri = rj = r,the edge associated with the arc is a concave edge; (3) for each arc Aij related to Ni

and Nj, if attribute ri = r and rj ≠ r, the edge associated with the arc is a convex edge.

Definition 3.3. A Convex Face is a face of which all the edges are convex.

Definition 3.4. A Combined Region, CR(r), is a subgraph of the graph G(N, A, R, E)given in Problem MCD, such that: (1) for every node Ni that belongs to the subgraph,related attribute ri = r; (2) for each arc Aij related to Ni and Nj, if attribute ri = r and rj

≠ r, the edge associated with the arc is a convex edge.

From the definitions, a combined region can be a concave region, or a region withseveral convex faces, or a combination of some concave regions and convex faces.

Based on Lemma 1~5, the combining process to get a good mold design can bepursued in two steps:

(1) Combine neighboring faces with concave edges into concave regions Ri’;(2) Combining concave regions and convex faces into combined regions.


85

Considering Properties 3.3 and 3.4 of the combining process, it is assumed that thefaces of Ri’ generated in Step (1) will not be divided into different regions in Step (2).That is, among all the combinations of R, those combinations, which have two facessharing a concave edge but belonging to different regions, will not be considered.

Without further dividing of Ri’, a concave region generated in Step (1) must satisfyLemma 1 in order to get a feasible mold piece. If no PD satisfies Lemma 1 for a regionRi’, we have the following lemma.

Lemma 3.6. Suppose faces F1, F2, …, Fk compose a region with only concave internaledges and convex boundary edges. If no direction PD makes an angle of at most 90o

with the outward normals of all the faces, no mold pieces to form them can beremoved individually without interference.

Proof. First according to Lemma 1, if no PD makes an angle of at most 90o with theoutward normals of all the faces, two or more mold pieces need to be used for regionR’. Accordingly, R’ can be divided into two or more regions. Each of them has a PDto satisfy Lemma 1.

For the situation of two regions R1’ and R2’, suppose faces F1 and F2 belong to themindividually with a neighboring edge E12, as shown in Figure 3.9.a. Accordingly, tosatisfy Equation 3.1, any PD1 for R1’ makes an angle from 0o to 90o with the outwardnormal ηηηη(F1); any PD2 for R2’ makes an angle from 0o to 90o with the outwardnormal ηηηη(F2), as shown in Figure 3.9.b. If only F1 and F2 are considered, the feasibleregion of PD for R’ is from ηηηη1 to ηηηη2. By adding a face (e.g. F3) in R1’ which shares aconcave edge with F1, the feasible range of PD1 will be reduced from ηηηη3 to ηηηη2

according to Equation 3.1. The feasible range is moving away from F1 toward F2.Similarly adding any faces in R2’ (e.g. F4, F6) will make the feasible region of PD2

move away from F2 toward F1. If PD1 and R1’ are in the same side of PD2 (that isCE1 • (PD1 x PD2) ≥ 0), any direction PD between PD1 and PD2 will satisfy Equation3.1 for both R1’ and R2’. Therefore it makes an angle of at most 90o with the outward

F1F2

F5

F3F4

F6

R 1'

E12

R 2'

R '

E12

F1 PD1

PD2

36 4R 1'

R 2'

F2

ηηηη1

ηηηη 2

F3

F4

F6

ηηηηηηηηηηηη

(a) Two neighboring regions (b) Relation of Parting Direction of the regions

Figure 3.9 – Dividing of Concave Regions.


86

normals of all the faces.

According to Lemma 3.3, two mold pieces related to R1’ and R2’ cannot be removedindividually without interference. !

Products designed for injection molding process seldom have a situation as the caseshown in Figure 3.9.b. More often, for a part that needs side cores or form pins, asshown in Figure 3.5, two concave regions R1 and R2 exist. Each has a PD that satisfiesLemma 1. However since PD1 and PD2 do not satisfy Lemma 3.3, a form pin is neededfor the part. So in this dissertation, we assume for a given part, any concave region musthave a PD to make an angle of at most 90o with the outward normals of all region faces.

The definition of concave region requires that all the internal edges of the region areconcave, and all the boundary edges are convex. From Lemma 3.7, we can see acombined region can always be divided into concave regions and convex faces.

Lemma 3.7. A combined region can be divided into concave regions and convex faces.

Proof. For a combined region R with F1, F2, …, Fk,

(1) If all the internal edges are concave, it is a concave region already;(2) If all the internal edges are convex, F1, F2, …, Fk are all convex faces;(3) Otherwise, suppose an internal edge E12 is a convex edge.

a. If the related faces F1 and F2 are in one plane, we can delete E12 and make a newface to replace F1 and F2 without affecting other faces and edges.

b. Otherwise, we can use the plane of F1 or F2 to split all the faces of the region Rinto two halves. Assign all the faces in the up side to region R1, and all otherfaces to region R2. Now F1 and F2 must be in different regions since E12 is aconvex edge. Therefore E12 is changed from an internal edge to a boundary edge.For all other edges, the splitting will not change the edges’ convex or concaveproperties. For all other faces Fi, (i) if the plane properly intersects Fi as shownin Figure 3.10.a, Fi is divided into Fi

+ and Fi- by adding a new edge Ei. Since Fi

+

and Fi- belong to different regions, the new convex edge Ei is a boundary edge.

P

Fi +

Fi -

Fi

Ei

+

-

+

-

Fi

Fj

P

Fi Fjconcaveedge

Fj '

Fi ' Fi Fi 'convexedge

Fj ' Fjconvexedge

concaveedge

concaveedge

R1 R 2

R1

R1

R 2

R 2

(a) Intersection of Face and Plane (b) Fake Faces

Figure 3.10 – Splitting of a Combined Region.


87

(ii) if the plane intersects a convex edge Eij, the edge is changed from an internaledge to a boundary edge; (iii) if the plane intersects a concave edge Eij as shownin Figure 3.10.b, a fake face Fi’ is introduced for Fi which has zero area and is inthe same plane as Fi. Similarly a fake face Fj’ is introduced for Fj. By adding Fi,Fj’ to R1 and Fj, Fi’ to R2, we can maintain the relationships that all edges within aregion are concave, and all edges between two regions are convex.

Continue the above process for regions R1 and R2, until all the internal edges of aregion are concave. !

By finding a loop of convex edges, a part can easily be divided into two combinedregions. According to Lemma 3.7, it is evident that a part can be divided into concaveregions and convex faces.

Based on the demoldability of a mold piece, the basic elements of MPMDM areconcave region, convex face and combined region. Compared with other approaches, theauthor believes these basic elements are more appropriate for multi-piece mold design. Amore detailed analysis is described in the next section.

3.4.2 Analysis of the Basic Elements

Given the faces of a part, concave/combined regions and convex faces are generatedas the basic elements for Problem MCD. Comparing the basic elements of MPMDMwith those of other approaches (refer to Section 2.2), the author lists similarities anddifferences as follows.

(1) Comparison with pockets

A “pocket” defined in (Chen, et al., 1993), or a “concave region” defined in(Weinstein and Manoochehri, 1996), is actually a combined region defined in thisdissertation. However by adding the concept of concave regions, our approach is able tohandle more general situations.

R1

R2

R4

R3

R

F

R1

R3R2

(a) Four concave regions (b) A combined region and three concave regions

Figure 3.11 – Concave Regions and Combined Regions.


88

A concave region is different from a combined region in its internal edges. If theinternal edges of a combined region R are all concave, R is a concave region also. Forexample, pockets R1~ R4 as shown in Figure 3.11.a are all concave regions. However, acombined region may consist of several concave regions. Two examples are given inFigure 3.11.b and Figure 3.12 for depression and protrusion features respectively. PocketR in Figure 3.11.b has two convex internal edges and composes concave region R1~R3.Similarly pocket R in Figure 3.12 has six convex internal edges and can be decomposedinto concave regions R1~R5. All the pockets are not considered in (Chen, et al., 1993)and (Weinstein and Manoochehri, 1996) since their corresponding V-maps are empty.By considering the concave regions of R instead of R itself, MPMDM can generate molddesigns for the parts.

(2) Comparison with features

Several approaches determine mold design for a part based on feature and relatedfeature recognition algorithms. The author believes that the dividing of part faces intoconcave regions and convex faces is more general than classifying faces into features.For example, features considered in (Gu, et al., 1999) are divided into through hole, blindhole, step, slot, pocket, boss, rib, etc. Using a different classification, (Fu, et al., 1999)considers inside internal undercut features, outside internal undercut features, insideexternal undercut features, and outside external undercut features. Based on the edgecharacteristics, each of them is further divided into three-edge, four-edge and more thanfour-edge. However, all these features are actually a special case of concave region. Soif we just consider the relation of parting direction and undercut, concave region andcombine region is more general and can handle more cases.

(3) Comparison with faces

Faces are the most basic elements of a 3D CAD model. The number of faces of apart is much more than that of regions, pockets, or features. For example, the pocket Rshown in Figure 3.12.a has 10 faces. The number of regions is 5 (Figure 3.12.b) and thenumber of features is 2. In the mold configuration design process, different combinationsof basic elements are explored to determine a mold design. Therefore using faces as

R2

R3

R4R5

R1R

F1 F2

F3

a) Before Splitting (b) After Splitting

Figure 3.12 – Combined Region Example.


89

basic elements may bring difficulties in the exploration process because too many facesexist.

In Sections 3.4.1 and 3.4.2, the basic elements of MPMDM are introduced. Withtheir definitions in mind, an approach to generate concave regions from the faces of a partis presented in the next section. Two algorithms and the analysis of the algorithms arealso presented.

3.4.3 The Generation Approach of Concave Regions

The generation of concave regions is central to the identification of the minimumnumber of mold pieces. In this section, the overall approach for generating concaveregions is presented first. Then the details of the third step in this approach are discussedwith two algorithms presented.

• Summary of the approachThree steps can generate concave regions of a part P. They are (1) classify all edges

of P into concave and convex; (2) Add neighboring faces with a concave edge to generatecombined regions; (3) Generate concave regions from the combined regions. Aftersummarizing Steps 1 and 2, Step 3 is discussed in detail as follows.

(1) Edge classification.

It is straightforward to determine if the dihedral angle of two neighboring faces isless than 180o. Suppose two faces (F1 and F2) intersect at edge BC. Point A and D aretwo points in F1 and F2 respectively (Figure 3.13). For edge BC,

i. if (AB x AC) • AD < 0, BC is a concave edge, where ‘x’ is cross product and ‘• ’is dot product of two vectors;

ii. otherwise, BC is a convex edge.

(2) Generation of combined regions.

This step is similar to the algorithm FIND_CVR for the generation of concaveregions given in (Weinstein and Manoochehri, 1996). Basically for any two neighboringfaces Fi and Fj, if the edge between them is concave, they should belong to a sameregion.

B

C

A

D

F1

F2

Figure 3.13 – Edge Classification.


90

The approach given in (Chen, et al., 1993) can also replace the above two steps.Their approach is to construct a convex hull CH(P) first, then generating pockets by theregularized difference between CH(P) and P.

(3) Generation of concave regions.

The proof of Lemma 3.7 provides a way to generate concave regions from acombined region. The essential step is to split the combined region into concave regionsby utilizing a convex edge that is internal to the combined region. It is likely that theseconvex edges either indicate the presence of undercuts or prevent mold pieces from beingdemoldable for other reasons. Step 3 is discussed in details as follows.

• Split Region AlgorithmsThe algorithm Split_Region (SR) uses a face bounded by a convex internal edge (i.e.,

a convex edge is internal to a combined region) to split a given region. For each newlygenerated region, the function is executed recursively.

Algorithm: Split_RegionInput: A combined region CR.Output: A set of concave regions SCVR.(1) Find all convex internal edges of CR;(2) If no convex internal edges exist, then add CR to SCVR, and return.(3) Find a face Fsplit which has convex internal edges, and construct a split surface

fsplit from Fsplit.(4) Split each face Fi of CR into Fi

+ and Fi- by fsplit.

(5) Generate a region CR+ by adding all faces Fi+.

(6) Generate a region CR- by adding all faces Fi-.

(7) Add connecting faces in CR+ and CR- to new regions CRi.(8) Call Split_Region for each CRi.

By running the algorithm for a region, at least two regions are generated (CR+ andCR-). Sometimes, more than two regions are generated according to the selected face andthe part geometry. For example, in Figure 3.12.a, if the plane of F1 is used to split R,three regions (R1, R2, and a combined region of R3 ~ R5) are generated in Step (7) since R2

and the combined region in CR+ are not connected.

Step (4), face splitting, is studied in (de Berg et al., 1997). All other steps are prettystraightforward except Step (3). In Step (3) of the algorithm, there are often manychoices in selecting a splitting face according to the internal edges. By using differentsplitting faces or different splitting orders, different split regions will result, as willdifferent mold designs.

Observing the combined region given in Figure 3.14.a, there are four convex edgeswithin the region. Related to the edges, we can select any of faces (F1, F2, F3, or F4) asthe splitting face. Totally there are 24 (4!) different combinations. For some of theselections, the splitting results of face F are shown in Figure 3.14.b. So some selections(like first F1, then F3) can generate concave regions by splitting regions twice. Othersmay require three splitting operations. Also since the generated concave regions aredifferent, the mold pieces generated for the part may be different. For example, if PD isselected for a mold piece M1 to form F (Figure 3.14.a), sub-face (1) may be added to M1


91

by using splitting faces F1, F3. However, if using splitting faces F1, F2, and F3, the sub-face (2) may be added to M1 (Figure 3.14.b).

A straightforward way to get a unique dividing result is to find all the faces that haveone or more convex internal edges, then use related planes to split all faces of the region(including the newly generated faces). So by using the splitting faces F1, F2, F3, and F4,face F in Figure 3.14.a can be divided as shown in Figure 3.14.b (Divide all). The resultis not related to the choice of splitting order or splitting face.

Therefore, a modified algorithm, Complete_Split_Region (CSR), for completesplitting is listed below:

Algorithm: Complete_Split_RegionInput: A combined region CR.Output: A set of concave regions SCVR.

(1) Find all convex internal edges of CR;(2) If no convex internal edges exist, then add CR to the set SCVR, and return.(3) Find faces related to the convex internal edges, add surfaces corresponding to the

faces to set SF.(4) for a surface fsplit in SF,(5) for each face Fi of CR,(6) Split Fi into Fi

+ and Fi- by fsplit, and replace Fi in CR.

(7) Add regions CRi generated by adding all neighboring faces of CR with onlyconcave edges.

However, by algorithm analysis of Complete_Split_Region presented below, it isclear that complete splitting is not a good approach. Readers can refer to the text of(Cormen, et al., 1990) for the standard notations used in the area of Algorithm Analysis.

Suppose a region R is composed by n faces. Among them, m faces are related toconvex internal edges (e.g. for region R given in Figure 3.14.a, n = 6, m = 4). By using aplane to split the region, some faces are divided into two faces. Others are either entirelyabove the plane, or entirely below the plane. Suppose the average number of dividing

A

B

F

C

F1

F2

F3

F4

R

A B

C

F A B

C

F

A B

C

F A B

C

F

(F1 -> F3) (F1 -> F2-> F4)

(F1 -> F2-> F3) (Divide all)

(1)

(2)

PD

(a) Example part (b) Different results

Figure 3.14 – Different Splitting Faces and Orders.


92

faces for each splitting plane is d, which is a small percentage of n. Assume a = d/n,which represents the percentage of faces that are split by the splitting plane, and b =1+a.

Algorithm Analysis of Complete_Split_Region:

In the algorithm CSR, the main steps are (4), (5) and (6). Other steps can be omittedcompared to them. For the first splitting, n faces will be handled and a•n will be divided.Therefore we will get n + a•n = b•n faces. For the second splitting, b•n faces will behandled and a• (b•n) will be divided. So for the third splitting, b•n + a• (b•n) = b2•nneed to be handled and a• (b2•n) will be split. Similarly, for the mth splitting, bm•n needto be handled. Suppose the analysis of each face is Θ(1), the running time of thealgorithm is:

T n m n b n b n b n b n n b b b

n b b n b

m m

m m

( , ) ... ( ... )

[( ) / ( )] ( )

= + ⋅ + ⋅ + ⋅ + + ⋅ = ⋅ + + + +

= ⋅ − − = ⋅

2 3 21

1 1 Θ

Suppose m= c•n, where c is a constant. So T n n bc n( ) ( )= ⋅ ⋅Θ

For any given part, 0 < a < 1 and 1 < b < 2. For large part with a big n, a algorithmwith Θ(n • bc•n) running time (b>1) is obviously not acceptable. Also, the number ofresulting faces will be very large (n•bc•n).

Algorithm Analysis of Split_Region:

Compared to CSR, the analysis of SR is a little more complicated because: (i) thenumber of new generated regions can be 2 or more, depending on the topology of partand the surface used to split. (ii) For each splitting operation, all internal convex edgesrelated to the splitting face are converted to boundary convex edges. So one or more facesthat share these edges are not potential splitting faces anymore. However, to simplify theanalysis of the algorithm, it is supposed that (1) each splitting generates only 2 regions;(2) the reduction in the number of splitting faces is not considered. The actual runningtime of the algorithm should be much better than the analysis given here because of thesetwo issues.

Follow the same representations of m, n, a, b, and c for a region R. In the algorithmSR, the main steps are (4), (7) and (8). For the first splitting, n faces will be handled. a•nand a•m faces will be divided, and the face used to split is not a splitting face anymore.Therefore for the next recursive execution, each region will get (n + a•n)/2 = b•n/2 facesand (m + a•m)/2 = (b•m)/2 splitting faces. So b•n/2 faces will be handled. a• (b•n/2)and a• (b•m/2) will be divided. Each region in the next iteration will have b2•n/22 facesand b2•m/22 splitting faces. This process will continue for m’ iterations until bm’m/2m’ =1, that is, only one splitting face exists in the region. No more recursive execution isneeded. So m’ = logb/2(1/m).

Suppose the analysis for each face is Θ(1), then the analysis for the algorithm is:

T n m n Tb

nb

m( , ) [ , ( )]= + ⋅ ⋅ ⋅ −22 2

1


93

= + ⋅ + ⋅ + + ⋅ + ⋅ ⋅− −n b n b nb

n Tb

nb

mm m m m2 1 122

22 2

... ( ) [( ) ,( ) )]' ' ' '

= + ⋅ + ⋅ + + ⋅ + ⋅ = ⋅ − −−n b n b n b n b n n b bm m m2 1 1 1... [( ) / ( )]' ' '

= ⋅ = ⋅Θ Θ( ) ( )' lg /

n b n bm mb 21

since bm

m m m T n m n mb

bbm

b

b

bb b

b

b

blglg

lg

lg

lglg lglg lg

lg lglg lg/

//

, ( , ) ( )2

22

1 1

1

2

12

121= = = = = ⋅

−−

−−Θ .

For any 1< b <1.412, 01

21< =

−− <r

b

b

lg lglg lg . So T n m n m n m c nr( , ) ( ) ( ) ( )= ⋅ = ⋅ = ⋅Θ Ο Ο 2 .

And the face number at the end of splitting is also O(c•n2).

Comparing the cost of algorithm SR and CSR, we can see algorithm SR is morefeasible from the algorithmic perspective. Therefore a good approach to select thesplitting faces and their orders for algorithm SR is necessary.

An ideal way to select a splitting face for a region is to check its neighboring regionsand use mold design knowledge to determine which splitting face is better or if furthersplitting is necessary. So some splitting cases shown in Figure 3.14.b may not besuitable. Characterizing this type of mold design knowledge is beyond the scope of thisdissertation. However, it is important to point out that if mold design knowledge isavailable, it should be applied to the selection of appropriate splitting faces and splittingorders.

According to the analysis of algorithm SR, an obvious set of candidates for thesplitting planes and orders is the set of faces and orders that can always divide the convexinternal edges of a region into two sides evenly. However, it may take a lot of time tofind the best face combination. Therefore in Step (3) of Split_Region, we actually use theheuristic: find a face composing the largest number of convex internal edges to constructfsplit. Comparing to the result of Complete_Split_Region, we may miss some facecombinations.

For parts that approximate a cylinder or a sphere, over-split regions may result.Also, even if all concave regions are generated, it is still impossible to explore all thecombinations of concave regions and convex faces in the region combination asdiscussed in Section 3.5. Therefore the author believes that it is unnecessary to split acombined region until all concave regions are generated.

Usually a combined region needs to be split only when it does not have a feasibleparting direction, as shown in the examples given in Figure 3.11, Figure 3.12 and Figure3.14. In the process of region combination, the main combining criterion is to test if aparting direction exists for both regions. Accordingly the parting direction of a regioncan determine if further splitting is necessary. So another step is added to algorithm SRbefore Step (1), that is:

(0) Get a parting direction PD of CR, if PD is not (0, 0, 0), add CR to SCVR and return.


94

The approach of MPMDM to analyzing CR parting directions is presented in the nextsection for the region combination process.

3.5 REGION COMBINATION OF MPMDM

After concave regions and convex faces are generated for a part, Problem MCD inSection 3.2.2 can be redefined for Step (2) in Figure 3.6.

Problem RMCD: Region-based Mold Configuration Design. Given a solidpart in the Boundary Representation, it can be transformed as a graph G(N, A, R, E), suchthat:

a. Each node in N is a convex face or a combined region;b. For every common edge between nodes ni, nj, there exists an arc aij-k connecting

them;c. An attribute ri, an integer number, is assigned to each node;d. For every arc aij, an attribute ek is assigned to represent the boundary edges of a

node. If ri ≠ rj, ek = 1; otherwise ek = 0.

Among all combinations of G(N, A, R, E), find a graph G(N, A, Ri, Ej) such that:

(1) for every pair of nodes ni and nj, if rsi = rsj = r, we can find a path to link ni and nj

with all nodes having the same r. In other words, all nodes with r are connected;(2) A direction PD makes an angle of at most 90o with the outward normals of all

faces of nodes with the same ri;(3) The total number of different ri is minimized.

The approaches related to Problem RMCD are presented in this section. They areorganized in the following manner. First the criteria considered in the combining processand approaches to evaluate parting direction are presented in Section 3.5.1. Theapplication of the approach in verifying draft angle of faces is introduced in Section3.5.2. Then the region combination process and related representations are described inSection 3.5.3. Finally an algorithm for combining regions is presented in Section 3.5.4with related design knowledge.

3.5.1 Combining Criteria and Their Evaluation Approach

The criteria that are important in the mold configuration design are mainly basedupon the fabrication cost of mold pieces, quality of molded part and productivity ofmolding operations. Three criteria that have been identified in Problem RMCD are faceconnectivity, parting direction and minimum number of mold pieces. Besides them,additional criteria should be considered for Rapid Tooling.

Rapid tooling has some additional properties. Since the material strength of themold (epoxy resin) is much lower than that of steel, some mold features are more easilybroken off in the ejection process, especially when the features are under-drafted (Palmer,1999). On the other hand, the mold fabrication cost is small since Rapid Prototypingtechniques are capable of producing virtually any shapes with rather low cost. Thereforethe ease of ejection plays a dominant role in choosing a good mold configuration forrapid tooling. Consequently, the only additional criterion to be considered in thisdissertation is the ease of ejection.


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The ease of ejection can be determined by the draft angle and the area in shearcontact between the molded part and tools during the mold-opening operation.Considering a face with normal ηηηη (ηx, ηy, ηz) which forms an angle α with a direction PD(dx, dy, dz), the value dp = ηηηη•PD = ηx dx + ηy dy + ηz dz = |ηηηη| |PD| cos(α). dp can evaluatethe value of α. If we take ηηηη and PD as unit vectors (|ηηηη|=1, |PD|=1), dp = cos(α). Alsosince a face Fi with bigger area (Ai) should have a bigger influence on the selection ofparting direction, Ai•dpi can be used to evaluate the ease of ejection for the face in PD.Ai•dpi is also the projected area of the face in PD because dp = cos(α).

The criterion face connectivity is also important in the region combination process.Since the faces of a region should be connected after combining, the face connectivity oftwo regions or a region and a face needs to be determined in the combining process. Fora part in the boundary representation, the face connectivity has been recorded in the datastructure of the part. Therefore the face connectivity can be accessed quickly and easily(refer to Section 4.2).

In the remainder of this section, the criterion parting direction will be discussed.Since an approach that can quickly determine the parting direction of several faces is notfound in literature (refer to Section 2.2), the author will focus on the evaluation of partingdirection in this section. An approach based on Linear Programming is presented fordetermining parting direction of two regions or a region and a face quickly and easily.

• Evaluation of Parting DirectionSuppose a combined region CR in Problem RMCD composes planar faces Fi (1≤ i ≤

n). Let ηηηη(Fi) = (ηix, ηiy, ηiz) be the outward normal of Fi and a direction PD = (dx, dy, dz).According to Lemma 3.1, PD is a parting direction of CR if it satisfies:

η1x dx + η1y dy + η1z dz ≥ 0η2x dx + η2y dy + η2z dz ≥ 0

…ηnx dx + ηny dy + ηnz dz ≥ 0

The set of feasible directions for the above inequalities can be null, one, or a range ofdirections. As stated in Section 2.2, the concept of V-maps can formulate the problem,and spherical algorithms can calculate the intersection of V-maps. However, for ProblemRMCD, a parting direction of a region is calculated for two reasons: (1) to determine if aremovable mold piece exists for a region to be combined; (2) to find good partingdirections to construct mold pieces. By further analysis, we can see:

i. In the process of region combination, the main concern is if a parting directionexists for two regions or a region and a face. The actual ranges of the feasibleparting directions are not important.

ii. Since different regions and faces are combined in trial and error, an approach todetermine the direction quickly and easily is essential.

iii. After region combination, a parting direction PD needs to be selected from thefeasible range of the parting directions according to some criteria. Therefore aproper direction is more important than the whole feasible range in constructing amold piece for a mold piece region.


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So in determining two regions or a region and a face to be combined, instead of thewhole range, only one PD needs to be calculated. Also if the criteria used for theselection from the feasible range are followed in the calculation of a direction, the samePD should be achieved for each mold piece after the region combination process.Therefore the evaluation of PD can be formulated as an optimization problem. Sinceonly one direction is calculated, it is quite obvious that the optimization approach is muchfaster and easier than the spherical algorithms, especially for regions with many faces.

Therefore for a region with face Fi (unit face normal ηηηηi and area Ai, 1≤ i ≤ n), anoptimization problem based on ease of ejection for determining a unit parting directionPD (dx, dy, dz) can be formulated as follows.

Problem PDOP: PD Optimization Problem.

Maximize: f d d d A dpx y z i ii

n

( , , ) = ⋅=∑

1

Subject to: dp d d di xi x yi y zi z= + + ≥η η η 0 for each face Fi (plane constraints).

x2 + y2 + z2 = 1 (sphere constraints)

An approach to solve Problem PDOP quickly is presented as follows.

• The Evaluation Approach based on Linear ProgrammingIn Problem PDOP, the sphere constraint is to get a unit vector. It can be

approximated by a set of linear surfaces as shown in Figure 3.15.a with acceptable errors.Suppose the equations of a planar surface Si are sxi x + syi y + szi z – si = 0 with facenormal (sxi, syi, szi) toward inside. We can formulate a new problem as:

Problem PDLP: PD Linear Problem.

Maximize: f d d d A dpx y z i ii

n

( , , ) = ⋅=∑

1

Subject to: dp d d di xi x yi y zi z= + + ≥η η η 0 for each face Fi (plane constraints).

s d s d s d sxi x yi y zi z i+ + ≥ for each face Si (sphere constraints).

The relations between Problem PDOP and Problem PDLP are:

RPD1

F1

F2Si

PD1

PD2

(0, 0, 0)

(a) Planar faces of a sphere (b) Relations between Problem PDOP and PDLP

Figure 3.15 – PD Evaluation Approach.


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(1) If problem PDOP has no solution, problem PDLP will yield the solution dx = dy =dz = 0;

(2) If problem PDOP has a solution PD(dx, dy, dz) such that f(dx, dy, dz) > 0, problemPDLP will yield a solution PD’ near PD within the sphere approximation error.

This is because if f(dx, dy, dz) > 0, there must be at least one dpi > 0. SupposeProblem PDLP gives us a solution PD1 as shown in Figure 3.15.b. It must be avector within the sphere because of the sphere constraints. Suppose we constructa ray from the origin to PD1. It will intersect with one planar surface Si at pointPD2. Since dpi > 0, f(PD2) > f (PD1). PD2 is a better solution than PD1. So thesolution given by Problem PDLP must intersect a planar surface Si. That is, thesolution is within the sphere approximation error.

(3) If problem PDOP has a solution PD(dx, dy, dz) such that f(dx, dy, dz) = 0, problemPDLP will yield the solution dx = dy = dz = 0. In this case, since all dpi = ηηηηi•PD =0, PD must be vertical to the normals of all faces (Refer to an example given inFigure 3.15.b). So if Problem PDLP gives a solution PD(0, 0, 0), two faces F1

and F2 which are not in the same surface should be considered. The unit vectorsPD1 = ηηηη1 x ηηηη2 and PD2 = ηηηη2 x ηηηη1 are to be checked by the plane constraints. If theconstraints are satisfied, PD1 or PD2 is also the solution of Problem PDOP.

Therefore solving problem PDLP can generate the solution of Problem PDOP.Since the solution given by Problem PDLP is only to be compared with (0, 0, 0), theapproximation of the sphere can be rather rough. An approximation of a sphere with 144faces is sufficient for determining a PD for the construction of mold pieces. Thesesurfaces can be pre-generated and used for any regions.

Problem PDLP is a linear optimization problem. It is well studied in operationsresearch (Reklaitis, et al., 1983). For a linear programming problem in only 3dimensions, several algorithms can solve it with O(n) time (n is the number ofconstraints) and linear storage (Megiddo, 1984; de Berg, et al., 1997).

Although the algorithm to get a PD for a region is limited to planar surfaces, quadricand parametric surfaces can also be handled by approximating them with a series ofplanar surfaces. By setting a smaller mesh size, a more accurate model results. Even fora region with a large number of faces, the running time to solve Problem PDLP is rathersatisfactory. A test example is shown in Figure 3.16. A cylinder face and a planar facecompose a region. Setting surface deviation with different values can approximate thecylinder face by different number of faces. The face number and corresponding runningtime on a PC-700 are listed below. The results obtained for the three cases are all thedirection (0.0, 0.0, 1.0). The testing time is based on calling LINGO system(www.lindo.com) in a PC with a 700 MHz Intel-III processor.

(a) Maximum deviation: -0.002”; Number of faces: 34; Time = 0.16 second.(b) Maximum deviation: -0.0001”; Number of faces: 143; Time = 0.17 second.(c) Maximum deviation: -0.00001”; Number of faces: 224; Time = 0.20 second.


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By using a linear program to determine a parting direction, it is evident that thesolution process becomes much faster and easier. Therefore it is feasible to explore morecombinations of regions and faces in less time. The approach for region combination ispresented in Sections 3.5.3 and 3.5.4.

In the next section, the approach described in this section is applied in verifying draftangles of a part. The author believes that it is an effective and efficient approach for theautomatic detection of non-drafted and under-drafted faces.

3.5.2 Verification of Draft Angle

One thing to be noticed in the plane constraints of problem PDOP and PDLP is that‘0’ in the right side of the inequality can be changed to a variable df. That is,dp d d d dfi xi x yi y zi z= + + ≥η η η . The value of df is actually related to the minimum draft

angle γ of a part by df = sin (γ).

For a part to be fabricated by the injection molding process, its surfaces parallel tothe parting direction must be drafted at least an angle γ in order to ease the ejection of thepart and reduce the damaging possibility of the part and molds (Rosato and Rosato,1995). The minimal draft angle γ of a part depends on many factors: molding process,material, wall-depth, depth of textured surface, etc. For a complex model, the designer

(a) (b) (c)

Figure 3.16 – A Region with Different Mesh Size.

Non-drafted

Under-drafted Drafted

Wrongly-drafted

F F F F

PD1

PD2

Figure 3.17 – Different Draft Angles.

R R R


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may lose track of which surfaces are drafted and which are not. Consequently a tool todetect those non-drafted and under-drafted surfaces is necessary.

The suitability of a draft angle for a face depends entirely on the parting direction ofthe face. As shown in Figure 3.17, the non-drafted face F for PD1 is a well-drafted facefor PD2. Therefore the verification of draft angle is actually an integrated problem thatshould be considered in the mold configuration design process.

Suppose the minimum draft angle for a part is given as γ. By setting df = sin (γ) toreplace 0 in the plane constraints of Problem PDLP, we can find all concave regions withnon-drafted or under-drafted faces. As shown in Figure 3.18.a, concave region R is foundas non-drafted because R has no solution for Problem PDLP. Without the help of acomputer system, it is rather difficult to find out all these features since the differencebetween non-drafted and drafted appropriately with γ = 1.5o is non-noticeable when theyare displayed on a monitor.

Also by following the same criteria in the region combination process, it isguaranteed that all regions have only well-drafted faces in their parting directions.However the assignment of minimum draft angle γ may influence the number of moldpieces. For example, if γ = 0, regions R1 and R2 in Figure 3.18.b can be combined intoone region in direction PD2. However, if γ =2.5o, R1 and R2 cannot be combined in PD2

any more. So one more mold piece needs to be constructed in PD1 for R1.

In the next section, the combining process, which is considered as a concave regiongrowing process, is discussed with the data structures of MPMDM presented.

3.5.3 Analysis of Region Combination Process and Related Representations

After getting combined regions (CR) and convex faces (CXF) from the faces (F) of apart, a combination that satisfies the requirements of Problem MCD needs to bedetermined. It is quite obvious that n(CR) + n(CXF) ≤ n(F), where n() denotes ‘numberof.’

(1) For a part P without CR, n(CR)= 0 and n(CXF) = n(F).

RR2

R1

PD2

PD1

(a) (b)

Figure 3.18 – Relationships Between Regions and Minimum Draft Angle.


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Since P is a convex polyhedron, in any two opposite directions d and –d, two moldpieces can be constructed to form P without any undercuts (Chen, et al., 1993). So therequirements presented in Problem RMCD can be satisfied quite easily. In this case,several other criteria need to be considered, like the flattest parting line (Majhi, et al.,1999), maximum projection area, etc. Industrial injection molding parts are rarelyconvex polyhedrons.

(2) For a part P with one or more CR, n(R) = n(CR) + n(CXF) < n(F).

Suppose n(R) = c•n(F) where c is a constant ratio depending on the given shape.Sometimes, c can be very small, that is n(R) is far less than n(F). As an example, the partgiven in Figure 3.16.c has 1 region with 224 faces and 6 convex faces. So n(F) = 230,n(R) = 1+6 = 7 and c =7/230 ≅ 0.03. But more often c is around 0.3 ~ 0.6.

Even if n(R) is less than n(F), an algorithm to explore all the combinations of CR andCXF for minimum mold piece number is still strongly NP-hard. Therefore someheuristics should be considered in combining regions. Since a CXF can be combined toany neighboring region if a PD exists for them, CXFs are more flexible to be combinedthan CRs. Therefore the focus of region combination is combined regions. In thisdissertation, an approach based on CR-growing process is developed. This approachallows CRs to grow individually by combining neighboring CXFs and CRs. The processcontinues until no further combining happens. The resulting regions correspond to amold design for P.

Some definitions of edges and faces related to the region combination are providedfirst. Suppose before CR-growing, a region CRi is given as shown in Figure 3.19.a. CRi

is composed by faces F1, F2, …, Fn.

Definition 3.5. A Core Face (COF) of CRi is one of the faces that compose CRi beforethe CR-growing process. A core face of CRi will always belong to CRi.

The edges of CRi consist of all the edges of faces F1, F2, …, Fn. They can be dividedinto three types: concave edges (CVE), convex edges (CXE), and parting edges (PE).

CR of Part P

NF1

NF2

NFm

NF3

PEs

COF

CXF1NF2

NFm

NF3

PEs

COFNF1NF5

NF6

CR of Part P

CXFCombining

NeighboringFaces

Core Faces

ConvexFaces

(a) Before combining (b) After combining

Figure 3.19 – Edges and Faces of a Region in a CXF Combining Step.


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Suppose Ei is an edge of CRi. In the boundary representation two faces own Ei (refer toSection 4.2). One of the owner faces must belong to CRi.

I If the two owner faces all belong to CRi,a. if their dihedral angle is less than 180o, Ei is a CVE;b. otherwise, Ei is a CXE.

II If one of the owner faces does not belong to CRi, Ei is a PE. And Ei must be aconvex edge based on the definition of combined region.

Definition 3.6. A Neighboring Face (NF) of CRi is one of the faces that does not belongto CRi but shares the ownership of an edge with CRi.

Therefore by checking the owner faces of all parting edges, all neighboring faces ofCRi can be identified.

There are two possible cases in the region growing process.

(i) CRi combined with a CXF.

Suppose NF1, … , NFk are neighboring faces of CRi. They are convex faces that donot belong to any regions. To determine if CRi can be combined with a face NF1, we canuse its unit normal (ηxNF1, ηyNF1, ηzNF1) to form an additional plane constraint:dp d d di xNF x yNF y zNF z+ = + + ≥1 1 1 1 0η η η and add it to the formulation of Problem PDLP for

CRi.

The linear program in Section 3.5.1 can solve the problem in O(n+1) time. We canalso simplify the solution approach according to the properties of an Linear Programmingproblem.

Since Problem PDLP has only 3 variables, each linear constraint is actually a halfplane in 3D space. So the feasible region is a polyhedron obtained by the intersection ofall the half planes. According to (Reklaitis, et al., 1983), one of the corner points of thefeasible region of a linear program is always an optimal solution. Suppose PD(dx, dy, dz)is the solution of Problem PDLP with n half plane constraints h1, …, hi. If a half planehi+1 is added as a new constraint, de Berg (1997) gives that:

(a) If PD satisfies the constraint hi+1, the new optimal solution PD’ = PD.(b) If PD does not satisfy the constraint hi+1, PD’ must be one of the intersection

points of hi+1 with h1 ~ hi, or the linear problem is infeasible.

For case (b), an algorithm running in linear time is also given to find the new optimalsolution. So the approach also runs in O(n) time, but it should have a smaller coefficientof n compared to that of the LP solver.

For some neighboring faces NF1, … , NFk, if PD’ exists, CRi can combine them intoone region. All the faces F1, …, Fn and NF1, …, NFk are called faces of CRi, and facesNF1, … , NFk are the Convex Faces (CXF) of CRi. So the faces of a combined region areactually composed by core faces and convex faces. One difference is that after a face isset as a core face, it will always belong to the region. On the contrary, a convex face mayleave the region to join another region as shown later.


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As convex faces switch among regions, some old PEs are changed to CXE of CRi

and some new PEs are generated related to the new combined faces. Similarly someneighboring faces are changed to convex faces of CRi and some new NFs are generatedrelated to the new PEs, which are shown in Figure 3.19.b.

(ii) CRi combined with another region.

As shown in Figure 3.20.a, suppose a neighboring face (NF3) of CR1 is also a face(CXF1) of another region CR2. That is, CR1 and CR2 are two neighboring regions and arecandidates for joining into a larger region.

Since the convex faces of CR1 and CR2 are more easily switched between regionsthan their core faces, CXFs are not considered in the test for region combination. That is,only COFs of CR1 and CR2, and the convex faces to connect COFs, are formulated asplanar constraints in Problem PDLP. If the problem has a non-zero solution PD, tworegions are combined into one region CR1’ (refer to Figure 3.20.b). And the core faces ofCR1’ are the faces of the planar constraints in Problem PDLP. All the convex faces ofCR1 and CR2 are not faces of CR1’ anymore. Instead they are CXF of the part and can becombined by any regions including CR1’.

To get convex faces to connect COFs of two regions, the original face of any convexface needs to be recorded also. That is, if F1 shares a parting edge PE1 with a region faceF2, besides recoding F1 as the neighboring faces of CR, origin(F1) is recorded as F2. Soas shown in Figure 3.20.a, if NF3 of CR1 and CXF1 of CR2 are the same face, the originalface of NF3 in CR1 is added to planar constraints of CR1. The same process can berecursively executed for the face of origin(NF3) until the original face is a core face.Similarly, the original faces of CXF1 can be added to CR2.

Therefore the main data structures for a combined region include:

Vector PD; // Parting direction dx, dy, dzBox Bound; // Bounding box of core facesEdge_List LPE; // Parting edges

NF2

NFm

NF3

PEs

NF1Region

Combining

CXF1NF2

NFm

NF3

PEs

COFNF1

CR1 and CR2 of Part P

COFCXF1

CR1' of Part P

CR1

CR2

CR1'

COF

CXF2

NeighboringFaces

Core Faces

ConvexFaces

(a) Before combining (b) After combining

Figure 3.20 – Edges and Faces of Two CRs in a Combining Step.


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Face_List LCOF; // Core facesFace_List LCXF; // Convex faces, origin of CXF is assigned as an attributeFace_List LNF; // Neighboring faces, origin of NF is assigned as an attribute

In light of the combining process based on CR-growing, an algorithm and itsanalysis are presented in the next section.

3.5.4 Region Combination Algorithm and Design Knowledge

Based on the data structures given in the last section, the algorithm Combine_Region(CR) of MPMDM is listed as follows.

Algorithm: Combine_RegionInput: A set of combined regions SCVR and convex faces of P.Output: A set of combined regions SCVR and convex faces of P.

(1) change ← FALSE.(2) for a region CRi in SCVR,(3) for each NFk in LNF, // decide the list of neighboring faces(4) rn ← region_num(NFk);(5) if rn = 0, then // NFk is not in any region(6) if combinable_convex_face(CRi, NFk) = TRUE, then(7) combine NFk with CRi, update LPE and LNF of CRi, change ← TRUE.(8) else // NFk belongs to region CRrn

(9) if combinable_region(CRi, CRrn) = TRUE, then(10) combine CRrn with CRi, update LPE and LNF of CRi, change ← TRUE.(11) if change = TRUE, then Go to Step (1).(12) else return.

The algorithm has a main loop to determine if any region combination should occur.Within each iteration, there are two loops in Step (2) and (3). So the running timedepends on the number of regions and the number of neighboring faces for each region.As small regions are combined into larger regions, the region number is decreasing andthe NF number for each region is increasing.

Although it is difficult to calculate the exact time cost of the algorithm, making somesimplifications can derive an approximate computational complexity measure. SupposeP has n faces, among them a•n faces are convex faces (0< a <1). If the input SCVR has m1

regions, and the output SCVR has m2 regions, we can use the average region number

mm m= +1 2

2to represent the region number in all the iterations. Each region will have at

most n/m faces. Similarly we can use NF to represent the average number of neighboringfaces for all regions in the whole process. Suppose the program exits when all convexfaces are combined into regions. Within each iteration, one region will combine afraction of its neighboring convex faces, suppose b•NF. So if algorithm CR will iterate k

times before exiting, we know a n k m b NF⋅ − ⋅ ⋅ ⋅ = 0 . That is k m NFa n

b⋅ ⋅ = ⋅

. Also


104

since the combining cost of a convex face is O(n

m) as shown in Section 3.5.3, the time

cost of algorithm CR is:

T n m k m NFn

m

a n

b

n

mO

n

m( , ) ( ).= ⋅ ⋅ ⋅ = ⋅ ⋅ =

2

So T n O n( ) ( ).= 2

For two regions or a region and a convex face, if the results in Step (6) or (9) arefalse in an iteration, there is no need to determine their combinability in any lateriterations. Therefore repeated executions can be avoided by adding a face array and aregion number array in each region. The arrays record neighboring convex faces andcombined regions that are not combinable with the region individually.

In Algorithm CR, Steps (6) and (9) need to be discussed further. Conceptually, byusing different combining orders and different rules in functionscombinable_convex_face and combinable_region, different mold designs can beobtained. A simple example is shown in Figure 3.21. Suppose a box has two similarcavities in its top (F1) and bottom (F2) sides. Faces 3 ~ 6 are vertical relative to faces 1and 2. No draft angle is considered. Several combining results for the part can beobtained by trying different combinations of faces 3~6 with regions 1 and 2. Since therelated mold designs have the same number of mold pieces and each mold piece can bedisassembled properly, they all satisfy the requirements given in Problem RMCD. Sojust from the geometric perspective, they are equally good designs.

Therefore, besides the requirements given in Problem MD, some mold designknowledge should also be considered in Functions combinable_convex_face andcombinable_region. The heuristic rules that are considered in this dissertation are listedwith some explanations.

(1) Core/Cavity property of a region.

A combined region CRi can be classified as an internal region or an external regionaccording to its parting edges. If a loop formed by some parting edges of CRi is aninternal loop of a neighboring face, CRi is an internal region. For example, the partingedges of R1 in Figure 3.21 form an internal loop of F1. So R1 is an internal loop.Otherwise CRi is an external region, like R2 in Figure 3.21.

R1

R2

F1

F2

F3

F4

F6

F5

Combining result 1:R1+F1+F3+F4+F5+F6, R2

Combining result 2:R1+F1+F3+F4+F5, R2+F6

Combining result 16:R1+F1, R2+F3+F4+F5+F6

...

Combining result 3:R1+F1+F3+F4+F6, R2+F5

Figure 3.21 – Different Combining Results of a Part.


105

Generally internal regions are related to core mold pieces, and external regions arerelated to cavity mold pieces. To facilitate the ejection of part from a core, convex faceswith vertical normals to PD usually go with cavity side instead of core side in the molddesign. So a normal mold design for the part given in Figure 3.21 is R1, F1 in core side,and R2, F2 ~ F6 in cavity side.

(2) Main parting direction of a region.

For a part with many combined regions CRi, we may get several parting directionsPDi related to the combined regions. Among them, a pair of opposite directions is themain parting direction (MPD) of the mold design. Others are all side directions.

Our approach to get the MPD is to find a pair of directions with the maximum regionvolumes among all PDi. That is, we find parting directions PDk which are in the same oropposite directions according to some tolerance. For each PDk, a volume of the relatedregion is calculated from its boundary box. The sum of all region volumes is assigned tothe direction. Finally we can get MPD by finding the direction with maximum volume.Accordingly we can assign a value if_main_pd to each region. If MPD can be set as aparting direction of a region, its if_main_pd is true. Otherwise it is false.

In general, it is preferred to combine a vertical face with a region in MPD than aregion in a side direction. So a convex face Fi is combinable with a region CRj if:

(a) the normal of Fi is not vertical to the PD of CRj;(b) Fi is a vertical face, if_main_pd of CRj is true, and CRj is an external region.

(3) Neighboring face number of a region.

Suppose face F is a neighboring face of region CR2 and also a face of CR1.According to the solution of Problem PDLP, suppose CR1 and CR2 are not combinable.If the neighboring face F is the only NF of region CR2, CR2 must be a cavity in face F asshown in Figure 3.22. So:

(a) if F is a convex face of CR1, delete F from CR1 and add it to CR2;(b) if F is a core face of CR1 (as shown in Figure 3.22.a), or if F is also the only

neighboring face of region CR1 after F is deleted (as shown in Figure 3.22.b), Fshould be divided into two faces F1 and F2 for CR1 and CR2.

CR1

CR2

PD1

PD2

CR1

PD1

CR2

PD2

F

F

O2

O1

C1

C2

M

F2

F1

Top view of F

L1

L2

(a) case 1 (b) case 2 (c) Dividing approach

Figure 3.22– Face Dividing for A Region of Cavity.


106

The approach to do the above face splitting is briefly described here and shown inFigure 3.22.c. Suppose internal loops L1 and L2 of F are related to CR1 and CR2. Thesmallest enclosing discs C1, C2 for each loop can be constructed respectively by usingalgorithm MiniDisc given in de Berg (1997). The algorithm runs in O(n) time, where n isthe number of points in a loop. Suppose O1, O2 are the centers of C1, C2, respectively,and M is the middle point of line segment O1O2. If we assume C1 and C2 do not overlap, aline can divide F into F1 and F2 by passing through M and being vertical to O1O2.

(4) Combining order of a region.

The order of regions in SCVR and the order of neighboring faces in LNF of a regionalso affect the combining results. For example, a part as shown in Figure 3.23.a has fourregions, R1 ~ R4. Depending on the order of R2, R3 related to R1, R4, a combining resultwith R1, R2, R3 as a region and R4 as another region may be generated. A result with R1,R4 as a region and R2, R3 as another region may also be generated. Similarly if R1 is thefirst region in SCVR, the above two results will be generated depending on the order ofneighboring faces in LNF of R1.

So the regions in SCVR can be reordered according to heuristic rules beforecombining regions. For example, regions can be ordered by their volumes; that is,moving regions with larger volumes to the beginning of SCVR. Hence, regions with largervolumes can be combined first. Regions can also be ordered according to whether or notthey are in the main parting directions. Regions in the main parting directions wouldcombine first. Also, regions of SCVR can be classified into two sets SCVR1 and SCVR2

according to some criteria, like the value of if_main_pd. Then let regions in SCVR1

combine with each other first. After the combining process finishes, regions in SCVR2 areadded to start a new combining process.

In Section 4.2, the design knowledge considered in MPMDM will be revisited in thediscussion of RTMDS and its implementations.

• AnalysisBy considering more mold design knowledge, a mold design that is more compatible

with good mold design practices can be achieved. Since different mold designscorrespond to different combinations of CR and CVX, the region-based approach is veryflexible in enabling the addition of new design knowledge. In algorithmCombine_Region, functions combinable_convex_face and combinable_region control thegeneration of different combinations of CR and CVX. So to add more design knowledge,related heuristic rules can be formulated and added into the two functions. Since othersteps remain unchanged, it is rather easy to implement.

Besides mold design knowledge, our approach can handle mold fabricationknowledge by following similar processes. For example, suppose mold pieces for the ribpart shown in Figure 3.23.b are to be fabricated with SLA machines. Assume surfacefinish requirements of 2 µm are specified for the two opposite faces F1 and F2.According to our knowledge of the SLA process, such high surface finish can be achievedonly by building the mold pieces such that F1 and F2 are the top surfaces of the part(West, 1999). Therefore to build mold pieces such that F1 and F2 are both at the topsurface, a constraint that F1 and F2 cannot be in the same region is added. By adding anadditional combining rule in functions combinable_convex_face and combinable_region,


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a better design that satisfies fabrication requirements is achieved. However, this may leadto a three-piece mold design for the part instead of a design with only two mold pieces.

Finally, besides one combining results, several results can be generatedautomatically, which allows the mold designer to select a mold configuration designamong them. In Section 9.4 this approach will be revisited.

After mold piece regions are generated for a part in this section, an approach forconstructing mold pieces effectively and efficiently will be presented in the next section.

3.6 MOLD PIECE CONSTRUCTION APPROACH BASED ON REVERSE GLUE

After region combination process presented in Section 3.5, several mold piece (MP)regions have been generated for a part with their parting directions and parting lines. Ifthere exist convex faces that do not belong to any regions, they can form one or more MPregions according to the connectivity. The parting directions of the new regions can beevaluated by solving problem PDLP (Section 3.5.1). An example is given in Figure 3.24.After region combination process, we got two regions (R1, R2) and four convex faces,

R1

R2

R3

R4PD1

PD2 F1

F2

(a) Different combining orders (b) Mold fabrication knowledge

Figure 3.23 – Two Example Parts for Combining Order.

R1

R2

PL1(PL2)

PL2(PL3)

R1

R2

R3

PL1(PL3)

PD1

PD2

PD3

M3

M2

M1

ConvexFace

(a) Regions and convex faces (b) Regions (c) Related Mold Pieces

Figure 3.24 – Regions and Mold Pieces of a Part.


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which are shown in Figure 3.24.a. The convex faces can form a new region R3. Theparting lines of the regions are shown in Figure 3.24.b. Correspondingly, three moldpieces need to be generated as shown in Figure 3.24.c.

This section focuses on Step (3) shown in Figure 3.6. An approach based on thereverse glue operation is developed for solving Problem MPC (Section 3.2.2) effectivelyand efficiently. The remainder of this section has been organized in the followingmanner. First, the reverse glue operation and the principle related to the operation arepresented in Section 3.6.1. A key step of the approach, generation of glue faces, is thendiscussed. A problem formulation and related algorithms are presented for the generationof glue faces in Section 3.6.2. Finally, the algorithms for constructing two-piece moldsand multi-piece molds are presented respectively in Section 3.6.3. The role of partingsurface in the approach is also described in this section.

3.6.1 Principle and Related Representations of the Approach

Although the approach of MPMDM is quite flexible for any mold base, for thediscussion below, it is supposed that a mold base (MB) is given as a rectangular blockthat can contain the injected part (P) entirely inside. The position and orientation of Pwithin MB is affected by gate design, runner design and ejection pin design. Theapproach of MPMDM is also flexible for the position and orientation of the part in MB.Suppose the main parting direction of P generated in Section 3.5.4 is rotated to the z-axisof MB, and the center point of the bounding box of P is translated to the center point ofthe bounding box of MB.

After P is positioned at a suitable position and orientation in MB, a Boolean moldbase M’ = MB – P (‘–’ is the subtraction operation) can be generated as shown in Figure3.25.a. From the equation, it is obvious that M’ contains two kinds of faces, outside facesand inside faces. Outside faces (Fout) of M’ are the faces corresponding to the faces ofMB. Inside faces (Fin) of M’ are the faces corresponding to the faces of P. Suppose allthe faces of a body or a region are denoted as F( ), and the union of two face sets isrepresented by +. So F(M’) = Fout-M’ + Fin-M’. (3.2)

The approach of MPMDM is also quite flexible for any parting surface, which willbe discussed in Section 3.6.3 in more details. Suppose a parting surface PS is given.According to the position related to PS, all outsides faces of M’ can be divided into twosets, faces above PS (denoted as ↑) and faces below PS (denoted as ↓). Therefore

Fout-M’ = ↑Fout-M’ + ↓Fout-M’. (3.3)

For a part P and MB’, suppose mold piece M1 is to be constructed for region R1 first.All other regions can be considered as a new region R2’. So only two mold pieces M1 andM2’ need to be constructed for regions R1 and R2’ respectively. Replacing P with R2’ andMB with M2’, all other mold pieces can be constructed by recursively executing the aboveprocess. Therefore, for discussion below, only the construction of two mold pieces (M1

and M2) for two regions (R1 and R2) is considered.

It is evident that each face of Fin in M’ has a corresponding face in P. Two faces arein the exact same position but with opposite face normals. We use ~ to denote this


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corresponding relationship. Since faces of R1 are to be formed by M1 and faces of R2 areto be formed by M2, all faces of Fin can be divided into two sets:

Fin-M’ = ~F(R1) + ~F(R2). (3.4)

From Equation (3.2), (3.3), and (3.4),

F(M’) = ↑Fout-M’ + ↓Fout-M’ + ~F(R1) + ~F(R2). (3.5)

For two bodies A, B in Boundary Representation, A = bA ∪ iA and B = bB ∪ iB,where b denotes the set of points on the boundary and i denotes the set of interior points.Therefore the union of A, B can be represented as: C = A ∪ B = bA ∪ bB ∪ iA ∪ iB(Mortenson, 1997). For the boundary of C, we have:

bC = b(bA ∪ bB ∪ iA ∪ iB) = bA ∪ bB. (3.6)

Gluing operation is a standard high-level Euler operation for half-edge datastructure, which is a de facto data structure of the boundary representation. More detailsabout the half-edge data structure including coedges are given in Section 4.2. The mainoperations of the gluing operation are to swap the partner coedges for all common edgesof two bodies, then merge all faces into one body. Mäntylä (1988) gave an equation torepresent union operation by gluing operation,

A ∪ B = (A out B) ⊕ (B out A), (3.7)

where ⊕ denotes gluing operation.

Considering the relation with bB, the boundary faces of A can be divided into threekinds of faces: faces outside of bB, faces on bB, and faces inside of bB. That is,

bA = (bA out bB) + (bA on bB) + (bA in bB). (3.8)

Similarly, bB = (bB out bA) + (bB on bA) + (bB in bA). (3.9)

Glue Face 2

Glue Face 1 Fout M’

Fin M’PF

Glue

ReverseGlue

Glue Face 1

Glue Face 2

M1

M ' = MB-P

↑Fout M’

M2

~F(R1)

~F(R2)

↓Fout M’

PF

MB

P

Part

(a) Boolean Mold Base M’ (b) Mold Pieces M1 and M2

Figure 3.25 – An Example of a Boolean Mold Base and Mold Pieces.


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In Problem MPC, M’ = M1 ∪ M2. From Equation (3.6) and (3.7),

bM’ = bM1 ∪ bM2 = (bM1 out bM2) ⊕ (bM2 out bM1). (3.10)

Since M1 and M2 are two different bodies that are assembled together, (bM1 in bM2)and (bM2 in bM1) in Equation (3.8) and (3.9) are null. So

bM1 = (bM1 out bM2) + (bM1 on bM2); (3.11)

bM2 = (bM2 out bM1) + (bM2 on bM1). (3.12)

If bM1 and bM2 are given, based on Equation (3.10) ~ (3.12), bM’ can be generatedby gluing operation (from Figure 3.25.b to Figure 3.25.a) with three steps.

(1) Finding faces (bM1 on bM2) and (bM2 on bM1) from bM1 and bM2, delete themfrom M1 and M2 respectively (Equation (3.11) and (3.12));

(2) Swap partner coedges for all common edges of M1 and M2;(3) Merge all faces of M1 and M2 into one body M’ with two shells (Equation (3.10)).

As shown in Figure 3.25.b, faces (bM1 on bM2) and (bM2 on bM1) are actually thecontact faces between M1 and M2 when they are assembled together. A definition forthem is given.

Definition 3.7. A Glue Face (GF) of a mold piece M1 related to another mold piece M2

is one of the faces (bM1 on bM2).

According to the definition, for each glue face of M1, there is a corresponding glueface of M2. These two faces are in the same position but with opposite face normals.

However, in Problem MPC, bM1 and bM2 are unknown. They are actually what is tobe generated. Since MB and P are given in Problem MPC, M’ and bM’ can be generatedquite easily based on the equation M’ = MB – P.

The boundary of a 3D model is faces. So bM’ = F(M’). From equation (3.5) and(3.10), it is evident that

↑↑↑↑Fout-M’ + ↓↓↓↓Fout-M’ + ~F(R1) + ~F(R2) = (bM1 out bM2) ⊕⊕⊕⊕ (bM2 out bM1)

Therefore, suppose

(bM1 out bM2) = ↑Fout-M’ + ~F(R1); (3.13)

(bM2 out bM1) = ↓Fout-M’ + ~F(R2). (3.14)

If all the glue faces (bM1 on bM2) and (bM2 on bM1) are also known, M1 and M2 canbe generated based on Equation (3.11) and (3.12) with three steps.

(1) Remove faces ↑Fout-M’ + F(R1) and ↓Fout-M’ + F(R2) from M’, and use them togenerate two new bodies M1 and M2 respectively (Equation (3.13) and (3.14));

(2) Generate glue faces (bM1 on bM2) and (bM2 on bM1), and add them to M1 and M2

respectively (approaches are discussed in Section 3.6.2 in more details);(3) Swap partner coedges for all common edges of M1 and M2 (Equation (3.11) and

(3.12)).

In this dissertation the above processes are named Reverse Glue operation.


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In the reverse glue operation, Step (1) and (3) are straightforward. However, Step(2), the generation of glue faces, may be rather difficult for some geometry. Theapproaches of MPMDM for the generation of glue faces are presented in the next section.

3.6.2 Generation Approach of Glue Faces

For equation M’ = M1 ∪ M2, if M1 and M2 are given, the result M’ obtained from thegluing process ( M M MGlue

1 2 → 'U ) is unique. However, if M’ is given, the result M1

and M2 from the reverse glue process ( M M MverseGlue' Re → 1 2U ) is not definitive.

Using different glue faces will give us different mold piece designs. For example, theapproach described in Section 2.3 generates parting surfaces by extending parting lines.These faces are also glue faces since they also satisfy (bM1 on bM2) and (bM2 on bM1)when assembling M1 and M2 together. So the key of problem MPC is actually to findappropriate glue faces. In this dissertation, only planar parting surfaces are consideredsince they can reduce mold fabrication cost and material flash in the injection moldingprocess (refer to Section 2.3).

In this section, a formal problem formulation for generating glue faces is presentedfirst. Three kinds of glue faces are identified. Algorithms for generating them aredescribed, with the focus on the generation of the inner glue faces.

• Problem Formulation of the Glue Face GenerationSuppose the given parting surface PS intersects with the outside faces of M’ at edges

EFout. The boundary edges of a region on the cavity of M’ are called its parting edges(PEs). They may compose one or several closed parting loops (CPL), as shown inFigure 3.26.a. One loop is the outer loop and others are inner loops.

F8

F2

F3

F7

F6

F1

F8

F1

F6

F5F4F1

F2

F8

F3

F4 F5

F6

F7

PF

CPL2

CPL1

EFout

PF

MB

P

M'

PFF1

F3 F6F2 F7

F8

M1

M2

(a) Parting edges and EFout in M’ (b) Glue faces (c) Generated mold pieces

Figure 3.26 – Glue Faces of an Example Part.


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To form mold piece M1 from ↑Fout-M’ and F(R1), and M2 from ↓Fout-M’ and F(R2), theglue faces should have the given parting edges and EFout as their boundary edges, asshown in Figure 3.26.b. So the generation of glue faces is actually a geometricreconstruction problem, that is, generating faces according to their boundary edges. Forgiven parting edges and EFout, there may exist several solutions. As an example, for thepart given in Figure 3.26.a, the glue face F3 shown in Figure 3.26.b is not required to beperpendicular to PS. So glue faces will be changed correspondingly, and result in anothermold piece design for the given part.

So the generation of glue faces considered in this dissertation can be formulated as:

Problem GFG: Glue Face Generation. A solid part MB’ is given in theBoundary Representation, which has two shells with outside faces (Fout) and inside faces(Fin) respectively. Suppose a planar parting surface PS and a parting direction PD aregiven. Further suppose a closed edge loop EFout, which is the intersection of Fout with PS,is given with several closed edge loops CPLi (1≤ i≤ k), which are composed by someedges of Fin. Generate faces F1 ~ Fm such that:

a. Fi are planar (1≤ i≤ m) .b. Suppose set E1(EFout, CPL1,…, CPLk) are all the edges of EFout and CPLi (1≤ i≤ k),

and set E2(F1,…,Fm) are the boundary edges of Union(F1, …, Fm). Two setsshould be the same, that is E1 = E2.

c. Fi (1≤ i≤ m) satisfy the disassembly requirements for a mold piece, that is,normal(Fi) •PD ≥ 0 (Equation (3.1)).

Among CPLi (1≤ i≤ k), suppose CPL1 is the loop with the biggest projection area inPS. To connect faces Fout and Fin, and separate faces ~F(R1) and ~F(R2) in M’, threekinds of glues faces, GFps, GFproj, and GFinner, exist.

(1) Glue face on the parting surface (GFps):

GFps connects the faces of Fout with the outer parting loops CPLi (if CPLi is on PS),or with project glue faces GFproj (if CPLi is not on PS). Usually GFps has only an outerloop (Lpso) and an inner loop (Lpsi). The outer loop L pso is formed by the intersection ofFout with PS, and the inner loop Lpsi is formed by the projection of the CPL1 onto PS. InFigure 3.26.b, F1 is a GFps.

(2) Glue faces by projection (GFproj):

Disassembling a mold piece in direction PD is similar to sweeping the faces of themold piece along PD. Thus we can generate glue faces for edges of CPL1 in the sameprinciple. GFproj connect CPL1 with GFps. So for each parting edge ei of CPL1, if it is noton PS, project it onto PS to get a new edge eip. A new glue face is generated by ei, eip andedges to connect them. In Figure 3.26.b, F2 ~ F7 are GFproj.

An algorithm to generate GFps and GFproj from CPL1 and EFout is presented below.

Algorithm: Glue_Faces_Outer_LoopInput: A linked parting loop CPL1, parting surface PS and M’.Output: A set SGF of glue faces GFps and GFproj.

(1) initialize edge array pe, se;


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(2) for each vertex vi of CPL1,(3) pv ← projection(vi); // project the vertex into PS(4) if(pvi ≠ vi) pei ← make_edge(vi, pv);(5) otherwise pei ← NULL;(6) for each edge ei of CPL1,(7) if edge_in_plane(ei, PS) = TRUE, sei ← ei; // edge is in PS(8) else // edge is not in PS(9) sei ← projection(ei, PS); // project ei onto PS for sei

(10) gf_proj ← make_face(ei, pei, sei, pei+1);(11) Add gf_proj to SGF;(12) Lpi ← make_loop(se); // generate inner loop of GFpf

(13) Lpo ← intersection (PS, Fout(M’)); // generate outer loop of GFpf 4(14) gf_pf ← make_face(Lpo, Lpi);(15) Add gf_pf to SGF.

There are only two individual loops in Algorithm Glue_Faces_Outer_Loop. So thecost of the algorithm is O(ne), where ne is the edge number of loop CPL1.

(3) Inner glue faces (GFinner):

Inner parting loops imply the existence of holes in part P. Correspondingly, innerglue faces should be generated to separate two mold pieces. If an inner parting loop CPLi

is in a plane, such as loop CPL2 shown in Figure 3.26.a, it is straightforward to generate aface GFinner according to the loop (F8 shown in Figure 3.26.b). However, in more generalcases, loop CPLi may not be in a plane. Therefore several glue faces are to be generatedfor the loop. The generation of inner glue faces according to a given loop is discussed inmore details as follows.

After generating the glue faces for region R1, the glue faces for region R2 can begenerated just by face copying, because glue faces for M1 and M2 are in same positionswith reversed directions.

• Generation of Inner Glue FacesThe shape of inner glue faces is determined by inner parting loops. For an arbitrary

f1

f2

f3

R1

R2

LR1

LR2

CPL2

E1

E2

E3E4

(a) Part with a snap fit (b) Linked parting loops (c) Resulted inner glue faces

Figure 3.27 – Generation of Inner Glue Faces.


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inner parting loop, the generation of inner glue faces may be much more complicated. Asan example, a part with a snap fit feature is shown in Figure 3.27.a. Suppose the partfaces are divided into two regions, R1 and R2. Accordingly two closed parting loops areformed. Between them, CPL2 is an inner parting loop, where the edges of CPL2 are notin one plane. For CPL2, there are two coedge loops LR1 and LR2 related to R1 and R2

respectively as shown in Figure 3.27.b.

The generation of inner glue faces from inner parting loops is actually a geometricreconstruction problem. Let A be a linked parting loop, and T a transformation such thatT(A) →B, where B is some planar faces whose boundary is defined by A. One commonreconstruction approach of surfaces from points consists of projecting the points to aplane, triangulating them, and converting this into a triangulated surface by projectingeach vertex back into three dimensions (Goodman and O'Rourke, 1997). The approach isoften used for cartographic problems. However, in our problem, a plane and theprojection of points are not defined. So it is difficult to find such a plane to avoid thecoincidence of point projections.

To construct inner glue faces for Problem GFG, new edges are to be generated andthey should satisfy two requirements: (1) each new edge is not within a face of M’; (2)each new edge is not an existing edge in M’. For requirement (1), if a new generated faceor edge is within a face of M’, the mold piece generated by reverse gluing operation willbe invalid because of face overlapping. Similarly, for requirement (2), if a new edge isan existing edge in M’, we will get a body with an edge shared by more than two faces.So the mold piece will be non-manifold.

Similar to many geometric reconstruction problems, faces that satisfy all the aboverequirements for a loop may be different. In this dissertation a greedy heuristic is used tominimize the number of generated faces. That is, starting from each coedge ei of a linkedloop, the number of succeeding coedges (nci) that can form a planar face with ei iscounted and assigned to ei. For example, for LR2 shown in Figure 3.27.b, the number foreach coedge is shown in Figure 3.28.a. So every time the approach starts from a coedgeemax with the biggest nci to generate a face. emax and all succeeding coedges are added toan empty set SE until a succeeding coedge is not in the same plane. If the first and the last

LR2

(2)

(1)

(0)

(0)

(2) (1)

(0)(0)

(0)

(2) (1)

(0)(2)

f1

(1)

(4)

(4)

f1

(4)(4)

f2

f3

(a) nc of coedges (b) f1 is generated (c) f2 and f3 are generated

Figure 3.28 – Process of Face Generation.


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coedges in SE are not linked, a new edge is generated with two related coedges to connectthem. One coedge is added to SE so that the coedges in SE can form a face, which is thenadded to a face set. Another coedge is added to the linked loop to replace all the coedgesin SE. Some edges in the loop may need to update their number nc. This processcontinues until no more coedges are in the loop. An example for the above process isshown in Figure 3.28 for the inner parting loop given in Figure 3.27.

When calculating the number of succeeding coedges to form a planar face, it isrequired to determine if a new edge to form a new face is valid. For example, in Figure3.27.b, E1 is an existing edge of the part, and E2 is within a face of R2. So they are allinvalid according to the requirements of new generated edges. For a face with an invalidnew edge, nc of all the coedges except the last one is assigned 0.

The algorithm to generate GFinner is presented as follows.

Algorithm: Glue_Faces_Inner_LoopInput: A linked parting loop CPLi.Output: A set SGF with glue faces GFinner.

(1) for each edge ei of CPLi,(2) nci ← edge_nc(ei, CPLi); // calculate nc for ei in CPLi

(3) while edges of CPLi are not null,(4) esn ← biggest_edge(nc, CPLi); // find the edge with biggest nc(5) m ← nc(esn), if m = 0, return error;(6) set Sce ← null;(7) add coedge esn, …, esn+ m to Sce;(8) if esn and esn+ m have a same vertex, ne ← null;(9) else ne ← make_edge (esn, esn+m); // new edge to form a closed loop(10) generate coedges cne1 and cne2 from ne; // if ne = null, cne1 and cne2 are null(11 ) gf_inner ← make_face(Sce, cne1); // generate new face(12) add gf_inner to SGF;(13) replace all coedges in Sce with cne2;(14) update_edge_nc(cne2, CPLi); // update nc for cne2 and its preceding edges

The main loops of the algorithm are Step (3) and (4). Since it takes O(lgn) to get anelement with the biggest number in an array with n elements, the cost of the algorithm isO(ne• lgne), where ne is the edge number of loop CPLi.

In this section the algorithms for generating glue faces are presented. Based on themthe algorithms for constructing two-piece molds and multi-piece molds are presentedrespectively in the next section.

3.6.3 Reverse Glue Algorithm and Parting Surface

Based on the steps of the reverse glue operation (Section 3.6.1) and AlgorithmsGlue_Faces_Outer_Loop and Glue_Faces_Inner_Loop (Section 3.6.2), the algorithms ofMPMDM for constructing mold pieces and the selection of parting surface are discussedin this section.


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• Algorithm for Two-piece MoldsBased on Equation (3.11) ~ (3.14), the algorithm for generating two mold pieces is

given as follows.

Algorithm: Two_Mold_Piece_GenerationInput: Parting surface (PS) and Boolean mold base M’ with faces Fout(M’), ~F(R1)

and ~F(R2).Output: Mold piece M1 and M2.

(1) for each vertex vi of the parting edges of ~F(R1),(2) get edges that share vi;(3) for each edge ei,(4) if ei intersects PS, split ei into edge ei

+ and ei- by PS;

(5) Use PS to cut Fout(M’) into faces ↑Fout-M’ and ↓Fout-M’.(6) generate parting loops CPLk from the parting edges of ~F(R1);(7) classify CPLk into an outer loop CPL1 and several inner loops CPLi;(8) initialize set SGF ← NULL;(9) Glue_Faces_Outer_Loop(CPL1, SGF); // generate GFps and GFproj

(10)For each inner loops CPLi,(11) Glue_Faces_Inner_Loop(CPLi, SGF); // generate GFinner

(12)set SGF’← copy_reverse_faces(SGF); // copy glue faces(13)Swap partner coedges for all common edges of SGF, ↑Fout-M’ and ~F(R1);(14)Swap partner coedges for all common edges of SGF’, ↓Fout-M’ and ~F(R2);(15)Generate body M1 with the faces of SGF, ↑Fout-M’ and ~F(R1), then Regulate M1.(16)Generate body M2 with the faces of SGF’, ↓Fout-M’ and ~F(R2), then Regulate M2.

Suppose the edge number of the parting edges is ne. Among them, the edge numberof the outer loop CPL1 is nLo, and the edge number of the inner loops CPLi is nLi. So thecost of Step (1) to (11) is O(ne+ nLi• lg nLi). The cost of Step (13) and (14) depends on ne

and the edge number of the part npe. One simple implementation to find all commonedges is to use two iterations. One checks all edges of ne, and another checks all edges ofnpe. So the cost of Step (13) and (14) is O(ne•npe). Therefore the cost of the algorithm is

E

V

PS

E1

E2

PE1

GFinter1E3

Figure 3.29 – An Example for Coedge Splitting.


117

O(ne+ nLi• lg nLi + ne•npe).

The edge splitting process in Step (1) ~ (4) is necessary. Without the splitting, somecoedges of GFproj may not be able to find a partner coedge. So the generated M1 and M2

will be invalid bodies. For example, if the parting surface PS is chosen for a part asshown in Figure 3.29, glue face GFinter1 is generated by projecting PE1 into PS. If edge Eis not divided into E1 and E2, edge E3 of GFinter1 will not be able to find a partner coedge.

A mold piece generated by the reverse glue approach may have several coplanarfaces. A body regulation operation is called in Step (15) and (16) in order to merge thesecoplanar faces.

• Algorithm for Multi-piece MoldsThe algorithm to generate two-piece molds can be extended for generating multi-

piece molds.

Algorithm: Multi_Mold_Piece_GenerationInput: Boolean mold base M’ with faces Fout(M’), and ~F(Ri) (1≤ i ≤ n).Output: Set of mold piece Mi (1≤ i ≤ n).

(1) rank region number i according to the main parting direction and region volumes;(2) for 1≤ i ≤ n-1, // only need to run n-1 times(3) assign attributes ~F’(R1) to faces ~F(Ri);(4) assign attributes ~F’(R2) to all faces ~F(Rj) (i+1≤ j ≤ n);(5) assign attributes Fout(M’) to all glue faces;(6) generate parting surface (PSi);(7) Two_Mold_Piece_Generation(PSi,M’,M1’, M2’); //call two-piece mold function(8) Mi ← M1’;(9) if i = n-1, Mn ← M2’;(10) else, M’ ← M2’.

Step (1) considers the generation order of mold pieces. It is quite obvious usingdifferent orders will give us different results. The two heuristics considered are (i)generating mold pieces for regions in main parting direction first; (ii) generating moldpieces for regions with bigger volumes (calculate by bounding box) first. Accordinglythe given regions can be ranked before generating mold pieces for them.

Suppose the number of mold piece is nm, the average number of parting edges foreach region is ne, the average edge number of CPLi is nLi, and the average edge number ofthe part is npe. The cost of the algorithm is O(nm•ne + nm•nLi• lg nLi+ nm•ne•npe).

The algorithm based on the Reverse Glue operation is very efficient. First it usesEuler operations directly rather than sweeping and Boolean operations. Second thealgorithm complexity can be only related to the parting edge number instead of the edgenumber of the part, if more advanced data structures are used. That is, for step (13) and(14) in Algorithm Two_Mold_Piece_Generation, there is no need to determine all edgesof the part if we record all candidate edges for the reverse glue operations in advance.The number of parting edges is usually far less than the edge number of a part, which isevident from the examples given in Section 4.4. Therefore the mesh sizes of quadricsurfaces or parametric surfaces can be refined without affecting the execution time ofAlgorithm Multi_Mold_Piece_Generation significantly.


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Step (6) in Algorithm Multi_Mold_Piece_Generation is generating a parting surfacefor region Ri. An approach for this step is discussed in more details as follows.

• Selection of Parting SurfaceFor the approaches based on extending parting lines, the parting surface is decided

entirely by the parting lines. For a given parting direction and some parting lines, there isonly one parting surface (refer to Section 2.3). However in the approach of MPMDM,parting surface (PS) is also a mold design variable, similar to parting direction (PD) andparting line (PL).

The principle of the reverse glue operation given in Section 3.6.1 does not have anyrequirement on the parting surface. So the approach can generate mold pieces for anarbitrarily chosen parting surface. For example, three mold designs for a simple part Pare shown in Figure 3.30.a ~ Figure 3.30.c. Their PD and PL are the same. The onlydifference is PF1 ~ PF3 are in different positions.

Although all the designs satisfy the requirements given in Problem MPC, thefabrication cost related to each design is different. For M1 in Figure 3.30.c, the shapeformed by e1 and e2 is a small protrusion, which may break easily in injection process.As a worse situation, if e1 becomes vertical to PD, e2 and e1 will form a protrusion withno volume as shown in Figure 3.30.d. Although theoretically M1 is still a valid shape, itis not manufacturable anymore. So the designs given in Figure 3.30.a and Figure 3.30.bare valid for the part, while the designs shown in Figure 3.30.c and Figure 3.30.d are not.

To generate a parting surface for a part with given PD and PL, the heuristic rule usedin the approach is given as follows. First all the vertices of the outer parting loop arerotated to make PD as the z-axis of the new coordination system. By adding different zvalue of each vertex to an array, zpf in the array that has the highest count number isidentified. So the parting surface to be generated is the plane defined by zpf and facenormal PD. User can also change it interactively in RTMDS, which will be discussed inSection 4.2.

3.7 SUMMARY OF CHAPTER 3

In this chapter a systematic method (MPMDM) is presented for the design of multi-piece molds for Rapid Tooling. An introduction to MPMDM is given first (Section 3.1).Then problem formulations and the steps of the method are presented in Section 3.2 andSection 3.3 respectively. Approaches associated with the different steps are presented inSection 3.4 ~ 3.6. This chapter provides foundations for developing a mold design

M1

M2

M1M1

M2M2

PF1

PF3

PF2

M1

M2

PF3

e1e2 e1e2P

P P P

(a) (b) (c) (d)

Figure 3.30 – An Example of Different Parting Surface.


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system that can be used to design molds to fabricate prototype parts of differentcomplexity. This mold design system (RTMDS) will be presented in Chapter 4.

Although not explicitly presented in the chapter, the discussions on the differentrequirements, information flow and information processing for the different steps ofMPMDM provide partial theoretical structural validation for Hypothesis 1, and thetheoretical analyses of the algorithms provide partial theoretical performance validationfor Hypothesis 1. These results are summarized below (Figure 3.31):

Hypothesis 1: Steps of MPMDM were presented in Section 3.3. The input and outputinformation of each step provides partial validity of the entire structure of MPMDM.From the description it is evident that the output from each of the steps are used insubsequent steps, which provides theoretical structural validation of MPMDM.Further the final output of the MPMDM is mold pieces that satisfy Problem MCD andProblem MPD (Section 3.2.2), which compose Problem MD. Therefore the results ofMPMDM were the targeted output of Problem MD. The theoretical structurevalidation of the three steps of MPMDM combined, also provides partial structurevalidation of MPMDM. The algorithms for each step were analyzed theoretically(Section 3.4~3.6). These algorithm analyses provide an insight on the performances

TheoreticalStructuralValidation

H1

H1.1

H1.2

H1.3

Information flow fordifferent steps of

basic elementgeneration phase

are consistent. Thefinal elements arethe desired output.

Information flow fordifferent steps ofregion combining

phase are consistent.The mold pieceregions are thedesired output.

Information flow fordifferent steps of

mold piececonstruction phaseare consistent. Themold pieces are the

desired output.

Information flow for differentphases of MPMDM are

consistent. Information inputfor the steps are either fromoutput of other steps or the

initial part informationprovided. The final mold

pieces are the desired output.

TheoreticalPerformance

Validation

The methodperformance inbasic element

generation phase isacceptable based

on algorithmanalysis.

The methodperformance in

region combiningphase is acceptablebased on algorithm

analysis.

The methodperformance in

mold piececonstruction phase

is acceptablebased on algorithm

analysis.

The methodperformance of

MPMDM isacceptable based on

the theoreticalanalysis of composing

algorithms.

Figure 3.31 – Partial Theoretical Validation.


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of the algorithms for general inputs. Therefore they also provide partial performancevalidation of MPMDM. The empirical structural and performance validation of theMPMDM will be presented in Chapter 4, 7 and 8.

Hypothesis 1.1: Face connectivity and demoldability of mold pieces are two mainconcerns in multi-pieces mold design. They were considered in Problem MCD. Fora given part, the convex and concave properties of its faces are a main factor todecide demoldability of mold pieces, and therefore its mold design. Correspondinglyconcave regions and convex faces are proper elements for mold design. Sevenlemmas were presented to provide partial theoretical structural validation ofHypothesis 1.1 (Section 3.4.1). Although complete splitting can generate a uniqueresult of concave regions and convex faces for a part, the running time of theapproach is not feasible. In the approach for generating concave regions and convexfaces, some heuristics were considered in selecting splitting surface and splittingcriterion in the region splitting. The approach gives a feasible solution with satisfiedrunning time. The algorithm analysis of SR (Section 3.4.3) provides partial theoreticalperformance validation of Hypothesis 1.1.

Hypothesis 1.2: By generating small regions, the efficiency to explore facecombinations is improved. As a criterion to determine feasible combinations ofregions and faces, parting direction of a region needs to be calculated. The process ofcalculating V-Map and making selection from it can be combined into a linearprogram (Section 3.5.1). Solving a linear program can give us a satisfactory solutionmuch more quickly and easily. The algorithm for Problem PDLP, which runs inlinear time and linear storage, provides partial theoretical performance validation ofHypothesis 1.2. Detection of non-drafted surfaces is an important step in mold design.Since draft type is tightly related to a parting direction and parting lines, the detectionshould be executed in the determination process of parting direction (Section 3.5.2).In region combination process, it is noticed that mold design knowledge and relatedheuristic rules are needed since it is infeasible to explore and generate all thecombinations of regions and faces (NP-hard problem). The algorithm based onregion combination is flexible in adding more design knowledge (Section 3.5.4). Theanalysis of the combining process and related algorithms provides partial theoreticalstructural validation of Hypothesis 1.2.

Hypothesis 1.3: Automatically splitting the core and cavity inserts is an important stepin the mold design process. Reverse glue operation is tightly related to glueoperation, which has sound theoretic basis in the area of geometric modeling. Themathematical proof of the reverse glue operation (Section 3.6.1) provides partialtheoretical validation of Hypothesis 1.3. Different glue faces will lead to differentmold piece design. It is better to generate a planar glue surface since it can reducemold fabrication cost and material flash in injection process. The generation of innerglue faces is a geometric reconstruction problem. A formal problem definition of theglue face generation is presented with two algorithms (Section 3.6.2). From thealgorithm analyses (Section 3.6.3), it is evident that the approach based on the reverseglue operation is very efficient. It provides partial theoretical performance validationof Hypothesis 1.3. The approach can provide instantaneous visual feedback on themold design results even for a part with rather complex geometries (Section 4.4.3).


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In the next chapter (Figure 3.32) the Rapid Tooling Mold Design system (Section4.1), an experimental system that can be used on the design of molds to fabricateprototype parts with widely varying complexities, will be presented. Two test examples(Section 4.4) and three industrial cases (Section 4.5) whose molds were automaticallygenerated by the system will be presented. These examples provide part of the empiricalstructural and performance validation of the hypotheses.


R1

R2

Rk

R3

F1

F2

F3

F4F5

F6

Fn

PL1

PL2

PD1

PD2

Part P

F1

F2

F3

F4

F5

F7

F8

Fn

Part P

(1) (2)

(3)

F1

F2

F3

F4

F5

F7

Fn

Part P

F8F8 F6

F6

F7

F9 F9

F9 M1

M2 Mk

Mold Base

PD1

F1

F2

F3

PD2

F4F5

F7 Fn


§2.2 §2.3 §2.4 §2.5 §2.6

MoldConfiguration

Design Methods


Tools

CADRepresentation


Techniques


Figure 3.32 - Summary of Chapter 3 and Preview of Chapter 4.

Chapter 4 – RTMDS and Its Usage for Mold Design of Industrial Parts

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CHAPTER 4

RTMDS AND ITS USAGE FOR MOLD DESIGN

Rapid Tooling and multi-piece molding can dramatically reduce the fabrication timefor complex injection molded prototypes. A systematic approach based on regions isdeveloped for automated design of multi-piece Rapid Tooling molds in Chapter 3. In thischapter, an experimental system, the Rapid Tooling Mold Design System (RTMDS)which is based on the MPMDM presented in Chapter 3, is described. First an overviewof the RTMDS is given in Section 4.1. Then the supporting modules and the relatedsoftware tools used in the system are presented in Section 4.2. The implementations ofthe mold design modules are discussed in Section 4.3. These modules are the cores of thesystem. The problems and related limitations in the implementations of the modules arealso presented in Section 4.3. With all the modules presented, two test examples andthree industrial examples are described in Section 4.4 and Section 4.5 respectively.Finally the relationship between the RTMDS and the validation of the hypotheses arediscussed in Section 4.6.



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4.1 OVERIEW OF RAPID TOOLING MOLD DESIGN SYSTEM

The method described in Chapter 3 has been employed to develop a Rapid ToolingMold Design System (RTMDS). The system is part of the RTTB project (Rapid ToolingTestbed) under development at Georgia Institute of Technology (Section 1.2.2). The roleof the system is to design mold pieces for a part whose prototypes are to be produced byRapid Tooling.

The RTMDS consists of several modules and related data (Figure 4.1). The mainmodules of RTMDS are (1) region generation, (2) region combination and (3) mold piececonstruction. They correspond to the three steps of the Multi-piece Mold Design methodshown in Figure 3.6. The modules are linked to a graphical user interface (GUI). TheGUI controls the execution order, set options for each module, and graphically displaysthe results. ACIS manipulation module and LINGO manipulation module are twosupporting modules which are used by these three main modules. Related to the givenCAD model, the ACIS manipulation module provides functions to evaluate geometry andtopology of the part, and to make necessary modifications to the part. Related to ProblemPDLP (Section 3.5.1), the LINGO manipulation module generates the LinearProgramming formulations that are to be solved by LINGO@. The module also transfersthe results into a parting direction and sends it back to the main modules. Related to thesystem modules, the data flow of RTMDS is from CAD models of a part and a mold baseto CAD models of mold pieces. Besides these CAD models, the main data structures ofRTMDS include several arrays for regions (Section 3.5.3) and convex faces. They aregenerated by the region generation module and modified by the region combination

User

Graphical User Interface

(1) RegionGeneration

Module

(2) RegionCombination

Module

(3) Mold PieceConstruction

Module

RTMDS

Input

MoldPiece

Bodies

Output

MoldBaseCAD

Model

Input

Face Loop Coedge Edge Vertex

LINGO

LPFormulation

Region Array

Convex FaceArrayetc.

ACIS ManipulationModule

LINGOManipulation

Module

Part CADModel

RTMDS Data Structures

Step 1 Step 2 Step 3

Processor

Data

Software

Figure 4.1 – The Organization and Data Flow of the RTMDS.


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module. Finally they are used by the mold piece construction module for generatingmold pieces.

In this chapter, RTMDS is presented in the following manners. First the supportingmodules (ACIS manipulation, LINGO manipulation, and GUI) and their related data areintroduced in Section 4.2. Their implementations and related software tools (ACIS,LINGO, and MS-MFC) are also presented Section 4.2. Based on the supportingmodules, the implementation and limitations of the three main modules (regiongeneration, region combination, and mold piece construction) are discussed in Section4.3. Finally two test examples and three industrial examples are presented in Section 4.4and Section 4.5 respectively. They illustrate the usage of RTMDS for mold designbesides verifying the system. Finally a summary of the chapter is given in Section 4.6.

4.2 SUPPORTING MODULES AND THEIR IMPLEMENTATIONS

As shown in Figure 4.1, three main support modules in RTMDS are ACISmanipulation, LINGO manipulation, and graphical user interface. They support theimplementations of MPMDM (Chapter 3) in the following manners. MPMDM isdeveloped mainly based on the geometric properties of a part. Consequently the abilityof evaluating and manipulating the part geometries is critical for the system. The authorchose ACIS as the geometry kernel for RTMDS because it is the most powerful solidmodeling kernel today (Section 4.2.1). The part and the mold base inputted to the systemare all represented in ACIS format. Correspondingly the ACIS manipulation module isdeveloped to support the operations that are related to geometries. To determine partingdirections in the generation and combining processes, a linear programming (LP)problem needs to be solved (Section 3.5.1). The LINGO manipulation module cangenerate the LP formulation related to the parting direction problem. It can also call theLINGO system to solve the problem and send back the results (Section 4.2.2). Finally,the Graphical User Interface module allows the user of the system to visually examine theresults of each step. The user can then interactively change design options, or the resultsdirectly based on the displayed results (Section 4.2.3).

4.2.1 ACIS Manipulation Module and Part Representation

Three-dimensional (3D) geometric modelers are central to applications that arerelated to 3D geometries. However it is difficult and usually impossible for thedevelopers of the applications to create a modeler from scratch. Therefore an existinggeometric modeler should be used in an application. In this section, the geometricmodeler used in the RTMDS, ACIS, is introduced briefly first. Then the preparationprocess to generate a part that can be read by ACIS is described. Finally theimplementation of the ACIS manipulation module is presented after the data structures ofACIS are briefly introduced.

• ACISACIS® 3D Geometric Modeler (ACIS) from Spatial Technology Inc. (renamed

Spatial Corporation recently, www.spatial.com) is a well-known solid modeling kernelfor creating an end user application. As the leading 3D geometric modeler in the world,ACIS has been used in over 200 applications. Some of them are today's leadingCAD/CAM/CAE software applications including AutoCAD 2000.


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The ACIS kernel consists of a set of C++ classes (including data and memberfunctions, or methods) and functions (Spatial Technology, 2000). These classes andfunctions provide a foundation of common modeling functionality (e.g. Booleanoperations). They also provide users the flexibility to be adapted and extended forparticular application requirements.

ACIS stores the information of a model to ACIS save files. ACIS also restores themodel information from these files. There are two types of ACIS save files: StandardACIS Text files (file extension .sat) and Standard ACIS Binary files (file extension .sab).The only difference between these two files is that the data is stored as ASCII text in a.sat file and as binary form in a .sab file. The organization of a .sat file and .sab file isidentical. Therefore the term SAT file is generally used to refer to both in thisdissertation.

To get a valid SAT file that can be handled by RTMDS, the converting process for aCAD model of a part or a mold base is presented as follows.

• The Preparation of Input FilesThe RTMDS can only handle the SAT file as the input. In addition, since MPMDM

only considers planar surfaces (Section 3.5), the boundary faces of a part or a mold basethat are inputted to the system must also be planar. Therefore the format for RTMDSshould be a SAT file with only planar faces (called PF-SAT file in this dissertation).

The process to generate a PF-SAT file for RTMDS is shown in Figure 4.2. A givenCAD model can be in any format, e.g. a SAT file, or a PRT file (ProEngineer), or aSLDPRT file (SolidWorks). The first step is to transfer the CAD file to a STL file. TheSTL or stereolithography format is an ASCII or binary file used in manufacturing. It isthe standard input for most rapid prototyping machines. In this process, quadric orparametric boundary surfaces are approximated with a series of planar faces. The secondstep is to transfer the STL file into a SAT file. In this process, all planar faces that are inthe same surface are combined into one face. The topology of faces, loops and edges arealso added to the file. Both STL and SAT are the formats that are supported in most

CADfile

STLfile

PF-SATfile

(1) (2)

Figure 4.2 – The Process to Generate an Input File for RTMDS.


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major commercial CAD systems.

In STL file, the shape of the object is defined by a list of the triangular surfaces thatmust meet up exactly with each other, without gaps or overlaps. The number of trianglesused can be user-defined. As the number of triangles increases and the relative trianglesize decreases, the shape begins to represent the outline of the quadric or parametric facesmore accurately. This is often referred to as the "facet resolution." CAD programs oftencontain optional settings that will give the designer some level of control over the qualityof the STL file output. The most common settings or variables in the CAD systems are"Chord Height" and "Angle Control." Their values are determined by the tradeoffbetween the accuracy requirements and the file size of the model.

• Part Representation and ImplementationACIS is a boundary-representation (B-rep) modeler, which means that it defines the

boundary between solid material and empty space. This boundary is made from a closedset of surfaces in some spatial relations. The geometry and topology entities in ACIS aregiven in Figure 4.3. Another entity class shown in the figure, which is extensively usedin RTMDS, is Attributes. An attribute is an object that is attached to an entity in an ACISmodel. It stores information about the entity that is not provided within the ENTITYclass itself. Therefore developers can attach application-specific data to entities byderiving their own attribute classes.

Figure 4.3 – Focus of the ACIS Entity Classes in RTMDS (Spatial Technology,2000).


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Since only straight lines and planar faces are considered in RTMDS, themanipulations related to the geometry entities are simple. However, in order to maintaina valid model, the manipulations related to the topology entities may be difficult.Therefore the ACIS manipulation module mainly focuses on managing the topologyentities. The relationship between them is shown in Figure 4.4. Most functions in themodule are related to face, loop, coedge, edge and vertex.

C++ applications may interface to ACIS through Application Procedural Interface(API), functions, C++ classes, and Direct Interface (DI) functions. C++ applications for

Focus ofRTMDS

Figure 4.4 – The Relationship of Topology Entity Classes in ACIS (SpatialTechnology, 2000).


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Microsoft Windows platforms may also take advantage of the ACIS interface toMicrosoft Foundation Classes (MFC), which is implemented in the ACIS MicrosoftFoundation Class Component (AMFC). The author adopted C++ classes, functions, andAMFC in the development of RTMDS. The detail files and classes of the module arelisted in Appendix A.

In the next section, another supporting module of RTMDS is introduced.

4.2.2 Problem PDLP and LINGO®

Problem PDLP was presented in Section 3.5.1 to evaluate a parting direction of somefaces. It is a linear optimization problem since the goal and the constraints are all linearfunctions of the variables. Many algorithms and systems can solve the problem. Amongthem the Simplex method is the most famous method to solve linear programs (Reklaitis,et al., 1983). Mainly based on the availability, a linear programming solver, LINGO®, isused in RTMDS for solving Problem PDLP, which is developed by Lindo Systems Inc(www.lindo.com).

LINGO provides both the Dynamic Link Library (DLL) and Object Linking andEmbedding (OLE) standards for interfacing with other applications. In RTMDS, theLINGO DLL is used to access LINGO's functionality. In the implementation of theLINGO manipulation module, the author found writing all constraints explicitly in amodel file is more robust than using a set with a template model file. The detailimplementation information is described in Appendix A.4.

4.2.3 GUI and Control Options

The Graphical User Interface (GUI) of RTMDS consists of menus, toolbars, and adrawing area as shown in Figure 4.5. The user can control the running of the system by

Menu

Toolbar

DrawingArea

Figure 4.5 – Screen Capture of the RTMDS Interface.


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using the menus and toolbars. The generated results are displayed in the drawing area forvisualization. Usually different colors are used to help the user to interpolate the results.All the figures shown in Section 4.4 and 4.5 are the screen captures of the results shownin the drawing area.

The GUI module of RTMDS is implemented based on Microsoft Foundation Classes(MFC). MFC consist of nearly 200 C++ classes, which are designed by Microsoft tomake Windows programming easier and quicker. Application programs inheritfunctionality from MFC as needed. In the development of RTMDS, the ACIS MicrosoftFoundation Class (AMFC) is also integrated with the MFC provided in the MicrosoftDeveloper Studio. The detail implementation information of the module is also given inAppendix A.

In the mold configuration design and mold piece construction processes, the usermay want to change some control options based on the running results (Section 4.3).These control options can be changed interactively in RTMDS. An example of settingtolerances is given in Figure 4.6.

In light of the supporting modules describe in this section, the implementations ofthe main modules, especially the limitations of the system, are discussed in the nextsection.

4.3 THE IMPLEMENTATION AND LIMITATIONS OF MOLD DESIGNMODULES

The three mold design modules discussed in this section are the region generationmodule, region combination module, and mold piece construction module (Figure 4.1).

Figure 4.6 – Setting Tolerance Values in RTMDS.


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All the codes of the modules are written in the C++ language using the Microsoft VisualC++ 6.0 compiler from personal computers (PC).

The algorithms of MPMDM presented in Chapter 3 are implemented in eight steps inRTMDS. They correspond to the eight buttons in the toolbar of the system. The stepsand the related descriptions in Chapter 3 are listed as follows.

Step 1: Find concave edges, and generate combined regions and convex faces(Section 3.4.3).

Step 2: For each region, find parting edges, neighboring faces and a parting direction(Section 3.5.3).

Step 3: For regions without a parting direction, divide them into several regions suchthat each one has a parting direction (Section 3.4.3 and Section 3.5.1).

Step 4: Get region properties (main parting direction and core/cavity) and changeregion orders accordingly (Section 3.5.4).

Step 5: Combine neighboring regions and faces into several regions (Section 3.5.4).Step 6: Judge parting surface for each region (Section 3.6.3).Step 7: Get suitable mold base for the part; Put it in a proper position and subtract the

part from the mold base (Section 3.6.1).Step 8: Generate mold pieces for each regions (Section 3.6.2 and Section 3.6.3).

Among them, Steps 1 ~ 3 belong to the region generation module; Steps 4 and 5belong to the region combination module; and Steps 6 ~ 8 belong to the mold piececonstruction module. From Section 4.3.1 to Section 4.3.3, the problems and limitationsidentified in the implementation of the three modules are presented individually.

4.3.1 Region Generation

The region generation module is to generate combined regions and convex facesfrom the faces of a part. The implementation of the module was based on the algorithmspresented in Section 3.4. The detail implementation information of the module isprovided in Appendix A. In testing the module with different cases, the author identifiedthree problems related to the region generation process. They also present the limitationsof the RTMDS.

• The Validity of Input ModelsFor a model input to RTMDS, it is assumed that each edge must be the border

between two and only two polygons. Related to the two polygons, two coedges arelinked to the edge and they should take each other as the partner coedge. Most CADmodels satisfy this requirement. However, in the testing of RTMDS the author observedsome CAD models with coedges whose partner coedges are null. Also since ACISsupport Non-manifold object, some CAD models may have edges that have more thantwo coedges. If these CAD models are input to RTMDS, the system may exit abnormallyor even generate a system error.

To generate a PF-SAT file, a CAD model is converted to a STL file first. Then theSTL file is converted to a SAT file. This converting process, especially the translationfrom the modeling format to STL, may lead to problems including missing geometry andloop-orientation inconsistencies. If different CAD systems are used in the process, some


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small edges of the model may be omitted due to the different tolerances used in thesystems.

The validation of a CAD model is a research area in solid modeling. The Euler’sRule can be used to check if a model is manifold (Mäntylä, 1988).

• The Difficulties in the Face SplittingIn Section 3.4.3, Algorithm Split_Region is presented to generate concave regions

from a combined region that does not have a feasible parting direction. The face splittingin Step (4) is a major operation in the algorithm. A splitting surface (SS) can divide allthe faces of the region into two sides. The faces in one side are added to region CR+, andthe faces in another side are added to region CR-. An illustrative example of the facesplitting of face F is shown in Figure 4.7.

In RTMDS, a face is represented by a polygon (outer loop) with zero or many insidepolygons (inner loops). Depending on the positional relationship of the polygons and thesplitting surface, lots of situations should be considered. For example, an edge of theouter loop can be in the plane of SS, which is shown in Figure 4.8.a. Also, an edge of aninner loop can be in the plane of SS as shown in Figure 4.8.b. The face splittingalgorithm in RTMDS is based on the intersection edges of two faces. It can handlegeneral cases as shown in Figure 4.7, and some special cases as shown in Figure 4.8.However, a face in a given part may be rather complicated with many special cases likeedge in SS or point in SS (Figure 4.9). Therefore it is rather difficult to develop a generalface splitting algorithm that considers all the special cases.

SS SS

F

F

F

+

-

Figure 4.7 – A Simple Example of Face Splitting.

F

SSSS

F

SSSS

F

F1 F2

F

+ +

-

F -

F+

(a) An edge of the outer loop in SS (b) An edge of an inner loop in SS

Figure 4.8 – Two Examples of Face Splitting.


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The face splitting is also a research problem in geometric modeling. It is actuallyone step of the Boolean operations. However, the author did not find a function in ACISto implement this operation. Also if the MPMDM is extended for a triangulated CADmodel, it is straightforward to develop a general face splitting algorithm for triangles.

• Data Structure and Memory UsageArrays instead of dynamic pointers are used in the current implementation of

RTMDS, since checking the values of an array is much easier in debugging. However,this may bring into a problem in the memory usage.

For a complicated part as shown in Section 4.5.3, there are more than 5000 faces.The size of several arrays, like array of concave edge, convex faces and combinedregions, should be bigger than 3000. Each element of the arrays may take 40 bytes for aconcave edge to 500KB for a combined region (Section 3.5.3). So the memory requiredfor running the part will be more than 3k x 500kB = 1500 MB. It is beyond thecapability of the computer used to test the system, which has 512 MB DRAM. So whenrunning the part in RTMDS, an error of low memory was observed and the programexited abnormally.

If changing all the arrays to pointers, the memory can be applied dynamically fromthe operation system (Windows). After finishing the usage of an array, its memory isreleased and returned back to the operating system. Therefore the above problem relatedto the low memory can be avoided.

4.3.2 Region Combination

From the regions and convex faces generated by the module of region generation, theregion combination module explores their different combinations to minimize the numberof mold pieces. The implementation of the module was based on the algorithmspresented in Section 3.5. The detail implementation information of the module isprovided in Appendix A. The problems and related limitations of the module arediscussed as follows.

SSF

Figure 4.9 – A Complicated Example of Face Splitting.


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• Different Combining Order and Options in RTMDSAs discussed in Section 3.5.4, the combining order of regions affects the combining

results, and correspondingly the generated mold pieces. However, it is rather difficult, oreven impossible, to get some general rules on the combining order. Sometimes oppositerules are observed for different parts. For example, in most cases, neighboring faces in asame plane should be combined into a region first. However, for a camera roller asshown in Figure 4.10, it is desired not to combine the neighboring faces in the same planefirst (F1, F2, and F3, F4).

Consequently several options were provided in RTMDS to control the combiningorders for different parts. These options include (1) if regions in the main partingdirection are combined first; (2) if the core and cavity properties of a region areconsidered in the combining process; (3) the combining iteration number before regionsthat are not in the main parting direction are added to the combining process; (4) if facesin the same plane are combined first. The user can set these options quite easily by themenus of RTMDS. Several interactive tools are also provided in the system to inquireand change the order of the generated regions.

Besides the order of regions, the order of neighboring faces of a region will alsoaffect the combining results (Section 3.5.4). However, their relations are not so evidentas the relations between the order of regions and the combining results. Therefore theorder of neighboring faces is not considered in the current implementation of RTMDS.

• Robustness and AccuracyA main combining criterion for the region combination process is whether a feasible

parting direction exists for all faces. The evaluation approach used in RTMDS is basedon solving Problem PDLP (Section 3.5.1). In the problem, ε1 is a tolerance variable forthe plane constraints: dp d d di xi x yi y zi z= + + ≥η η η ε1 for each face Fi. After solving the

problem, ε2 is another tolerance variable for judging if faces are combinable according tothe solution. That is, suppose the solution of Problem PDLP is v(x, y, z). Thecombinability of the faces can be determined by judging the length of v with ε2.

if (v.x•v.x + v.y•v.y + v.z•v.z < ε2) thenFaces are combinable;

elseFaces cannot be combined.

F1

F2

F3 F4

Figure 4.10 – The Combining Order of Faces in a Same Plane.


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Setting proper values of ε1 and ε2 is important for the combining process.Considering the approximation errors of quadric or parametric surfaces, it is desirable toset ε1 and ε2 smaller to make the combining process more robust. However, this may alsolead to some undesired results. As an example, a camera roller is shown in Figure 4.11,in which F is a planar face for the approximation of a cylindrical face. It is desired tocombine face F with region R1. However, if ε1 is set as –0.0174, and ε2 is set as 0.85,face F is also combinable with region R2 based on Problem PDLP. The plane constraintsand the solution of problem PDLP are also given in Figure 4.11. From the planeconstraints, especially the inequality of F, it is quite obvious that ε1 makes x of v non-zero.

F

R1

R2

z

xy

Problem PDLP for Determining the Combinability of R1 and F: MAX= (-0.134028*(x1-x2))+(0.380045*(y1-y2))+(-0.000000 *(z1-z2));: !The plane constraints;: (0.000000*(x1-x2))+(1.000000*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.980783*(x1-x2))+(0.195102*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.000000*(x1-x2))+(1.000000*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.000000*(x1-x2))+(0.000000*(y1-y2))+(-1.000000*(z1-z2))>=(-0.017400);: (-0.980783*(x1-x2))+(0.195102*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.000000*(x1-x2))+(1.000000*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.000000*(x1-x2))+(0.000000*(y1-y2))+(1.000000*(z1-z2))>=(-0.017400);: (0.000000*(x1-x2))+(1.000000*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (-0.831473*(x1-x2))+(0.555566*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (-0.555566*(x1-x2))+(0.831473*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (-0.195102*(x1-x2))+(0.980783*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.195102*(x1-x2))+(0.980783*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.555566*(x1-x2))+(0.831473*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (0.831473*(x1-x2))+(0.555566*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400);: (-0.987689*(x1-x2))+(-0.156430*(y1-y2))+(0.000000*(z1-z2))>=(-0.017400); ← F

Solution of Problem PDLP: v = (-0.2057, 0.94488, 0.0). Length |v| = 0.935.

Figure 4.11 – An Example for Setting Tolerance Value.


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The combining errors in the beginning stages will affect the determinationsthereafter. Therefore the errors may spread and undesired results may be generated. Forexample, as the result given in Figure 4.11, the parting direction of region R1 is v1 (-0.2057, 0.94488, 0.0) instead of v2 (0.0, 1.0, 0.0). So some faces that are not combinablein v2 can now be combined in v1; on the contrary, some faces that are combinable in v2

cannot be combined in v1.

In RTMDS, interactive tools are provided for setting values of these tolerancevariables (Figure 4.6). Picking tools are also provided to change faces of a region to adifferent region.

4.3.3 Mold Piece Construction

The mold piece construction module is to generate mold pieces from the regions of apart and a given mold base. The implementation of the module was based on thealgorithms presented in Section 3.6. The detail implementation information of the moduleis provided in Appendix A. After testing the module, the author discusses somelimitations of the module as follows.

• Setting Parting SurfaceParting surface is important in the mold piece construction (Section 3.6.3). There are

cases that the parting surfaces automatically generated by the RTMDS were not the sameas what the user expected. The RTMDS also provides interactive tools for the user tochange the parting surface. Two approaches are provided in the system. The user canpick an existing face or two existing edges. A parting surface will be constructed by thesystem accordingly.

• Position the Part Relative to the Mold BaseAs discussed in Section 3.6.1, the RTMDS will translate a part to the center point of

a mold base automatically. However, the position of the part in the mold base may needto be changed in the actual mold design. For example, the positions of possible ejectorpins are fixed by the ejection pattern of an injection molding machine. Therefore the partshould be positioned in the mold base based on the geometry of the part and the positionsof the pins. RTMDS allows the user to change the position of the part interactively.

• Generation of Inner Glue FacesIn Section 3.6.2, Algorithm Glue_Faces_Inner_Loop was presented for the

generation of inner glue faces. In the algorithm, the greedy heuristic is used to minimizethe number of generated faces. The algorithm can handle many cases, including the casesshown in Figure 3.25 and Figure 3.26. However, it is also observed that an inner loop ofa part can be rather complicated. Consequently the generation of inner glue faces for theloop can be difficult. An illustrative example is given in Figure 4.12. The inner partingloop is shown in the figure with the parting surface. RTMDS was unable to generateinner glue faces from the loop that satisfies Problem GFG given in Section 3.6.2.

It is noticed that no matter how complicated the inner loop can be, the generatedinner glue faces should satisfy Problem GFG. Therefore the author believes ProblemGFG is a general problem formulation for the generation of inner glue faces. Nopublications in the areas of computational geometry and CAD were found that discussed


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Problem GFG or similar geometric problems. However, in order to construct moldpieces for an injection molded part, the problem should be studied further.

• Mold Piece Construction OrderIn Algorithm Multi_Mold_Piece_Generation (Section 3.6.3), mold pieces are

constructed in the order of region R1, R2, …, Rn. Before the algorithm is executed, theuser of the RTMDS can also interactively change the construction order via the toolsprovided by the system.

In light of the RTMDS and its modules described in the last section and this section,some testing examples of the RTMDS will be presented in Section 4.4 and 4.5. Theseparts have widely varying complexities as shown in Table 4.1.

4.4 INTUITIVE EXAMPLE PARTS

The main purpose of the examples given in this section is to test the methodspresented in Chapter 3. Since the examples are intuitive, it is easier to verify theimplementation of the system described in the last two sections. There are four steps that

Inner PartingLoop Parting

Surface

Figure 4.12 – Mold Pieces for a Pocket with Empty V-Map.

Table 4.1 – The Complexities of the Example Parts in Section 4.4 and 4.5.

Complexity Example Part1 (Section

4.4.1)

Example Part2 (Section

4.4.2)

Industrial Part1 (Section

4.5.1)


4.5.2)


4.5.3)Low X X

Medium X XHigh X


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are taken in each example to verify the results from the system are correct and makesense. These steps or tasks are:

Task 1: Verify the regions and the convex faces generated by the system are thesame as what are expected from the method;

Task 2: Verify the regions after combining are the same as what are expectedfrom the method;

Task 3: Verify the generated mold pieces can form the combined regions insideand the mold base outside;

Task 4: Examine the running time of each module to determine whether they arein accordance with the algorithm analysis of MPMDM (Chapter 3).

Besides industrial parts as shown in Section 4.5, the author constructed more than 10example parts and tested them in the RTMDS. The two parts presented in this section arechosen because (1) they represent the two most common undercuts, protrusion andextrusion; (2) they are the simplest parts that are related to two-piece molds and multi-piece molds respectively.

A detail description of the results for each step of the RTMDS is given in Section4.4.2 because the complexity of the test example 2 is suitable for explaining the runningresults. It also illustrates the correctness of the implementations of the system. For otherexamples given in this and the next sections, the results of each example are organized inthe following manners. First, the information of the part (the size of the file, face number,etc.) is presented in a table. The screen captures of the graphical results of some keysteps given by the RTMDS are given in figures. The information related to these steps isalso provided in the table. Finally the running time of each step and the total time arelisted in the table. All the tests in this dissertation are based on a personal computer witha 700 MHz Intel-III processor.

4.4.1 Test Example 1: A Box with a Rib

Input SAT file size: 11 KB.

The first test example is a rib part as shown in Figure 4.13. It is a simple part withonly 11 faces. The part has only one protrusion feature, and two mold pieces aresufficient for producing the part. Table 4.2 lists the information regarding the part,generated regions, and the execution time of each step. Figure 4.14 shows the graphicaloutputs of the RTMDS. Different regions and convex faces (CVX) are marked withdifferent colors.


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Besides the results automatically generated by the RTMDS, the user can also changesome faces of a region to the set of convex faces interactively. In Figure 4.15.a, theregion and convex faces of the part are shown, which are slightly different from theresults shown in Figure 4.14.b. Correspondingly, different mold designs are generated bythe RTMDS, which are also shown in Figure 4.15.

Figure 4.13 – A Test Example of a Rib Part.

Table 4.2 – The Information for Test Example 1.

Face No. Concave face No. Concave edge No.Part Info.11 5 4

Initially generated After dividing After combiningRegion Info.1 1 1

CPL No. Edge # of CPL1 Edge # of CPLi

1 4 0GFps No. GFproj No. GFinner No.

Reverse glue Info.

1 0 0Step 1 Step 2 Step 30.01 0.44 0.00

Step 4 Step 5 Step 60.12 0.38 0.01

Step 7 Step 8 Total Time

Running Time (s)

0.02 0.29 1.27


139

ConvexFaces

Region

ConvexFace

Region

(a) Region Generation (1 region+ 6 CXFs) (b) Region Combination (1 region+ 1 CXF)

(c) Mold Base and the Parting Surface

(d) Two Generated Mold Pieces

Figure 4.14 – Graphical Results of a Mold Design for Test Example 1.


140

ConvexFaces

Region

PartingSurface

(a) Region Combination Result with the Parting Surface (1 region + 5 CXFs)

(b) Mold Base and the Parting Surface


Figure 4.15 – Graphical Results of Another Mold Design for Test Example 1.


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4.4.2 Test Example 2: A Box with a Through Hole and Two Grooves


The second test example is a box with a through hole and two grooves as shown inFigure 4.16. It is a simple part with only 18 faces. The part has three extrusion features.It cannot be made by two mold pieces. Therefore multi-piece mold design is needed forproducing the part. Table 4.3 lists the information regarding the part, generated regions,reverse glue operation, and the execution time of each step.

A detail description of the results generated in each step is given as follows. It canfamiliarize the reader with the notions given in the result table. It also illustrates thecorrectness of the implementations of the RTMDS.

Figure 4.16 – A Test Example of a Box with a Through Hole and Two Grooves.

Table 4.3 – The Information for Test Example 2.





Reverse glue Info. For R1

1 0 1CPL No. Edge # of CPL1 Edge # of CPLi


Reverse glue Info. For R2

1 2 0Step 1 Step 2 Step 30.01 0.49 0.0

Step 4 Step 5 Step 60.05 0.45 0.00


Running Time (s)

0.02 0.32 1.34


142

Step 1: Based on the criterion of judging convex and concave edges (Section 3.4.3),there are 14 concave edges (5+5+4) in the part as shown in Figure 4.17. If a facecontains one or more concave edges, it is a concave face. Therefore 12 concave faces(4+4+4) are identified. Based on their connectivity, three regions are generated forall the concave faces (Figure 4.17). All the remaining faces are convex faces.Therefore the total face number is 12+6 = 18.

Step 2: For each region, the parting edges and neighboring faces are recorded in the datastructure of the RTMDS. Also a parting direction for each region is generated by theLINGO system. They are: PD1 for R1 is (0.0, 1.0, 0.0), PD2 for R2 is (-0.707, 0.0,0.707), PD3 for R3 is (0.707, 0.0, 0.707).

Step 3: Since all regions have a parting direction, this step is skipped. However, forindustrial example 2, this step is critical since a generated region R has no feasibleparting direction (Figure 4.24).

Step 4: The main parting direction of the part is in y-axis because the volume of thebounding box of R1 is the biggest. Also R1 is a core since one of its edge loops is aninner loop of a face. Instead R2 and R3 are cavities since all their edge loops are outerloop of faces.

Step 5: Based on the connectivity of faces and combining criterion (Section 3.5.1), R1

and a convex face are combined into a region. R2, R3 and a convex face are combinedinto a region (Figure 4.18.b). In the figures, different regions and convex faces(CVX) are marked with different colors. After the region combination process, 2regions (R1, R2) and 4 convex faces are generated. For the four convex faces, a newregion R3 is generated.

R2

5 concave edges4 concave faces

1 region

R3


1 region

R1


1 region

6 convex faces

z

x

y

Figure 4.17 – Illustration of the Running Results of Test Example 2.


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Step 6: The parting surfaces generated for R1 and R2 are shown in Figure 4.18.d andFigure 4.19.c respectively. The parting surface for R3 is not used in the mold piececonstruction process, therefore it is not shown here.

Step 7: A mold base is generated based on the size of the part. After it is positioned asshown in Figure 4.18.d, the Boolean operation is executed with the part.

ConvexFaces

R1

R2

R3

R1

R2

ConvexFaces

(a) Region Generation (3 regions + 6 CXFs) (b) Region Combination (2 regions + 4 CXFs)

ConvexFaces

R2

R1

(c) Another View of the Region Combination Results (d) Mold Base and the Parting Surface

Figure 4.18 – Graphical Results of Mold Design for Test Example 2. (Step 1~7)


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(a) Mold Piece M1 for R1

(b) Mold Piece M2’ for R2 and R3 (c) Mold Base and the Parting Surface

(d) Mold Piece M2 for R2 (e) Mold Piece M3 for R3

(f) Putting Three Mold Pieces Together

Figure 4.19 – Graphical Results of Mold Design for Test Example 2 (Step 8).

GFinner

GFinner

GFps

GFps

GFprojGFps GFproj

GFps


145

Step 8: Since three regions are generated in the region combination process (Figure4.18.b and c), there are two phases in Algorithm Multi_Mold_Piece_Generation.Figure 4.19.a and b show the graphical outputs of the first phase. Figure 4.19.d and eshows the graphical outputs of the second phase. In the generated mold pieces, thegenerated glue faces are in the same color as those of the related regions. In the firstphase, there are two parting loops (related to R1). Each of them has 4 edges. Sinceall the edges of CPL1 are in the parting surface, there is no GFproj (Section 3.6.2).Related to the two parting loops, faces GFps and GFinner are generated as shown inFigure 4.19.a and b. In the second phase, there is only one parting loops with 12edges (related to R2). Among them, six of the parting edges are not in the partingsurface, and they form two GFproj and one GFps (Figure 4.19.d and e). There is noGFinner in the second phase.

After the correctness of the implementations of the RTMDS is illustrated, therunning time of each step is also given in Table 4.3. From the results, it is evident thatthe RTMDS is also very efficient. Since the running time for this example is too short,further analysis on the computation time of the RTMDS is given in Section 4.5.3.

In the next section, three industrial parts are presented to further verify the molddesign method, and also to demonstrate that the RTMDS is capable to handle morecomplex cases.

4.5 INDUSTRIAL CASES

In this section three industrial parts whose molds were designed by the RTMDS arepresented. For each example the four tasks described in Section 4.4 are also followed toverify that the results from the system are correct and make sense. In addition, the authoralso physically validated the mold design of industrial example 1 and producedprototypes for the part design.

The examples were chosen as the case studies of the RTMDS mainly for the reasonsgiven as follows.

(1) The three example parts are typical injection molded parts;(2) The complexity of the parts covers a typical range of part complexities based on

the face number of a part. The three examples have face number from 330 to5000, which correspond to medium complexity to high complexity.

(3) The size of the first example is suitable for an existing mold base of a Morgan-Press injection molding machine, which is located at RPMI. Therefore it can beused to test the RTMDS in loading an existing CAD model of a mold base.

(4) The second example has a combined region which does not have a feasibleparting direction. So the region needs to be divided into several concave regions.Also the part has several inner parting loops. Therefore the algorithms of dividingregions and generating inner glue faces can be tested by the second example.

(5) The third example has more than 5000 faces. It is used mainly to test if theRTMDS can handle parts with high complexity.

The three examples are organized based on their complexities. From industrialexample 1 to 3, the part complexity is increased. The results of each example areorganized in the same way as that of the examples given in Section 4.4. The information


146

of the part and the mold design steps is presented in a table. The screen captures of thegraphical results of some key steps given by the RTMDS are given in figures. All thetests are also based on a personal computer with a 700 MHz Intel-III processor.

4.5.1 Industrial Example 1: A Housing


The first industrial example is a housing for a telephone adapter as shown in Figure4.20. The original part was produced by the injection molding process. There are bothprotrusion and extrusion features in the part. Table 4.4 lists the information regarding thepart, generated regions, and the execution time of each step. Figure 4.21 shows thegraphical outputs of the RTMDS. Different regions are marked with different colors.The standard mold base used in a Morgan-Press injection molding machine is also shown

Figure 4.20 – A Housing for A Phone Adapter.

Table 4.4 – The Information for Industrial Example 1.





Reverse glueInfo.

1 23 0Glue_Faces_Outer_Loop Glue_Faces_Inner_Loop Multi_Mold_Piece_Generation

0.03 0.0 1.57Step 1 Step 2 Step 30.06 2.40 0.0

Step 4 Step 5 Step 60.04 7.15 0.36


Running Time(s)

0.94 1.75 12.7


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in Figure 4.21.c.

The mold design generated by the RTMDS (Figure 4.21.d) is further verified byphysical validations. First the mold design is built directly by a SLA-3500 machine. Aphoto of the built mold pieces is given in Figure 4.22.a. Using the SLA mold pieces,more than 10 functional prototypes of the part are produced by the Morgan-Pressmachine, which are shown in Figure 4.22.b. The material of the prototypes ispolystyrene. The parameters used in the injection molding machine are:

Temperature (Barrel / Nozzle): 430 / 450 oF.Pressure (Injection / Pilot): 2450 / 70 psi; Clamp Force: 13 tons.Time (injection / Cooling): 13 / 390 second.

Therefore, it is validated that the mold design for the housing (Figure 4.21.d) canproduce the part by the Rapid Tooling process.


148

R1

R2

(a) Region Generation (28 regions) (b) Region Combination (2 regions)

(c) Mold Base of the Morgan-Press


Figure 4.21 – Graphical Results of a Mold Design for Industrial Example 1.


149

(a) SLA Mold Pieces

(b) Injection Molded Parts

Figure 4.22 – Physical Validation of the Mold Design for Industrial Example 1.


150

4.5.2 Industrial Example 2: A Thin Wall Part


The second industrial example is a thin wall part as shown in Figure 4.23. Both thetop view and the bottom view are given. The part is a component of the telephoneadapter whose housing is presented in Section 4.5.1. The original part was produced bythe injection molding process. The part is rather complicated with several through holesand slots. Table 4.5 lists the information regarding the part, generated regions, and theexecution time of each step. Figure 4.24 shows the graphical outputs of the RTMDS.Different regions are marked with different colors. A combined region shown in Figure4.24.a is divided into eight concave regions as shown in Figure 4.24.b. In the dividing,face F1 and F2 (Figure 4.24.a) are the splitting surfaces because they have the biggestnumber of convex internal edges (Section 3.4.3). Finally the generated mold pieces forthe part are given in Figure 4.25.

(a) Top View (b) Bottom View

Figure 4.23 – A Thin Wall Part for A Phone Adapter.

Table 4.5 – The Information for Industrial Example 2.





Reverse glueInfo.

1 14 11Glue_Faces_Outer_Loop Glue_Faces_Inner_Loop Multi_Mold_Piece_Generation

0.03 18.0 21.0Step 1 Step 2 Step 30.18 2.84 5.15

Step 4 Step 5 Step 60.13 10.4 0.02


Running Time(s)

1.08 21.04 40.84


151

F2

F1

R

(a) Region Generation (31 regions)

R1R2

R3R4

R5

R6R7

R8

(b) Region Dividing (38 regions, R is divided into R1 ~ R8)

R2

R1

(c) Region Combination (2 Regions)

Figure 4.24 – Graphical Results of a Mold Design for Industrial Example 2.


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4.5.3 Industrial Example 3: A Complex Housing

Input SAT file size: 5.3 MB.

The third industrial example is a complex housing as shown in Figure 4.26. Theoriginal part was produced by the injection molding process. As discussed in Section4.3.1, the current implementation of the RTMDS requires more than 1500MB memoryfor this part. Therefore in the PC used for testing the system, an error of low memorywas given by the operation system when loading the part in the system. However, theauthor used interactive tools that are provided by the RTMDS to divide the faces into tworegions. The two regions are marked with different colors and shown in Figure 4.27.The par was used to test the module of mold piece construction in the RTMDS. Table4.6 lists the information regarding the part, the reverse glue operation, and the executiontime of the algorithms used in the module (Section 3.6). Finally the generated moldpieces for the part are given in Figure 4.28.a. The author also built the mold pieces usinga SLA-3500 machine. A photo of the built mold pieces is shown in Figure 4.28.b.

Figure 4.25 – Generated Mold Pieces for Industrial Example 2.


153

Table 4.6 – The Information for Industrial Example 3.Face No. Edge No.

5493 9344Concave face No. Concave edge No.

Part Info.

4880 4448CPL No. Edge # of CPL1 Edge # of CPLi


Reverse glueInfo.

1 13 0Glue_Faces_Outer_Loop Glue_Faces_Inner_Loop Multi_Mold_Piece_GenerationComputation

Time (s) 0.18 0.0 40.5

Figure 4.26 – A Complex Housing.

R2

R1

Figure 4.27 – Region Combination (2 regions).


154

From the running results shown in Table 4.6, it can be observed that the computationtimes of algorithms Glue_Faces_Outer_Loop and Glue_Faces_Inner_Loop are mainlyaffected by the edge number of CPL1 and CPLi respectively. The industrial part 2 is aninteresting example with several inner parting loops. From its results, it is noticeable thatour algorithm to generate the inner glue faces GFinner, which is O(ne• lgne) (refer toSection 3.6.2), is more time-consuming than the algorithm to generate GFps and GFproj,

(a) CAD models of two mold pieces

(b) Two mold pieces made by SLA

Figure 4.28 – Mold Pieces for Industrial Part 3.


155

which is only O(ne) (Section 3.6.2). Besides the edge number of CPL1 and CPLi, thecomputational time of algorithm Multi_Mold_Piece_Generation is also affected by theedge number of the part npe. These observations are all in accordance to our algorithmanalysis presented in Section 3.6.

Based on the test examples and the industrial examples presented in Section 4.4 and4.5, an evaluation of the algorithms of the RTMDS is provided in the next section.

4.6 EVALUATION OF EXAMPLES AND CASES

In Chapter 3, algorithm analyses of several key steps of the Multi-Piece Mold DesignMethod were presented. Comparing the theoretic analysis results with the experimentresults obtained from the RTMDS, we can partially validate the implementation of theRTMDS. Based on the examples presented in this chapter and the case studies presentedin Chapter 7 and 8, the running time of the three stages is discussed individually.

4.6.1 Region Generation Process

The region generation process is the first step of the Multi-Piece Mold DesignMethod (Section 3.3). For the algorithms presented in Section 3.4.3, the running time ofthe region generation process is O(ne + nf

2), where ne and nf are the edge number and theface number of a given part respectively. In the RTMDS, steps 1 to 3 are related to theregion generation process. By adding the running time of step 1 ~ 3, we get the runningtime of the region generation process as shown in Table 4.7. It is noticed that industrialexample 2 is not considered here because it is the only part which has regions to bedivided in step 3.

Table 4.7 – Experimental Data of Region Generation Process.

Total FaceNo. (nf)

Concave FaceNo.

Total EdgeNo. (ne)

Concave EdgeNo.

Region GenerationTime (second)

Test Exp1 11 5 24 4 0.45Test Exp2 18 12 48 14 0.5Robot Arm 92 66 206 40 1.84

Industrial Exp1 330 207 712 223 2.46Camera Roller 544 336 1274 446 3.05

The relations of the running time of the region generation process with the face andedge number of parts are shown in Figure 4.29. The lines in the graph are in accordancewith the algorithm analysis result.


156

4.6.2 Region Combination Process

The approaches for the region combination process utilized in the RTMDS arepresented in Section 3.5. Based on the algorithm analysis (Section 3.5.4), the runningtime of the region combination process is O(nf

2), where nf is the face number of a givenpart. In the RTMDS, steps 4 to 6 are related to the region combination process. Byadding the running time of step 4 ~ 6, we get the running time of the region combinationprocess as shown in Table 4.8. It is noticed that the camera roller is not considered herebecause its combination process has three iterations instead of one (Figure 8.5).

Table 4.8 – Experimental Data of Region Combination Process.

Total FaceNo. (nf)

ConcaveFace No.

Total EdgeNo.

ConcaveEdge No.

Region CombinationTime (second)

Test Ex1 11 5 24 4 0.51Test Ex2 18 12 48 14 0.5

Robot Arm 92 66 206 40 0.72Industrial Exp1 330 207 712 223 7.55Industrial Exp2 606 489 1337 543 10.55

The relations of the running time of the region combination process with the faceand edge number of parts are shown in Figure 4.30. The lines in the graph are inaccordance with the algorithm analysis result.

Region Generation

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500

Face/Edge No.

Run

nin

gTi

me

(sec

)

Total Face No.

Concave Face No.

Total Edge No.

Concave Edge No.

Figure 4.29 – Relations of Region Generation Time with Face/Edge Number.


157

4.6.3 Mold Piece Construction Process

The region generation process is the last step of the Multi-Piece Mold DesignMethod (Section 3.6). For the algorithms presented in Section 3.6.3, the running time ofthe region generation process is O(npe + nLi • lg nLi + npe • ne), where ne, nLi and npe arethe total edge number, the inner loop edge number, and the parting edge number of agiven part respectively. In the RTMDS, steps 7 and 8 are related to the mold piececonstruction process. By adding the running time of step 7 and 8, we get the runningtime of the mold piece construction process as shown in Table 4.9.

Table 4.9 – Experimental Data of Mold Piece Construction Process.

TotalFace No.

ConcaveFace No.

Total EdgeNo. (ne)

PartingEdge No.

(npe)ne * npe

K* ne *npe

Mold PieceConstruction

Time (Second)Test Exp1 11 5 24 4 96 0.48 0.31Test Exp2 18 12 48 8 384 1.92 0.34Industrial Exp1 330 207 712 54 38448 192.24 2.69Industrial Exp2 606 489 1337 234 312858 1564.3 22.12Industrial Exp3 5493 4880 9344 234 2186496 10932.5 40.5

The relations of the running time of the mold piece construction process with theface and edge number of parts are shown in Figure 4.31. A constant K is multiplied toeach value in the column of ne• npe to make the value of ne• npe fit in the figure. The linesin the graph, especially k• ne• npe, are in accordance with the algorithm analysis result.

Region Combination

0

2

4

6

8

10

12

0 500 1000 1500

Face/Edge No.

Run

ning

Tim

e(s

ec)

Total Face No.

Concave Face No.

Total Edge No.

Concave Edge No.

Figure 4.30 – Relations of Region Combination Time with Face/Edge Number.


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4.6.4 The Whole Process of RTMDS

Finally, the total running time of the mold design process is shown in Table 4.10. Itcorresponds to the sum of the running time of steps 1 ~ 8 in the RTMDS. The relationsof the total running time with the face and edge number of parts are shown in Figure4.32. The lines in the graph illustrate the running time behavior of the RTMDS for agiven part.

Mold Piece Construction

0

5

10

15

20

25

30

35

40

45

0 2000 4000 6000 8000 10000 12000

Face/Edge No.

Ru

nnin

gT

ime

(sec

)

Total Face No.

Concave Face No.

Edge No.

K*ne*npe

Figure 4.31 – Relations of Mold Piece Construction Time with Face/Edge Number.

Total Running Time

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500

Face/Edge No.

Run

ning

Tim

e(s

ec)

Total Face No.

Concave Face No.

Total Edge No.

Concave Edge No.

Figure 4.32 – Relations of Mold Design Running Time with Face/Edge Number.


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Table 4.10 – Experimental Data of Mold Design Process.

TotalFace No.

ConcaveFace No.

Total EdgeNo. (ne)

ConcaveEdge No.

Total RunningTime (Second)

Test Exp1 11 5 24 4 1.27Test Exp2 18 12 48 14 1.34Industrial Exp1 330 207 712 223 12.7Industrial Exp2 606 489 1337 543 40.84

In the next section, a brief summary is given which discusses the relevance of theseresults with regard to the hypotheses of the dissertation.


Mold design is a laborious process that requires significant time from the molddesigner. Automated mold design significantly reduces the mold design time, andtherefore reduces the lead-time of Rapid Tooling process in producing functionalprototypes. In this chapter, a Rapid Tooling Mold Design System (RTMDS) is presentedfor the automated design of multi-piece molds for Rapid Tooling. An introduction to theRTMDS is given first (Section 4.1). Then the supporting modules and the software toolsused in the system are presented in Section 4.2. The mold design modules are theimplementation of the Multi-piece Mold Design Method (Chapter 3). Problems in theimplementation of the modules are described in Section 4.3. Finally two test examplesand three industrial examples are presented in Section 4.4 and Section 4.5 respectively.

The RTMDS and the given case studies provide partial empirical structural andperformance validation of Hypothesis 1 (Figure 4.33). The results from testing thehypothesis are summarized as follow.

Hypothesis 1: The implementations of the RTMDS were presented in Section 4.2 and4.3. The case studies for testing the system are described in Section 4.4 and 4.5.Although not explicitly presented in the sections, the discussions on the relations andthe implementations of the modules of the RTMDS provide partial empiricalstructural validation of Hypothesis 1, and the discussions on the results and therunning time of different examples provide partial empirical performance validationof Hypothesis 1. The summary of testing the sub-hypotheses (H1.1 ~ H1.3) presentedbelow will provide more details.

Hypothesis 1.1: The region generation module is developed based on the basic elementsof concave regions and convex faces (Section 4.3.1). The module can generate thebasic elements of mold design for several test parts and industrial parts in acceptabletime (Section 4.4 and 4.5). The generated elements can be used in the regioncombination process for mold configuration design.

Hypothesis 1.2: The region combination module is developed based on the concaveregions and convex faces generated in the region generation module (Section 4.3.2).The module can generate mold configuration design for several test parts andindustrial parts in acceptable time (Section 4.4 and 4.5). The generated moldconfiguration design can be used in constructing mold pieces for the parts.


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Hypothesis 1.3: The mold piece construction module is developed according to thecombined regions of a part and a given mold base (Section 4.3.3). The module cangenerate mold pieces for several test parts and industrial parts in acceptable time(Section 4.4 and 4.5). The generated mold pieces can be used in the Rapid Toolingprocess to produce functional prototypes.

Automatic generation of mold design is also a key step of Design-for-Manufacturingsystem for injection molding process (Chapter 6). After CAD models of molds aregenerated, detailed Design-for-Manufacturing feedback can be provided to the designer,and RP process planning and Injection molding simulation is enabled. In the next chapter(Figure 4.34), geometric tailoring, which is part of design-for-manufacture (Figure 1.14),will be presented. After discussing the principles of functional testing (Section 5.2) anddesign decision templates (Section 5.3), the usage of the template for Rapid Tooling ispresented in Section 5.4. Three case studies, a tensile bar, a rib part and a ring gear, areused to test the presented scenario (Section 5.5). They also provide partial validation ofHypothesis 2.1.

EmpiricalStructuralValidation

H1

H1.1

H1.2

H1.3

Develop the RTMDS basedon the MPMDM. The

system can be used togenerate mold design for

several test parts andindustrial parts.


Validation

Mold designs of thecase stuides aregenerated by the

RTMDS in acceptabletime, and they can be

used to produceprototpyes in the Rapid

Tooling process.

Develop the regiongeneration module basedon the basic elements of

concave regions andconvex faces. The

module can generate thebasic elements of molddesign for several test

parts and industrial parts.

Basic elements of thecase stuides aregenerated by the

module in acceptabletime, and they can be

used in the regioncombining process of

mold configurationdesign.

Develop the regioncombining module

based on the concaveregions and convex

faces. The module cangenerate mold

configuration design forseveral test parts and

industrial parts.

Mold configurationsof the case stuides

are generated by themodule in acceptabletime, and they can beused in constructingmold pieces for the

parts.

Develop the mold piececonstructing module forthe combined regionsand a mold base. Themodule can generate

mold pieces for severaltest parts and industrial

parts.

Mold pieces of thecase stuides aregenerated by the

module in acceptabletime, and they can be

used in the RapidTooling process to

produce prototypes.

Figure 4.33 – Empirical Structural and Performance Validation for Hypothesis 1.


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R1

R2

Rk

R3

F1

F2

F3

F4F5

F6

Fn

PL1

PL2

PD1

PD2

Part P

F1

F2

F3

F4

F5

F7

F8

Fn

Part P

(1) (2)

(3)

F1

F2

F3

F4

F5

F7

Fn

Part P

F8F8 F6

F6

F7

F9 F9

F9 M1

M2 Mk

Mold Base

PD1

F1

F2

F3

PD2

F4F5

F7 Fn



GivenAnalternative tobe improvedthroughmodification;Assumptions usedtomodel thedomainof interest.

Thesystemparameters:n number of systemvariables p+q number of systemconstraintsp equalityconstraints q inequalityconstraintsm number of systemgoals Gi(X) systemconstraint functionfk(di ) functionof deviationvariables tobeminimizedat priority levelkfor thepreemptivecase.


-, di+ i = 1, ... , m


gi(X) = 0; i = 1, ..., p gi(X) ≥ 0; i = p+1, ..., p+qSystemgoals (linear, nonlinear)

Ai(X) + di- - di

+= Gi ; i = 1, ..., mBounds

Ximin≤ Xi ≤ Xi

max; i= 1, ..., nDeviationvariables

di-, di

+ ≥ 0; di- . di

+= 0; i = 1, ..., mMinimize

Preemptivedeviationfunction(lexicographicminimum)Z=[f

1(di-,di

+ ),..., fk(di

-, di+)]


i(d

i− +d

i+) where W

i=1, W

i≥0∑∑

§2.2 §2.3 §2.4 §2.5 §2.6

MoldConfiguration

Design Methods


Tools

CADRepresentation


Techniques



Chapter 5 – Formulating Design Requirements for Rapid Tooling as Geometric Tailoring Problem

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CHAPTER 5

FORMULATING DESIGN REQUIREMENTS FOR RAPIDTOOLING AS GEOMETRIC TAILORING PROBLEM

In the current usage of Rapid Tooling, the iterations of design changes between thedesigner and manufacturer may take a long time before production-representativeprototypes are produced. In this chapter, a geometric tailoring approach, materialgeometric tailoring, is presented to reduce the iterations due to the material propertydifferences between the products and prototypes. First the properties of Rapid Tooling,especially the part properties of the AIM tooling, are introduced to provide a context ofmaterial geometric tailoring (Section 5.1). Related to the principle of functional testingand similarity methods, the fundamentals of geometric tailoring are presented in Section5.2. The material geometric tailoring decision template is introduced in Section 5.3,which enables a “clean digital interface” between design and fabrication, effectivelyseparating design activities from manufacture activities. The usage of the MGT decisiontemplate, including formulating function properties and solving approaches, is presentedin Section 5.4. Finally three test examples are discussed in Section 5.5 to demonstrate ascenario of design-manufacture collaboration with the MGT decision template.


GivenAnalternative tobe improvedthroughmodification;Assumptions usedtomodelthedomainof interest.

Thesystemparameters:n number of systemvariables p+q number of systemconstraintsp equalityconstraints q inequalityconstraintsm number of systemgoals Gi(X) systemconstraint function

fk(di ) functionof deviationvariables tobeminimizedatprioritylevelkfor thepreemptivecase.


-, di+ i = 1, ... , m


gi(X) = 0; i = 1, ..., p gi(X) ≥ 0; i = p+1, ...,p+qSystemgoals (linear, nonlinear)

Ai(X) + di- - di

+= Gi ; i = 1, ..., mBounds

Ximin≤ Xi ≤ Xi


di-, di

+ ≥ 0; di-. di

+= 0; i = 1, ..., mMinimize

Preemptivedeviationfunction(lexicographicminimum)Z=[f

1(d

i-,d

i+),..., f

k(d

i-,d

i+)]


i(d

i

− +di

+) where Wi=1, W

i≥0∑∑


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5.1 PROPERTIES OF RAPID TOOLING

As discussed in Section 1.2, the two problems in the current usage of Rapid Tooling(RT) that are addressed in this dissertation are:

(1) The mold design step in the process of using RT may take a long time for partswith a wide variety of geometries;

(2) The iterations of design changes may take a long time for parts with a widevariety of design requirements.

In Chapter 3 and 4, the Multi-piece Mold Design method (MPMDM) and the relatedRapid Tooling Mold Design system (RTMDS) were presented, which addressed the firstproblem. In this chapter and the next chapter, approaches that address the secondproblem will be presented. The properties of Rapid Tooling are discussed first in thissection to provide a context for Chapter 5 and 6. They also foster a better understandingof Geometric Tailoring which is to be presented in the remainder of this chapter.

In the RTTB project (Section 1.2.2), the direct AIM tooling is the focus of theresearch on design for RT (Section 1.1.3). Therefore the properties of direct AIMTooling and its differences from the steel tooling are discussed in this section. In otherliterature like (Karapatis, et al., 1998) and (Barlow, et al., 1996), the properties of otherRT processes are also discussed.

Compared with the conventional steel tooling, the direct AIM tooling has threeunique properties: (1) the mold material properties, (2) the fabrication method to get thetools, and (3) the part material properties. They are discussed in detail as follows.

5.1.1 Mold Material Properties

In the Direct AIM tooling, tools are made of epoxy-based Stereolithographyphotopolymers. Currently a series of materials are available for SLA machines (e.g. SL7540, SL 7510, SL 5530, SL 5520, SL 5510, SL 5190, refer to www.3dsystems.com).Each material has slightly different thermal and mechanical properties. However, ifcompared with the properties of steel, they are pretty similar. Some of the chief physicalproperties of the stereolithographic materials that may have an influence in the injectionmolding process are:

! Compressive strength! Tensile Strength! Flexural Strength! Shear Strength! Impact Strength! Wear Resistance! Surface Hardness! Coefficient of thermal expansion! Thermal Conductivity! Specific Heat! Thermal Diffusivity! Heat Deflection Temperature & Glass Transition Temperature * (Unique to

plastic mold. It is a measure of temperature up to which the material can be usedwithout significant degradation in strength).


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! Thermal Fatigue! Creep behavior under load for extended periods of timeNot all the properties are discussed in this section. Instead the author will focus on

the properties of thermal conductivity, tensile strength, and shear strength, and discusstheir effects on the injection molding process as follows.

Stereolithography material has a thermal conductivity of 0.185 W/m-K, while thevalue of steel mold is typically 50 W/m-K (Dawson, 1998). Because this value is 300times lower than the thermal conductivity of steel, longer cycle times must be employedto allow the part to cool before ejection.

The maximum shear and tensile strengths of epoxy resins are much lower than thoseof steel (e.g. the tensile strength of a typical alloy steel is 650 MPa). Furthermore, theshear and tensile strengths of epoxy resins are tightly related to temperature of thematerial. As the temperature increases, both the tensile and shear strengths of thematerial decrease. The relation of strength versus temperature of epoxy SL5170 is givenin Figure 5.1 (Rahmati and Dickens, 1997). As hot plastic which is above 200 oC isinjected into the mold, the mold loses much of its strength. However, due to the lowthermal conductivity of the mold material and the short period of injection, this toolingmethod is able to produce parts. Cooling time requirements must be adjusted to deal withthe reduction in material strength at higher temperatures.

Because of the low shear and tensile strengths of the mold materials, lower injectionpressures should be used to keep the mold from breaking. Also while the stress-straindata are normally used for metal tool design, the creep-rupture data are more suitable forcomposites and plastics (Jayanthi and Hokuf, 1997). Hence the design strength of themold is dependent upon both the magnitude of the applied load and the duration of itsapplication. The strength required must be adequate to resist the compressive, bending

Figure 5.1– Maximum Tensile and Shear Strength of SL5170 Versus Temperature.


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and shearing stresses set up by the molding material under pressure as it moves into themold cavity and hardens.

In summary, the mechanical properties of the mold materials are quite different fromthose of conventional steel molds. The low thermal conductivity and limited materialstrength of the mold need to be considered in the injection molding process of the directAIM tooling. Recently Dawson discussed the rapid mold selection and injection moldingconditions in (Dawson, 2001). It provides more details of the relations between theinjection molding conditions and the mold material properties.

5.1.2 Mold Fabrication Properties

The layer based SL fabrication induces sidewall surface imperfections due to thestair-stepping effects as shown in Figure 5.2.a. This is due to the parabolic profile of thecured region in the photopolymer, as shown in Figure 5.2.b. The effect can act aslocalized undercuts in the cavities which can make ejection of the mold part difficult.

Also the algorithm that generates the build file from the solid model CAD data turnsall surfaces into a series of triangles. This approximation of the CAD model leads toinaccuracies in built geometry, as well as to the above “stair step” roughness on curvedsurfaces. Inaccuracies of building geometry also are caused by part shrinkage as thestereolithographic polymer cures.

The direct AIM tooling has different mold material and fabrication properties fromsteel tooling. Therefore the mold life of direct AIM tooling is much lower than that of thesteel tooling. The actual failure of a feature may occur during injection or ejection in thedirect AIM tooling (Palmer, 1999). In the injection process, failures are caused by afeature’s inability to resist the force of the polymer flow. In the ejection process, failuresare caused when a feature cannot overcome the shrinkage forces of the part uponejection. Therefore the three main types of failure are flow, pullout, and chipping(Palmer, 1999), which are introduced briefly as follows.

(a) Flow failure: When the polymer is injected into the cavity, the pressure, which isa factor of injection pressure, gate dimensions, and shot size, may cause the featureto bend or break.

(a) (b)

Figure 5.2 – Stair Stepping Effect of Layer Manufacturing and Reason.


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(b) Pullout failure: When the parts are ejected, the shrinkage force of the polymeronto the mold causes the frictional force between the mold and the part. Themaximum tensile strength of the mold is reached before the part can be ejected,and the mold features break off.

(c) Chipping failure: The most probable cause of chipping occurs due to a crack inthe material. Chipping failures usually do not occur on the first shot. It is assumedthat the failure is due to a defect or crack within the bulk material of the feature.

The low mold life of the direct AIM tooling will be considered in Chapter 6, whichwill address more general issues of the design for rapid tooling.

5.1.3 Part Properties of the AIM Tooling

Thermoplastic materials that can be used in the direct AIM tooling includepolypropylene, polyethelene, delrin, polystyrene, ABS, polycarbonate, glass filled nylon,glass filled PBT, etc. All of these materials are used in production. Although the partsproduced from the direct AIM tooling can be in the same material as that of theproduction parts, their mechanical behavior and performance may be different (Dawson,1998). The mechanical properties of an injection molded part are affected by themicrostructure of the part. In polymers the microstructures that affect the properties arecrystallinity, molecular orientation, and defects such as voids. Since the use of SL insertsmay lead to longer cycle time, different properties of the injection molded parts isunavoidable. Dawson (1998, 2001) gave a comparison of the mechanical properties ofthe molded materials that were simultaneously injected in a SL mold and a steel mold.Some results taken from his work are given in Table 5.1 and

Table 5.2. The results indicate a variation in the material properties of atacticpolystyrene parts produced from steel and SLA molds. Another result in (Dawson,1998) is that the residual stress of the parts obtained from the Direct AIM tooling issmaller than that of the parts obtained from the steeling tooling.

From the part properties of the AIM tooling, it is noticed that the prototypesproduced by the RT process have properties different from those of the production parts.This may lead to difficulties in the prediction of product behaviors through the functionaltesting of the prototypes, which is discussed in the next section in more details. Toaddress this problem, the principles of functional testing are presented first. Based on the

Table 5.1– Tensile Properties Comparison for Atactic Polystyrene (Dawson, 1998).

Tooling Ultimate Stress(Mpa)

Young’sModulus (Gpa)

UltimateElongation (%)

Density(g/cc)

Steel 37.4 3.2 1.3 1.045Direct AIM 32.8 3.4 1.1 1.042

Table 5.2 – Flexural Properties Comparison for Atactic Polystyrene (Dawson, 1998).

Tooling Ultimate Stress(Mpa)

Young’s Modulus(Gpa)

UltimateElongation (%)

Steel 67.5 3.0 2.6Direct AIM 71.0 4.0 1.9


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principles, a design-for-prototyping approach, geometric tailoring, is introduced in thenext section.

5.2 PRINCIPLES OF FUNCTIONAL TESTING AND GEOMETRIC TAILORING

Although RT has the potential to be used in low-volume manufacturing (Section1.1.4), the parts produced by rapid tooling are still mainly used for functional testing. Todevelop reliable functional tests, systematic methods based on sound principles arenecessary.

5.2.1 Principle of Functional Testing – Buckingham ΠΠΠΠ Theorem

Currently the Buckingham Π theorem is the basis in prototype testing to get thecorrelation between physical models and products (Baker, et al., 1991). The basic idea ofthe theorem is using dimensionless variables in the analysis. A brief description of theBuckingham Π theorem is given as follows, which is adapted from (Cho, et al., 1999).

Consider two systems (a test model and a product) that can be described as

Model: Xm = f (dm,1, dm,2,…, dm,n),

Product: Xp = f (dp,1, dp,2, …, dp,n),

where X is the state of interest, and di is a system parameter. In the equations,subscripts m and p denote the model and the product respectively. From the BuckinghamΠ theorem, the above system equations can be equivalently represented as

Model: πm,x = F (πm,1, πm,2, …, πm,N),

Product: πp,x = F (πp,1, πp,2, …, πp,N),

where πi is a dimensionless parameter and N = n-k. The πi is a power function ofsystem parameter set D = d1, d2, …, dn, and the πx is a power function of systemparameter set D and the state X.

From the non-dimensional equations, πm,x and πp,x are identical if πm,i = πp,i for any i= 1, 2, …, N. As a consequence, one can correlate the model and product states from

πp,x (Dp) = πm,x (Dm), (5.1)

if πp,i (Dp) = πm,i (Dm), ∀ i = 1, 2, …, n, (5.2)

where D denotes the full-set of system parameters.

Equations 5.1 and 5.2 are the fundamental basis of the functional testing. The formeris called the prediction equation, and the latter are conditions for scaled models orsimilarity constraints. One should design prototypes not to violate any of the similarityconstraints, in order to predict the product states through the prediction equation.

A simple example is given as follows to aid the reader to understand the theorem,and the limitations of the similitude method which will be presented in the next section.This example is adapted from web site www.mech.uwa.edu.au/courses/e101/.

Suppose a car travels straight with a velocity U ms-1. The car has a mass M kg. Ifthe driver suddenly applies the brakes so that a constant force F kgms-2 is applied, one


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wants to know how long will the car take to stop. To design a prototype to test thedistance, a brief analysis based on Buckingham Π theorem is given.

Suppose L is the distance the car takes to stop. It is the variable to test.

So one can write f (L, U, M, F) = 0. To get dimensionless variables, one can knowLF

MU 2 1LNM

OQP = from the units of the variables. Therefore π = LF

MU 2 , which is a

dimensionless variable.

Suppose π0 is the solution or root of the equation f (π) = 0, one can know

LMU

F= π0

2

.

Therefore an experiment can be designed with M’, U’ and F’ to test L’, and calculateπ0 accordingly. After substituting M’, U’ and F’ with the given values we are interested,the value of L can be calculated from the equation. One thing to be noticed is that in thetesting, M’, U’ and F’ can be much smaller than M, U, and F.

Based on the Buckingham Π theorem, a complex system can be constructed andtested by geometrically scaling down, or changing materials, or simplifying models. Bydoing that one can dramatically reduce the cost and time in building prototypes whilegetting reasonable predictive values.

5.2.2 Similarity Methods

The similarity method, which is based on the Buckingham Π theorem, is widely usedfor both effective empirical modeling and scale testing (Baker, et al., 1991). It canexperimentally predict the behavior of a target system through an indirect scaled testingprovided the two systems are ‘similar’.

To use the similarity method, one should be able to list all dominant systemparameters, which is shown in the example given in the last section. If any dominantsystem parameters are not considered, the system states cannot be well represented, andthe correlation between the systems may become erroneous. Also the parameter valuesof the model and product should be known beforehand. An additional assumption madein the method is that the unknown governing equations f of the model and the productshould be identical. However, the assumption may not be satisfied especially forprototypes that are built from different materials by various rapid prototyping techniques.This is also the main reason why rapid prototyping is mostly utilized in early designphases for visualization, while rapid tooling is utilized in later design phases forfunctional testing.

Motivated by using RP techniques in functional testing, which presents significantbenefits through the material changes, a group of researchers at University of Texas atAustin presented an empirical similarity method (ESM) to improve the accuracy of scaletesting (Cho, et al., 1998; Cho, et al., 1998; Cho, et al., 1999; Cho, et al., 1999). ESMutilizes specimens to derive the similarity transformation, which correlates model andproduct states, empirically. As shown in Figure 5.3, the model and the productspecimens are the geometrically simplified versions of the scaled model and the product.


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In the ESM, the product state is predicted through the specimen pair (the model and theproduct specimens) and the scaled model.

The ESM is proposed for the scale testing with RP. In the case studies given in thepapers, some tested prototypes had widely different material and geometry scale fromthose of the production parts. However, ESM requires additional effort to fabricate aspecimen pair, although the additional effort is much smaller than that required tofabricate geometrically complex test products.

Although guided by the similarity method, the reliability of scale testing results hasbeen challenged frequently (Baker, et al., 1991). This may be because some of the modelparameters or loading conditions are difficult to control, or there are some uncertainnoises that affect the accuracy. Considering the properties of rapid tooling, a design-for-prototyping approach, geometric tailoring, is proposed. It is also based on theBuckingham Π theorem. Its fundamentals are presented in the next section.

5.2.3 Fundamentals of Geometric Tailoring

A mechanical designer may describe a new product with many attributes, which mayinclude the shape of the component, material, tolerance information, cost and time. Someof the attributes are related to the design functions, such as geometric variables andmaterial properties. Some of the attributes are related to the fabrication processconditions such as cost and time. In this chapter, the author only considers the attributesthat are related to the design functions. Other attributes will be considered in the nextchapter.

Figure 5.3 – Fundamental Terms and Concept of the ESM (Cho, et al., 1999).


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With the representations given in Section 5.2.1, suppose two parts (a test model anda product) that are described as

Xm = f (dm,1, dm,2,…, dm,n),

Xp = f (dp,1, dp,2, …, dp,n).

Where X is the state of interest, and di is a system parameter. In the equations,subscripts m and p denote the model and the product respectively. Therefore twoobservations on X and di are the basis of this research.

Observation 5.1. In order to get accurate prediction results of XP, it is generally desiredto control the system parameters dm,i and dp,i identically.

This is because in functional testing, the representation of the function f is usuallyunknown. Therefore there may exist some unknown system parameters, or uncertainnoises, or uncontrollable loading conditions. All these factors will affect the accuracyof scale testing results. Therefore it is desired to make dm,i = dp,i.

Observation 5.2. System parameters di (1≤ i≤ n) are not equally important to X.

For an attribute Xj, some parameters are rather important. Other parameters may haveonly negligible effects. For example, to test the fatigue of a gear design, themaximum stress of the gear under the designed constraints and loads is important.However, the small changes of the gear dimension may have only negligible effectson the fatigue.

Therefore the system parameters can be divided into two categories based on theirrelations with X. Suppose parameters d1, d2, …, dk are important to X, while parametersdk+1, dk+2, …, dn are negligible to X. Therefore the principle of the approachdeveloped in this research is to change parameters dm,j (k+1≤≤≤≤ j ≤≤≤≤ n) to make dm,i asclose to dp,i (1≤≤≤≤ i≤≤≤≤ k) as possible. And these parameter changes are named GeometricTailoring in this dissertation.

A formal definition of geometric tailoring given in Section 1.2.4 is revisited asfollow.

Geometric tailoring, in this dissertation, is to change the geometry of a part to lowerfabrication cost and time, and to produce functional prototypes to mimic theproduction functions, because the material and fabrication process to get molds andparts are different from those in producing products.

The manufacturability of a given design depends on the following three factors: (1)the ability to produce the design within the specifications; (2) the ability to produce thedesign with a low production cost; (3) the ability to produce the design within a shortproduction time. Related to these three factors, two kinds of geometric tailoring areconsidered in this dissertation. Material Geometric Tailoring (MGT), which is thegeometric tailoring to mimic the production functions related to material properties, ispresented in this chapter. Material Process Geometric Tailoring (MPGT), which is thegeometric tailoring to lower the fabrication cost and to mimic the production functionsincluding accuracy and surface finish, is discussed in Chapter 6. MGT is actually aspecial case of MPGT. However, because the cost and some functions such as accuracy,


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which are related to the fabrication process, are also considered in MPGT, the processplanning should also be included in the solving process of MPGT. Therefore its solvingstrategy is much more complicated (Section 6.4).

Since a part design may have a wide variety of design requirements for its prototypes(Section 1.2.3), both MGT and MPGT have their applications in actual situations. Forexample, for a part design, if the functional properties are much more important than thecost and time, MGT should be used. However if the fabrication cost and time are alsoconcerns for the prototypes, MPGT should be used instead. Correspondingly processrequirements such as mold life are added in the problem formulation of MPGT (Section6.2).

Although the geometric tailoring approaches in this dissertation are developed forrapid tooling, they also have implications in design for rapid prototyping. For example,SOMOS 8120 and SOMOS 9120 are two materials used in Stereolithography Apparatus(SLA). They have properties close to polyethylene and polypropylene respectively. Thegeometric tailoring of prototypes may be necessary in order to get better results by testingthe prototypes that are built with SOMOS 8120 and SOMOS 9120 for polyethylene andpolypropylene parts (Sambu, 2001).

Parts produced from the direct AIM tooling have slightly different mechanicalproperties from those of the production parts produced with steel tools (Section 5.1.3).Although these differences are much smaller than those between the parts built by rapidprototyping and the production parts, the accuracy of the functional tests will be affected.Lots of design functions such as the maximum stress, fatigue, deflection of a rib, arerelated to the material properties. A systematic approach that is developed for handlingthe differences of material properties in using Rapid Tooling is presented in the nextsection.

5.3 DESIGN DECISION TEMPLATE FOR MGT

Based on the Buckingham Π theorem, it is important for a prototype to behavesimilarly to a product in order to verify the design with more confidence. However sincethe relationship between the attributes and parameters is usually complex, it may not bestraightforward to choose some parameters and tailor them to get the same values of theattributes. Also a designer may have several functions of interest in a prototype.Therefore several attributes of the prototype need to be tailored for the same values asthose of the product. Usually the changing of a system parameter for an attribute mayaffect some other attributes. More importantly, not all system parameters are in thecontrol of the designer. For example, the material properties of the prototype are actuallydetermined by the manufacturing process, and therefore are in the control of themanufacturer. Considered all these factors, a scenario of design-manufacturecollaboration based on the MGT decision template is proposed for the material geometrictailoring. The steps in the method are illustrated in Figure 5.4.

As stated in Section 1.2.1, the designer and manufacturer are usually distributedgeographically and organizationally in the current usage of Rapid Tooling. The RTTBproject is trying to develop a “clean digital interface” to divide the designer andmanufacturer of RP and RT (Allen and Rosen, 1997; Rosen, 2000). The DFMapproaches presented in this dissertation are actually only one scenario of separating the


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designer and manufacturer in a distributed computing environment. However the focusof this work is how to formulate and solve the DFM problem provided the scenario hadbeen chosen (e.g. material and process had been selected by other modules).

There are eleven steps to the scenario based on the MGT decision template asillustrated in Figure 5.4. The input to the problem is the part design to be tested. Basedon it, the designer first needs to determine which and how many functions are to be testedin a prototype. Sometime several functions can be integrated into one prototype in orderto reduce cost. However this may bring difficulties in producing production-

Part design to be tested

Designer Manufacturer

Step 1Determine properties of interest.

Step 2Determine loading conditions, attributes,and system parameters that are relevant

to the properties of interest.

Step 3Determine the importance of the attributes

and their relations with systemparameters.

Step 4Formulate attributes, system parameters

and their requirements in MGT designtemplate.

Step 6Determine attributes affected by the

material differences.

Step 7Formulate additional relations in the MGTtemplate, hence complete deisgn decision

formulation for MGT.

Step 8Solve the formulated MGT problem.

Step 9Produce prototypes for tailored part

design.

Step 5Send MGT design template, CAD model.and/or FEA model to the manufacturer.

Chosen Materialand Process

CAD Model,MGT template

Step 10Send prototypes to the designer.

Step 11Execute the functional testing of the

prototypes.

Figure 5.4 – Steps of a Scenario based on the MGT Decision Template.


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representative prototypes for all the attributes.

According to the properties of interest (e.g. fatigue), the designer can determinedesign attributes and system parameters that are related to the properties (e.g. maximumstress, refer to Section 5.5 for more details). Also the loading conditions in the tests are tobe determined. Generally the loads and boundary conditions should be the same as thoseof the production design. They may have already been formulated in a Finite ElementAnalysis (FEA) model in the part design process. Otherwise in Step 7 the manufacturermay build response surface models according to analysis based on the loads andboundary conditions (Section 5.4).

Step 3 is to determine the importance of the attributes and related system parameters.Some attributes and parameters are important for the prototypes. For example, theattributes that are related to the properties of interest should be production-representative.Some geometry parameters that are related to assembly are also important. They shouldalso remain unchanged in the prototypes. Other geometry and material parameters aremuch less important for the purpose of the functional testing. Therefore they providemore design freedoms for the geometric tailoring.

Currently Steps 1 to 3 mainly depend on the knowledge and experience of thedesigner. Tools to help the designer in these steps are important. However they arebeyond the scope of this dissertation.

Based on the MGT decision template, which is to be presented in this section, thedesigner can formulate the requirements of the attributes and parameters in a compromiseDSP format (Step 4). Then the formulated decision template and CAD model can be sentto the manufacturer (Step 5). If a FEA model is constructed in the part design process forthe production material, it can also be sent to the manufacturer and used in the analysis ofthe prototype with its material properties. The rapid development of the Internet providesa distributed computing environment for sending these models. The formats andapproaches developed in the RTTB for this step are presented in the theses of (Gerhard,2001) and other on-going students.

The approaches of selecting the material and process to produce the prototypes in theRTTB are presented in (Herrmann and Allen, 1999). For a chosen fabrication materialand process, the manufacturer first needs to determine which attributes and parametersare needed in the geometric tailoring (Step 6). Based on the decision template and CADmodel, additional relations or missing values are added to the template to formulate acomplete decision problem (Step 7). Then the formulated compromise DSP can besolved (Step 8). Accordingly the manufacturer can change the part design and produceprototypes for it (Step 9). After the prototypes are produced, they are sent back to thedesigner for the functional tests (Step 11).

Of these steps, the author will focus on Steps 4, 7, and 8, which are enclosed withthicker lines in Figure 5.4. The MGT decision template, which is discussed in thereminder of this section, is related to Step 4 and 7. Its methodology and formulation arepresented in Section 5.3.1 and 5.3.2 respectively. The approaches to generate therelations in the MGT template and to solve the MGT problem are presented in Section5.4. They are related to Step 7 and 8 respectively.


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5.3.1 MGT Decision Template and its Methodology

For the design-manufacture collaboration problem (transfer design information fromthe designer to the manufacturer), a “Solid Interchange Format” (SIF) has been proposedas a “clean digital interface” between design and manufacturing in the area of RP (Sequinand McMains, 1995). SIF contains part geometry and possibly surface finish, tolerance,material, and perhaps other related information types. Although this information is asignificant extension beyond STL, IGES, or even STEP file formats, it is still insufficientto support design for functional prototyping (Section 5.2.3).

In this research, a design decision template is proposed as an effective digitalinterface. It incorporates a series of compromise decisions. Using the template, enoughdesign information is transferred to enable the manufacturer to explore alternative partdesigns in the event that the functional testing requirements cannot be achieved.Effectively, this transfers the burden of design for manufacturing to the manufacturerwith more design freedom. Therefore the time of iteration between the designer andmanufacturer may be reduced.

The methodology of using the decision template as the digital interface between thedesigner and manufacturer is further illustrated in Figure 5.5. Suppose the designfunctions that the designer plans to test in the prototype are Fi. The design variables ofthe part design that are related to Fi are Di. As stated before, only the design functionsthat are related to the material properties are considered in the MGT. So supposematerial properties that are related to Fi are Mi. The relations between Fi, Di, ard Mi canbe formulated as Fi = f(Di, Mi). This relation f can be rather simple and may be given as asimple equation (Section 5.4.1). It can also be complicated and unknown to the designer.In this case, the response surface model is employed in this dissertation to formulate theirrelation (Section 5.4.2). One thing to be noticed is that in Rapid Tooling, the material

DesignFunctions

Fi

DesignVariables

Di

MaterialProperties

Mi

ProcessVariables

Pi

Designer

Manufacturer

Fi = f (Di, Mi)

FunctionSpace

DesignSolutionSpace

MaterialProperties

Space

DesiredRange

DesiredRange

SolutionPoint

DesiredRange

DesiredPoint

Figure 5.5 – The Decision Template for the MGT and Design Freedom.


175

properties of the prototypes are determined by the fabrication process variables Pi.Therefore Mi is actually controlled by the manufacturer, while Di is controlled by thedesigner.

In the current usage of Rapid Tooling, a STL file or a SIF file is transferred from thedesigner to the manufacturer. In the file, the designer has already set the values of thedesign variables and functions (by the requirements on material properties). For aselected fabrication process and material, the only design freedom for the manufacturer isthe process parameters, which are adjusted to achieve the desired material properties.However, the relation of Mi and Pi may be complicated especially for Rapid Toolingwhich composes several steps. It is rather difficult, sometimes even impossible toachieve the desired material properties (Section 5.1.3). Therefore the design is sent backto the designer and a new iteration begins. In most cases, tradeoffs between differentfunctions have to be made if the desired values cannot be achieved for all Fi. To reachsuch a “satisfying” solution for the designer and manufacturer, several iterations and lotsof negotiations may be necessary. This process may take a long time for somecomplicated part designs.

By using the decision template, the author believes that the designer andmanufacturer can make tradeoffs more quickly to reach the “satisfying” solution. This isbecause the designer has formulated the design requirements on the prototype in thedecision template. Based on the template, the manufacturer will have more designfreedom in producing prototypes. Besides the selection of the process parameters, theadditional design freedom may come from the following two sides.

(1) Design variables of the part design are given in desired ranges instead of specifiedpoints.

As discussed in Section 5.2.3, the principle of geometric tailoring is to changesome unimportant parameters to make the more important variables moreproduction-representative. Therefore based on the decision template, themanufacturer has the freedom to change the unimportant parameters in thespecified ranges.

(2) Design functions of the part design are given in desired ranges instead ofspecified points.

The decision template is based on the feasible and satisficing regions of thedesign space. The term satisficing was coined by (Simon, 1996) to describe aparticular form of less-than optimal solutions (Section 2.4). Instead of sending anoptimal solution based on the design information, the designer can send thedecision template to the manufacturer for a satisficing solution. Therefore themanufacturer will have more design freedom in producing prototypes.

The additional design freedom enables the manufacturer to slightly change thematerial properties of the prototypes. That is, if the material properties Mi are changed toMi’, the manufacturer can adjust Pi to Pi’ to achieve the same function Fi. In many cases,the manufacturer may feel much more confident in producing prototypes with propertiesMi’ instead of Mi (Section 5.1.3). Therefore it is more likely that the required prototypescan be produced without being sent back to the designer for a redesign. Three examples


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and two case studies are given in Section 5.5, and Chapter 7 and 8 respectively. Theyillustrate the applications of the decision template and its advantages.

Related to the MGT and MPGT, two kinds of design decision templates areintroduced in this dissertation. They have the same methodology. The decision templatesare:

• MGTDT – Material Geometric Tailoring Decision Template, a compromisedecision in which the component’s dimensions are modified to provide similarfunctional performance when a prototype material replaces the productionmaterial.

• MPGTDT – Material-Process Geometric Tailoring Decision Template, acompromise decision in which the component’s dimensions are modified to suit aprototype material and fabrication process.

They embody the relevant design information and designer preferences. Theformulation of the MGTDT is presented in the next section. MPGT and MPGTDT willbe discussed in Chapter 6.

5.3.2 Formulation of the MGTDT

The compromise Decision Support Problem (DSP) is a general framework forsolving multi-objective, non-linear optimization problems (Section 2.4). In this researchit was employed to formulate the MGTDT. The compromise DSP is central to modelingmultiple design objectives and assessing the tradeoffs pertinent to the design for RapidTooling. The compromise DSP word formulation of the MGTDT is presented in Table5.3.

Table 5.3 – Word Formulation of the MGTDT.

GIVEN:

• Parametric CAD model of part • Material properties • Functional property models • Goal preferences as weights • Target values for functional properties

FIND:

System Variables: Deviation Variables:

Geometry variables Deviation of goals from targets

SATISFY:

Goals: Constraints:

Meet target functional properties Meet geometry and/or assembly constraints

Meet targets of geometry variables Bounds:

Bounds for all system variables

MINIMIZE:

Deviation Function: Weighted sum of goal deviations


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Mathematically, the compromise DSP is a multi-objective decision model based onMathematical Programming and Goal Programming. A general mathematicalformulation of MGTDT is presented in Table 5.4, which is an extension of the wordformulation given in Table 5.3.

Table 5.4 – Mathematical Formulation of the MGTDT.

GIVEN:

• nf - number of functions;ng - number of geometry variables; nm - number of material variablesp – equality constraints; q – inequality constraints.• Geometry variables Gp, i i = 1, …, ng

• Material properties Mp, i i = 1, …, nm

• Design functions Fp, i = fi (Gp,j, Mp,n) i = 1, …, nf; j = 1, …, ng; n = 1, …, nm

• Material properties Mm, i** i = 1, …, nm

• Design functions Fm, i = fi (Gm,j, Mm,n)** i = 1, …, nf; j = 1, …, ng; n = 1, …, nm

• Weight Wi i = 1, …, ng+nf

FIND:

System Variables:

Gm, i i = 1, …, ng

Deviation Variables:

di+, di

- i = 1, …, ng+nf

SATISFY:

Goals:

Fp, i / Fm,i - di+ + di

- = 1 i = 1, …, nf [5.3]

Gp,i / Gm,i - dnf+i+ + dnf+i

- = 1 i = 1, …, ng [5.4]Constraints:

di+ • di

- = 0, di+ ≥ 0, di

- ≥ 0 i = 1, …, ng+nf

gj(Gi) = 0, gk(Gi) ≤ 0 i = 1, …, ng; j=1, …, p; k=1, …, q Bounds:

Gimin ≤ Gi ≤ Gi

max i = 1, …, ng

Fimin ≤ Fi ≤ Fi

max i = 1, …, nf

MINIMIZE:

Deviation Function: Wi • (di+ + di

-) where ΣWi = 1, Wi ≥ 0 (i= 1, …, ng+nf)

Note:1. Subscripts m and p denote the prototype model and the product respectively.2. Symbol ‘**’ denotes the entries that are to be completed by the manufacturer.


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The math formulation contains systems variables, goals and constraints. The systemvariables in the MGTDT are the geometry variables (Gj) and material properties (Mn).The geometry variables typically include modifying dimensions (e.g. lengths and angles)of features in a part. In some cases geometry variables could be a whole feature. In suchcases the manufacturer has freedom to redesign the feature, e.g. use a rib instead of bossto fulfill the same function. Material properties Mi include Young’s modulus, tensilestrength, and elongation yield, etc.

The goals in the MGTDT include functional and geometry goals. The functionalgoals (Equation 5.3) include matching the functional properties of prototype parts to theirtarget values. Target values correspond to functional properties of the production part.These functional properties Fi can be stress, deflection, weight, or some others. Thegeometry goals (Equation 5.4) include matching the geometry variables of the tailoredpart to their target values. Target geometry values are the most desirable values of partdimensions provided by the designer.

The constraints in the MGTDT include geometry constraints that arise due to spaceand weight limitations. While performing geometric tailoring of an assembly, theassembly requirements are also considered in the constraints.

One thing to be noticed is that in the formulation, the designer does not have all theinformation (the material properties of the prototypes are out of his/her control). Theentries indicated by ‘**’ in Table 5.4 denote information that the manufacturer mustsupply in order to complete the problem formulation and generate a solution. However,based on his/her information, the designer can instantiate the MGTDT to create theproblem formulation suitable for communicating to the manufacturer.

In light of the methodology and formulation of the MGTDT presented in this section,the usage of MGTDT is discussed in the next section, especially the approaches toformulate the relation f in the template.

5.4 USAGE OF MGTDT

In the MGT decision template, the designer provides target values for geometryvariables and design functions, and also their preferences. The Manufacturer providesmaterial properties of prototype parts and related quantitative relations between goals andsystem variables. Therefore the information required to solve the material geometrictailoring problem comes from the design and manufacturing organizations. Essentiallythe MGTDT is actually a method to organize the design and manufacturing informationinto one formulation. After the MGT problem is formulated, it can be solved with the aidof engineering optimization software (e.g. OptdesX). The approaches to formulate therequired quantitative models for functional properties are presented in Section 5.4.1. Theprocess of solving the MGT problem is discussed in Section 5.4.2.

5.4.1 Formulating Functional Properties in the MGTDT

The functional properties Fi in Table 5.4 are related to the geometry variables (Gj)and material properties (Mn). Suppose the relationship can be stated as Fi = fi(Gj, Mn).Here fi is the function of Fi for a set of design variables Gj and Mn.


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For simple geometry, it is possible to obtain an analytical equation for the designfunctions such as stress or weight. But for complicated geometry it is usually notpossible. In such circumstances, design functions can be represented as a responsesurface generated by fitting a surface through experimental data points. Both approachesare described in more details as follows. They correspond to the two types of“quantitative” models in common use: Analytical models and Simulation models.

(1) Functions Represented as Analytic EquationsIn numerous handbooks of mechanical design, different design functions are usually

represented as analytic equations of geometric variables and material properties. Forexample, a cantilever beam is shown in Figure 5.6. If a force ‘F’ is applied at one end ofthe beam, the equations of the maximum stress (σ), maximum strain (ε) and maximumdisplacement (δ) in the beam can be calculated by Equations 5.5 ~ 5.7 respectively(Shigley and Mischke, 1989).

σ = FLc

I(5.5)

ε = FLc

YM I*(5.6)

δ = FL

I YM

3

3 *(5.7)

Therefore for a cantilever beam, if the design functions of interest in the MGTDT arethe maximum stress, or maximum strain, or maximum displacement, the functions fi maybe represented as analytic equations directly. In the equations, L, c and I are geometryvariables (refer to Figure 5.6, I is the moment of inertial of the cross-section area about zaxis). YM is material property, which represent Young’s modulus.

Depending on loading conditions, the same design functions may be represented indifferent equations. For example, if a displacement δ, instead of a force ‘F’, is applied inat one end of the cantilever beam, the maximum stress, maximum strain and forcerequired to cause this displacement are calculated by Equations 5.8 ~ 5.10 respectively.

σ δ= 32

c YM

L

*[5.8]

Figure 5.6 - A Cantilever Beam.


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ε δ= 32

c

L[5.9]

FI YM

L= 3

3

δ *[5.10]

Therefore in the MGT problem the loads and boundary conditions for testing the designfunctions are also very important. The designer should formulate them in functions fi inthe MGTDT, or in a FEA model that is to be transferred to the manufacturer. Based onthe loading conditions, the manufacturer can tailor the part and produce production-representative prototypes more quickly.

For more complicated geometry, the analytical equations are usually not accurateenough for the MGT problem. Therefore, the response surface models are employed torepresent function fi in this dissertation.

(2) Functions Represented as Response Surface EquationsResponse surface methodology, or RSM, is a collection of mathematical and

statistical techniques that are useful for the modeling and analysis of problems in which aresponse of interest is influenced by several variables and the objective is to optimize thisresponse (Section 2.5). The result response surface equations (RSEs) allow for a betterunderstanding of the relationships between the inputs and the response. In the examplesgiven in Section 5.5.2 and 5.5.3, second order response surfaces are used (k = 2). So theRSE can be written in the form of a polynomial function as:

y = b0 + bixii =1

k

∑ + biixi2 + bi xix j

j =1, i≠ j

k

∑i=1

k

∑i =1

k

∑ [5.11]

In much of this work, response surfaces are used to replace computationallycomplex, but high fidelity analyses for usage during design synthesis. Response surfacesapproximate the actual design space and are based on the high fidelity analyses, butenable much faster syntheses. After synthesis, a check is performed to ensure that theperformance indicated by the synthesis result is not too far off. This type of synthesis isparticularly useful during concept exploration (Chen, 1995). The method is called theRobust Concept Exploration Method (RCEM) (Section 2.5).

In order to create the response surfaces for the system goals, a number ofexperiments must be run to gather data for an empirical model. When the system goalsare dependent on two or more factors (system variables), Design of Experiments (DOE)techniques can be practiced to determine the experiment sequence for the empiricalmodel (Montgomery, 1991). Factorial experiment designs involve testing a number ofvariables, or factors, at different values, or levels. In the examples given in this chapter,the experiments used to construct response surfaces (model building experiments) werefractional factorial experiment designs with a face centered central composite design.Three levels of each factor were considered. Other techniques for constructing responsesurface equations can be found in (Box and Draper, 1987; Khuri and Cornell, 1987;Myers and Montgomery, 1995).

Based on the experiment design, the experiments can be performed either on analysispackages or through physical experimentation. In the examples given in Section 5.5.2


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and 5.5.3, ANSYS (www.ansys.com), a finite element analysis (FEA) software system, isused to get the responses of maximum stress and maximum deflection. After theexperiments, MINITAB (www.minitab.com), a statistical software system, is used to fitresponse surface equations through the experimental data. These response surfaceequations are incorporated into the MGTDT for the MGT problem.

5.4.2 Formulating and Solving the MGT Problem

The mathematical formulation of the MGTDT presented in Table 5.4 is a generalformulation that can be used for a specific case. For a part design, the designer firstidentifies all the functional requirements of the prototype. Once the functionalrequirements are identified, the geometry variables that have some design freedom andthat affect the functional requirements should be identified. Appropriate bounds shouldbe chosen for the geometry variables. The other requirements such as space or weightconstraints should also be identified. Once all the problem specific information isidentified, the quantitative relationship between goals and system variables should begenerated. For simple factors like weight, analytical equations can be developed. Forcomplicated factors like stress, experimental approach should be used. Design ofexperiments can be used to identify the list of experiments required to generate aresponse surface model. The experiments could be performed on analysis packages suchas ANSYS. After all the design information is filled in the MGTDT, the designer cansend it to the manufacturer with other information such as a CAD model and FEA model.Based on the template, CAD model, and FEA model, the manufacturer can identify theproblem specific information and generate quantitative models in the same way. Threeexamples are given in Section 5.5 to demonstrate the above process to formulate theMGT problem based on the MGTDT.

After completing the formulation of the MGT problem, the manufacturer can solve itwith the aid of computer. Since the MGT problem is formulated in compromise DSPformat, it can be solved using the ALP algorithm (Mistree, et al., 1993), which is part ofthe DSIDES (Decision Support in Designing Engineering Systems) software. Theexamples given in Section 5.5.2 and 5.5.3 are comparatively simple, which are used toverify the idea of material geometric tailoring. Since the efficiency of the solving processis not the main concern in these examples, the author employed the exhaustive searchmethod to solve the compromise DSP. In the two case studies to be presented in Chapter7 and 8, OptdesX (www.et.byu.edu/~optdes), an engineering optimization software system,was employed in the solving process.

With the MGT and MGTDT presented, three examples are given in the next sectionwhich verify the idea of the material geometric tailoring for Rapid Tooling.

5.5 INITIAL CASE STUDIES

In this section, the author went through three case studies to verify the correctness ofthe material geometric tailoring (MGT). They also illustrate the application of thematerial geometric tailoring decision template (MGTDT). In Section 5.1, a tensile barexample is presented which is used to familiarize the reader with the basic idea of theMGT and the formulation of the MGTDT. It is the simplest example, in the author’sopinion, that can illustrate the MGT problem. In Section 5.2 a rib part problem is


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considered for both design requirements (maximum deflection) and manufacturingrequirements (mold life). Tradeoffs are made by solving a compromise DSP based on theMGTDT. In Section 5.3, a ring gear part, which is taken from a cordless drill, is studiedin order to get functional prototypes using the AIM tooling.

The examples were planned to test different aspects of the MGT problem (Table5.5). In the tensile bar problem a simple equation is derived, while simulation software(ANSYS) is used to study the design requirements with relation to geometry variables inthe rib and gear problems. The number of design functions considered in the problems oftensile bar and rib is only one. However two design functions are considered in the ringgear example. In the rib problem, the author also added the mold life consideration informulating and solving the MGT problem. It sets the stage for the research to bepresented in Chapter 6, which will consider more manufacturing requirements of theRapid Tooling process.

5.5.1 Building Prototypes of a Tensile Bar

In this section the author uses a simple example to illustrate the principle of the MGTand MGTDT.

• Problem IntroductionSuppose the designer finished a product design, within which a tensile bar is used to

hold some other components to the housing of the product (Figure 5.7). In the designprocess, the designer was not sure about the weight of these components (F). So he/shehad to estimate a value for F and continued the design of the tensile bar. The designequation for the tensile bar is pretty simple (Shigley and Mischke, 1989). From

σ =•F

h t, the designer can know F h t= • •σ , where σ is the stress of the part under the

load. It should be smaller than the ultimate yield strength of the material, Su.

The value of Su is related to the material and fabrication process for the part design.Although different scenarios for the selection of material and fabrication process arefeasible and explored in the RTTB project, the author assumes that the designer willselect them in this example. Suppose atactic polystyrene and H13 steel tools are selected

Table 5.5 – Experimental Plan For Testing the MGT.

Exp 1: Tensile Bar Exp 2: Rib Part Exp 3: Ring GearFunction Representation

Analytic EquationsResponse Surface Equations

XX X

Number of Design FunctionsOneMany

X XX

Design FunctionsForceDeflectionStress

XX

XOther Manufacturing Requirements

YesNo X

XX


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for the production parts. By referring to design handbooks, the designer can get the valueof Su. Suppose Su = 37.4 MPa (Dawson, 1998).

Accordingly the designer can finish the tensile bar design by using a safe factor

γ σ= max

Su

. Suppose the values of the variables are h = h0 and t = t0. After finishing the

initial design, the designer hopes to do functional testing to verify that the design works.In industries prototypes are also used as the milestones for different design phases.

Instead of waiting for production injection mold tooling for the tensile bar, thedesigner decides to fabricate a prototype tensile bar using the AIM tooling with the samematerial atactic polystyrene. Since the designer is unsure about the maximum load Fwhich may vary widely in the running of the product, the designer would like theprototype tensile bar to have the same safe factor γ as that of the production tensile bar.Therefore problems (e.g. the value of the maximum load is assumed too small in thedesign) can be identified in the functional testing of the product.

Base on the equations of σ and γ, one can get

γ =• •F

S h tu

max . [5.12]

Therefore the functional property of interest in this example is the maximum forceFmax. Since Su of the prototypes is unknown for the designer, he/she can formulate theabove design information in a MGTDT and let the manufacturer to take care how toproduce qualified prototype tensile bars.

• Designer ActivitiesTo begin, the designer must define the problem from his/her perspective, ensuring

that the design intent is communicated and design freedom is properly specified. For thetensile bar, the designer specifies three design variables, length (L), width (h), andthickness (t). For the product design as shown in Figure 5.7, L and t also affect theassembly of the product. Therefore the designer may do not want the manufacturer tochange their values. However, h can be tailored by the manufacturer in a range specifiedby the designer without affecting other components. Therefore the designer can

L

t

h

F

Fixed

Figure 5.7 – The Illustration of a Tensile Bar Example.


184

formulate L and t into the MGT formulation but given them tight ranges. The designercan also formulate h as the only geometry variable in the formulation that can be changedby the manufacturer. The latter way is used in the formulation shown in Table 5.6.Suppose the range of h is 0.4 to 0.6 inch, and the range of Fm is 0.99•Fp to 1.01•Fp.These ranges also specify the allowable design freedom within which the manufacturermay tailor the design. One thing to be noticed is that if the design freedom given to themanufacturer is too small, the manufacturer may not be able to find a satisfying solutionfor the MGT problem. Therefore the design will be sent back to the designer to begin anew iteration.

In the MGT formulation (Table 5.6), the two goals include one goal on meetingwidth value of the production tensile bar (Equation 5.13) and the other goal on achieving

Table 5.6 – MGT Tensile Bar Problem Formulation.

Given:! Length L = 5.0 inch, Thickness t = 0.2 inch, Width h = 0.5 inch! Ultimate yield strength of the product Su, p = 37.4 Mpa! Ultimate yield strength of the prototype Su, m

**

! The maximum force of the product, Fp= Su,p • h • t! Weight W1, W2

Find:! The geometric variables:

• Width, h! The maximum force of the prototype, Fm

! Deviation variables• d1

+, d1-, d2

+, d2-

Satisfy:! Goals:

• Width:0 5

11 1.

hd d+ − =− + [5.13]

• Maximum Force:F

Fd dp

m

+ − =− +2 2 1 [5.14]

! The Force Equation:• Fm= f(Su,m, h, t) **

! 0.99 • Fp ≤ Fm ≤ 1.01 • Fp

! di-, di

+ > 0, i = 1, 2di

- • di+ = 0, i = 1, 2

! The bounds on the system variables:0.4 inch ≤ h ≤ 0.6 inch

Minimize:! The deviation function (Archimedean formulation):

• Z W d d W d d Wi= + + + =− + − +1 1 1 2 2 2 1( ) ( ),( )Σ

Note: Symbol ‘**’ denotes the entries that are to be complemented by the manufacturer.


185

the same maximum force in the prototype tensile bar as will exist in the productiontensile bar (Equation 5.14). An analytic equation for the last goal is also given in theformulation based on Equation 5.12. The designer can assign different weight for thegoals based on the requirements. In this example, the force goal is much important thanthe width goal. Therefore a weight scenario can be W1=0.05 and W2=0.95. In Chapters 7and 8, a method using preference model to generate weights for different goals isemployed in the two case studies (Hernandez and Mistree, 2001).

With this information, the designer instantiates the MGT template to create theproblem formulation that is suitable for communicating with the manufacturer. Thisinstantiated template is shown in Table 5.6. The entries indicated by ‘**’ denoteinformation that the manufacturer must supply in order to complete the problemformulation and generate a solution.

• Manufacturer ActivitiesWith the aid of agents in a distributed computing environment (Gerhard, 2001), the

manufacturer can receive the MGT formulation sent by the designer, and also thematerial and process that are selected for the prototypes. In the MGT formulation, twopieces of information from the manufacturer are required: the prototype material propertySu,m, and the force behavior as a function of system variables. Suppose the AIM toolingfor polystyrene is chose for the prototype. Based on his/her knowledge and experienceon the material and process, the manufacturer knows:

Su,m=32.8 Mpa

Fm= Su,m • h • t.

Therefore the empty entries in Table 5.6 can be completed. And the completeproblem formulation can be solved using an exhaustive search algorithm. The results forthe weight scenario W1=0.05 and W2=0.95 are:

h = 0.57 inch, Max Force = 3.739, Deviation Z= 0.6%.

Comparing to the target values h = 0.5 inch and Fp = 3.74, the errors of h and Fm are14% and 0.2% respectively. Considering the weigh W2 = 0.95, this result makes sense.

Another comparison given here is to compare the error of Fm with that of the tensilebars produced without geometric tailoring. For this example, Fm will be 3.28 if nogeometric tailoring is executed. Therefore the error of Fm is 12.3%, which is muchbigger than 0.2%.

Besides the weight scenario given by the designer, the manufacturer may alsoexplore other scenarios. Three weight scenarios and the related results are shown in Table5.7. In these scenarios, the dimension goal is weighted more highly. As can be seen, the

Table 5.7– Scenarios of Goal Weights and Related Results.

Scenario Width Goal Max Force Goal h Fm

1 0.1 0.9 0.57 3.7392 0.5 0.5 0.57 3.7393 0.9 0.1 0.564 3.703


186

results across all scenarios are almost identical, indicating a solution that is stable withrespect to preferences.

Based on the results, the manufacturer can reconstruct a CAD model of the tensilebar using the variable values. The designer may be notified about the results by amessage from the manufacturer before the prototypes are produced. In the fabricationprocess, the manufacturer should also make efforts to make the prototypes have thematerial properties that are formulated in the MGT (Su,m=32.8 Mpa for this example).

• Physical ValidationThe research presented in this section is mainly based on the work given in (Dawson,

1998). The author also did some physical validation for the example. The validationresults presented in this section will also used in Section 7.5 for an example of a robotarm.

First a pair of SLA molds was built in the tooling mode in a SLA-3500 machine(Figure 5.8). Then a Morgan-press injection-molding machine (G-100T) from MorganIndustries Inc. was used to fabricate the prototypes of the tensile bar. The material wasgeneral-purpose polystyrene from the Dow Chemical Company (www.dow.com). Themain injection molding parameters used in the process were:

Barrel/Nozzle temperature: 430 / 450 oFClamping Force: 11 tonsInjection Pressure: 2.5 x 103 psiPilot Valve Pressure: 6x10 psiInjection Time: 35 secondCooling Time: 200 secondCycle Time: 6 minutes

Figure 5.8 – Photo of SLA Tools and Prototype Tensile Bars.


187

More than 10 parts were produced. Some of them are shown in Figure 5.8.

Among the prototypes, four tensile bars are used to determine material properties(Figure 5.9). The obtained values are presented in Table 5.8. In the table, the values in‘estimated’ row correspond to the material properties from (Dawson, 1998) and are usedin the MGT problem; ‘mean’ row corresponds to the mean values of the materialproperties obtained from physical experimentation. The variation of the materialproperties (maximum % deviation) for different specimens is 4 – 9%. This is a smallerror and hence the values can be considered to be consistent. Comparing the actual andestimated values, tensile strength and Young’s modulus have an error of 1.7% and 2.5%respectively. These values are very small indicating a very good match between theestimated and actual values. Strain values are not used in the MGT problem and henceno comparisons are provided.

Table 5.8– Material property validation results for polystyrene.

ReplicationTensile Strength

(MPa)Young's Modulus

(MPa) % Strain @ Yield1 30.7 3392 1.012 32.2 3730 0.973 33.6 3698 1.104 32.5 3123 1.15

Mean 32.3 3486 1.06Max. Deviation Percentage 4.81% 7.03% 8.51%

Estimated 32.8 3400Deviation Percentage 1.71% 2.46%

Figure 5.9 – Photo of Prototype Tensile Bars Used for Tensile test.


188

• DiscussionThe design freedom given by the designer to the manufacturer is important for

reducing the iterations between the designer and manufacturer. In this example, theauthor noticed that if the bounds of h given by the designer are changed from 0.4 ≤ h ≤0.6 to 0.45 ≤ h ≤ 0.55, no solution could be found for the given requirements. Thereforethe design may be sent back to the designer to begin a new iteration.

From the testing results shown in Table 5.8, it can be noticed that the materialproperties of the prototypes are actually in a certain range instead of a point. This isbecause of some inevitable noise factors in the rapid tooling process (Figure 5.10).Suppose the material property of interest is Y and its distribution can be represented asthe normal distribution as shown in Figure 5.10. The mean and standard deviation of Yare µY and σY. They are mainly determined by the capability of the manufacturingprocess. For a given target value y, bias(y) = y-µY. If bias(y) is larger than the giventolerance, the produced parts will have very low quality. Therefore the principle ofgeometric tailoring is to change the target value y to y’ in order to make bias(y’) < bias(y).It is obvious that the best value of y’ is µY. In the current usage of Rapid Tooling, y is setby the designer who may have no experience of the process and material. Therefore ymay have a big bias. However, by using the MGTDT, the manufacturer can set the valueof y with a lower bias based on his/her understanding of the process and material.Therefore geometric tailoring can help the manufacturer to produce qualified prototypesmuch more easily. The designer can also benefit from it because less iteration is needed.

With the detail usage of the MGTDT presented in this section, two other examplesare presented in the next two sections. Instead of analytic equations, the ResponseSurface Equations (RSEs) are used to represent functional properties.

5.5.2 Building Prototypes of a Rib Part

• Problem IntroductionSuppose a rib part as shown in Figure 5.11 is used to support a load F = 25 pounds.

Similar to the tensile bar example presented in Section 5.5.1, atactic polystyrene and H13

µ ,σ

RT ProcessPart P

ControlFactors

Performance

NoiseFactors

y'

x

z

z z

µ ,σY Y

ProbabilityDistribution

µYy Mean

Standarddeviation

σY

Tolerancedeviation

Quality withinspecification

Target y

Qualitydistribution

bias

Y

y'

GeometricTailoring

Figure 5.10 – Quality Distribution and Geometric Tailoring.


189

steel tools were chosen for producing the production parts. The designer finished the ribdesign as b = 0.192 inch, h = 0.75 inch, w= 0.852 inch. No draft angle was considered (α= 0). According to the design, the deflection of the rib under the load will be 0.02 inch,which satisfies the assembly requirements of the product.

However since this deflection is critical to the performance of the product, thedesigner needs 50 prototypes to validate his/her design within one week. Suppose theerror of the deflection of the prototypes is 5%. Since the lead-time to get the productiontools for the part is too long, the designer decides to use the AIM tooling in producing theprototypes.

• Designer ActivitiesIn this example, suppose the designer is only interested in the deflection of the rib

when testing the prototypes. Therefore the functional property of interest in this exampleis the maximum deflection of the rib, which is related to the flexural Young’s modulus ofthe material. The flexural Young’s modulus of a production part made of atacticpolystyrene with H13 steel molds is YMsteel = 3.0 Gpa = 436710 psi (Dawson, 1998).

In the design process, the designer had already used a Finite Element Analysissoftware system (e.g. ANSYS) to simulate the maximum deflection related to the load.This simulation was used by the designer in his/her design. Suppose in the productionpart the maximum deflection of the rib is 0.02 inch. Based on this value and the errorsthat are allowed in the functional testing, the designer may set the range of the maximumdeflection for the prototypes as 0.019 ≤ MDm ≤ 0.021.

Among the three design variables (b, h, w) of the rib, assume w is fixed because ofother assembly requirements. Therefore b and h are the design variables that can bechanged by the manufacturer. Their ranges are shown as follows.

0.182 ≤ b ≤ 0.202;0.71 ≤ h ≤ 0.79.

F

h

b

w

Figure 5.11 – The Illustration of a Rib Example.


190

In the rib design, the designer did not consider any draft angles because he/she wasnot aware of its necessity. Therefore based on all the information, the designer mayinitiate a MGT rib problem formulation as shown in Table 5.9.

Table 5.9 - MGT Rib Problem Formulation By the Designer.

Given:! b = 0.192 inch, h = 0.75 inch, w= 0.852 inch! Load F = 25 lb! Flexural Young’s modulus of the product YMsteel = 436710 psi! Flexural Young’s modulus of the prototype YMSLA

**

! The maximum deflection of the rib MDp = 0.02 inch! Weight W1, W2, W3

! 50 prototypes are neededFind:! The geometric variables:

• Thickness, b• Height, h

! The maximum deflection, MDm


+, d1-, d2

+, d2-, d3

+, d3-

Satisfy:! Goals:

• Thickness:0192

11 1.

bd d+ − =− + [5.15]

• Height:0 75

12 2.

hd d+ − =− + [5.16]

• Maximum Deflection:0 02

13 3.

MDd d

m

+ − =− + [5.17]

! The maximum deflection equation:• MDm = f(b, h) **

! 0.019 inch ≤ MDm ≤ 0.021 inch! di

-, di+ > 0 i = 1, 2, 3

di- • di

+ = 0 i = 1, 2, 3! The bounds on the system variables:

0.182 inch ≤ b ≤ 0.202 inch0.71 inch ≤ h ≤ 0.79 inch


• Z W d d W d d W d d Wi= + + + + + =− + − + − +1 1 1 2 2 2 3 3 3 1( ) ( ) ( ),( )Σ


In the MGT formulation (Table 5.9) three goals are considered (Equations 15 ~ 17).The designer can assign different weights to them based on his/her requirements.Different weight scenarios will be discussed further in the solving process of the MGT.


191

The MGT formulation and CAD model of the part can be sent to the manufacturer.In addition, the designer can also send the FEA model that was used in determining themaximum deflection of the rib to the manufacturer.

• Manufacturer ActivitiesAfter receiving the above design information, the manufacturer can begin the

geometric tailoring for the part. Suppose the AIM tooling for polystyrene is chose for theprototype. Correspondingly the manufacturer knows that the flexural Young’s modulusof atactic polystyrene of parts produced with SLA molds are YMSLA = 4.0 Gpa = 582280psi (Dawson, 1998). The manufacturer can also employ the FEA model to understand therelation of the maximum deflection with the material differences of the prototype ribs.After getting an equation of MDm, all the empty entries in Table 5.9 can be completed.The solving process will be very similar to that of the tensile bar example (Section 5.5.1).

However, suppose the manufacturer notice that there is no draft angle in the ribdesign. This may lead to some difficulties in the injection molding process (Rosato andRosato, 1995). Especially the mold life of SLA molds will be significantly lower for apart design without draft angle (Palmer, 1999). Suppose this consideration needs to beadded to the MGT formulation.

To tailor the draft angle based on the mold life, the manufacturer must have ananalytical equation to represent the relation of the mold life and the draft angle. Based onthe work of (Palmer, 1999), suppose a response surface equation is generated as:

Number of Shot = 23.13 – 14.54•hr + 16.06•α + 1.4•hr•hr + 0.67•α•α - 2.24•hr•α ,

where hr is the height ratio (h/b) and α is the draft angle. Although this model isdeveloped for a mold protrusion and may be rather crude, the author will use it in thisexample to demonstrate how to integrate the design and manufacturing considerations.

The manufacturer can add draft angle α as an additional geometry variable in theMGT formulation with its target value as 0.5o. Accordingly there are two thicknessvariables, b1 and b2, which are related to the top and bottom thickness of the rib (Figure5.12). They have the relation as:

b1 = b2 - 2•h• tan(α)

where h is the height and α is draft angle.

b1

b2

ha

Figure 5.12 – Geometry Variables of the Rib by Adding Draft Angle.


192

Furthermore another constraint is related to the value of b1. In the injection moldingprocess, the top surface of the rib will be used as the ejection position. After thecalculation, mold designer decided that the ejection pins need to be 0.125 inch indiameter. Therefore, the top thickness b1 should be greater than 0.125 inch.

Correspondingly the relation MDm = f(b, h) in Table 5.9 is changed to MDm = f(b2, h,α). This relation can be represented by a Response Surface equation. Based on the FEAmodel given by the designer, the manufacturer can simulate the maximum deflectionrelated to the load and boundary conditions for different geometries. This process isdescribed in more details as follows.

The three design factors chosen for the experiments and their ranges are given inTable 5.10. The experimental design that is used in this example is CCF (CentralComposite Faced) design (Montgomery, 1991). There are totally 15 experiments. Theexperimental design and related results are shown in Table 5.11.

Table 5.10 – Design Factors and Their Ranges.Design Factor Low Band (-1) Middle (0) High Band (1)

Bottom Thickness (b2) (in.) 0.182 0.192 0.202Height (h) (in.) 0.71 0.75 0.79

Draft Angle (αααα) (o) 0 1.25 2.5

Table 5.11 – Results of the Experiments for Rib Part.Design Factors Response

Exp. b2 (in.) h (in.) αααα (o) Max. Deflection (inch)1 0.202 (1) 0.79 (1) 2.5 (1) 0.02102 0.202 (1) 0.79 (1) 0 (-1) 0.01713 0.202 (1) 0.71 (-1) 2.5 (1) 0.01844 0.202 (1) 0.71 (-1) 0 (-1) 0.01365 0.182 (-1) 0.79 (1) 2.5 (1) 0.02956 0.182 (-1) 0.79 (1) 0 (-1) 0.02047 0.182 (-1) 0.71 (-1) 2.5 (1) 0.02438 0.182 (-1) 0.71 (-1) 0 (-1) 0.01549 0.202 (1) 0.75 (0) 1.25 (0) 0.0178

10 0.182 (-1) 0.75 (0) 1.25 (0) 0.020611 0.192 (0) 0.79 (1) 1.25 (0) 0.020512 0.192 (0) 0.71 (-1) 1.25 (0) 0.015413 0.192 (0) 0.75 (0) 2.5 (1) 0.021114 0.192 (0) 0.75 (0) 0 (-1) 0.015415 0.192 (0) 0.75 (0) 1.25 (0) 0.0178

In the FEA analysis, the element type that is used is Brick 20 node (solid 95). Theauthor also noticed that the maximum stress given by ANSYS is around 4000 psi, whichis less than 6000 psi. Therefore the stress for the rib design is not a main concern. Twoscreen dumps given by ANSYS for two experiments (experiment 6 and 7) are shown inFigure 5.13.


193

A statistical software system, MINITAB (www.minitab.com), was used to generatethe response surface model for the above experiments. The response surface equationgiven by MINITAB is:

Max_Deflection = 0.43 - 4.799•b2 + 0.093•h + 0.021 • α + 14.722•b2•b2 + 0.139•h•h –

1.281•b2•h – 0.093•b2•α - 0.002•h•α

(R2 = 98.8%, R2 (adj) = 96.7%, Max. dev = 4.5%, Avg. dev = 1.98%)

Figure 5.13 – Two Screen Dumps of Experiment Results for Rib Part.


194

The response surface of the maximum deflection has very high R2 and R2 (adj)values. This indicates that the response surface fits very well through actual data. Thegraphical relations of the response and the variables given by MINITAB are shown inFigure 5.14. They are provided for a better understanding of the response surfaceequation.

Therefore the manufacturer can formulate the MGT problem by adding all themanufacturing requirements. The complete MGT problem formulation is shown in Table5.12. In the formulation five design goals are considered instead of the three goals given

(a) Relation of Deflection with h and b2

(b) Relation of Deflection with b2 and α

Figure 5.14 – Graphical Relations of The Maximum Deflection and Variables.

inch

inch

inch

inch

inch

inch


195

in Table 5.9. The goals of draft angle (Equation 5.18) and shot number (Equation 5.19)are added by the manufacturer.

Table 5.12 – Complete MGT Rib Problem Formulation.Given:! b2 = 0.192 inch, h = 0.75 inch, w= 0.852 inch! Load F = 25 lb.! Flexural Young’s modulus of the product YMsteel = 436710 psi! Flexural Young’s modulus of the prototype YMSLA = 582280 psi! The maximum deflection of the rib MDp = 0.02 inch! Weight W1, W2, W3, W4, W5

! 50 prototypes are needed.Find:! The geometric variables:

• Bottom Thickness, b2

• Height, h• Draft angle, α

! The maximum deflection, MDm

! Shot number of a mold before failure, NS.! Deviation variables

• d1+, d1

-, d2+, d2

-, d3+, d3

-, d4+, d4

-, d5+, d5

-

Satisfy:! Goals:

• Bottom thickness:0192

12

1 1.

bd d+ − =− +

• Height:0 75

12 2.

hd d+ − =− +

• Maximum Deflection:0 02

13 3.

MDd d

m

+ − =− +

• Draft Angle:0 5

14 4.

α+ − =− +d d [5.18]

• Shot number:50

15 5NS

d d+ − =− + [5.19]

! The maximum deflection Response Surface Model:• MD = 0.43- 4.799•b2+ 0.093•h +0.021•α +14.722•b2•b2+ 0.139•h•h- 1.281•b2•h –

0.093•b2•α - 0.002•h•α! 0.019 inch ≤ MDm ≤ 0.021 inch! Top thickness: b1 = b2 – 2•h• tan(α)! Requirement of ejector pin: b1 > 0.125! Height ratio: hr = h/b2

! Number of shot: NS = 23.13 – 14.54•hr + 16.06•α + 1.4•hr•hr + 0.67•α•α - 2.24•hr•α! di

-, di+ > 0 i = 1, …, 5

di- • di

+ = 0 i = 1, …, 5! The bounds on the system variables:


196

0.182 inch ≤ b2 ≤ 0.202 inch0.71 inch ≤ h ≤ 0.79 inch0o ≤ α ≤ 2.5o


• Z W d d W d d W d d W d d W d d= + + + + + + + + +− + − + − + − + − +1 1 1 2 2 2 3 3 3 4 4 4 5 5 5( ) ( ) ( ) ( ) ( )

( )ΣWi =1

A C program using the exhaustive searching method is developed to solve theformulated problem. Four different goal preference scenarios were investigated. Allgoals were weighted evenly in Scenario 1. In Scenario 2, the dimensional goals wereweighted more highly. Scenarios 3 weighted the shot number goal more highly. InScenario 4 the shot number goal was not considered and all other four goals wereweighted evenly. Weights for the four scenarios are shown in Table 5.13.

Results of the MGT problem are shown in Table 5.14 for each of the four scenarios.As can be seen, the results across all scenarios are almost identical, indicating a solutionthat is stable with respect to preferences. However, the deviations of the solutions arerather high. This is because the goal for shot number is set too high (50) and the responsesurface equation of the shot number is crude. However, the author believes thesesolutions can help the designer and manufacturer to understand the possible difficulties inmaking tradeoffs. Therefore the negotiations between them may be speeded up.

Since the main goal of this example is to illustrate the capability of the MGTDT inintegrating the design and manufacturing requirements, no further errors are made inimproving the accuracy of the solutions for the example. Based on the results, themanufacturer selects the following rib part dimensions:

Table 5.13– Scenarios of Goal Weights.

Dimension GoalsScenario Thickness

GoalHeight Goal Draft Angle

Goal

Max.Deflection

Goal

ShotNumber

Goal1 0.2 0.2 0.2 0.2 0.22 0.3 0.3 0.3 0.05 0.053 0.1 0.1 0.1 0.1 0.64 0.25 0.25 0.25 0.25 0

Table 5.14– Rib Part MGT Results.

Scenario BottomThicknessb2 (inch)

Heighth (inch)

DraftAngle α

(o)

Max.Deflection

MDm (inch)

ShotNumber

NSDeviation

1 0.19 0.74 2.5 0.0191 10 0.972 0.19 0.74 2.5 0.0191 10 0.453 0.19 0.74 2.5 0.0191 10 2.484 0.195 0.79 1.7 0.0191 1 0.21


197

b2 = 0.19 inch, h = 0.74 inch, α= 2.5o, MDm = 0.0191, NS = 10.

• DiscussionBased on the simulation of ANSYS, the maximum deflection of the rib part

produced by SLA molds is 0.0178 inch if no geometric tailoring is executed. Themaximum deflection of the production part is 0.02 inch. Therefore the error between theprototype and production parts is (0.02-0.0178)/0.02 = 11%, which is bigger than theerror requirement (5%). In some cases, it may bring problems in the design verificationbased on the functional testing. For example, the maximum deflection of the prototypesin this example is 0.0178 inch, which is less than what the production design will perform(0.02 inch). Therefore even the running results given by the functional testing aresatisfactory, the product design may still fail to achieve the design specifications.However by using the tailored dimensions, the error between the prototype andproduction parts is only (0.02-0.0191)/0.02 = 4.5%, which is within the error requirement(5%).

Although the response surface equation of the shot number is not accurate, it canfoster a better understanding of the necessity of the geometric tailoring. When the moldlife requirement is considered, the draft angle given by solving the MGT problem is 2.5o.Correspondingly the SLA molds can produce 10 parts before they fail. However if thismanufacturing requirement is not considered, draft angle is 1.7o and the shot number forthe SLA molds is only 1 (Scenario 4). This will be not acceptable in producing theprototypes. Even worse, the draft angle given by the designer is 0o. The manufacturermay not be able to get a single part for this design. In Chapter 6, the author will presentan approach for Rapid Tooling in order to consider more requirements that are related tothe manufacturing process.

Figure 5.15 – Parts Produced by SLA Molds.


198

• Physical ValidationFor the tailored part design, the author used a SLA-3500 and Morgan-press injection-

molding machine in producing molds and parts respectively. The material is a general-purpose polystyrene from the Dow Chemical Company (www.dow.com). Over 18 partswere made before a pullout failure happened in a SLA mold piece. Two parts are shownin Figure 5.15. The SLA molds and the pullout failure are shown in Figure 5.16.

However, because of the lack of experiment device, the deflection of the prototypesis not tested under the load. In the next section a ring gear example is presented, in whichtwo functional properties are considered in one prototype.

5.5.3 Building Prototypes of a Ring Gear

• Problem IntroductionThe design problem under consideration here is the design of ring gears for a speed

reducer in a family of cordless drills. These speed reducers are planetary gear trains,

Figure 5.16 – SLA Molds and Pullout Failure.

Layer 1 Layer 2 Layer 3

FromMotor

ToChuck

Ring gear ofinterest

Figure 5.17 – The Ring Gear and the Speed Reducer of a Cordless Drill.

PulloutFailure


199

typically with three stages, as shown in Figure 5.17. Suppose ring gears for layer 1 and 2are made of injection molded, atactic polystyrene. In this problem the author onlyconsidered the ring gear for layer 1.

For different drill with voltage of 9.6V, 12V, 14.4V and 18V, the maximum torqueto chuck is given in Table 5.15. A gearshift is used for providing two different outputspeeds (1400 rpm and 400 rpm) for the user by engaging and disengaging the secondplanetary layer.

Suppose a gear train has been designed and a prototype gear train is required forfunctional testing. Rather than waiting for production injection mold tooling for the ringgear to be machined, the designer wants to fabricate prototype ring gears using a RapidTooling process. In the cordless drill design, the reliability of its transmission is ratherimportant. Therefore suppose the prototype ring gears are to be used for fatigue testing.The error requirement of the fatigue is 5%.

• Designer ActivitiesA spur ring gear can be determined by three design variables, teeth number (N), pitch

diameter (PD), and face width (W). They are shown with other terminologies in Figure

Table 5.15 – Maximum Torque of The Speed Reducer.

Voltage (V) Maximum Torque (in-lb)9.6 15612 200

14.4 23718 330

Figure 5.18 – Some Terminology Describing a Spur Gear (Shigley and Mischke,1989).


200

5.18 (Shigley and Mischke, 1989). The diameter of the pitch circle is called pitchdiameter.

Suppose the ring gear design in layer 1 are: N=54, PD = 1.335inch, F= 0.215 inch.Since the speed reducing ratio of the gear train is related to the teeth number by equationN

N2

1

1

2

= ωω

. The designer may not want the manufacturer to change the number of N

because of the speed requirements. Therefore face width (F) and pitch diameter (PD) arethe design factors that can be tailored by the manufacturer. Suppose their ranges are:

0.204 ≤ F ≤ 0.226;

1.268 ≤ h ≤ 1.402.

Then because the fatigue property of the ring gear is investigated in the functionaltesting, the designer should determine an approach to evaluate the fatigue of the ringgear. An approach based on the maximum stress is presented as follows.

When the cordless drill is in work, the ring gear is under fluctuating stresses asshown in Figure 5.19.a. Suppose the maximum Von Mises stress is σmax. Since σmin = 0for the ring gear, one can get

mean stress (σm) = (σmax + σmin ) / 2 = σmax /2,

stress amplitude (σa) = (σmax - σmin ) / 2 = σmax /2.

The fatigue, or gear life, can be represented by the number of cycles (N) that theteeth can run before it breaks. The gear life is related to the fluctuating stresses in thegear teeth (Shigley and Mischke, 1989). Suppose a factor is c, nominal stress Sa = cσa.The relation of Sa and N can be illustrated in Figure 5.19.b.

Dowling (1993) gave a way to calculate the number of cycles for given fluctuatingstresses. For different materials including polymer,

N = (Sa/a)1/b,

where a= k1 • Sut, b = k2.

In the equations, Sut is the tensile strength of the material, k1 and k2 are all factors.

Since Sa = c•σ a = c•σmax /2, for a teeth of the ring gear,

Time

σ

0 N

Sa

0

σmax σa

(a) Fluctuating Stress in the Gear (b) The Stress-Life Curve

Figure 5.19 – Fatigue of Ring Gear.


201

N = [Sa / (k1• Sut)]1/k2 = k• (σmax / Sut)

1/k2. [5.20]

Suppose k = kakbkckdkekf, where ka is surface factor, kb is size factor, kc is load factor,kd is temperature factor, ke is material factor, kf is miscellaneous-effect factor (corrosion,residual stress, metal spraying, cyclic frequency, stress concentration, etc.). Since theproduction and prototype gears have similar surface, size, load, temperature, andmaterial, ka, kb, kc, kd, ke, kf in the above equations can be thought as equal.

Based on Equation 5.20,

Nprototype = kprototype• (σmax/Sut)prototype1/k2prototype,

Nproduct = kproduct • (σmax/Sut) product1/k2product,

Therefore, similar fatigue performance (Nproduct = Nprototype) can be achieved bymaking the ratio σσσσmax/Sut of prototype gears and production gears as similar aspossible.

In this example, since the same ring gear is operated in two different speeds, twodifferent loads are added to the ring gear. In the functional testing, it is desired that theprototype ring gears will behavior similar to the performance of the production ring gearsin both loads. Therefore two functional properties (MS1, MS2) should be considered inthe MGT formulation.

Related to different output speeds, two speed ratios are 0.0613 (use layer 1, 3) and0.019 (use layer 1, 2, 3). Suppose the number of planet gear is 5. For the given design,

High torque T1= 156/5*0.0613=1.913 lb-in;

Low torque T2= 156/5*0.019=0.593 lb-in.

The maximum stresses of the product ring gear for low and high speed given byANSYS are 3282 psi and 1017 psi respectively. The tensile strength of a production ringgear made of atactic polystyrene is 37.4 Gpa (Dawson, 1998). That is, Sut, product = 37.4Gpa = 5430 psi.

Based on all the information, the designer may initiate a MGT ring gear problemformulation as shown in Table 5.16.

Table 5.16 - MGT Ring Gear Problem Formulation By the Designer.

Given:! N = 54, PD = 1.335 inch, W= 0.215 inch! Load T1 = 1.913 lb-in, T2 = 0.593 lb-in! Yield Strength of the product Su, p = 5430 psi! Yield Strength of the prototype Su, m

**

! The maximum stress of the product gear under T1: MS1, p = 3282 psi! The maximum stress of the product gear under T2: MS2, p = 1017 psi

! Target maximum stress of the prototype gear under T1: TMSMS S

Smp u m

u p1

1,

, ,

,

=•


Smp u m

u p2

2,

, ,

,

=•


202

! Weight W1, W2, W3, W4

! 50 prototypes are neededFind:! The geometric variables:

• Pitch Diameter, PD• Face Width, W

! The maximum stress under T1, MS1,m



+, d1-, d2

+, d2-, d3

+, d3--, d4

+, d4-

Satisfy:! Goals:

• Pitch Diameter:1335

11 1.

PDd d+ − =− + [5.21]

• Face Width:0 215

12 2.

Wd d+ − =− + [5.22]

• Maximum Stress under T1:TMS

MSd dm

m

1

1

3 3 1,

,

+ − =− + [5.23]


MSd dm

m

2

2

4 4 1,

,

+ − =− + [5.24]

! The maximum deflection equation:• MS1,m = f1(PD, W, T1)

**

• MS2,m = f2(PD, W, T2)**

! 2733 psi ≤ MS1,m ≤ 3021 psi846.9 psi ≤ MS2,m ≤ 936.1 psi

! di-, di

+ > 0 i = 1, 2, 3, 4di

- • di+ = 0 i = 1, 2, 3, 4

! The bounds on the system variables:1.268 inch ≤ PD ≤ 1.402 inch0.204 inch ≤ W ≤ 0.226 inch


• Z W d d W d d W d d W d d Wi= + + + + + + + =− + − + − + − +1 1 1 2 2 2 3 3 3 4 4 4 1( ) ( ) ( ) ( ),( )Σ


In the MGT formulation (Table 5.16), four goals include two goals on geometry(Equations 5.21 and 5.22) and two goals on maximum stress (Equations 5.23 and 5.24).The MGT formulation and CAD model of the part can be sent to the manufacturer. Inaddition, the designer can also send the FEA model that was used in determining themaximum stress of the ring gear to the manufacturer.


203

• Manufacturer ActivitiesAfter receiving the design information, the manufacturer can begin the geometric

tailoring for the part. Suppose the AIM tooling for polystyrene is chose for the prototype.Correspondingly the manufacturer knows the yield strength of atactic polystyrene ofgears produced with SLA molds are Su,m = 32.8 Gpa = 4760 psi (Dawson, 1998).

The manufacturer can also employ the FEA model to understand the relation of themaximum stress with the material and geometry differences of the prototype ring gear.The design variables given by the designer are PD and W. Correspondingly the designfactors and their ranges for the experiments are given in Table 5.17. CCF (CentralComposite Faced) design is used in the experimental design in this example(Montgomery, 1991). There are totally 9 experiments for each load. The experimentaldesign and related results given by ANSYS are shown in Table 5.18.

In the FEA analysis, the element type that is used is Tet-10 node (solid 72). Twoscreen dumps given by ANSYS for two experiments (experiment 1 and 7) are shown inFigure 5.20.

Based on the results of the experiments, the response surface equations for themaximum stress given by the statistical software (MINITAB) are:

MS1,m = -50522 + 565972*W - 3537*PD -1249312*W*W + 1968*PD*PD - 31208*W*PD;(R2 = 99%, R2 (adj) = 97.3%, Max. dev = 1.5%, Avg. dev = 0.8%)

MS2,m = -15634 + 175337*W - 1119*PD - 386502*W*W + 631*PD*PD - 9837*W*PD.(R2 = 99.0%, R2 (adj) = 97.3%, Max. dev = 1.5%, Avg. dev = 0.8%)

Table 5.17 – Design Factors and Their Ranges of Ring Gear.

Design Factor Low Band (-1) Middle (0) High Band (1)Thickness W (inch) 0.204 0.215 0.226

Pitch diameter PD (inch) 1.268 1.335 1.402

Table 5.18 – Results of the Experiments for Ring Gear.

Design Factors ResponseMax. Stress (psi)

Exp. W (in.) PD (in.) Torque =1.913 lb-in

Torque =0.593 lb-in

1 0.226 (1) 1.402 (1) 2581 8002 0.226 (1) 1.268 (-1) 3325 10313 0.204 (-1) 1.402 (1) 2939 9114 0.204 (-1) 1.268 (-1) 3591 11135 0.215 (0) 1.402 (1) 2924 906.46 0.215 (0) 1.268 (-1) 3535 10967 0.226 (1) 1.335 (0) 2956 916.48 0.204 (-1) 1.335 (0) 3183 986.89 0.215 (0) 1.335 (0) 3282 1017


204

The response surfaces of the maximum stresses have very high R2 and R2 (adj)values. This indicates that the response surface fits very well through actual data. Thegraphical relation of MS1,m and the design variables (PD and W) given by MINITAB isshown in Figure 5.21. It is provided here for a better understanding of the responsesurface equation. Although not shown, MS2,m has a similar relation with the designvariables.

Based on all the information, the manufacturer can formulate a complete MGTproblem as shown in Table 5.19.

Figure 5.20 – Two Screen Dumps of Experiment Results for Ring Gear.


205

Table 5.19 – Complete MGT Ring Gear Problem Formulation.

Given:! N = 54, PD = 1.335 inch, W= 0.215 inch! Load T1 = 1.913 lb-in, T2 = 0.593 lb-in! Yield Strength of the product Su, p = 5430 psi! Yield Strength of the prototype Su, m = 4760 psi! The maximum stress of the product gear under T1: MS1, p = 3282 psi! The maximum stress of the product gear under T2: MS2, p = 1017 psi


Smp u m

u p1

1,

, ,

,

=•


Smp u m

u p2

2,

, ,

,

=•

! Weight W1, W2, W3, W4

! 50 prototypes are neededFind:

! The geometric variables:• Pitch Diameter, PD• Face Width, W


Figure 5.21 – Graphical Relation of the Maximum Stress with PD and W.


206



+, d1-, d2

+, d2-, d3

+, d3--, d4

+, d4-

Satisfy:! Goals:

• Pitch Diameter:1335

11 1.

PDd d+ − =− +

• Face Width:0 215

12 2.

Wd d+ − =− +


MSd dm

m

1

1

3 3 1,

,

+ − =− +


MSd dm

m

2

2

4 4 1,

,

+ − =− +

! The maximum deflection equation:• MS1,m = -50522 + 565972*W - 3537*PD -1249312*W*W + 1968*PD*PD - 31208*W*PD

• MS2,m = -15634 + 175337*W - 1119*PD - 386502*W*W + 631*PD*PD - 9837*W*PD

! 2733 psi ≤ MS1,m ≤ 3021 psi846.9 psi ≤ MS2,m ≤ 936.1 psi

! di-, di

+ > 0 i = 1, 2, 3, 4di

- • di+ = 0 i = 1, 2, 3, 4

! The bounds on the system variables:1.268 inch ≤ PD ≤ 1.402 inch0.204 inch ≤ W ≤ 0.226 inch


• Z W d d W d d W d d W d d Wi= + + + + + + + =− + − + − + − +1 1 1 2 2 2 3 3 3 4 4 4 1( ) ( ) ( ) ( ),( )Σ

A C program using exhaustive searching method is developed to solve theformulated problem. Three scenarios of goal preference were investigated. All goalswere weighted evenly in scenario 1. The two other scenarios weighted the dimensiongoals more highly, as in scenario 2, or weighted the maximum stress goals more highly,as in scenario 3. Weights for the three scenarios are shown in Table 5.20.

Table 5.20– Scenarios of Goal Weights.

Dimension GoalsScenario Face Width

GoalPitch Diameter

Goal

Maximum Stress 1Goal

Maximum Stress 2Goal

1 0.25 0.25 0.25 0.252 0.4 0.4 0.1 0.13 0.1 0.1 0.4 0.4


207

Results of the MGT problem are shown in Table 5.21 for each of the three scenarios.As can be seen, the results across all scenarios are almost identical, indicating a solutionthat is stable with respect to preferences.

Based on the results, the manufacturer selects the following ring gear dimensions:

W = 0.216 inch, PD = 1.401 inch, MS1 = 2837.5, MS2 = 900.

Discussion:

If no geometric tailoring is executed, the maximum stress of the prototype ring gearis 3282 psi to transfer T1, and 1017 psi to transfer T2. Therefore the errors between theprototype and production gears are (3282 – 3282•4760/5430)/ (3282•4760/5430) = 14%for T1 and (1017 - 1017•4760/5430)/ (1017•4760/5430) =14% for T2. Therefore thefatigue property of the prototype gears will be different from that of the product gears.After tailoring the dimensions, the error reduced to 1.4% (T1) and 1% (T2) respectively.

• Physical ValidationFor the tailored part design, the author used a SLA-3500 and a Sumitomo (75 Ton)

injection-molding machine (www.sumitomopm.com) in producing molds and partsrespectively. The material is general-purpose polystyrene from the Dow ChemicalCompany. Over 10 shots were made. The SLA mold is still in good condition after theinjection-molding process was stopped. The main injection molding parameters used inthe process were:

Temperature setting:Zone 1: Throat of Hopper 220 oCZone 2: Melt Zone 225 oCZone 3: Transition Zone 230 oCZone 4: Metered Zone 235 oCZone 15 Nozzle 235 oC

Clamping Force: 50 tonsHold Time: 15 secondHold Pressure: 8.0 kgf/cm2

Shot Size: 40.5 mmCooling Time: 450 secondCycle Time: 8 minuteInjection Speed: 25 mm/sec

Some of the produced ring gears are shown in Figure 5.22. One of the SLA moldpieces installed in the standard mold plate of the injection-molding machine is shown inFigure 5.23. However, because of the lack of experiment device, the maximum stressesof the prototypes are not tested under the loads.

Table 5.21– Ring Gear MGT Results.

Scenario Face Width w(inch)

Pitch Diameterpd (inch)

Max. Stress1 MS1 (psi)

Max. Stress2 MS2 (psi)

Deviation(%)

1 0.216 1.401 2837.5 900.1 12 0.215 1.402 2885 903.7 23 0.216 1.401 2837.5 900 1.6


208

Figure 5.22 – A Photo of the Prototype Ring Gears.

Figure 5.23 – A SLA Mold Piece in a Standard Mold Plate.


209

After the test examples are presented for the MGT and MGTDT, a brief summary isgiven in the next section, which discusses the relevance of these results with regard to thehypotheses of the dissertation.


In the current usage of Rapid Tooling, the iterations of design changes between thedesigner and manufacturer may take a long time before production-representativeprototypes are produced. In this chapter and the next chapter, methods on geometrictailoring for Rapid Tooling are presented to address the problem. One geometrictailoring problem, material geometric tailoring, is considered in Chapter 5. Theproperties of Rapid Tooling, especially the part properties of the AIM tooling, areintroduced in Section 5.1 to provide a context of material geometric tailoring. Related tothe principle of functional testing and similarity methods, the fundamentals of geometrictailoring are presented in Section 5.2. The material geometric tailoring decision templateis introduced in Section 5.3, which enables a “clean digital interface” between design andfabrication, effectively separating design activities from manufacture activities. Theusage of the MGT decision template, including formulating function properties andsolving approaches, is presented in Section 5.4. Finally three test examples are discussedin Section 5.5 to demonstrate a scenario of design-manufacture collaboration with theMGT decision template.

The research question and hypothesis that are related to this chapter are Q2.1 andsub-hypothesis 2.1, which are repeated here:

Q2.1. How to reduce the iterations between the designer and manufacturer inproducing functional prototypes that have different material propertiesfrom products?

Sub-Hypothesis 2.1: The designer can initiate a material geometric tailoring(MGT) formulation based on a MGT decision template; therefore themanufacturer, who completes and solves the MGT problem, can produceproduction-representative prototypes more quickly.

Although not explicitly presented in the chapter, the discussions on the properties ofRapid Tooling, the Buckingham Π theorem, information flow and information processingfor the different steps in using MGTDT provide partial theoretical structural validationfor Hypothesis 2.1. In Section 5.5, the usage of MGTDT in designing a prototype tensilebar, rib part and ring gear is presented. The discussions on the problem requirements ofthe examples and the usage of the MGTDT provide partial empirical structure validationfor Hypothesis 2.1, and the discussions on solving process and physical validation of theexamples provide partial empirical performance validation for Hypothesis 2.1. Theseresults are summarized in Figure 5.24.

The author also made an effort on providing theoretical performance validation forHypothesis 2.1. Some explanations on the performance validation of the hypothesis aregiven as follows.

Currently the decisions on part design are made by the designer. In the designprocess, the designer formulates goals, constraints and preferences to make the decisions.However these decision factors are not transferred to the next stages (e.g. Design-for-


210

Manufacture). So design freedom decreases quickly. Consequently the cost and time tomake decisions in the later stages increase dramatically. In this research, the MGTDT isproposed to formulate sufficient design information for the material geometric tailoringproblem. By communicating the formulated design information to later stages, theknowledge about design increases without decreasing design freedom dramatically. Withthe maintained design freedom, some difficulties in the later stages can be solved easilyand quickly (Figure 5.24). This was illustrated in the discussions of the tensile barexample (Section 5.5.1).

The validation of hypothesis 2.1 also partially supports Q2 and hypothesis 2, whichare repeated here:

Q2. How to reduce the time of iteration between the designer and manufacturerin the usage of Rapid Tooling for a wide variety of design requirements?

Theoretical Structure Validation Theoretical Performance Validation

Empirical Structure Validation Empirical Performance Validation

Hypothesis 2.1

The requirements on producing functionalprototypes of a tensile bar, a rib part, anda ring gear using the AIM tooling arerepresentative of the problems. TheMGTDT can be used to formulate thedesign and manufacturing requirements.

Production-representative prototypesof the tensile bar, rib part and ringgear were produced based on thescenario of using MGTDT to transferDFM to the manufacturer withoutiterations.

The MGT is based on Buckingham PItheorem; the MGTDT is based on decisionbased design and other design technologiesincluding compromise DSP and RSM.

CU

MU

LA

TIV

E

100%

0%

Potential TimeSavings

Knowledgeabout Design

DesignFreedom

Design Timeline

IncreaseKnowledgeMaintain

Freedom

Figure 5.24– Summary of Hypothesis Validation.


211


Among different design requirements considered in Q2, only functional propertiesthat are related to material properties are considered in Q2.1. Accordingly, for thegeometric tailoring considered in hypothesis 2, only material geometric tailoring is testedin hypothesis 2.1.

With the principles of geometric tailoring introduced in this chapter, a design forRapid Tooling system is presented in the next chapter (Figure 5.25). Besides therequirements that are related to material properties, other requirements that are related toprocess planning are also considered in the system (Section 6.1). The Material-ProcessGeometric Tailoring (MPGT) problem and the MPGT decision template are described inSection 6.3. A solving approach for the MPGT problem is presented in Section 6.4.


R1

R2

Rk

R3

F1

F2

F3

F4F5

F6

Fn

PL1

PL2

PD1

PD2

Part P

F1

F2

F3

F4

F5

F7

F8

Fn

Part P

(1) (2)

(3)

F1

F2

F3

F4

F5

F7

Fn

Part P

F8F8 F6

F6

F7

F9 F9

F9 M1

M2 Mk

Mold Base

PD1

F1

F2

F3

PD2

F4F5

F7 Fn



GivenAnalternative tobeimprovedthroughmodification;Assumptionsusedtomodel the domainof interest.

Thesystemparameters:n numberofsystemvariables p+q number of systemconstraintsp equalityconstraints q inequalityconstraintsm numberofsystemgoals Gi(X) systemconstraint functionfk(di ) functionofdeviationvariables tobeminimizedatpriority levelk for thepreemptive case.

FindValues for thesystemvariables Xi i = 1, ... , nValues for thedeviation variables di

-, di+ i = 1, ... , m


gi(X) = 0; i = 1, ..., p gi(X)≥ 0 ; i = p+1, ..., p+qSystemgoals (linear, nonlinear)

Ai(X) + di- - di

+= Gi ; i = 1, ..., mBounds

Ximin≤ Xi ≤ Xi


di-, di

+ ≥ 0; di-. di

+= 0; i = 1, ..., mMinimize

Preemptive deviationfunction(lexicographicminimum)Z=[f1(di

-,di+),..., f

k(di-, di

+)]


i(d

i

− + di

+) where Wi=1, W

i≥0∑∑

§2.2 §2.3 §2.4 §2.5 §2.6

MoldConfiguration

Design Methods


Tools

CADRepresentation


Techniques



ParametricCAD Model

of Part


Part DesignFind


ConstraintsGoals

MinimizeDeviation








Surface Fin ishAccuracyCostTime


Part Design

(Chp3 & 4)


GivenMold Design



MinimizeDeviation


GivenMold Design



MinimizeDeviation




Input and Output

Processor

C-DSP Template


(Section 6.2.3)

(Section 6.3)

(Section 6.4& 6.5)


Chapter 6 – A Decision-Based Design for Rapid Tooling System

212

CHAPTER 6

A DECISION-BASED DESIGN FOR RAPID TOOLING SYSTEM

In the current usage of Rapid Tooling, the iterations of design changes between thedesigner and manufacturer may take a long time before production-representativeprototypes are produced. Based on the principle of geometric tailoring presented in thelast chapter, a design for Rapid Tooling system (DFRTS) was presented for material-process geometric tailoring (MPGT) to address the problem in this chapter. First theinfrastructure and scope of the DFRTS are introduced in Section 6.1. The components ofthe DFRTS that are related to the process planning of the AIM tooling are described inSection 6.2. The MPGT decision template for the designer and the integrated MPGTproblem formulation are presented in Section 6.3. A solution strategy for the MPGTproblem and a three-stage solution process for the DFRTS are described in Section 6.4.Finally a comparison of the DFRTS and the current usage of RT is given from theperspective of decision-making.


ParametricCAD Model

of Part


Part DesignFind


ConstraintsGoals

MinimizeDeviation










Part Design

(Chp3 & 4)


GivenMold Design



MinimizeDeviation


GivenMold Design



MinimizeDeviation




Input and Output

Processor

C-DSP Template


(Section 6.2.3)

(Section 6.3)

(Section 6.4& 6.5)


213

6.1 OVERVIEW OF DESIGN FOR RAPID TOOLING SYSTEM

As stated in Section 1.2.3, the “customers” of the RTTB (Section 1.2.2) have a widevariety of design requirements. These requirements for the prototype parts may includethe shape of the products, function properties, tolerances, surface finish, batch size, costand time. Among these requirements for Rapid Tooling, the requirements on functionproperties are related to the material properties of the prototypes. In Chapter 5, thematerial geometric tailoring (MGT) problem and MGT decision template were presentedto address the differences of the material properties between the products and prototypes.The principles of geometric tailoring (Section 5.2.3) and MGTDT (Section 5.3.1) areapplicable to the material-process geometric tailoring (MPGT) and MPGT decisiontemplate that are to be presented in Section 6.3. The example of the rib part (Section5.5.2) also sets the stage for the research in this chapter with more manufacturingrequirements considered.

Some requirements for Rapid Tooling, such as tolerances, surface finish, batch size,cost and time, are tightly related to the fabrication process. Therefore the processplanning of the Rapid Tooling should also be considered. Consequently a Design forRapid Tooling System (DFRTS) is proposed in this chapter to address the geometrictailoring problem related to these requirements. Its relation with geometric tailoring andDFM is described in Section 1.2.4 and repeated here in Figure 6.1.

In Chapter 5, the approach based on the MGTDT formulated the decisions in designstages into a compromise DSP, and then transferred it to later stages. The requirementsof the later stages were integrated into the problem formulation, which was then solvedfor a satisficing solution (Section 5.3.1). In this chapter, the DFRTS employs the sameprinciple. That is, the decisions of the part design, RP process planning, and IJM processplanning are formulated into compromise DSPs. Instead of solving them for ‘optimal’solutions for each stage, the formulations are transferred to later stages and integratedinto a bigger problem. Finally a compromise DSP solver is developed in the DFRTS toget a satisficing solution for all the stages. The process is illustrated in Figure 6.2.

Mold Design

RP ProcessPlanning

IJM ProcessPlanning

GeometricTailoring

Design forRapid Tooling

(Chp 6)

Design forManufacture

Considerationson Topology,

Assembly,etc.

Figure 6.1 – Relations of DFRT, Geometric Tailoring and DFM.


214

The related sections in this dissertation for the components of the DFRTS are alsoshown in Figure 6.2. The input to the system is the part design whose functionalprototypes are to be produced by the Rapid Tooling. For the given CAD model of thepart, the method and related system for generating CAD models of mold pieces arepresented in Chapter 3 and 4 respectively. The MPGT decision template for design

ParametricCAD Model

of Part


Part DesignFind


ConstraintsGoals

MinimizeDeviation






Cooling TimeDraft AngleRib Height/width RatioPart Thickness


Building DirectinLayer ThicknessFill OvercureHatch Overcure

E. Rapid ToolingCost EstimatorTimeCost

Part Design

(Chp3 & 4)

The RP ProcessPlanning FormulationGiven

Mold DesignFind

RP Process ParametersSatisfy

ConstraintsGoals

MinimizeDeviation

The IJM ProcessPlanning Formulation

GivenMold Design



MinimizeDeviation

D. MPGT ProblemSolver



Input and Output

Processor

C-DSP Template

(§6.2.1) (§6.2.2)

(§6.3.1)

(§6.2.3)

(§6.3.2, §6.4 )

Figure 6.2 – Infrastructure of the DFRTS and Related Sections.


215

requirements is presented in Section 6.3.1. The RP process planner (B) and injectionmolding process analyzer (C) are discussed in Section 6.2.1 and Section 6.2.2respectively. The MPGT problem formulation and the related MPGT problem solver (D)are presented in Section 6.3.2 and 6.4. By solving the integrated problem, the tailoredpart design and the related mold design, Rapid Prototyping and injection molding processparameters are determined.

The DFRTS is a decision-based system because the decisions and their formulationsprovide a unified framework for integrating design and manufacturing requirements fordesign-for-manufacturing (DFM) problem. In this chapter, the author will focus onformulating the decisions in the part design and solving the integrated problem.

Before the components of the DFRTS are introduced, the research scope of thesystem is described as follows. As shown in Figure 6.3, four phases in using the directAIM tooling to build prototypes are (1) design part, (2) design mold, (3) build mold, and(4) build part. For each phase, there are several variables. The variables that areconsidered in the DFRTS are marked by underlines in the figure. Some other variables inphases II, III, and IV that are not considered are also shown in the figure. Thedetermination of the research scope was mainly based on the importance of the variables,and also the research work that had done at our lab (Rapid Prototyping andManufacturing Institute). In the next section, the computational modules that aredeveloped for RP and Injection molding (IJM) process are presented. Related to thesemodules, the software systems of SLA process planner, mold life predictor, and RT costestimator are also described.

Parting DirectionParting Lines

Parting SurfacePart OrientationEjection Pin No.

Ejection Pin PositionSprue TypeGate Size

Cooling Channelsetc.

Layer ThicknessBuilding Orientation

Hatch OvercureFill Overcure

Boder OvercurePre-dip delayPost-dip delay

etc.

Cooling TimeThermal Cure

Nozzle TemperatureClamping Force

Injection PressureInjection TimeHolding Time

Packing Pressureetc.

CostTime

Part NumberStressStrainWeight

Surface finishFlatness tolerance

Parallelism tolerancePositional toleranceCircularity tolerance

Concentricity tolerancePerpendicularity tolerance

PartCAD

Model Mold CAD ModelMold build by RP

Mold after Epoxy Back Fill Injection Part

Phase I:Design Part



Phase IV:Build Part

Z-level waitSweep period

Figure 6.3 – Research Scope of the DFRTS.


216

6.2 PROCESSING PLANNING OF THE AIM TOOLING

The process planning of the AIM tooling includes the process planning of RapidPrototyping (RP) and injection molding (IJM) processes. The computational modules ofbuilding time, tolerances, and surface finish that are related to the RP process arepresented in (McClurkin, 1997; Lynn, 1998; West, 1999). The computational modules ofmold life that are related to the IJM process are presented in (Cedorge, 1999; Le Baut,1999; Palmer, 1999; Pham, 2001; Rodet, 2001). The software systems that weredeveloped based on these computational modules are presented in (Sambu, 2001). In thework of the DFRTS the author worked closely with Shiva Sambu, a M.S. student at ourlab. We also used the same case studies, robot arm and camera roller, which will bepresented in Chapter 7 and 8 respectively.

To foster a better understanding of the DFRTS, the word and mathematicalformulations of the RP and IJM processes shown in Figure 6.2 are presented in Section6.2.1 and 6.2.2. The software systems of SLA process planner, mold life predictor, andRT cost estimator are briefly discussed in Section 6.2.1, 6.2.2, and 6.2.3 respectively.

6.2.1 SLA Process Planner

Related to the RP process used in the AIM tooling (Section 1.1.3), SLA processplanner corresponds to the RP process planner (B) shown in Figure 6.2. SLA processplanner develops a plan that is used to fabricate a part in the stereolithography process(Section 1.1.2). These plans consist of a set of parameter values that influence or controlhow the part is to be fabricated.

Many properties of the fabricated parts, such as surface finish and tolerances, dependon the values of the process variables (e.g. build orientation and layer thickness).However changing the value of a process variable may affect several properties in theways that may conflict with each other. For example, decreasing the layer thickness inthe fabricating process can build parts with better surface finish. However, it will alsoincrease the building time and cost significantly. Therefore tradeoffs are necessary inorder to generate a good plan.

As a multi-objective problem, the SLA process planning can be formulated using thecompromise Decision Support Problem (cDSP). The word formulation of the SLAprocess planning is given in Table 6.1.

Table 6.1 – SLA Process Planning Word Formulation (Sambu, 2001).

GIVEN:

• CAD model of part • SLA material property models • Target values for SLA process goals • Target mold life

• SLA process models• Goal preferences as weights

FIND:


SLA process variables Deviation of goals from targets

SATISFY:


217

Goals: Constraints:

Meet target material properties Meet SLA process constraints

Meet targets of SLA process goals Bounds:


MINIMIZE:


As shown in Figure 6.3, the SLA process variables that are considered are PartOrientation (PO), Layer Thickness (LT), Hatch Overcure (HOC) and Fill Overcure(FOC). The SLA process goals are Surface Finish (SF), Accuracy (AC), Build Time(BT), and material properties of Young’s Modulus (YM) and Tensile Strength (TS). Forthe process variables and goals, the mathematical formulation of SLA process planningproblem is presented in Table 6.2. The formulation is a general math formulation whichcan be extended for other RP processes. The exact quantitative relationship betweengoals and system variables is given in (West, 1999) and (Sambu, 2001).

Table 6.2 – SLA Process Planning Mathematical Formulation (Sambu, 2001).

GIVEN:

• CAD model of part • ACTi, SFTi • YMT, TST, EYT • AC = f (PO, LT, HOC, FOC, ZL, SP) • YM = f (LT, HOC) • SF = f (PO, LT) • TS = f (LT, HOC) • BT = f (PO, LT, HOC, FOC, ZL, SP)

FIND:


Part Orientation in vat (PO)

Slicing scheme (LT)

Hatch Overcure (HOC)

Fill Overcure (FOC)

Z-Level wait (ZL)

Sweep Period (SP)

d1+, d1

-, d2+, d2

-, d3+, d3

-, d4+, d4

-,dk

+, dk-, dj

+, dj-

SATISFY:

Goals: Constraints:

(YMT / YM) - d1+ + d1

- = 1 di+ • di

- = 0

(TST / TS) - d2+ + d2

- = 1 Large horizontal planes(BTmax - BT) / (BTmax - BTmin) - d4

+ + d4- = 1 Support structures

(ACTk / ACk) - dk+ + dk

- = 1 Bounds:

(SFTj / SFj) - dj+ + dj

- = 1 Bounds for all system variables

MINIMIZE:

Deviation Function: g (d1+, d1

-, d2+, d2

-, d3+, d3

-, d4-, dk

-, dj-)


218

A software system based on the ACIS solid modeling kernel was developed in C++on a PC by (West, 1999) and refined by (Sambu, 2001). A screen capture of the SLAprocess planner is given in Figure 6.4.

6.2.2 Stereolithography Mold Life Predictor

Related to the SLA molds that are used in the injection molding (IJM) process of theAIM tooling, stereolithography mold life predictor corresponds to the injection moldingprocess analyzer (C) shown in Figure 6.2. As discussed in Section 5.1, the mold life ofthe direct AIM tooling is much lower than that of the steel tooling because of the materialand fabrication properties of stereolithography molds. Therefore the mold life is a mainconcern in the IJM process planning, and is formulated in the word formulation for theIJM process planning (Table 6.3).

Table 6.3 – IJM Process Planning Word Formulation (Sambu, 2001).

GIVEN:

• CAD model of molds • Mold life models • Target mold life • Goal preferences as weights

FIND:


IJM Process variables Deviation of goals from targets

Figure 6.4 – Screen Capture of the SLA Process Planner.


219

SATISFY:

Goals: Constraints:

Meet target mold life Meet IJM process constraints

Bounds:


MINIMIZE:


Many factors affect the mold life of SLA molds (Cedorge, 1999; Le Baut, 1999;Palmer, 1999; Pham, 2001; Rodet, 2001). These factors can be broadly classified intofour categories: geometry variables, injection molding parameters, SLA parameters andmaterial properties. Material properties include the properties of both part and moldmaterial. The factors that are investigated in the mold life predictor are shown in Table6.4. These factors are chosen based on the data obtained from experiments performed atGeorgia Tech. Some of these factors are investigated qualitatively while the others areinvestigated to obtain quantitative models.

Table 6.4 – Factors Selected for Mold Life Prediction (Sambu, 2001).

Geometry variablesInjection molding

parameters SLA parameters Material properties

Feature type Injection pressure Layer Thickness Young’s modulus

Feature size Injection velocity Hatch Overcure Tensile strength

Draft angle Injection temperature Border Overcure Heat deflection temperature

Surface area Ejection temperature Laser beam size Glass transition temperature

Hydraulic radius Ejection force Degree of cure Friction coefficient

Bonding Area Cooling time Thermal cure Poisson’s ratio

Wall thickness Hold pressure Thermal expansion coefficient

Gating type

Among the injection molding parameters, only the cooling time is related to thefabrication cost and time directly. Although the other variables are also related to themold life, they are mainly determined by some other requirements. For example, theinjection pressure should be large enough to enable the polymer to fill the whole cavity.These requirements are not considered in the IJM process planning formulation of theDFRTS. Therefore there is only one system variable (cooling time) in the cDSPmathematical formulation of IJM process planning problem (Table 6.5).

In the formulation, cooling time (CT) is the time from injection of molten plastic intothe mold to the ejection of solidified part from the mold. The cooling time has a lowerbound of 180 seconds and an upper bound of 330 seconds. The lower bound is to ensurethat the parts are solidified before they are ejected out of the mold. The mold lifepredictor is applicable for CT ≤ 330 and hence this value is chosen as the upper bound.The sub-script ‘k’ corresponds to the number of molds (NM) required to injection mold


220

the desired number of parts (NP). So, there is a total of NM system variables in Table 6.5.The formulation has two goals: cost and time. A modified cDSP formulation proposed in(Hernandez and Mistree, 2001) is used to formulate the goals and their weights, which isdiscussed in Section 6.3.2. The method to evaluate cost and time is presented in Section6.2.3.

Table 6.5 – IJM Process Planning Mathematical Formulation (Sambu, 2001).

GIVEN:

• CAD models of mold pieces • RP variables used to build the mold • Mold life ML = f (CT) • Number of parts NP • Time T = f (ML, CT, NP) • Cost C = f (ML, CT, NP)

FIND:


CTk d dj p j p, ,,+ −

SATISFY:

Goals: Constraints:

Cost:C C t

td dp

pp p

− −+ − =+

+

+− +max( , ),

,, ,

1 1

11 1

01

C t≤ +1 5,

Time:T T t

td dp

pp p

− −+ − =+

+

+− +max( , ),

,, ,

2 1

22 2

01

T t≤ +2 5,

Bounds: d dj p j p, ,*+ − = 0

180 ≤ CT ≤ 330 d dj p j p, ,,+ − ≥ 0

MINIMIZE:

Deviation Function: w d dj pjp

j p j p, , ,==

+ −∑∑ +1

2

1

4

c h

A software system based on the mathematical modes was developed in C++ on a PCby (Sambu, 2001). A screen capture of the SL mold life predictor is given in Figure 6.5.


221

6.2.3 Rapid Tooling Cost Estimator

The cost and time of the fabrication process are related to the process planning.Developing quantitative models of cost and time requires analysis of the whole rapidtooling process. The steps involved in obtaining injection molded parts from the CADmodel of the part in rapid tooling process are listed in Table 6.6. The estimated values oftime and cost for each step are also given in Table 6.6.

Table 6.6 – Cost and Time Estimates for Different Steps in Rapid Tooling Process(Sambu, 2001).

Step Time (hr) Cost ($ / hr)Mold design 0.5 50

SLA Preparation 0.25 50SLA 250 - 35

SLA Building BTSLA 3500 - 65

Human – 0.25 Human – 20RP Cleaning

TPM – 0.5 TPM – 5Postcuring 1 10

Thermal curing TC * 8 10Human – 0.25 Human – 20

BackfillingSetting – 12 Setting – 0

Human – 20Machining 0.25

Machine – 20Human – 20

Assembling 0.25Machine – 30

Molding parts CyT * NP 30

Figure 6.5 – Screen Capture of SL Mold Life Predictor.


222

The time and cost of all the steps is straightforward except the SLA building step.The build time of prototypes using a SLA machine depends upon the geometry of theparts and also the SLA process variables. The SLA build time estimator, presented in(McClurkin and Rosen, 1998), reads the vector (.v) and range (.r) files created by3Dlightyear or Maestro, 3D System’s software, and calculates the build time of theprototype to within roughly 2%. There are three common types of build vectors used inthe stereolithography process: fill vectors, hatch vectors, and border vectors. These threetypes of vectors may be directly related to the properties of the sliced CAD model.

A software system based on the data in Table 6.6 and the SLA build time estimatorwas developed in C++ on a PC by (Sambu, 2001). The Rapid Tooling cost estimator canreceive RT process parameters and return the time and cost related to the processparameters.

After the process planning modules of the DFRTS are presented, MPGT and MPGTdecision template are described in the next section.

6.3 MPGT DECISION TEMPLATE AND MPGT PROBLEM FORMULATION

The designer may have different requirements for the prototypes of a part design. Asstated in Section 5.3, if only functional properties (e.g. maximum stress and deflection)are the main concerns for the prototypes, the designer can initiate a problem formulationbased on the material geometric tailoring decision template (MGTDT), and transfer it tothe manufacturer. Since the process variables are not needed in the formulation, theproblem can be solved quickly and easily (refer to the example given in Section 5.5).However, if the requirements of surface finish, tolerances, cost and time are alsoconsidered for the prototypes, the material-process geometric tailoring decision template(MPGTDT) should be used instead (Figure 6.6). Since more goals are added into theformulation, the tradeoffs have to be made between the functional properties and theother requirements, which are related to the process planning of RT. It is more difficult

FunctionalProperties

Tolerances

Flatness toleranceParallelism tolerancePositional tolerance

Circularity toleranceConcentricity tolerancePerpendicularity tolerance

StressStrainLoadetc.

Surface Finish

Cost and Time

Designer's Requirements for Prototypes

MGTDT

MPGTDT

(§5.3)

Figure 6.6 – Relations of Designer’s Requirements and MGTDT/MPGTDT.


223

because of the coupling between sub-problems (Section 6.4).

The word and mathematical formulations of the MPGTDT are presented in Section6.3.1. Based on the MPGT formulation given by the designer and the process planningformulations given in the last section, an integrated MPGT problem formulation isdescribed in Section 6.3.2.

6.3.1 MPGTDT

As stated in Section 5.3.1, the definition of Material-Process Geometric TailoringDecision Template is:

• MPGTDT – Material-Process Geometric Tailoring Decision Template, acompromise decision in which the component’s dimensions are modified to suit aprototype material and fabrication process.

The MPGTDT has the same principle as that of the MGTDT (Section 5.3.1). Theyalso have similar formulations, except more goals and constraints are added in theMPGTDT. The compromise DSP word formulation of the MPGTDT is presented inTable 6.7, in which tolerances and surface finish are called process goals.

Table 6.7– Word Formulation of the MPGTDT.

GIVEN:

• Parametric CAD model of part • Functional property models • Target mold life • Material Properties • Target values for functional properties • Prototype material properties • Target values for process goals • Target cost and time

FIND:


Geometry variables Deviation of goals from targets

SATISFY:

Goals: Constraints:


Meet targets of geometry variables

Meet targets of process goals Bounds:

Meet targets of mold life Bounds for all system variables

Meet target cost and time

MINIMIZE:


Correspondingly to the word formulation, a general mathematical formulation ofMPGTDT is presented in Table 6.8.


224

Table 6.8 - Mathematical Formulation of the MPGTDT.

GIVEN:

• nf - number of functions;ng - number of geometry variables; nm - number of material variablesnsf - number of surface finish variables; ntol - number of tolerance variablesp – equality constraints; q – inequality constraints.

• Gp, i i = 1, …, ng

• Mp, i i = 1, …, nm

• Fp, i = fi (Gp,j, Mp,n) i = 1, …, nf; j = 1, …, ng; n = 1, …, nm

• Mm, i** i = 1, …, nm

• Fm, i = fi (Gm,j, Mm,n)** i = 1, …, nf; j = 1, …, ng; n = 1, …, nm

• SFi,T , SFi** i = 1, …, nsf

• Toli,T , Toli** i = 1, …, ntol

• MLT, ML** • Time: [Tmin , Tmax], T** • Cost: [Cmin , Cmax], C** • Wi i = 1, …, nf+ng+nsf+ntol+2

FIND:

System Variables:

Gm, i i = 1, …, ng

Deviation Variables:

di+, di

- i = 1, …, nf+ng+nsf+ntol+2

SATISFY:

Goals:

Fp, i / Fm,i - di+ + di

- = 1 i = 1, …, nf [6.1]

Gp,i / Gm,i - dnf+i+ + dnf+i

- = 1 i = 1, …, ng [6.2]

SFi,T / SFi - dnf+ng+i+ + dnf+ng+i

- = 1 i = 1, …, nsf [6.3]

Toli,T / Toli - dnf+ng+nsf+i+ + dnf+ng+nsf+i

- = 1 i = 1, …, ntol [6.4]

(Tmax–T)/(Tmax–Tmin)- di+ + di

- = 1 i = nf+ng+nsf+ntol+1 [6.5]

(Cmax–C)/(Cmax–Cmin)- di+ + di

- = 1 i = nf+ng+nsf+ntol+2 [6.6]

Constraints:

di+ • di

- = 0, di+ ≥ 0, di

- ≥ 0 i = 1, …, nf+ng+nsf+ntol+2

gj(Gi) = 0, gk(Gi) ≤ 0 i = 1, …, ng; j=1, …, p; k=1, …, q Bounds:

Gimin ≤ Gi ≤ Gi

max i = 1, …, ng

Fimin ≤ Fi ≤ Fi

max i = 1, …, nf

SFimin ≤ SFi ≤ SFi

max i = 1, …, nsf


225

Tolimin ≤ Toli ≤ Toli

max i = 1, …, ntol

Tmin ≤ T ≤ Tmax Cmin ≤ C ≤ Cmax

MINIMIZE:

Deviation Function: Wi • (di+ + di

-) where ΣWi = 1, Wi ≥ 0 (i= 1, …, nf+ng+nsf+ntol+2)

Note:1. Subscripts m and p denote the prototype model and the product respectively.2. Symbol ‘**’ denotes the entries that are to be completed by the manufacturer.

Compared with the mathematical formulation of the MGTDT given in Table 5.4,more variables and goals (Equations 6.1 ~ 6.6) are considered in the MPGTDT besidesgeometry variables (Gj), material properties (Mn), and functional properties (Fi). Thesevariables and goals include surface finish (SFj), tolerance (Toli), mold life (ML), time (T)and cost (C). As stated before, they are tightly related to the process planning. Thereforemore entries are indicated by ‘**’, which denotes information that the manufacturer mustsupply in order to complete the problem formulation and generate a solution.

System Variables

Goals & Constraints

RP Process Planning

Given

AC = f (PO, LTi, HOCi, FOCi)SF = f (PO, LTi)

ML = f (LT, TA)

Nm = f (ML)BT = f (PO, LTi, HOCi, FOCi, Nm)Cm = f (BT, TA)Tm = f (BT, TA)

PO, LTi, HOCi, FOCi, TA

AC, SF, Cm, Tm

System Variables

Goals & Constraints

MPGTDT

Given

Gi, θj

System Variables

Goals & Constraints

IJM Process Planning

GivenCp = f (CT)Tp = f (CT)

CT

Cp, Tp

System Variables

Goals & Constraints

MPGT Problem

Given

AC = f (PO, LTi, HOCi, FOCi)SF = f (PO, LTi, Gj, θθθθk)ML = f (LT, TA, Gj, θθθθk, CT)Nm = f (ML)BT = f (PO, LTi, HOCi, FOCi, Nm)Cm = f (BT, TA)Tm = f (BT, TA)Cp = f (CT)Tp = f (CT)C = Cm + Cp

T = Tm + Tp

Gj, θk, PO, LTi, HOCi, FOCi, TA, CT

AC, SF, C, T,

Analysis

Synthesis

Fh = f (Gj, MPg)

Fh

Fh = f (Gj, MPg)

Fh

coupling

coupling

coupling

coupling

Figure 6.7 – Relations of the MPGTDT and MPGT Problem.


226

Based on the design information, the designer can instantiate the MPGTDT to createthe problem formulation suitable for communicating to the manufacturer. The usagescenario of the MPGTDT can be the same as the one presented in Section 5.3, thereforewill not be repeated here. The problem formulation given by the designer and theformulations of the process planning can be integrated into a MPGT problem formulation(Figure 6.7), which is to be discussed in the next section.

6.3.2 MPGT Problem Formulation

In the MPGTDT problem, the requirements on production-representative functions,accuracy, surface finish, and fabrication cost and time are considered. By integrating theproblem formulations of the MPGTDT (Table 6.7), RP process planning (Table 6.1) andIJM process planning (Table 6.3), one can obtain a Material Process Geometric Tailoring(MPGT) problem formulation. The word formulation of the MPGT problem is shown inTable 6.9.

Table 6.9 – MPGT Problem Word Formulation.

GIVEN:

• Parametric CAD model of part • Functional property models • Target values for process goals • RP process goals models • Target mold life • Mold life models

• Target values for functional properties • RP process models • Target material properties • Prototype material properties • Target cost and time • Goal preferences as weights

FIND:


Geometry variables

RP process variables

IJM process variables

Deviation of goals from targets

SATISFY:

Goals: Constraints:


Meet targets of geometry variables Meet RP process constraints

Meet targets of process goals Meet IJM process constraints

Meet target mold life Bounds:

Meet target cost and time Bounds for all system variables

MINIMIZE:


In the MPGT formulation, the system variables include geometry variables, RPprocess variables and IJM process variables. The goals include meeting the targetfunctional properties, geometry variables, mold life, cost/time, and process goals. Theprocess goals are the RP process goals. Experiments performed by (Cedorge, 1999)


227

showed that the surface finish of the part surfaces is very close to the surface finish of themold surfaces. Therefore the accuracy and surface finish of the part depends on theaccuracy and surface finish of the mold, and hence these goals are considered for themold instead of the part. The mathematical models of accuracy and surface finish for RPprocess can be used here directly. The material properties of RP process are notconsidered in the formulation because they are related to the molds instead of the parts.The constraints include meeting the geometry constraints, RP process constraints and IJMprocess constraints.

Related to the above word formulation, a MPGT problem mathematical formulationis presented in Table 6.10 by integrating the math formulations given for the designprocess (Table 6.8), RP process (Table 6.2) and IJM process (Table 6.5). One thing to benoticed in Table 6.10 is that the modified cDSP formulation proposed in (Hernandez andMistree, 2001) is used (Section 2.5). Hernandez and coauthors modified the objectivefunction (and correspondingly the goals) in cDSP formulation according to the LinearPhysical Programming (LPP) formulation developed by Messac, et al. (1996). In LPP,all the goals are classified into class 1S, 2S, 3S and 4S. Class 1S corresponds tominimization goals, class 2S corresponds to maximization goals, class 3S corresponds totarget matching goals and class 4S corresponds to range matching goals. Range matchinggoals have range of values as target instead of a single point. Hernandez and coauthorsproposed to split all the goals of class 3S and class 4S into two independent goals of class1S and class 2S. Therefore the designer can express preferences of each goal throughvarious degrees of desirability: unacceptable, highly undesirable, undesirable, tolerable,desirable, and ideal.

Table 6.10 – MPGT Problem Mathematical Formulation.

GIVEN:

• Parametric CAD model of part, GTi, θTs • Function F h = fh (Gj, MPg) • Accuracy AC = f (PO, LT, HOC, FOC, ZL, SP) • Material properties MPg • Surface finish SF = f (Gi, θs, PO, LT) • Mold life ML = f (Gi, θs, LT, TC, CT) • Build time BT = f (Gi, PO, LT, HOC, FOC, ZL, SP) • ACTq, SFTr, FTh, NP, CT, TT, MPTg • Total time T = f (BT, CT, TC, ML, NP) • Cost C = f (BT, CT, TC, ML, NP)

FIND:


Gi, θS

POkm, LTkmn, HOCkm, FOCkm, ZLkml, SPkml

TCk, CTk

d dj p j p, ,,+ −

SATISFY:

Goals: Constraints:

F F t

td dh h h p

h ph p h p

− −+ − =+

+

+− +max( , ),

,, ,

1 01 [6.7] h (Gi) = 0

F F t

td dh h h p

h ph p h p

+ −+ − =+

−

−− +min( , ),

,, ,

1 01 [6.8] t F th h h, ,5 5

− +≤ ≤


228

G G t

td di i i p

i pi p i p

− −+ − =+

+

+− +max( , ),

,, ,

1 01 [6.9] Large horizontal planes

G G t

td di i i p

i pi p i p

+ −+ − =+

−

−− +min( , ),

,, ,

1 01 [6.10] Support structures

θ θs s s p

s ps p s p

t

td d

− −+ − =+

+

+− +max( , ),

,, ,

1 01 [6.11] AC tq q≤ +

,5 , SF tr r≤ +,5

θ θs s s p

s ps p s p

t

td d

+ −+ − =+

−

−− +min( , ),

,, ,

1 01 [6.12] C t≤ +

4 5, , T t≤ +5 5,

AC AC t

td dq q q p

q pq p q p

− −+ − =+

+

+− +max( , ),

,, ,

1 01 [6.13] d dj p j p, ,

+ −• = 0

SF SF t

td dr r r p

r pr p r p

− −+ − =+

+

+− +max( , ),

,, ,

1 01 [6.14] d dj p j p, ,,+ − ≥ 0

C C t

td dp

pp p

− −+ − =+

+

+− +max( , ),

,, ,

1 1

11 1

01 [6.15]

Bounds:

T T t

td dp

pp p

− −+ − =+

+

+− +max( , ),

,, ,

2 1

22 2

01 [6.16] Bounds for all system variables

MINIMIZE:

Deviation Function: w d dj pj

h i o q r

pj p j p,

( )

, ,=

+ + + + +

=

+ −∑∑ +1

2 2

1

4

d i

Where,

g – index for material property variablesh – index for function variablesi – index for geometry dimension variablesj – running index for all deviation variablesk – index for the number of moldsl – index for the number of blocks of different Z-level wait (ZL) and Sweep

period (SW) in a mold piecem – index for the number of mold pieces in each mold designn – index for the number of blocks of different layer thicknesses in a mold piecep – 1,…4; used for goal formulation in LPP (linear physical programming)q – index for the number of accuracy requirementsr – index for the number of surface finish requirementss – index for draft angle variables

The system variables in the MPGT problem (Table 6.10) include the systemvariables from the MPGTDT (geometry variables: part dimensions Gi and draft angle θs),RT process planning problem (RP process variables: part orientation PO, layer thickness


229

LT, hatch overcure HOC, fill overcure FOC, z-level wait ZL, sweep period SP) and IJMprocess planning problem (IJM process variable: thermal cure TC and cooling time CT).

If single mold assembly cannot produce the desired number of parts (NP), multiplemold sets should be built. It is possible to build different mold sets with different valuesof geometry variables. But, in this formulation, it is assumed that all the molds arefabricated with same values of geometry variables to ensure that all the prototypes haveidentical properties.

There is a total of 2H+2I+2S+Q+R+2 goals in Figure 7.6 that include ‘2H’ functiongoals (Equations 6.7 and 6.8), ‘2I’ part dimension goals (Equations 6.9 and 6.10), ‘2S’draft angle goals (Equations 6.11 and 6.12), ‘Q’ accuracy goals (Equation 6.13), ‘R’surface finish goals (Equations 6.14) and one of each of time and cost goals (Equations6.15 and 6.16). The goals of part dimension, draft angle, and function are targetmatching goals and in LPP formulation of cDSP, each of these goals is divided into twoindependent goals of class 1S (minimization) and class 2S (maximization). The goals ofaccuracy, surface, cost and time are minimization goals (class 1S).

Function goals are affected by only geometry and material variables. Accuracy goalis affected by only the RP variables. Surface finish goal is affected by geometryvariables (e.g. draft angle) and RP variables (e.g. part orientation). Cost and time areaffected by all the system variables. The constraints in MPGT for RT problem includegeometry and assembly constraints affected by geometry variables, and RP constraints.The other constraints arise due to the LPP formulation of cDSP problem. Objectivefunction is formulated as a weighted sum of goal deviations (archimedean formulation).For each deviation, weight wj,p can be determined from target values of the related goal.For example, the target values of cost goal can be $98, $125, $150, $170, and $180corresponding to the target levels of ideal, desirable, tolerable, undesirable, andunacceptable. An algorithm that can be used for the calculation was presented in(Hernandez and Mistree, 2001).

After the MPGT problem formulation is given, a solution approach for the DFRTS ispresented in the next section.

6.4 SOLVING THE MPGT PROBLEM

After the MPGT problem was formulated in the last section, a solution strategy and arelated solution process are presented in Section 6.4.1 and 6.4.2 for the problemrespectively.

6.4.1 Solution strategy

The formulating and solution approaches for the MGT problem (Table 5.4) arepresented in Section 5.4.2. To get the quantitative relationship between goals and systemvariables, analytical equations or the response surface equations can be employed in theformulation (Section 5.4.1). Software systems such as DSIDES or OptdesX can be usedto solve the completed problem formulation. However the approaches cannot be directlyused for the MPGT problem.


230

Different from the MGT problem, the MPGT problem (Table 6.10) has geometricand discrete variables besides continuous variables. This is because the MPGT problemhas the requirements that are tightly related to the process planning (e.g. surface finishand build time), hence related process variables are added in the formulation. Someprocess variables may be discrete or related to geometry. In the research scope of theDFRTS, part orientation (PO) and layer thickness (LT) are two discrete variables. Theyare also two important process variables in the RP process planning. For a part as givenin Figure 6.8.a, some possible part orientations are shown in Figure 6.8.b. From the

(a) An Example Part

Orientation 1Orientation 2 Orientation 3 Orientation 4

Orientation 5 Orientation 6

(b) Different Part Orientations (PO) in SLA Building Process

Slicing scheme 1 Slicing scheme 2 Slicing scheme 3 Slicing scheme 4

(c) Different Slicing Schemes (LTi) for a Part Orientation (PO)

Figure 6.8 – Possible Values of PO and LTi for a Example Part (Sambu, 2001).


231

figure, it is evident that the effects of PO on accuracy (AC) or surface finish (SF) of facesare not continuous. Small changes of PO can cause large changes of AC and SF. Twoconstraints, large horizontal planes and support structures, are also related to the partorientations (West, 1999). For one of the part orientations, different slicing schemes areshown in Figure 6.8.c. For SLA process the values of layer thickness (LT) can only be0.002, 0.004 and 0.008 inch.

Besides the discrete variables, the number of variables LTi in the MPGT is not fixed.This is because the adaptive slicing is used in the RP process planning. Compared touniform slicing process, which uniformly slices the CAD model into a finite number ofslices at a certain thickness, the adaptive slicing can reduce the stair-step effect andimprove the surface quality, without greatly sacrificing the amount of fabrication time.However because different layer thickness is used, the number of LTi depends on thelocal surface geometry. For different part orientations, there may be different layerthickness.

Based on the above considerations, a solution strategy is developed for the MPGTproblem as shown in Figure 6.9. Suppose in a problem P, X is discrete variables and Y iscontinuous variables. The solution strategy divides the problem P into three sub-problems P1, P2, and P3. In the first sub-problem P1, a set of candidate values of thediscrete variables (X) are determined based on the default values of Y. For each candidatevalue of X, satisficing values of the continuous variables (Y) are determined in the secondsub-problem P2. Finally a solution is chosen among all the solutions given by P1 and P2

in the third sub-problem P3. The solution is then returned as the solution of problem P.

In many cases the solution given by the solution strategy is not optimal for problemP. However, it is a satisficing solution which is evident from the formulations of P1, P2,and P3. When several sub-problems are integrated into a problem P, the problem mayhave several variables, goals, and constraints (refer to Table 6.10). Therefore it will berather difficult, or even infeasible, to find the optimal solution of problem P. Instead an

Problem PFind

X, YMinimize

Z=f(X, Y)

P1Given

Y'Find

XMinimize

Z=f(X, Y')

P2Given

X'Find

YMinimize

Z=f(X', Y)

P3Given

Xi', Yi'Select

X, YMinimize

Z=f(Xi', Yi')

Figure 6.9 – The Solution Strategy for the MPGT Problem.


232

approach to find a satisficing solution may be more appropriate. The case studies of arobot arm and a camera roller (Chapter 7 and 8) will further illustrate it.

Considering the discrete variables PO and LT in Table 6.10, the MPGT problem isdivided into modified RP process planning problem (A) and modified MPGT problem(B) (Figure 6.10). Modified RP process planning is a sub-set of RP process planner andhas only part orientation (PO) and slicing scheme (LTi) as system variables. The purposeof modified RT process planning problem is to obtain a set of candidate orientations and

System Variables

Goals & Constraints

MPGT ProblemGiven

Fh = f (Gj, MPg)AC = f (PO, LTi, HOCi, FOCi)SF = f (PO, LTi, Gj, θk)ML = f (LT, TA, Gj, θk, CT)

Nm = f (ML)BT = f (PO, LT i, HOCi, FOCi, Nm)

Cm = f (BT, TA)Tm = f (BT, TA)Cp = f (CT)Tp = f (CT)C = Cm + Cp

T = Tm + Tp


AC, SF, C, T, Fh

System Variables

Goals & Constraints

A. Modified RP Process Planning

GivenAC = f (PO, LTi)SF = f (PO, LTi)BT = f (PO, LTi)

PO, LTi

AC, SF, BT

B. Modified MPGT Problem

System Variables

Goals & Constraints

Given

AC = f (PO, LTi, HOCi, FOCi)SF = f (PO, LTi, Gj, θk)ML = f (LT, TA, Gj, CT, θk)

Nm = f (ML)BT = f (PO, LTi, HOCi, FOCi, Nm)Cm = f (BT, TA)Tm = f (BT, TA)Cp = f (CT)Tp = f (CT)C = Cm + Cp

T = Tm

+ Tp

Gj, θk, HOCi, FOCi, TA, CT

AC, SF, C, T, Fh

Fh = f (Gj, MPg)PO=PO', LTi=LTi'

x1

x2

y

OneSolution

Large horizontal planesSupport structures

Large horizontal planesSupport structures

Figure 6.10 – Two Sub-problems of the MPGT.


233

a set of promising slicing schemes for all the mold pieces. The goals considered in theproblem are surface finish, accuracy and build time. The default values of other RPprocess variables (e.g. hatch overcure HOC and fill overcure FOC) are used in computingthe values of the goals. The constraints of large horizontal planes and support structuresare also considered in determining part orientations. The results of modified RP processplanning include a set of slicing schemes for a set of part orientations of each mold piece(Sambu, 2001).

The solutions of PO and LTi are then used to solve modified MPGT problem. Theonly difference between MPGT problem and modified MPGT problem is that moldorientation and slicing scheme are fixed in modified MPGT problem. Therefore theproblem is simplified because of less system variables and constraints. By generatingequations of goals and the continuous variables, one can get the solution of thecontinuous variables related to the given PO’ and LTi’.

Finally a selection is performed to determine the best of the obtained solutions. Thissolution is accepted as the solution for the MPGT problem.

From the infrastructure of the DFRTS (Figure 6.2), one can see the MPGT problemis actually formulated for one mold piece generated by the RTMDS. Because the molddesign variables (parting direction, parting line, parting surface) in the RTMDS are alsovariables that are related to part geometries (Chapter 3 and 4), they can be treated in thesame way as PO and LTi. The solution processes of the DFRTS and relatedimplementations are presented in the next section.

6.4.2 Solution Process and Implementations

Related to the solution strategy presented in Figure 6.9, the solution process of theDFRTS can be divided into three phases, modeling design functions and fabricationprocesses, solving modified MPGTs, and selecting among different solutions (Figure6.11). They are described in more details as follows.

• Phase I: Modeling Design Functions and Fabrication Processes.To make tradeoffs between different goals, the quantitative relationships between the

goals and system variables should be generated first. In the MPGT problem (Table 6.10),equations are needed for function (Fh), accuracy (AC), surface finish (SF), mold life(ML), build time (BT), total time (T) and cost (C). These equations can be divided intotwo categories, models of design functions and models of fabrication processes.

- Models of Design Functions.

Design function Fh in the MPGT problem can be any functional properties that thedesigner is interested in. For example, it can be maximum stress (refer to Section 5.5.3),maximum deflection (refer to Section 5.5.2), or load (refer to Section 5.5.1). Theapproaches to formulate equations for the MPGT problem are the same as those for theMGT problem (Section 5.4.1). In the case studies to be presented in Chapter 7 and 8, anapproach similar to RCEM (Chen, 1995) was used to formulate the design functions ofstress, deflection, and rotation. The approach is shown in Figure 6.11. It has five mainsteps, which are (1) identify factors and ranges of system variables; (2) design of


234

experiments; (3) execute simulations (ANSYS); (4) analyze experiment results; and (5)build response surface models.

It is obvious that models of design functions are related to design requirements.Therefore the above steps should be executed for different parts and design requirements.

- Models of Fabrication Processes.

Models of process properties are important in the process planning. The processproperties in the scope of DFRTS are accuracy, surface finish, mold life, build time ofSLA, time and cost of RT. One thing to be noticed is that these models are needed fordifferent fabrication processes; however, for different parts using the same fabricationprocess, the models can be reused.

A significant amount of work is done at Georgia Tech. in obtaining quantitativemodels for accuracy, surface finish and build time on SLA-250 (SOMOS 7110) andSLA-3500 (SL 7510) (McClurkin, 1997; Lynn, 1998; West, 1999; Sambu, 2001). Theyare introduced briefly to foster a better understanding of the capabilities and limitationsof the DFRTS.

Accuracy AC = f (PO, LT, HOC, FOC, ZL, SP):

Mold life

Variables

Objective funcitonand constraints

Systemvariables

A. Rapid ToolingMold Design

System

B. Refined RPProcess Planner

A set of molddesigns

A set of POand LT

D1. OptdesX

D2. C DataModule

C. Mold LifePredictor

E. RT CostEstimator

Cost andTime

Variables

x1

x2

y

x1

x2

y

RSEs ofFh, AC, SF

RSE of ML

Solution for amold designand PO, LT

Selection

Solution ofMPGT

...

...

DesignFreedom

x1

x2

y

Part Design

1.1 Identifyfactors and

ranges

1.2 Design ofExperiments

1.3 PhysicalExperiment /SimulationPrograms

1.5 BuildResponse

Surface Models

1.4Analyze

ExperimentResults

Phase I: Modeling designfunctions and fabrication

processes

Phase II: Solving modified MPGTs Phase III:Selecting

amongdifferentsolutions

(othersolutions)

Solutionsfor differentmold designand PO, LT

Figure 6.11 – The Solution Process of the DFRTS.


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Lynn (1998) performed a number of experiments on SLA 250 – SOMOS 7110 anddeveloped mathematical models to predict accuracy of SLA prototypes. The first set ofexperiments performed by Lynn was screening experiments. From the screeningexperiment results, Lynn identified hatch overcure, fill overcure, sweep period, z-levelwait, layer thickness and part orientation to be the RP variables that have significanteffect on part accuracy. A Face Centered Composite design of experiment is used todetermine the next set of experiments (main experiments) used to generate responsesurface models. Different types of accuracies are studied, which include flatness,parallelism, perpendicularity, concentricity, circularity, and positional tolerance. Byusing the similar experimental methodology, (Davis, 2001) generated quantitative modelsfor accuracy of the parts built on SLA 3500 – SL 7510.

Surface finish SF = f (Gi, θs, PO, LT):

West (1999) performed surface finish experiments on SLA 250 – SOMOS 7110 anddeveloped quantitative models to predict surface finish. The RP variables that affectsurface finish are part orientation and layer thickness. Experiments were run for threedifferent layer thickness (2, 4 and 8 mils) for a test piece as shown in Figure 6.12. Thetest piece consists of ten squares each rotated in an increment of 5° across the length ofthe piece. This geometry allows a surface finish measurement to be taken from each ofthe rectangular planar surfaces. In performing the surface finish experiments, effect ofsupport structure was eliminated by taking measurements from the region of the surfacethat is not affected by supports. Additional experiments were performed to predictsurface penalty due to support structures. By using the similar experimental method,Sambu (2001) generated quantitative models for surface finish of the parts built on SLA3500 – SL 7510.

Build time BT = f (Gi, PO, LT, HOC, FOC, ZL, SP):

McClurkin (1997) developed a Build Time Estimator (BTE), which reads the vector(.v) and range (.r) files created by Maestro, 3D System’s software for stereolithography,and calculates the build time of prototypes with errors of roughly 2%. Since these filesare still not available in the SLA process planning, West (1999) developed quantitativemodels based upon empirical data collected from the BTE. In West’s experiments, the

Figure 6.12 – Surface Finish Test Piece.


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build time was calculated from hatch time, fill time, border time and recoat time. Layerthickness, cross-section area and perimeter of the layer were considered as the variablesthat affect build time. Based on West’s work, Sambu (2001) performed sets of build timeexperiments on SLA 250 and SLA 3500, and generated better build time models that areapplicable over larger range of RP variables.

Total time T = f (BT, CT, TC, ML, NP) and Cost C = f (BT, CT, TC, ML, NP):

Sambu (2001) developed quantitative models of cost and time based on the analysisof the whole Rapid Tooling process. The whole direct AIM tooling process is dividedinto ten steps as shown in Table 6.6. The estimated values of time and cost /hr for eachstep are also presented in the table. They were determined based on the practice at ourlab and literature survey.

Mold life ML = f (Gi, θs, LT, TC, CT):

A significant amount of work is done at Georgia Tech. in obtaining quantitativemodels for stereolithography mold life on direct AIM tooling (Cedorge, 1999; Le Baut,1999; Palmer, 1999; Crawford, 2001; Pham, 2001; Rodet, 2001). Different types offailures and factors that affect mold life were identified (Section 6.2.2). Sambu (2001)developed a software system based on the quantitative models from these work to predictejection force and stereolithography mold life.

• Phase II: Solving Modified MPGTs.For a part design, the software modules in this phase include Rapid Tooling mold

design system, refined RP process planning, mold life predictor, optimization software(OptdesX), a C data module, and Rapid Tooling cost predictor. For a part design, a set ofmold designs is generated by the Rapid Tooling mold design system. For each molddesign, a set of part orientations and slicing schemes is generated by the refined RPprocess planner. For each of them, a modified MPGT problem can be formulated byconsidering the equations of goals and system variables. It is then solve by OpdesX withthe aid of the C data module, mold life predictor, and RT cost predictor. The softwaremodules in phase II are described in more detail as follows.

A. Rapid Tooling Mold Design System.

The RTMDS generates a set of mold designs for the CAD model of a part. TheCAD models of mold pieces are generated by the system (refer to Chapter 4).

B. Refined RP Process Planner.

Refined RP-PP software generates a set of promising process plans for each moldpiece for each mold design. The process plans obtained are only partial, as they do nothave information about RP process variables (HOC and FOC). Along with the processplans, an information file is also generated by RP-PP software. This file contains theinformation regarding the scan and recoat times for each of the mold pieces. Thisinformation is used in the C module to calculate the build time for the required quantitiesof all the mold halves (Sambu, 2001).


237

C. Mold Life Predictor.

According to given process variables, the mold life prediction software module canpredict the shot number of stereolithography molds before they fail (Section 6.2.2). Themodule was implemented in C and can be called by the C data module (D2).

D1. OptdesX.

OptdesX (www.et.byu.edu/~optdes/) is an engineering optimization software systemwhich uses Simulated Annealing (SAN) algorithm. Since it was used in the case studiesin Chapter 7 and 8, a detail description of the algorithm is given as follows in order tofoster a better understanding of the results given in the case studies.

SAN is a heuristic based optimization algorithm that does not use gradients infinding the optimum solution. SAN models a phenomenon in nature – the annealing ofsolids – to optimize a complex system. From a given starting point, the algorithm makessmall random changes in the design variables in the problem, causing a change in thevalue of objective function. If the change is negative (i.e., if objective function isreduced), then the current solution is better and it is accepted as the starting point for thenext iteration. If the change is positive (i.e., if the objective function is increased), thenew solution is worse; however, it may still be accepted according to the Boltzmannprobability factor presented in equation 6.1.

PE

k Tb

= −FHG

IKJexp

∆[6.17]

Where,

∆E is the change in objective function value,kb is Boltzmann constant, andT is the temperature (corresponds to actual annealing process).

This equation is used in annealing process. The Boltzmann probability ‘P’ iscompared to a random number between 0 and 1 drawn from a uniform distribution; if it ishigher, the solution is accepted. Accepting worse solutions in the earlier phases ofproblem execution, SAN escapes some local minima.

In the OptdesX implementation of SAN, temperatures T in equation 6.17 are notspecified. Instead, starting and final probability are specified. Apart from these values,number of cycles and maximum perturbation values should also be specified. Number ofcycles is the number of SAN iterations performed (number of different solution pointsinvestigated) before the SAN execution is stopped. Each cycle has different temperatureand hence the Boltzmann probability is calculated for each cycle. Maximum perturbationis the maximum allowable variation in the value of a system variable (per iteration).

SAN is an effective algorithm in solving discrete and mixed problems but it can alsobe used to solve continuous problems. As SAN is a heuristic based algorithm that usesrandom number generation, it is possible to obtain different solutions for differentattempts of solving a problem with the same starting point.


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D2. C Data Module.

As an analysis module for OptdesX, the C data module is used to calculate goals andintermediate responses for OptdesX. The values of the variables are passed fromOptdesX to the C module and the values of goals, constraints and objective function arecalculated in C module and passed to OptdesX (Figure 6.11). Some equations areembedded in the C program of the module. The C data module can also call the mold lifepredictor and the RT cost estimator for the values of mold life and cost/time respectively.

E. RT Cost Estimator.

According to given process variables, the rapid tooling cost estimator can predict thecost and time for the whole RT process (Section 6.2.3). The module was implemented inC and can be called by the C data module (D2).

• Phase III: Selecting the Solution of the MPGT.For each mold design, part orientation and slicing scheme, OptdesX gives a solution

with its deviation value Zi. After all the solutions are generated, the solution with leastvalue of Zi is selected as the solution for the MPGT problem.

The MPGT problem and the above solution processes are further illustrated in thecase studies in Chapter 7 and 8. In the next section, the author compares the solutionprocess of the DFRTS with the current usage of RT, and explains the advantages of theDFRTS base on decision order and design freedom of variables.

6.5 COMPARISON OF THE CURRENT USAGE AND DFRTS

In the current usage of Rapid Tooling, sub-problems of part design, RP processplanning, and IJM process planning are formulated and solved sequentially (Figure 6.13).That is, decisions on design variables are made first based on the design requirements.Then mold variables are determined and mold pieces for the part are generated. For themold pieces, decisions on RP process variables are made based on the requirements forRP process. Finally decisions on IJM process variables are made based on the remainingdesign freedoms.

However, the sub-problems of using Rapid Tooling are coupled. The arrows in Figure6.13 indicate the coupling between geometric tailoring, RP process planning, and IJMprocess planning. In the geometric tailoring formulation, geometry variables Gi affect themold life of stereolithography molds (Section 6.2.2). They could also affect surfacefinish if they affect surface orientation. θj is the draft angles for different features. Draftangle affects surface finish of drafted surfaces (Section 6.4.2). In the RP processplanning problem, ML is mold life. As explained in Section 6.2.2, mold life of a SL moldfeature is affected by its height, width, draft angle (geometric variables), layer thicknessused to build the feature (RP process variables) and the cooling time (IM processvariable) used in the injection molding of the parts. Also the goals of ‘meeting targets oftime and cost’ are coupled because the specified time and cost are for the wholefabricating process, which is the sum of those for each individual process.


239

The couplings between different sub-problems may cause the sequential solutionprocess unable to find solutions in some later stages. Therefore iterations are necessarywhich may take a long time.

In the DFRTS, the sub-problems are synthesized into a MPGT problem, and thecouplings between the sub-problems are formulated in the synthesis problem formulation.The coupled goals/responses are bolded in the problem formulation as shown in Figure6.14. Two additional goals, total cost (C) and total time (T), are also included in theformulation. These goals are not part of any of the sub-problem formulations but aresynthesized from mold cost/time (RP-PP) and part cost/time (IJM-PP).

For the MPGT problem, an approach of obtaining satisficing solution is developedby decomposing the problem into two sub-problems: modified RP process planningproblem and modified MPGT problem (Section 6.4.1). The decomposition of MPGTproblem is also shown in Figure 6.14. By solving the two new sub-problems, one can getthe solution for the MPGT problem.

So in the design for Rapid Tooling process, what is the benefit to synthesize the sub-problems into a combined problem, and then decompose the synthesis problem into sub-problems again in order to solve it?

System Variables

Goals & Constraints

RP Process Planning

Given


ML = f (LT, TA)



AC, SF, Cm, Tm

System Variables

Goals & Constraints

Geometric Tailoring

Given

Gi, θj

System Variables

Goals & Constraints



CT

Cp, Tp

PartCAD

Model Mold CAD ModelMold build by RP

Mold after Epoxy Back Fill Injection Part

Phase I:Design Part



Phase IV:Build Part

couplingcoupling

Fh = f (Gj, MPg)

coupling

coupling

(Chp 3 &4)

Fh

DecisionOrder: θj CTPO, LTi, HOCi, FOCi, TAGi,

MoldVariables

Figure 6.13 – The Current Usage of RT and Related Decision Order of Variables.


240

System Variables

Goals & Constraints

RP Process Planning

Given


ML = f (LT, TA)



AC, SF, Cm, Tm

System Variables

Goals & Constraints

MPGTDT

Given

Gi, θj

System Variables

Goals & Constraints



CT

Cp, Tp

System Variables

Goals & Constraints

MPGT Problem

Given

AC = f (PO, LTi, HOCi, FOCi)SF = f (PO, LTi, Gj, θθθθk)ML = f (LT, TA, Gj, θθθθk, CT)Nm = f (ML)BT = f (PO, LTi, HOCi, FOCi, Nm)Cm = f (BT, TA)Tm = f (BT, TA)Cp = f (CT)Tp = f (CT)C = Cm + Cp

T = Tm + Tp


AC, SF, C, T,

System Variables

Goals & Constraints

Modified RP-PP

GivenAC = f (PO, LTi)SF = f (PO, LTi)BT = f (PO, LTi)

PO, LTi

AC, SF, BT

System Variables

Goals & Constraints

Modified MPGT Problem

Given

AC = f (PO, LTi, HOCi, FOCi)SF = f (PO, LTi, Gj, θk)ML = f (LT, TA, Gj, CT, θk)Nm = f (ML)BT = f (PO, LTi, HOCi, FOCi, Nm)Cm = f (BT, TA)Tm = f (BT, TA)Cp = f (CT)Tp = f (CT)C = Cm + Cp

T = Tm + Tp

Gj, θk, HOCi, FOCi, TA, CT

AC, SF, C, T, Fh

Analysis

Synthesis

SolvingApproach

DecisionOrder:

MoldVariables PO, LTi θjGi, ,HOCi, FOCi, TA, CT

Fh = f (Gj, MPg)

Fh

Fh = f (Gj, MPg)

Fh

Fh = f (Gj, MPg)

coupling

coupling

coupling

coupling

Figure 6.14 – The DFRTS and Related Decision Order of Variables.

Solution


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Comparing the sub-problems before and after the synthesis (Figure 6.14), one cannotice that the decisions on system variables are reordered. That is, although the decisionon mold variables, PO, and LTi is still sequential, the decisions on all other variables (Gi,θi, HOCi, FOCi, TA, CT) are concurrent in the DFRTS. Concurrently formulating andsolving the variables enables us to explore more design spaces because the variables arenot fixed in the former stages. Consequently it is more likely to find a satisfying solution;hence the iterations and lead-time in the current usage of RT can be reduced. Thereforeusing a different decision order based on identified conflicting variables and goals is thekey for the above question.

Beside less iteration and time, other research also indicates that modelingconcurrency may result in better designs. Karandikar (1989) studied the design of acylindrical pressure vessel. He formulated the design and manufacturing requirementsrelated to the pressure vessel and solved the problems sequentially and concurrently. Bycomparing the results, Karandikar concluded that “coupling design and manufacture andsolving for the design and manufacturing variables concurrently, as opposed tosequentially, result in an increase in design freedom and possibly in the quality of theresulting design.” Sobieszczanski-Sobieski (1989) also gave a good example tographically illustrate the argument that the sequential approach may lead to a suboptimaldesign. The example is presented here to foster a better understanding of the advantagesof the DFRTS over the current usage of RT.

Suppose a two-variable design and manufacturing problem with X1 as designvariable and X2 as manufacturing variable. A certain performance measure P, expressedas the single goal, has to be maximized. Suppose the designer has constraints C1 and C2.The design space based on the design requirements can be sketched as shown in Figure6.15.a. It is obvious that the optimal solution is at O1 for the designer.

(a) Design Requirements (b) Design and Manufacturing requirements

Figure 6.15 – A Two Variable Design Space (Karandikar and Mistree, 1991).


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Assume the manufacture has an additional constraint C3 due to manufacturingconsiderations. Therefore the design space by considering all the requirements can besketched as shown in Figure 6.15.b. Since constraint C3 makes the earlier solution O1

infeasible, the new solution for the problem is at O3. With the concurrent formulation,one would get this solution. However using the sequential formulation and solutionapproach, the value of design variable X1 is frozen in the design stage, thus only X2 canbe changed in the formulation, and O2 is obtained as the solution. It is seen that O2 is ona lower P contour than O3 and, hence, represent a poorer design.

In the DFRTS, the decisions on continuous design and process variables are madeconcurrently by the manufacturer. By modeling the concurrency, a better design (withless cost) is obtained in the case study of a camera roller (Section 8.7). It can also beexplained by the analysis shown in Figure 6.15. Two case studies are presented in thenext two chapters to illustrate the usage of the DFRTS. In the next section, a briefsummary is given for discussing the hypotheses of the dissertation and the validationstrategy to be used in the case studies.


In the current usage of Rapid Tooling, the decisions are made in the same order asthe information flow. In each design stage, goals, constraints and preferences areformulated to make the decision. But these decision factors are not transferred to the nextstages, so design freedom decreases quickly. Consequently the cost and time to make thedecisions in the later stages increase dramatically. To address the problem, the decisionsin different design stages may be reordered so that the decisions of some couplingvariables can be formulated and solved concurrently. Based on this idea, a design forRapid Tooling system (DFRTS) was presented for material-process geometric tailoring(MPGT) in this chapter. First the infrastructure and scope of the DFRTS are introduced inSection 6.1. The components of the DFRTS that are related to the process planning ofthe AIM tooling are introduced in Section 6.2. The MPGT decision template for thedesigner and the integrated MPGT problem formulation are presented in Section 6.3. Asolution strategy for the MPGT problem and a three-stage solution process for theDFRTS are described in Section 6.4. Finally a comparison of the DFRTS and the currentusage of RT is given from the perspective of decision-making.

The research question and hypothesis that are related to this chapter are Q2.2, Q2.3and sub-hypothesis 2.2, 2.3. They are repeated as follows.

Q2.2. How to formulate the design for Rapid Tooling problem which integratesdecisions on design and manufacturing variables and other design andmanufacturing requirements including goals, constraints, and preferences?

Q2.3. How to solve the design for Rapid Tooling problem effectively andefficiently?

Sub-Hypothesis 2.2: The design for Rapid Tooling problem can be formulated byseveral compromise DSPs and tasks, which can then be integrated into adesign for Rapid Tooling system (DFRTS).


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Sub-Hypothesis 2.3: A three-stage solution process can be utilized to get asatisficing solution effectively and efficiently based on design andmanufacturing models and continuous/discrete variables.

Based on the validation strategy presented in Section 1.3.2, the DFRTS and therelated hypotheses were validated in the following categories.

! Theoretical Structural Validity

The structural validity includes validating the ingredients, the components, and theintegrity. They are described in more details as follows.

- Validating the ingredients

Validating the ingredients includes validating specific modules of a cDSPformulation (like response surfaces) and validating the functions in a softwareimplementation. Different aspects of validating the response surface models arepresented below.

a) Verifying that the R2, R2 (adj), maximum and average deviations of the responsesurface models are reasonably small.

b) Verifying the behavior of the response surface models with respect to the designvariables and studying if the behavior is intuitively reasonable.

c) Verifying if the response surface is a good estimation of the actual design space.Additional validation experiments should be performed to do this validation.

The response surfaces of the fabrication processes (Section 6.4.2) are problem-independent. Sambu (2001) described their validations in more details. The responsesurfaces of design functions are problem-dependent and are formulated and validated foreach problem separately (refer to Chapter 7 and 8).

Validating the functionality of a software implementation involves identifying thepotential problem-causing functions and testing their functionality for different sets ofinputs. The validation of RTMDS is discussed in Chapter 4. The validations of RPprocess planner, mold lifer predictor, and RT cost estimator are also presented in (Sambu,2001).

- Validating the components

Validating the components includes validating the formulation and implementationof different cDSP’s and tasks. Different aspects of validating the cDSP formulations arepresented as follows.

a) Verifying that the quantitative relationships between all goals /constraints and thesystem variables are available.

b) Verifying that the response surfaces used in the cDSP have reasonable fits.Otherwise the cDSP problem should be solved again with the response surfacesgenerated in a smaller design space that better approximates the actual designspace.


244

In the DFRTS, the cDSP’s are solved in OptdesX using Simulated Annealingalgorithm. The aspects of the validation of the implementation of cDSP’s are presentedbelow.

a) Verifying that the cDSP implementation satisfies the constraints on deviationvariables.

a. di+, di

- ≥ 0b. di

+ • di- = 0

b) Verifying that the Linear Physical Programming (LPP) formulation of cDSP hasresulted in a solution with reasonable values of goal achievements. This isachieved by observing the region of the goals in which the solution falls.

c) Verifying that the SAN algorithm is functioning correctly for the MPGT problemfor the chosen algorithm parameters.

d) Verifying the effect of starting point on the final solution of the problem.

e) Verifying the accuracy of the SAN algorithm by comparing its results to gridsearch results.

The above verifications were tested for the two case studies and are presented inChapter 7 and 8. In the case studies, the result of the cDSP was also validated bycomparing the values of estimated (response surfaces) and actual (experiment) values ofgoal achievements at the obtained solution.

- Validating the integrity

The integrity between different modules of the DFRTS is to test if the informationflow between the components is complete and correct. The integrity between differentsub-problems of MPGT problem is presented in 6.4.2. It is also validated through thetwo case studies in Chapter 7 and 8.

! Empirical Structural Validity

Two case studies are chosen for the validation of the DFRTS. They are a robot arm(Section 7.1) and a camera roller (Section 8.1). Both of them are industrial parts that areproduced by polymer injection molding. The functional requirements of the parts are themaximum stress, deflection and rotation, which are all typical in the functional testing ofprototypes. Other manufacturing requirements (e.g. surface finish, tolerances) are alsorequired for the prototypes because of assembly requirements. The case study of therobot arm is comparably simple, which makes the physical validation of each designrequirement feasible. The case study of the camera roller is rather complex. It is used totest that the DFRTS can be utilized for complex cases. Therefore the author believes thatthe case studies are appropriate to verify the performance of the system.

For each case study, the validations of ingredients, components and integrity asstated earlier were tested and are presented in Chapter 7 and 8 respectively.

! Empirical Performance Validity

The solutions given by the RTMDS for each case study were checked with theachieved values of design requirements. Physical prototypes were also produced for the


245

tailored part designs and the related process planning. For the case study of the robotarm, further physical validations are performed on testing the given requirements (surfacefinish, accuracy, material properties, weight, etc.) and performances of process planning(mold life, build time, etc.). All of these tests validate the usefulness of the DFRTS inproducing functional prototypes for Rapid Tooling.

! Theoretical Performance Validity

The discussions on decision order and related design freedoms presented in Section6.5 provide partial theoretical performance validation for Hypothesis 2.2 and 2.3.

The validations of hypothesis 2.2 and 2.3 also partially validate hypothesis 2, whichis repeated as follow:


In addition to the material geometric tailoring problem that was tested in hypothesis2.1 (Chapter 5), the material-process geometric tailoring was tested in hypothesis 2.2 and2.3 in this chapter. Therefore these two kinds of geometric tailoring problems for RapidTooling were all tested for hypothesis 2.

With the DFRTS introduced in this chapter, a robot arm case study is presented inthe next chapter (Figure 6.16). In the case study, both the RTMDS (Chapter 4) and theDFRTS (chapter 6) will be tested.


Chp 4: RTMDS and its UsageChp 6: Design for Rapid Tooling

ParametricCAD Model

of Part


Part DesignFind


ConstraintsGoals

MinimizeDeviation










Part Design

(Chp3 & 4)


GivenMold Design


SatisfyConstraintsGoa ls

MinimizeDeviation


GivenMold Design



MinimizeDeviation




Input and Output

Processor

C-DSP Template


(Section 6.2.3)

(Section 6.3)

(Section 6.4& 6.5)


Chapter 7 – Functional Prototypes of a Robot Arm

246

CHAPTER 7

FUNCTIONAL PROTOTYPES OF A ROBOT ARM

The RTMDS (Chapter 4) and DFRTS (Chapter 6) are applied to determine the molddesign and geometric tailoring for a robot arm design in this chapter. A problem ofproducing functional prototypes of a robot arm is introduced in Section 7.1. By using theRTMDS, two mold designs are generated for the robot arm (Section 7.2). A standardmold base for Morgan Press injection molding machine is used in the mold piececonstruction process. The formulation of the geometric tailoring problem for the robotarm is presented in Section 7.3. Among the three design functions, two of them arerepresented by response surface equations, and one is represented by an analyticalequation. The generation of the equations is also presented in Section 7.3. The solutionprocess of the MPGT problem, which consists of three stages, is described in Section 7.4.The modified MPGT problems that are related to eight slicing schemes are solved byOptdesX. The results of physical experiments for the validation of the problem solutionare presented in Section 7.5. Finally an evaluation of sequential and concurrent solutionprocesses for the robot arm case is provided in Section 7.6, and the validation of thehypotheses is discussed in Section 7.7.

Chp 7: Prototypes of a Robot Arm


247

7.1 A ROBOT ARM DESIGN – PROBLEM DESCRIPTION

A challenging task in many robot designs is to design and control a robot armequipped with a hand for skillful manipulation tasks. Accurate position, velocity andforce are required in the running of the robot arm. Because they are related to so manyfactors, besides the calculations and simulations, prototypes are usually required to testthe robot design. A case study of producing functional prototypes for a robot arm designis presented in this chapter. In performing the case study, the author worked togetherwith Shiva Sambu, a M.S. student in our lab, who also studied the Rapid Prototypingscenario for the robot arm case (Sambu, 2001).

Suppose a new robot arm design as shown in Figure 7.1.a is used in a roboticmechanism. Several of them are needed in order to obtain desired degrees of freedom.Usually, robot arms are made of steel (metal) to bear high loads with little deflection. Forlow load applications (e.g. toy), plastic arms are used. The loading conditions on therobot arm are shown in Figure 7.1.b. The larger cylindrical hole is completely fixed. A5N tensile load and a 20N bending load are applied uniformly on the smaller cylindricalsurface. For the loading conditions, the designer determined that the production robotarms are injection molded in atactic polystyrene material. Since the two cylindrical holesin the robot arm are connected with other components, they are required to have goodsurface finish (ideal – 20 µin; desirable – 40 µin; tolerable – 80 µin; undesirable – 130µin; unacceptable – 200 µin). Also the top and bottom surfaces (flat surfaces of the robotarm) have parallelism and flatness tolerance requirements according to assemblyrequirements. Suppose the parallelism tolerance is (0.002, 0.004, 0.008, 0.013, 0.020)inch and the flatness tolerance is (0.001, 0.002, 0.004, 0.007, 0.010) inch, whichcorrespond to (ideal, desirable, tolerable, undesirable, unacceptable) (refer to (Hernandezand Mistree, 2001) for the goal formulation).

D

d

t

(a) A Robot Arm and the Geometry Variables for Geometric Tailoring

(b) Loading Conditions on the Robot Arm

Figure 7.1 – A Robot Arm Design.

Fixed

Loads


248

After finishing the design, the designer required fifty functional prototypes of therobot arms fabricated in the production material for functional testing. Suppose in thetesting the designer is interested in the stress, displacement and weight of the prototypeparts under the loading conditions. Because the length of the robot arm affects theorientation of the links in the robotic mechanism, its dimension cannot be modified. Alsothe diameter of the holes affects the size of the bushings and hence cannot be changed.After considering the dimensions of the robot arm, the designer determined that thegeometry variables of D, d and t have some design freedom. As shown in Figure 7.1.a,‘D’ is the diameter of the link at the larger end, ‘d’ is the diameter of the link at thesmaller end, and ‘t’ is the thickness of the robot arm. Within the ranges given by thedesigner, the manufacturer can modify them in the produced prototypes.

Finally considering the time and budget constraints, the designer wants to get theprototypes within a week for the maximum cost of $1500.

7.2 MOLD DESIGN WITH AID OF RTMDS

The Rapid Tooling Mold Design System (RTMDS) was presented in Chapter 4. It isbased on the Multi-piece Mold Design Method (MPMDM) which was presented inChapter 3. The RTMDS is developed to reduce the design time in several importantmold design steps, including determining parting directions, parting lines, and partingsurfaces, and constructing mold pieces.

The size of the robot arm CAD file (.sat) is 103 KB. After loading it in the RTMDS,the mold designer can generate a mold design with two mold pieces as shown in Figure7.3.d within 18 seconds. The information regarding the part, generated regions, reverseglue operation, and the execution time of each step is listed in Table 7.1. The reader canrefer to the descriptions given in Section 4.4.2 for the meaning of each items. Therunning time given in the table is based on a personal computer with a 700 MHz Intel-IIIprocessor. One thing to be noticed is that the running time of step 7 includes the time forthe user to interactively select the input file name of the mold base. Similarly the runningtime of step 8 includes the time for the user to interactively select the output file names ofthe mold pieces.

Table 7.1 – The Information for Robot Arm.

Face No. Concave face No. Concave edge No.Part Info.

92 66 40Initially generated After dividing After combining

Region Info.26 26 1 region and 1 CXF


3 24 18+16GFps No. GFproj No. GFinner No.

Reverse glue Info.

1 0 2Step 1 Step 2 Step 30.01 1.83 0.00

Step 4 Step 5 Step 60.33 0.38 0.01


Running Time (s)

4.5* 10.1* 17.16*

Note: ‘*” denotes that the time of interactively selecting files is included in the measurement.


249

The graphical outputs given by the RTMDS are shown in Figure 7.2, Figure 7.3, andFigure 7.4. Different regions and convex faces (CVX) are marked with different colorsin the figures. A brief description of the figures is given as follows.

Based on the criterion of judging convex and concave edges (Section 3.4.3), thereare 40 concave edges in the robot arm. They are mainly related to the approximation ofthe cylindrical holes. Therefore 66 concave faces are identified for the approximation.Based on their connectivity, 26 regions are generated initially. These regions are shownin Figure 7.2.a, with a closer view shown in Figure 7.2.b. In the figure, region R1 consistsof F1 and F2. Because F2 and F3 are in the same plane, they are convex with each other.Therefore F3 and F4 will generate a new region R2 based on the region generation

(a) Generated Regions

(b) A Closer View of the Regions

(c) Two Combined Regions

Figure 7.2 – Generated Regions for the Robot Arm.

F1

F3F2

R1 R2

F4


250

algorithms (Section 3.4.3). They are combined into two regions during the regioncombination process as shown in Figure 7.2.c.

(a) Region Combination (1 region+ 1 CXF)

(b) A Mold Base for Morgan Press (c) Part with the Parting Surface


Figure 7.3 – Graphical Results of a Mold Design for the Robot Arm.

R1

CXF

GFinner1GFinner2 GFps


251

One region combination result is shown in Figure 7.3.a. The mold base used for therobot arm is given in Figure 7.3.b. It is one of the standard mold bases for the MorganPress@ injection molding machine. The generated mold pieces for the regioncombination result are shown in Figure 7.3.d. The generated glue faces (GFps, GFinner1,GFinner2) are also shown in the figure.

The user can change the combination results by setting different control options. Forthe robot arm, if the option of not combining vertical faces is set, a different combinationresult will be given as shown in Figure 7.4.a. Correspondingly, different mold pieces aregenerated by the RTMDS (Figure 7.4.b) with the glue faces (GFps, GFinner1, GFinner2) forthe mold pieces.

The generation of CAD models of mold pieces is an important task in the molddesign process. After it is finished, other tasks can be started. In IronCAD system

(a) Region Combination (1 region + 25 CXFs)


Figure 7.4 – Graphical Results of Another Mold Design for the Robot Arm.

R1

CXFs

GFinner1GFinner2 GFps


252

(www.ironcad.com), the author added ejector-pin holes, gate and runner in the moldpieces. A complete mold design for the robot arm is shown in Figure 7.5. The molddesign is further verified by physical validations. A photo of the built mold pieces andprototype parts is given in Figure 7.6.

Figure 7.5 – A Complete Mold Design for the Robot Arm.

Figure 7.6 – Physical Validation of the Mold Design for the Robot Arm.

Gate

Ejector Pin Holes

Runner


253

7.3 GEOMETRIC TAILORING WITH AID OF DFRTS – MODELING

The requirements given by the designer for the prototypes of the robot arm includefunctional properties (stress, displacement, weight), surface finish, accuracy (parallelismtolerance, flatness tolerance), cost and time. It is a material-process geometric tailoring(MPGT) problem because of the process-related requirements. The MPGT decisiontemplate (MPGTDT) can be used to formulate the design requirements, and the DFRTScan be used to solve the MPGT problem for the robot arm.

The geometric tailoring of the robot arm will be presented in Section 7.3 and 7.4,which corresponds to the modeling and solving stages in the solution process of theDFRTS (Figure 6.10).

7.3.1 MPGT Decision Template for the Robot Arm

As stated in Section 7.1, the functional prototypes of the robot arm should havesimilar performances of maximum stress, deflection and weight as those of theproduction parts which are 5.99 MPa, 0.51mm and 3.40g respectively. There are alsoother requirements such as surface finish, accuracy, time and cost, which are shown inFigure 7.7. The design variables that can be tailored by the manufacturer are D, d, t.Therefore based on the mathematical formulation of the MPGTDT (Table 6.8), thedesigner may initiate a MPGT robot arm problem formulation for the above designinformation (Table 7.2).

Table 7.2 – MPGT Robot Arm Problem Formulation by the Designer.

PROBLEM STATEMENT:Design requirements given by the designer for producing functional prototypes of the robot armas shown in Figure 7.1.GIVEN:! Geometry variables that affect part functionality: D, d, t! Tensile load = 5N, bending load = 20N! Production parts injection molded in Atactic polystyrene with steel molds

• Young’s modulus YMp = 3200 Mpa• Tensile strength TSp = 37.4 Mpa• Density Denp = 1.04 g/cc

! Prototype parts **

• Young’s modulus YMm**

D

d

t

Figure 7.7 – Process-related Goals of Robot Arm Design.

Surface Finish SFParallelismtolerancePTOL

Cost ≤ $1500

Time≤ 7 days

Flatness tolerance FTOL


254

• Tensile strength TSm**

• Density Denm**

! 50 prototypes are needed! Goals of prototypes under loads:

• Von-mises stress requirement (S),• Weight requirement (W)• Y-displacement requirement (YD)• Geometry variables requirements (D, d, t)

! Goals of produced prototypes:• Parallelism tolerance between top and bottom flat surfaces (PTOL)• Flatness tolerance for top and bottom flat surfaces (FTOL1, FTOL2)• Surface finish for the surfaces of the cylindrical holes (SF1, SF2)• Total cost (C)• Total time (T)

! Targets for the goals: (Ideal, desirable, tolerable, undesirable, unacceptable) (19 goals)• S (1S) # 5.99, 6.29, 6.89, 7.79, 8.99 (MPa) [7.1]• S (2S) # 5.99, 5.69, 5.09, 4.19, 3.00 (MPa)• W (1S) # 3.40, 3.57, 3.91, 4.42, 5.10 (g)• W (2S) # 3.40, 3.23, 2.89, 2.38, 1.70 (g)• YD (1S)# 0.51, 0.54, 0.59, 0.66, 0.77 (mm)• YD (2S)# 0.51, 0.48, 0.43, 0.36, 0.26 (mm)• D (1S) # 20.32, 20.83, 21.84, 23.37, 25.40 (mm)• D (2S) # 20.32, 19.81, 18.80, 17.27, 15.24 (mm)• d (1S) # 10.16, 10.41, 10.92, 11.68, 12.70 (mm)• d (2S) # 10.16, 9.91, 9.40, 8.64, 7.62 (mm)• t (1S) # 3.048, 3.099, 3.200, 3.353, 3.557 (mm)• t (2S) # 3.048, 2.996, 2.896, 2.743, 2.539 (mm)• PTOL # 0.002, 0.004, 0.008, 0.013, 0.020 (in)• FTOL1, FTOL2 # 0.001, 0.002, 0.004, 0.007, 0.010 (in)• SF1, SF2 # 20, 40, 80, 130, 200 (µin)• C # 150, 300, 600, 1000, 1500 ($)• T # 3, 4, 5, 6, 7 (days)

FIND:! The geometric variables:

• D, d, t! The requirements: S, YD, W, PTOL, FTOL1, FTOL2, SF1, SF2, C, T! Deviation variables

• d din in+ −, i = 1,…,19, n = 1,…,4

SATISFY:! Goals:

• C, T, PTOL, FTOL, SF are minimization goals (class 1S)• S, W, YD, D, d and t are target-matching goals (class 3S) and each of these goals is split

into two independent goals of class 1S and class 2S.• Class 1S goals formulation:

A x max A x t

td d

q q q n

q nq n q n

( ) ( ) ,,

,, ,

− −+ − =+ − +1 0

1c h

q = 1,…,13, n=1,…,4

• Class 2s goals formulation:


255

A x min A x t

td dr r r n

r nr n r n

( ) ( ) ,,

,, ,

+ −+ − =+ − +1 0

1c h

r = 1,…,6, n=1,…,4

Where Ap(x) is the pth goal and tp,n is the target for the pth goal in nth region (LPP).! Requirement equations:

• Maximum stress S = f1(D, d, t)** [7.2]• Maximum Y-displacement YD = f2(D, d, t)** [7.3]• Volume V = f3(D, d, t)** [7.4]• Weight W = Denm * V [7.5]• PTOL, FTOL, SF, T, C**

! Constraints:• 3.00 Mpa ≤ S ≤ 8.99 Mpa• 1.72 g ≤ W ≤ 5.15g• 0.34 mm ≤ YD ≤ 1.02 mm• PTOL ≤ 0.020• FTOLj ≤ 0.010 j = 1, 2• SFj ≤ 200 j = 1, 2• C ≤ 1500• T ≤ 7

• d din in+ −• = 0

• d din in+ − ≥, 0

! Bounds:• 15.24 mm ≤ D ≤ 25.40 mm• 7.62 mm ≤ d ≤ 12.70 mm• 2.539 mm ≤ t ≤ 3.557 mm

MINIMIZE:! The deviation function (Archimedean formulation):

Z w d di nin

i n i n= ++==

+ −∑∑ , , ,11

19

1

4

c hNote: Symbol ‘**’ denotes the entries that are to be complemented by the manufacturer.

The Problem formulation in Table 7.2 contains the specific information regarding thegoals and targets but does not contain the quantitative models of the response surface ofgoals (Equations 7.2 ~ 7.5) in terms of system variables. In the problem formulation, thetargets for the goals given by the designer are formulated in Linear PhysicalProgramming formulation (Section 2.6.2). Each goals has five target values which arecorresponds to idea, desirable, tolerable, undesirable, and unacceptable. For example, inEquation 7.1, S (1S) # 5.99, 6.29, 6.89, 7.79, 8.99 (MPa) where 1S means this is aminimization goals, and the ideal value of S is less than 5.99 Mpa. It is desirable if S is inrange (5.99, 6.29] MPa; it is tolerable if S is in range (6.29, 6.89] MPa; it is undesirable ifS is in range (6.89, 7.79] MPa; it is highly undesirable if S is in range (7.79, 8.99] MPa;and it is unacceptable if S is bigger than 8.99 MPa. The targets of other goals have thesame meanings. As discussed in Section 2.6.2, it is more straightforward for the designerto assign target values for some goals than to assign weights for the goals.

After the MPGT problem is formulated, the designer can send it with the CADmodel of the part, and the FAE model for analyzing stress and deflection to the


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manufacturer. After being produced, the prototypes of the robot arm will be sent backfrom the manufacturer for the functional testing.

7.3.2 Modeling Design Functions

After receiving the design information, the manufacturer can begin the geometrictailoring for the robot arm. First the design functions and the fabrication processes needto be modeled. Suppose the direct AIM tooling (SLA 3500 – SL 7510 molds) for Atacticpolystyrene is chosen for the prototypes. In this section the models of design functionsare introduced. The models of the AIM tooling process for the robot arm are presented inthe next section.

The three design requirements for the robot arm are the maximum stress (S), themaximum Y-displacement (YD), and volume (V). In this study, the equations of S andYD are represented by response surface models, and the equation of V is represented by aanalytic equation. The steps of getting the equations are described as follows.

• Generating Maximum Stress and Y-Displacement EquationsAs shown in Figure 6.10, the approach used in this study is similar to the RCEM

(Section 2.5). Five steps are used to generate the response surface equations of S and YD.

(1) Identify factors and ranges

The design factors considered in the response surface equations of S and YD are D, dand t. Their ranges are given in Table 7.3.

(2) Design of Experiments

A three factor – three level central composite response surface design with α =1.6818 (for rotatability) is used as the design for the experiments. In this design, onlyone replication is used because the experiments are performed on a software system(ANSYS) instead of physical experiments. Table 7.4 lists the values of the designvariables for each of the experiments.

Table 7.4 - List of Experiments for Response Surface Generation.

Expt. No. D (mm) d (mm) T (mm)1 15.24 7.62 2.542 25.4 7.62 2.543 15.24 12.7 2.544 25.4 12.7 2.545 15.24 7.62 3.5566 25.4 7.62 3.5567 15.24 12.7 3.5568 25.4 12.7 3.556

Table 7.3 – Design Factors and Their Ranges of Robot Arm.

Design Factor Low Band (-1) Middle (0) High Band (1)D (mm) 15.24 20.32 25.40d (mm) 7.62 10.16 12.70t (mm) 2.539 3.048 3.557


257

9 11.7856 10.16 3.04810 28.8544 10.16 3.04811 20.32 5.8928 3.04812 20.32 14.4272 3.04813 20.32 10.16 2.1945614 20.32 10.16 3.9014415 20.32 10.16 3.048

(3) SimulationThere are a total of 15 experiments. For each experiment, the analysis of the robot arm isperformed by a FEA software system (ANSYS). The parametric FEA models are firstgenerated in a CAD software system, Pro/Engineer (www.ptc.com), and then executed inANSYS. The loading conditions shown in Figure 7.1.b are applied on the robot arm.Representative stress and y-displacement plots obtained from ANSYS are given in Figure7.8. The distributions of stress and displacement in the robot arm for all the cases aresimilar to the plots shown in Figure 7.8.

For the distributions, one can see that the Von-Mises stress is highest near the largecylindrical hole and at the edges of the robot arm, and the y-displacement is highest at thesmaller end of the robot arm. The stress concentration at the large cylindrical hole is veryhigh due to modeling conditions (the cylindrical surface is completely constrained). Inpractice, these stress concentrations do not exist and only arise due to modelinglimitations. Hence, the stress values around the large cylindrical surface are discardedwhile determining the maximum Von-Mises stress in the robot arm. The consideredregions in the robot arm are shown in Figure 7.8.a. They are caused by the bending andtensile forces instead of the boundary conditions, and therefore in our interests. Thedisplacement distribution is more uniform. The maximum displacement in the end of thesmaller hole is used as the results. The maximum stresses and y-displacements given byANSYS for each experiment are shown in Table 7.5.

(a) Maximum Von-Mises Stress (b) Maximum y-displacement

Figure 7.8 – Example ANSYS Output of Analysis for the Robot Arm.

Considered Regions


258

Table 7.5 – Results of the Experiments for Robot Arm.

Expt. No. D (mm) d (mm) T (mm) S (Mpa) YD (mm)1 15.24 7.62 2.54 12.781 1.20202 25.4 7.62 2.54 5.599 0.44183 15.24 12.7 2.54 11.32 0.86414 25.4 12.7 2.54 4.625 0.34435 15.24 7.62 3.556 9.075 0.85766 25.4 7.62 3.556 4.023 0.31847 15.24 12.7 3.556 8.070 0.61728 25.4 12.7 3.556 3.274 0.24699 11.7856 10.16 3.048 15.66 1.377410 28.8544 10.16 3.048 3.369 0.253411 20.32 5.8928 3.048 7.429 0.695212 20.32 14.4272 3.048 5.485 0.396313 20.32 10.16 2.19456 8.325 0.662714 20.32 10.16 3.90144 4.712 0.376815 20.32 10.16 3.048 5.999 0.4808

(4) Build Response Surface Models

A statistical software system, MINITAB, was used to perform regression analysisand ANOVA of the obtained data. The response surfaces generated by the system arepresented in equations 7.6 and 7.7 along with their R2, R2 (adj), maximum deviation andaverage deviation values.

Max_Stress (MPa) = 68.5522 - 3.1668 • D - 0.8671 • d - 9.7250 • t + 0.0457 • D • D +0.0150•d• d + 0.4571• t • t + 0.0072 • D • d + 0.1952 • D • t + 0.0661 • d • t [7.6]


Ydisp (mm) =7.22889 -0.33009•D -0.20477•d -0.84273• t +0.00434•D•D + 0.00254•d•d+ 0.02793 • t• t + 0.00397•D • d + 0.01794• D • t + 0.01196• d• t [7.7]


(5) Validation of response surfaces

The response surfaces of von-mises stress and y-displacement (equations 7.1 and7.2) have very high R2 and R2 (adj) values. This indicates that the response surface fitsvery well through actual data. The average errors of the fitted surfaces of stress and y-displacement are 4.4% and 5.0%. These are reasonably small values considering thedesign space considered in the problem. The maximum errors are high (16% and 18%).It is observed that the same data point results in the maximum errors for both stress andy-displacement response surfaces. Excluding this data point, the maximum error forstress is decreased to 6.7%, and the maximum error for y-displacement is decreased to8.5%. These values are considerably smaller than the original values.

To gain a better understanding between the responses and design variables, theresponse surfaces are plotted in Figure 7.9.a and Figure 7.9.b. The variation of the stressand displacement with D, d and t are almost the same. The values of stress and


259

displacement decrease with increase in any of the three design variables. This isintuitive, as the stress reduces with increase in stress bearing area. The increase in any ofthe variables (D, d and t) increases the stress bearing area. The behavior of displacementis also similar. This argument provides qualitative validation for the response surfacemodels.

In order to validate the quantitative models, additional experiments are designed andperformed to determine the error between actual design space (experimental data) andapproximated design space (response surface model). The values of the geometryvariables for the validation experiments are selected such that they spread evenlythroughout the design space. This approach of selecting the validation experiments isillustrated in Figure 7.10 for a two factor- three level central composite design. Thepoints A1 – A9 are the experiments corresponding to the two factors - three levels centralcomposite response surface design. The points B1 – B4 are the validation experiments forthem. These points are evenly spaced with respect to each other and with respect to theoriginal set of experiments that are used to generate the response surface. For the robotarm example which uses a three factor – three level central composite design, thisapproach results in 8 experiments. A fractional factorial experimental design with 4experiments is used in this case to obtain validation experiments.

For the four validation experiments, the values of D, d and t along with the FEAresults obtained from ANSYS are presented in Table 7.6. The maximum and averageerrors of stress response surface are 7.0% and 4.3% respectively. These values for

(a) Relations of Stress with D, d, t

(b) Relations of Y-displacement with D, d, t

Figure 7.9 – Graphical Relations of Responses and Variables.

MPa

mm

mmmm mm

MPa MPa

mmmm

mm

mm

mmmm

mmmm

mmmm

mm


260

displacement response surface are 8.1% and 5.1%. These error values are comparable tothe values obtained from the original set of experiments (used for response surfacegeneration). This indicates that the response surface is in fact an approximation of theactual design space and is not merely a fitted line through the data points considered inthe DOE.

Table 7.6 - Results of Validation Experiments for the Robot Arm.

Design Variables Stress YDispExp.No. D (mm) d (mm) t (mm) Actual Estimated % Error Actual Estimated % Error1 22.86 11.43 3.302 4.38 4.07 7.0 0.51 0.48 6.52 22.86 8.89 2.794 5.57 5.49 1.5 0.69 0.69 0.03 17.78 11.43 2.794 8.05 8.41 4.5 0.97 1.03 5.94 17.78 8.89 3.302 7.27 7.58 4.3 0.95 1.03 8.1

As a summary, the stress and displacement response surfaces are validated by:

a) The response surfaces have a reasonably good fit through the actual data points.This is tested by checking the R2, R2 (adj), maximum deviation and averagedeviation values for these response surfaces.

b) It is qualitatively verified (from the response surface plots) that the behavior ofthe response surfaces is in accordance with the expected behavior.

c) The low values of maximum deviation and average deviation for the validationexperiments indicate that the response surface models are in fact a reasonablygood approximation of the original design space.

• Generating Volume and Weight EquationsAn analytical equation was developed to determine the volume of the robot arm.

Based on the geometry of the robot arm (Figure 7.1), an equation is given in Equation7.8.

Volume (mm3)= π/4•D2 + [(L/2) • (D+d) • Sin (cos-1((D-d)/2L))] – [(1/4) • (D2 – d2) •cos-1((D-d)/2L)] – [(π/4) • (D0

2 – d02)] • t [7.8]

A1

A2A3

A4

A9

A6

A8

A7 A5

B1

B4B3

B2

Figure 7.10 – Validation Experiment Design for the Response Surfaces.


261

Where,L = 63.5 mm, is the center-to-center distance between the holes in the robot arm,D0 = 10.32 mm, is the diameter of the larger hole,d0 = 5.16 mm, is the diameter of the smaller hole.

Based on the volume, the weight of prototypes made of SL7510 can be calculated byequations 7.9.

Weight (kg) = 1.22 * 10-3 * Volume [7.9]

The models of volume and weight represent the actual design space. They arederived from the geometry relations, therefore valid.

7.3.3 Modeling Fabrication Processes

For the fabrication process of prototypes (direct AIM tooling), suppose SLA 3500 –SL 7510 molds are used in the injection molding process. As discussed in Section 6.4.2,the models of surface finish, accuracy, mold life, build time and cost for the fabricationprocess were formulated based on other research work. These models are the same fordifferent parts. Therefore they were used in the geometric tailoring for the robot armdirectly. The models of process that are related to the robot arm case are listed asfollows.

Surface finish SF = f (PO, LT, θ1, θ2)Surface finish of parting surface SFP = f (PO, LT)Parallelism tolerance PTOL = f (PO, LT, HOC, FOC)Flat tolerance FTOL = f (PO, LT, HOC, FOC)Mold life ML = f (LT, CT, TA, D, d, t, θ1, θ2)

Mold number NN

MLmp=

−+L

NMOQP

11

SLA build time BT = f (PO, LT, HOC, FOC, Nm)Mold cost Cm = f (BT, TA)Part cost Cp = f (CT)Total cost C = Cm + Cp

Mold time Tm = f (BT, TA)Part time Tp = f (CT)Total time T = Tm + Tp

For the direct AIM tooling, the RP process variables are mold orientation (PO),slicing scheme (LTp), hatch overcure (HOCp), fill overcure (FOCp), and thermal aging(TA). In the variables, the sub-script ‘p’ corresponds to the block number of differentlayers. TA is a boolean variable, which related to if the mold pieces are thermal curedbefore they are used in the IJM process. The IJM process variable is cooling time (CT).The bounds on the cooling time depend on part geometry and are different for differentparts. Experimentally it is verified that a cooling time of 5 – 7 minutes is required toinjection-mold the robot arm with SLA molds on Morgan press injection-moldingmachine. Attempts to mold parts with lower cooling times resulted in deformed parts.Beside the process variables, the manufacturer may add two draft angle variables θ1, θ2

(Figure 7.11) for mold life consideration.


262

The goals of the robot arm are transferred to goals of mold pieces directly based onthe accuracy and surface finish of the part surfaces are very close to those of the moldsurfaces (Cedorge, 1999). For a mold design given by RTMDS (Figure 7.3.d), someprocess-related goals are shown in Figure 7.11. The manufacturer may add an additionalsurface finish requirement (SFP ≤ 100 µin) for the parting surfaces of the mold pieces.


Based on the information given by the designer, and the models of design functionsand fabrication processes, the manufacturer can formulate a complete MPGT problem asshown in Table 7.7.

Table 7.7 – MPGT Robot Arm Problem Formulation.PROBLEM STATEMENT:Find a tailored part design and related process parameters for producing functional prototypes ofthe robot arm as shown in Figure 7.1. In the problem the mold design of the robot arm is alreadydetermined (Figure 7.12).GIVEN:! Geometry variables that affect part functionality: D, d, t! Tensile load = 5N, bending load = 20N! Production parts injection molded in Atactic polystyrene with steel molds

• Young’s modulus YMp = 3200 Mpa• Tensile strength TSp = 37.4 Mpa

Figure 7.11 – Some Process Planning Goals.

Parting Surface Finish SFP1,2Flatness tolerance FTOL1~3

Surface Finish SF

Draft Angle αParallelism tolerance PTOL


263

• Density Denp = 1.04 g/cc! Prototype parts injection molded in Atactic polystyrene with SLA 3500 – SL 7510 molds

• Young’s modulus YMm = 3400 MPa• Tensile strength TSm = 32.8 MPa• Density Denm = 1.04 g/cc

! Required number of prototype parts Np = 50! Goals of prototypes under loads:


! Goals of produced prototypes:• Parallelism tolerance between top and bottom flat surfaces (PTOL)• Flatness tolerance of mold pieces (FTOL1, FTOL2, FTOL3)• Surface finish of mold pieces (SF1, SF2)• Surface finish of parting surfaces (SFP1, SFP2)• Total cost (C)• Total time (T)• Draft Angle (θ1, θ2)

! Targets for the goals: (Ideal, desirable, tolerable, undesirable, unacceptable) (12 goals)• S (1S) # 5.99, 6.29, 6.89, 7.79, 8.99 (MPa)• S (2S) # 5.99, 5.69, 5.09, 4.19, 3.00 (MPa)• W (1S) # 3.40, 3.57, 3.91, 4.42, 5.10 (g)• W (2S) # 3.40, 3.23, 2.89, 2.38, 1.70 (g)• YD (1S)# 0.51, 0.54, 0.59, 0.66, 0.77 (mm)• YD (2S)# 0.51, 0.48, 0.43, 0.36, 0.26 (mm)• D (1S) # 20.32, 20.83, 21.84, 23.37, 25.40 (mm)• D (2S) # 20.32, 19.81, 18.80, 17.27, 15.24 (mm)• d (1S) # 10.16, 10.41, 10.92, 11.68, 12.70 (mm)• d (2S) # 10.16, 9.91, 9.40, 8.64, 7.62 (mm)• t (1S) # 3.048, 3.099, 3.200, 3.353, 3.557 (mm)• t (2S) # 3.048, 2.996, 2.896, 2.743, 2.539 (mm)• θ1 # 0.0, 0.5, 1.5, 3.0, 5.0 (degree)• θ2 # 0.0, 0.5, 1.5, 3.0, 5.0 (degree)• PTOL # 0.002, 0.004, 0.008, 0.013, 0.020 (in)• FTOL1, FTOL2, FTOL3 # 0.001, 0.002, 0.004, 0.007, 0.010 (in)• SF1, SF2 # 20, 40, 80, 130, 200 (µin)• SFP1, SFP2 # 10, 20, 40, 65, 100 (µin)• C # 150, 300, 600, 1000, 1500 ($)• T # 3, 4, 5, 6, 7 (days)

! Allowable layer thickness (LT): 2, 4 and 8 milsFIND:! The system variables:

• D, d, t• Draft angle variables: θ1, θ2

• RP process variables: PO, LTP, HOCp, FOCp, TA• IJM process variables: CT

! The requirements: S, YD, W


264

! Deviation variables• d din in

+ −, i = 1,…,24, n = 1,…,4SATISFY:! Goals:

• C, T, PTOL, FTOL, SF, SFP, θ1, θ2 are minimization goals (class 1S)• S, W, YD, D, d and t are target-matching goals (class 3S) and each of these goals is split


A x max A x t

td d

q q q n

q nq n q n

( ) ( ) ,,

,, ,

− −+ − =+ − +1 0

1c h

q = 1,…,18

• Class 2s goals formulation:A x min A x t

td dr r r n

r nr n r n

( ) ( ) ,,

,, ,

+ −+ − =+ − +1 0

1c h

r = 1,…,6


• S (Mpa) = 68.5522 - 3.1668 • D - 0.8671 • d - 9.7250 • t + 0.0457 • D • D + 0.0150•d• d+ 0.4571• t • t + 0.0072 • D • d + 0.1952 • D • t + 0.0661 • d • t

• YD (mm) = 7.22889-0.33009•D-0.20477•d-0.84273• t +0.00434•D•D + 0.00254•d•d +0.02793 • t• t + 0.00397•D • d + 0.01794• D • t + 0.01196• d• t

• V (cc) = 0.7854•D2 + [1.25 • (D+d) • sin (cos-1((D-d)/5))] – [0.25 • (D2 – d2) • cos-1((D-d)/5)] – 0.1571 • t• 16.39

• W (g) = 1.04 * V• SFi = f (PO, LT, θ1, θ2), i =1, 2• SFPi = f (PO, LT), i =1, 2• PTOL = f (PO, LT, HOC, FOC)• FTOLi = f (PO, LT, HOC, FOC) , i =1, 2,3• ML = f (LT, CT, TA, D, d, t, θ1, θ2)

• NN

MLmp=

−+L

NMOQP

11

• BT = f (PO, LT, HOC, FOC, Nm)• Cm = f (BT, TA)• Cp = f (CT)• C = Cm + Cp

• Tm = f (BT, TA)• Tp = f (CT)• T = Tm + Tp

! Constraints:• 3.00 Mpa ≤ S ≤ 8.99 Mpa• 1.72 g ≤ W ≤ 5.15g• 0.34 mm ≤ YD ≤ 1.02 mm• C ≤ $1500• T ≤ 7 days• PTOL ≤ 0.020 inch• FTOLj ≤ 0.010 inch, j = 1, 2, 3• SFj ≤ 200 µin j = 1, … ,4• SFPj ≤ 100 µin j = 1, 2


265

• d din in+ −• = 0

• d din in+ − ≥, 0

! Bounds:• 15.24 mm ≤ D ≤ 25.40 mm• 7.62 mm ≤ d ≤ 12.70 mm• 2.539 mm ≤ t ≤ 3.557 mm• 0 degree ≤ θ1 ≤ 5 degree• 0 degree ≤ θ2 ≤ 5 degree• PO (θx, θy, θz) – Discrete variable• TA (0 or 1) – Discrete variable• 2 ≤ LT ≤ 8 (mils) – Discrete variable• LT=2, 0.002 (mils) ≤ HOC ≤ 0.006 (mils), 0.012 (mils) ≤ FOC ≤ 0.016 (mils)• LT=4, 0.003 (mils) ≤ HOC ≤ 0.007 (mils), 0.004 (mils) ≤ FOC ≤ 0.008 (mils)• LT=8, 0.001 (mils) ≤ HOC ≤ 0.005 (mils), 0.002 (mils) ≤ FOC ≤ 0.006 (mils)• 300 second ≤ CT ≤ 420 second


Z w d di nin

i n i n= ++==

+ −∑∑ , , ,11

24

1

4

c h

After the MPGT problem for the robot arm is formulated, the solution process forsystem variables are presented in the next section.

7.4 GEOMETRIC TAILORING WITH AID OF DFRTS – SOLVING

In the DFRTS, the solution process of the MPGT problem has three stages, solvingdiscrete variables, solving continuous variables, and selecting a solution (Figure 6.8). Forthe robot arm, these steps are introduced in Section 7.4.1~7.4.3 respectively.

7.4.1 Solving Discrete Variables

The discrete variables in the DFRTS are mold design variables, part orientations andlayer thickness. They are determined with the aid of RTMDS and refined RP processplanner.

• Mold Design VariablesAs presented in Section 7.2, the RTMDS can aid the manufacturer to generate

feasible mold designs for the robot arm. Two mold designs generated by the system arefurther considered in this case. They are shown in Figure 7.12.


266

• Part Orientation in SLA BuildingFor each mold design, the mold pieces are inputted to the refined RP process planner

to generate a set of part orientations (PO) and layer thickness (LT). As describe inSection 6.4.2, only the discrete variables (PO and LT) are considered in the modified RPprocess planner. Hatch overcure (HOC) and fill overcure (FOC) are set as default values.Its problem formulation is similar to RP-PP problem (Table 6.2). Sambu (2001) gave adetail description of the refined RP process planner. Solving the modified RT-PPproblem results in a set of promising mold orientations and slicing schemes for each ofthe mold pieces.

Two part orientations are further considered for each mold design. This results in 4possible results as shown in Figure 7.13. For mold design 1, one orientation is obtainedfor mold cavity and two are obtained for mold core. For mold design 2, two orientationsare obtained for mold cavity and one is obtained for mold core.

(a) MD1

(b) MD2

Figure 7.12 – Two Considered Mold Designs for the Robot Arm.

RobotArm


267

(a) MD1-PO1

(b) MD1-PO2

(c) MD2-PO1

(d) MD2-PO2

Figure 7.13 – Four Considered Part Orientations for the Mold Designs.

MoldDesign

MoldDesign


268

Each of these orientations yielded two promising slicing schemes. This results in atotal of 8 slicing schemes. All the slicing schemes obtained are shown in Figure 7.14. Inthe figure, the dark shade corresponds to 2-mil layer thickness, light shade corresponds to8-mil layer thickness and the medium level shade corresponds to 4-mil layer thickness.

Slicing schemes 1 and 2 correspond to mold orientation shown in Figure 7.13.a. Thecylindrical surfaces and the parting surfaces have surface finish specifications. Theparting surface is horizontal for this orientation. Hence, in these slicing schemes, lowerlayer thickness (2 and 4-mils) are used for the block with the cylindrical surface and theremaining blocks (top and bottom) are built with the maximum allowable layer thicknessof 8-mils. The same reasoning applies for slicing schemes 5 and 6. They have the sameslicing schemes but correspond to mold orientation in Figure 7.13.c. Slicing schemes 3and 4 correspond to mold orientation in Figure 7.13.b. The parting surface of the moldcore is vertical in this orientation and only a layer thickness of 2-mils satisfies the surfacefinish constraint. Hence, most of the regions of the molds have 2-mil layer thickness.

Slicing Scheme 4Slicing Scheme 3

Slicing Scheme 1 Slicing Scheme 2


Slicing Scheme 7Slicing Scheme 8

Figure 7.14 – Eight Considered Slicing Schemes for the Part Orientations.


269

The slicing schemes 7 and 8 are similar to 3 and 4 but correspond to mold orientation inFigure 7.13.d.

For each slicing scheme, a modified MPGT problem can be formulated (Section6.4.1) and solve with the aid of OpdesX.

7.4.2 Solving Other Variables

The values of other variables of the MPGT problem (including a discrete variablesTA) are determined by solving a modified MPGT problem. The formulation of themodified MPGT problem is identical to the MPGT problem (Table 7.7) except for onedifference. The set of promising mold orientations and the slicing schemes are availablein the modified MPGT problem. As these values are already determined, PO and LT aredropped from the system variables. The modified MPGT formulation for robot armproblem is presented in Table 7.8.

Table 7.8 – Modified MPGT Robot Arm Problem Formulation.

PROBLEM STATEMENT:Find a tailored part design and related process parameters for producing functional

prototypes of the robot arm as shown in Figure 7.1. In the problem the mold design of the robotarm is already determined as well as the part orientation (PO) and the layer thickness (LTi) usedin SLA process.GIVEN:! Parametric CAD models of mold pieces! Geometry variables that affect part functionality: D, d, t! Required number of prototype parts Np = 50! Mold orientation (PO)! Slicing Schemes (LT)! Goals of prototypes under loads:


! Goals of produced prototypes:• Parallelism tolerance between top and bottom flat surfaces (PTOL)• Flatness tolerance of mold pieces (FTOL1, FTOL2, FTOL3)• Surface finish of mold pieces (SF1, SF2)• Total cost (C)• Total time (T)• Draft Angle (θ1, θ2)

! Targets for the goals: (Ideal, desirable, tolerable, undesirable, unacceptable) (12 goals)• S (1S) # 5.99, 6.29, 6.89, 7.79, 8.99 (MPa)• S (2S) # 5.99, 5.69, 5.09, 4.19, 3.00 (MPa)• W (1S) # 3.40, 3.57, 3.91, 4.42, 5.10 (g)• W (2S) # 3.40, 3.23, 2.89, 2.38, 1.70 (g)• YD (1S)# 0.51, 0.54, 0.59, 0.66, 0.77 (mm)• YD (2S)# 0.51, 0.48, 0.43, 0.36, 0.26 (mm)• D (1S) # 20.32, 20.83, 21.84, 23.37, 25.40 (mm)• D (2S) # 20.32, 19.81, 18.80, 17.27, 15.24 (mm)


270

• d (1S) # 10.16, 10.41, 10.92, 11.68, 12.70 (mm)• d (2S) # 10.16, 9.91, 9.40, 8.64, 7.62 (mm)• t (1S) # 3.048, 3.099, 3.200, 3.353, 3.557 (mm)• t (2S) # 3.048, 2.996, 2.896, 2.743, 2.539 (mm)• θ1 # 0.0, 0.5, 1.5, 3.0, 5.0 (degree)• θ2 # 0.0, 0.5, 1.5, 3.0, 5.0 (degree)• PTOL # 0.002, 0.004, 0.008, 0.013, 0.020 (in)• FTOL1, FTOL2, FTOL3 # 0.001, 0.002, 0.004, 0.007, 0.010 (in)• SF1, SF2 # 20, 40, 80, 130, 200 (µin)• SFP1, SFP2 # 10, 20, 40, 65, 100 (µin)• C # 150, 300, 600, 1000, 1500 ($)• T # 3, 4, 5, 6, 7 (days)

FIND:! The system variables:

• D, d, t• Draft angle variables: θ1, θ2

• RP process variables: HOCp, FOCp, TA• IJM process variables: CT

! The requirements: S, YD, W! Deviation variables

• d din in+ −, i = 1,…,24, n = 1,…,4

SATISFY:! Goals:

• C, T, PTOL, FTOL, SF, SFP, θ1, θ2 are minimization goals (class 1S)• S, W, YD, D, d and t are target-matching goals (class 3S) and each of these goals is split


A x max A x t

td d

q q q n

q nq n q n

( ) ( ) ,,

,, ,

− −+ − =+ − +1 0

1c h

q = 1,…,18


td dr r r n

r nr n r n

( ) ( ) ,,

,, ,

+ −+ − =+ − +1 0

1c h

r = 1,…,6


• S (Mpa) = 68.5522 - 3.1668 • D - 0.8671 • d - 9.7250 • t + 0.0457 • D • D + 0.0150•d• d+ 0.4571• t • t + 0.0072 • D • d + 0.1952 • D • t + 0.0661 • d • t

• YD (mm) = 7.22889-0.33009•D-0.20477•d-0.84273• t +0.00434•D•D + 0.00254•d•d +0.02793 • t• t + 0.00397•D • d + 0.01794• D • t + 0.01196• d• t

• V (cc) = 0.7854•D2 + [1.25 • (D+d) • sin (cos-1((D-d)/5))] – [0.25 • (D2 – d2) • cos-1((D-d)/5)] – 0.1571 • t• 16.39

• W (g) = 1.04 * V• SFi = f (PO, LT, θ1, θ2), i =1, 2• SFPi = f (PO, LT), i =1, 2• PTOL = f (PO, LT, HOC, FOC)• FTOLi = f (PO, LT, HOC, FOC) , i =1, 2,3• ML = f (LT, CT, TA, D, d, t, θ1, θ2)


271

• NN

MLmp=

−+L

NMOQP

11

• BT = f (PO, LT, HOC, FOC, Nm)• Cm = f (BT, TA)• Cp = f (CT)• C = Cm + Cp


! Constraints:• 3.00 Mpa ≤ S ≤ 8.99 Mpa• 1.72 g ≤ W ≤ 5.15g• 0.34 mm ≤ YD ≤ 1.02 mm• C ≤ $1500• T ≤ 7 days• PTOL ≤ 0.020 inch• FTOLj ≤ 0.010 inch, j = 1, 2, 3• SFj ≤ 200 µin j = 1, … ,4• SFPj ≤ 100 µin j = 1, 2

• d din in+ −• = 0

• d din in+ − ≥, 0

! Bounds:• 15.24 mm ≤ D ≤ 25.40 mm• 7.62 mm ≤ d ≤ 12.70 mm• 2.539 mm ≤ t ≤ 3.557 mm• 0 ≤ θ1 ≤ 5• 0 ≤ θ2 ≤ 5• TA (0 or 1) – Discrete variable• LT=2, 0.002 (mils) ≤ HOC ≤ 0.006 (mils), 0.012 (mils) ≤ FOC ≤ 0.016 (mils)• LT=4, 0.003 (mils) ≤ HOC ≤ 0.007 (mils), 0.004 (mils) ≤ FOC ≤ 0.008 (mils)• LT=8, 0.001 (mils) ≤ HOC ≤ 0.005 (mils), 0.002 (mils) ≤ FOC ≤ 0.006 (mils)• 300 second ≤ CT ≤ 420 second


Z w d di nin

i n i n= ++==

+ −∑∑ , , ,11

24

1

4

c h

The problem is solved by an engineering optimization software system to determinea solution for the related slicing scheme. In this case study, the modified MPGT problemis solved by OptdesX (with SAN algorithm) for the robot arm. A C module is developedto calculate the values of stress, Y-displacement, volume, weight, surface finish, andaccuracy in the formulation (Table 7.8). The values of SLA mold life and cost/time arecalculated by a mold life predictor and a RT cost estimator respectively (Section 6.4.2).

As explained in Section 6.4.2, OptdesX uses Simulated Annealing (SAN) algorithmin this study. SAN algorithm runs for 5000 cycles with a maximum perturbation value of0.3. Then, the obtained solution is further optimized by running SAN several times with


272

same maximum perturbation value (1000 cycles each time) till the solution is converged.The similar procedure is then repeated for maximum perturbation values of 0.1, 0.03 and0.01 (in that order) to obtain the final solution. Multiple values of maximum perturbationcan ensure problem convergence. For each modified MPGT problem, three differentstarting points are also investigated to test the convergence of the solution. The startingpoints used in the study are presented in Table 7.9.

Table 7.9 - Starting Points Investigated for Each Slicing Scheme.

Starting PointVariable 1 2 3

D 15.24 20.32 25.4d 7.62 10.16 12.7t 2.539 3.048 3.557

TC 0 0 1draft1 0 2.5 5draft2 0 2.5 5

CT 300 300 300HOC1 0.001 0.001 0.001HOC2 0.002 0.002 0.002HOC3 0.001 0.001 0.001FOC1 0.002 0.002 0.002FOC2 0.012 0.012 0.012FOC3 0.002 0.002 0.002

Starting points 1, 2 and 3 correspond to low, medium and high values of thevariables respectively. It can be observed that same cooling time and HOC and FOCvalues are used for all the starting points. The available models indicate that using thelowest possible value of cooling time would reduce the value of objective function andhence this value is used for all the starting points. Also, HOC and FOC do not affect theaccuracy of parts /molds built on SLA 3500 – SL 7510 and hence these values affect onlythe build time. Build time is minimized by using the minimum possible values for thesevariables and hence the lower bounds (for the corresponding layer thickness) are chosenfor all HOC and FOC variables.

The complete solutions (system variables) obtained from OptdesX for all the slicingschemes are presented in Table 7.10. The related goal for all the slicing schemes areprovided in Table 7.11 (PTOL= 0.0016 inch, FTOL1 = FTOL2 = FTOL3 =0.0018 inch).

Table 7.10 – Solutions of System Variables for Different Slicing Schemes.

Slice#

StartPoint

D(mm)

d(mm)

t(mm)

TA(0/1)

draft1(o)

draft2(o)

CT(s)

HOC1(mil)

HOC2(mils)

HOC3(mils)

FOC1(mils)

FOC2(mils)

FOC3(mils)

1 20.62 9.9 2.987 0 0.61 0 300 0.001 0.002 0.001 0.002 0.012 0.0022 20.68 9.92 2.990 0 0.61 0.01 300 0.001 0.002 0.001 0.002 0.012 0.0021

3 20.46 9.91 2.980 0 0.61 0.03 300 0.001 0.002 0.001 0.002 0.012 0.0021 20.47 10 2.997 0 0.5 0 300 0.001 0.002 0.001 0.002 0.012 0.0022

2 20.18 9.91 3.035 0 0.5 0 300 0.001 0.002 0.001 0.002 0.012 0.002


273

3 20.31 9.97 3.042 0 0.5 0 300 0.001 0.002 0.001 0.002 0.012 0.0021 20.69 9.82 2.936 0 0.59 0.01 300 0.001 0.002 0.001 0.002 0.012 0.0022 20.68 9.9 2.990 0 0.61 0.02 300 0.001 0.002 0.001 0.002 0.012 0.0023

3 20.69 9.89 2.988 0 0.6 0 300 0.001 0.002 0.001 0.002 0.012 0.0021 21.12 9.48 2.901 0 0.73 0.5 300 0.001 0.002 0.001 0.002 0.012 0.0022 20.64 9.8 2.901 0 0.74 0.5 300 0.001 0.002 0.001 0.002 0.012 0.0024

3 20.88 9.41 2.896 0 0.72 0.5 300 0.001 0.002 0.001 0.002 0.012 0.0021 20.30 9.9 2.989 0 0.61 0 300 0.001 0.002 0.001 0.002 0.012 0.0022 20.47 9.87 2.963 0 0.6 0 300 0.001 0.002 0.001 0.002 0.012 0.0025

3 20.35 9.89 2.957 0 0.6 0 300 0.001 0.002 0.001 0.002 0.012 0.0021 20.26 10.2 2.986 0 0.5 0 300 0.001 0.002 0.001 0.002 0.012 0.0022 20.32 9.99 2.997 0 0.5 0 300 0.001 0.002 0.001 0.002 0.012 0.0026

3 20.29 10.2 3.021 0 0.5 0 300 0.001 0.002 0.001 0.002 0.012 0.0021 20.56 9.93 2.973 0 0.6 0 300 0.001 0.002 0.001 0.002 0.012 0.0022 20.31 9.89 3.000 0 0.61 0.04 300 0.001 0.002 0.001 0.002 0.012 0.0027

3 20.46 9.9 2.980 0 0.6 0.05 300 0.001 0.002 0.001 0.002 0.012 0.0021 21.46 9.43 2.904 0 0.72 0.5 300 0.001 0.002 0.001 0.002 0.012 0.0022 21.21 9.4 2.915 0 0.72 0.5 300 0.001 0.002 0.001 0.002 0.012 0.0028

3 20.90 9.51 2.920 0 0.73 0.51 300 0.001 0.002 0.001 0.002 0.012 0.002

Table 7.11 – Solutions of Goals for Different Slicing Schemes.

Slice#

StartPoint

Stress(Mpa)

y-disp(mm)

Weight(g)

SF1(µin)

SF2(µin)

SFP1(µin)

SFP2(µin)

Cost($) Mold #

Obj.Function

1 6.05 0.49 3.34 65 64 6 6 570 1 0.0353982 5.99 0.49 3.36 65 64 6 6 570 1 0.035421

3 6.16 0.5 3.31 65 64 6 6 570 1 0.0354581 6.09 0.49 3.35 132 131 6 6 595 2 0.4327272 6.21 0.51 3.33 132 131 6 6 595 2 0.4327772

3 6.1 0.5 3.37 132 131 6 6 595 2 0.4327151 6.14 0.5 3.29 65 64 6 64 1108 1 0.823242 6 0.49 3.36 65 64 6 64 1108 1 0.82323

3 6 0.49 3.35 65 64 6 64 1108 1 0.8231681 6.04 0.49 3.27 132 132 6 64 1059 1 1.0392922 6.26 0.51 3.24 132 132 6 64 1059 1 1.0397924

3 6.21 0.51 3.22 132 132 6 64 1059 1 1.0386921 6.24 0.51 3.3 65 64 6 6 563 1 0.0351792 6.2 0.5 3.29 65 64 6 6 563 1 0.0351935

3 6.29 0.51 3.27 65 64 6 6 563 1 0.0351291 6.2 0.5 3.33 132 131 6 6 596 2 0.4328182 6.2 0.5 3.32 132 131 6 6 596 2 0.4328066

3 6.1 0.49 3.37 132 131 6 6 596 2 0.4327691 6.11 0.5 3.32 65 64 6 64 1144 1 0.9803522 6.22 0.51 3.31 65 64 6 64 1144 1 0.9803897

3 6.17 0.5 3.31 65 64 6 64 1144 1 0.98041 5.84 0.48 3.32 132 132 6 64 1119 1 1.3056712 5.97 0.49 3.29 132 132 6 64 1119 1 1.3053118

3 6.12 0.5 3.26 132 132 6 64 1119 1 1.30588


274

The analysis and validation of the solutions are presented in Section 7.4.4. From thesolutions for the eight slicing schemes, one solution should be selected as the solution ofthe MPGT problem.

7.4.3 Selecting A Solution

From Table 7.11, it can be seen that slicing schemes 1 and 5 have lowest values ofobjective function. Slicing schemes 2 and 6 use 4-mil layer thickness for the cylindricalsurfaces and hence have worse surface finish than slicing schemes 1 and 5. This resultsin slightly higher objective function values for these slicing schemes. All other slicingschemes (3, 4, 7 and 8) have bad surface finish and build time (cost) goal achievementsand hence have very high values of objective function.

The slicing schemes 1 and 5 correspond to orientations in Figure 7.13.a and Figure7.13.c, and mold design in Figure 7.12.a and Figure 7.12.b respectively. Both the moldorientations result in same value of objective function (Table 7.11) indicating that the neteffect of the achievement of all the goals considered in the problem formulation is thesame. However, there are additional concerns that are not quantifiable (for which noquantitative models are available) and hence are not incorporated in the problemformulations. For the robot arm example, one such concern is the positional tolerance ofthe cylindrical holes. The location of the cylindrical holes with respect to the partboundary should be accurate but no quantitative models are available to estimate thiserror. For the mold orientation in Figure 7.13.a, the positional error (of the cylindricalsurface with respect to the part boundary) occurs due to shrinkage and SLA buildinginaccuracies. For mold orientation in Figure 7.13.c, apart from these errors, alignmenterror also exists. When the two mold pieces come together while injection molding, it ispossible to have some alignment errors between the mold halves. This misalignmentresults in higher (worse) values of the positional tolerance.

Therefore, it is determined that solution corresponding to slicing scheme 1 issuperior to other solutions, and it is regarded as the solution of the MPGT problem. Withthe tailored robot arm design, the values of mold design variables, RP process variables,and IJM process variables are also generated. They are summarized as shown in Table7.12.

Table 7.12 – Solutions of the MPGT Problem for the Robot Arm.

D(mm)

d(mm)

t(mm)

TA(0/1)

Draft1(o)

draft2(o)

CT(s)

HOC1(mil)

HOC2(mils)

HOC3(mil)

FOC1(mil)

FOC2(mil)

FOC3(mil)

20.62 9.9 2.987 0 0.61 0 300 0.001 0.002 0.001 0.002 0.012 0.002

7.4.4 Post-Solution Analysis

The objective function values of the solutions obtained for all the starting points arevery close to each other (Table 7.11). This indicating that all the starting points convergeto same value. In Table 7.10, TA is a Boolean variable to indicating if thermal aging isperformed for the molds. Thermal aging is an eight hours curing process for SLA molds,which may slightly increase the strength of SLA molds (Rodet, 2001). However it alsoincrease the mold cost and time. For the robot arm, its affect on the mold life cannot


275

compensate for the increase of cost. Therefore TA has a value of 0 for all the slicingschemes, which indicates that the mold pieces should not be thermally aged.

CT is the cooling time used in the injection molding process. With longer coolingtime, the injection molded part will become strong enough for the ejection operation.However, as the part further shrinks onto the molds, the SLA mold life decreases.Therefore the value of cooling time depends on the above tradeoff and varies for differentparts. For the robot arm, a lower CT bigger than 300 seconds will increase the mold lifebased on the model used in the study. It also reduces part cost and time. Therefore thevalue of CT will always decrease to the lower bound of the given range.

For SLA 3500, HOC and FOC only affect build time (they also affect tolerance forSLA 250). Therefore, in the obtained solution they decrease to their lower bounds, whichresults in minimum possible build time for each mold piece for the slicing schemes. Theinter-relationships between all goals and variables are shown in Figure 7.15, which canfoster a better understanding of the results listed in Table 7.10 and Table 7.11.

Before we can accept these solutions, we need to confirm that the operation of SANalgorithm in solving MPGT problem for robot arm is accurate. The variation of the valueof objective function with iteration (cycle) number for slicing scheme 1 for starting point3 is shown in Figure 7.16. The fluctuation in the value of objective function is high in thebeginning and small in the later iterations. Also the objective function value hasgradually reduced. This indicates the proper functioning of SAN algorithm in solvingmodified MPGT problem.

LT

HOC

FOC

PO

TA

θ1

θ2

CT

Mold Life

Part Cost

Mold Cost

Number of Molds

Mold Time

Part Time

Build Time

Number of Parts

ZL

SP

BOC

Surface Finish

Total Cost

Total Time

Weight

Stress

Y-Disp

ParallelismTolerance

FlatnessTolerance

D

d

t

Goal

Systemvariables

Others are intermediatevariables

Figure 7.15 – Inter-Relationship Diagram of Goals and Variables.


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The effect of modifying the target ranges on the obtained solutions is also studied.In the modified MPGT problem, all the goals can be divided into four categories:geometry, functional, RP process and cost. The goals in RP process category areaccuracy and surface finish goals. Accuracy goals are not affected by hatch and fillovercures (for SLA 3500 – SL 7510) and hence will not be affected by modifying targetsin MPGT problem. Also, the only variable in MPGT problem that affects surface finishis draft angle. For small changes, draft angle does not have a significant effect on surfacefinish. Hence, the effect of modifying the target ranges on RP process goals is notsignificant and therefore not considered.

The experiments by modifying target ranges of the goals in the other three categoriesare presented in Table 7.13. In each of these experiments, the target range for all thegoals in the corresponding category is reduced by 40%. By running OptdesX for the newvalues of the goals, the results for the experiments are presented in Table 7.14. Theimprovement and deterioration of the goals due to this modifying in target ranges arerepresented by positive and negative percentages respectively.

Table 7.13 - Target Modification Experiments.

Original Values Modified ValuesGoalsMinimum ideal Maximum Minimum ideal Maximum

D (mm) 15.24 20.32 25.4 17.272 20.32 23.368d (mm) 12.7 10.16 12.7 11.684 10.16 11.684

GeometryExperiment

t (mm) 2.539 3.048 3.557 2.7426 3.048 3.3534stress (Mpa) 3.00 5.99 8.99 4.196 5.99 7.79y-disp (mm) 0.26 0.51 0.77 0.36 0.51 0.666

FunctionalExperiment

weight (g) 1.70 3.4 5.10 2.38 3.4 4.42Cost Exp. cost ($) 150 150 1500 150 150 960

Figure 7.16– Objective Function vs. Iteration No. for Modified MPGT Problem.


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Table 7.14 – Results of Target Modification Experiments.

GeometryExperiment

FunctionalExperiment

CostExperiment

GoalsTargetValues

Values ofCurrentSolution

Newvalues Improvement



D (mm) 20.32 20.62 20.52 33.3% 20.56 20% 20.65 -10%d (mm) 10.16 9.9 10.02 46.2% 9.91 3.8% 9.94 15.4%t (mm) 3.048 2.987 3.012 41.0% 2.988 1.6% 2.977 -16.4%

stress(MPa) 5.99 6.05 6.027 38.3% 6.079 -48.3% 6.04 16.7%y-disp (mm) 0.51 0.49 0.487 -15% 0.491 5% 0.49 0%Weight (g) 3.4 3.34 3.37 50% 3.34 0% 3.34 0%

Cost ($) 150 570 570 0% 570 0% 570 0%

From the values in Table 7.14, it can be seen that all the geometry goals improve (bymore than 30%) when their target ranges are reduced. The stress goal is worsenedsignificantly and displacement goal is improved a little by reducing the target ranges offunctional goals. The weight goal hasn’t changed. Reducing the target range of cost goalhas not changed its value.

It is noticed that reducing the target range of functional goals has worsened theachievement of stress goal. To identify the reason for this behavior, additionalexperiments by reducing the target range of one goal at a time by 80% are performed forall the functional goals. The new experiments are presented in Table 7.15. The obtainedresults are presented in Table 7.16. They confirm that the achievements of all the goalsare improved by reducing their target ranges. This indicates that the deterioration ofstress goal when all the functional goal target ranges are reduced is due to internal trade-offs (trade-off within the functional goals).

Table 7.15 - Individual Functional Target Modification Experiments.


stress (Mpa) 3.00 5.99 8.99 5.392 5.99 6.59y-disp (mm) 0.26 0.51 0.77 0.46 0.51 0.562

IndividualFunctionalExperiment weight (g) 1.70 3.4 5.1 3.06 3.4 3.74

Table 7.16 – Results of Individual Functional Target Modification Experiments.

Stress Experiment Ydisp Experiment Weight ExperimentGoals

TargetValues

Values ofCurrentSolution




D (mm) 20.32 20.62 20.82 -66.6667 20.48 46.66667 20.72 -33.3333D (mm) 10.16 9.9 9.93 11.53846 9.9 0 9.88 -7.69231T (mm) 3.048 2.987 2.95 -60.6557 2.941 -75.4098 2.994 11.47541

stress(MPa) 5.99 6.05 6 83.33333 6.24 -316.667 5.97 66.66667y-disp (mm) 0.51 0.49 0.485 -25 0.507 85 0.484 -30Weight (g) 3.4 3.34 3.34 0 3.27 -116.667 3.37 50

Cost ($) 150 570 570 0 570 0 570 0


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For the obtained solution (with nominal target values), all the RP process variables(HOC and FOC) are already at their lower bounds resulting in minimum possible buildtime for each mold piece for the current slicing scheme (#1). Also, the number ofrequired molds is the minimum possible value (1). Therefore the obtained build time isthe minimum possible build time for the slicing scheme. As the solution (Table 7.12)indicates that no thermal aging is needed, the build time is already at its minimumpossible value. The cost is also at its minimum possible value and cannot be reducedfurther for this slicing scheme. Hence, no change is detected in its value in spite ofreducing its target range.

After the solving process of the MPGT problem is validated, the results of physicalexperiments are presented in the next section. They further validate the solution of theproblem.

7.5 PHYSICAL VALIDATION

Several physical experiments were performed including build time, accuracy(flatness tolerance and parallelism tolerance), surface finish, material properties,geometry, weight, and mold life. They verify the values of the goals given in the solution(Table 7.12), and are described as follows.

7.5.1 Build Time Validation

The mold pieces are built on SLA 3500 – SL 7510. The layout of the mold pieces onthe SLA platform (corresponding to the obtained solution from the modified RP-PPproblem) is shown in Figure 7.17. For this layout, the estimated build time from buildstation is 4:39 hours and that from the RS models is 4:53 hours. The actual build time is4:50 hours. The values indicate the close correspondence between the actual andestimated (RS models) values. A summary of build time validation is given in Table7.17.

Table 7.17 - SLA Build Time for the Mold Pieces of Robot Arms.

Estimated build time (RS models) 4:53 hrActual build time 4:50 hrError 1.0%


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7.5.2 Accuracy Validation

The accuracy of the robot arms is measured on a Brown & Sharpe’s Microval PFxTM

454 moving bridge Coordinate Measuring Machine (CMM). The repeatability of themachine is 0.16 mils in all the three directions. A spherical probe with a 5 mm tipdiameter is used to take the measurements. The CMM machine is calibrated beforemaking the measurements. The robot arm is clamped in a vise and 10 data points aretaken from each of the two flat surfaces to determine the flatness and parallelismtolerances. One of the surfaces has ejector pin marks but they are carefully avoided whiletaking the measurements.

The measurements resulted in a flatness tolerance of 0.7 mil for the top surface(without ejector pin narkss) and 1.9 mil for the bottom surface. The parallelism errorbetween the two surfaces is 2.2 mils. The estimated flatness and parallelism tolerancesfor these surfaces (obtained from RP-PP software) are 1.7 mils and 1.6 mils respectively.Though the flatness error of the first surface is small, the other values are close to theexpected values of 1.7 (flatness) and 1.6 mils (parallelism). A summary of accuracyvalidation is given in Table 7.18.

Table 7.18 – Accuracy of the Robot Arms.

FTOL1 (mils) FTOL2 (mils) PTOL (mils)Estimated (RP-PP) 1.7 1.7 1.6

Actual (CMM) 0.7 1.9 2.2

Error 143% -10.5% -27.3%

Figure 7.17 – Mold Piece Layout on SLA 3500 Platform (Sambu, 2001).


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7.5.3 Surface Finish Validation

The surface finish of the robot arm is measured on a Taylor-Hobson Form Talysurf(a surface profilometer) with a 60 mm stylus and a 0.002in tip. The repeatability of theTalysurf is 0.86 µin. The robot arm has surface finish specification on the two innercylindrical surfaces. As the probe does not fit in these holes, the robot arms were brokenat the cylindrical holes, as shown in Figure 7.18, and the surface finish of the innersurfaces (both larger and smaller cylindrical surfaces) is measured. Two measurementswere made on each surface, one with a stroke length of 1 mm and another with a stokelength of 1.5 mm. For both measurements, a value of 0.25 mm is used as the filter(limiting value between waviness and roughness). The two measurements serve asreplications to determine sudden variations in the surface. After ensuring that there areno sudden variations in the surface (by comparing the results of both replications), themeasurement with stroke length of 1.5 mm is used as the surface finish value, as it haslonger trace and hence is a better representation of the complete surface.

The surface finish measurements of the large and small cylindrical surfaces for 1 mmand 1.5 mm stroke length are presented in Table 7.19 along with the estimated surfacefinish values. The measurements are made for two purposes: 1) to study the effect ofshot number on the surface finish, and 2) to compare the estimated and actual surfacefinish values. From the obtained data, no trend is seen in relating surface finish values tothe shot number. Hence, it is concluded that the shot number does not affect surfacefinish. This indicates that either the part or the mold stair-steps deform elastically duringthe part ejection to retain the surface profile.

Table 7.19 - Surface Finish Experiment Results on Prototype Parts.Large Cyl. Surface

(µin)Small Cyl. Surface

(µin)Stroke length 1 mm 1.5 mm 1 mm 1.5 mm

2 70 75 - -

7 75 63 83 80ShotNo.

46 82 73 71 68Average 75 70 77 74

Max % Error 9.0 10.5 7.5 8.7Estimated 64 64 65 65

% Error 15.2 8.7 15.7 12.0

Figure 7.18 – Pieces of Robot Arm Used in Surface Finish Experiments.


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The surface finish values for different stroke lengths serve as replications but thevalue corresponding to 1.5 mm stroke length is a better representative of the wholesurface and hence is used for comparison purposes. The errors (between estimated andactual values of surface finish) for the large and small cylindrical surfaces are 9% and12% respectively. These values are small considered the experimental errors.

7.5.4 Material Property Validation

Experiments are performed on a screw driven instron testing machine to determinethe Young’s modulus and tensile strength of the injection molded polystyrene material.The same process parameters as the obtained solution are used in the fabrication process.However the robot arms are not good candidates for testing the material properties.These parts have a varied cross-section and hence determining the stress and strain wouldbe difficult. Also, the length is too small to insert an extensometer on the part to measurethe displacement. Due to these reasons, it is decided to use standard tensile testingspecimens instead of robot arms. The same process parameters are used in producing thetensile bars in order to ensure that the material properties of the robot arm and the tensilebar are the same.

Four tensile testing specimens (for four replications) are used to determine materialproperties (Figure 5.8). The obtained values are presented in Table 5.8 and also repeatedin Table 7.20. In the table, ‘estimated’ row corresponds to the values of the materialproperties used in the MPGT problem and ‘mean’ row corresponds to the mean values ofthe material properties obtained from physical experiments. The variation of the materialproperties (maximum % deviation) for different replications is 4 – 9%. This is a smallerror and hence the values can be considered to be consistent. Comparing the actual andestimated values, tensile strength and Young’s modulus have an error of 1.7% and 2.5%respectively. These values are very small indicating a very good match between theestimated and actual values. Strain values are not used in MPGT problem and hence nocomparisons are provided.

Table 7.20 - Material Property Validation Results for Polystyrene.

ReplicationTensile Strength

(MPa)Young's Modulus

(MPa) % Strain @ Yield1 30.7 3392 1.012 32.2 3730 0.973 33.6 3698 1.104 32.5 3123 1.15

Mean 32.3 3486 1.06Max % Deviation 4.81 7.03 8.51

Estimated 32.8 3400 -% Deviation 1.71 2.46 -

7.5.5 Geometry and Weight Validation

The dimensions of the fabricated robot arm (D, d, and t) are measured with verniercalipers. The weight of the robot arm is measured on a weighting machine with anaccuracy of 0.0001 gm. The dimensions and the weight of the injection-molded robot


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arm along with the expected values (from CAD model) are presented in Table 7.21. Thepresented values are obtained for one of the prototype parts but all the parts are observedto be similar.

Table 7.21 - Injection-molded part dimensions and weight.

D (mm) d (mm) t (mm) Weight (g)CAD model 20.64 9.9 2.99 3.34

Actual 20.71 10.04 3.65 4.09Error 0.3% 1.4% 18.1% 18.3%

The error of the D and d values is very small (< 1.5%). The outer surfaces of the partare drafted (by 1.50) and hence it is difficult to measure the dimensions D and daccurately. Hence, these values of error are reasonable (considering the mold building,injection-molding shrinkage and measurement errors). The error in part thickness ‘t’ isvery high (18%), and cannot be considered as experimental error. It has been observedthat this error is due to flashing, which is “extra plastic enter between or under moldparts” (Rees, 1995). The flashing occurred due to the mismatch between the partingsurfaces and not due to the opening of the mold. It is observed that the mold pieces arecurved in shape due to warpage. This resulted in a gap between the two mold pieces thatinevitably causes flashing. Also, it is observed that the clamping force did not reduce thisgap as the parting surfaces are below the mold base height, hence the clamping force isnot exerted on the molds. Flashing is observed on all the molded parts and its thicknessis measured to be around 0.6 mm. This value explains most of the error in the value ofthickness. The misalignment between the parting surfaces (cause of flash) of the moldpieces is shown in Figure 7.19.

A probable reason for the considerable warpage in the mold could be due to the lowvalues of hatch overcure used in building the mold. From the solution obtained for theMPGT problem (Table 7.12), it is noted that minimum possible hatch and fill overcurevalues are used in building both the mold pieces. Using small values of overcuredecrease the building time. However, it also leads to smaller degree of cure of the moldsbuilt on the SLA machine. Later, when the parts are thermally cured, all the uncuredresin is cured, which leads to bulk shrinkage and warpage. Higher quantities of uncured

Mold Core

Mold Cavity

Gap due towarpage

Figure 7.19 - Gap Between the Parting Surfaces due to Mold Warpage.


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material lead to higher warpage. Hence, a probable solution to reduce warpage is to usehigher hatch and fill overcures. In order to add this consideration in the DFRTS,quantitative models are needed to predict warpage as a function of RP process variables.Then this problem can be prevented.

The error in the weight of the robot arm is ≈18%. This error is also very high. But itis primarily due to the flashing and the related increase in part thickness. Using thevalues of dimensions and weights, the actual and estimated density values are computed.The estimated density (used in MPGT problem formulation) is 1.04 gm/cc. The densitycomputed from the experimental values of dimensions and weight is 1.025 gm/cc. Theerror in the value of density is 1.5%. This value is very small indicating that the error inweight is primarily due to variation in thickness and not due to density.

7.5.6 Mold life validation

To verify the mold life of the generated molds, several robot arms are injection-molded from the molds. The injection-molding parameters used are presented in Table7.22. The value of the cooling time obtained from the MPGT solution is used in physicalexperimentation. The other parameters are adjusted to obtain good parts.

Table 7.22 - Injection Molding Parameters Used for the Robot Arm.

Parameter Value UnitsClamping Force 11 TonsInjection Pressure 3 MpsiPilot Valve Pressure 70 psiPlastic Temp. in barrel 430 0FPlastic Temp. in nozzle 450 0FInjection Time 20 SecondCooling Time 300 SecondRelease Agent Used No units

A total of 50 parts are molded from one set of molds. These include the partiallyfilled shots obtained during the initial phases of parameter fine-tuning and theintermittent unexpected failed shots. Out of the 50 parts, 26 are the complete shots.Samples of complete and incomplete shots are shown in Figure 7.21. All the 50 partsmolded have plastic solidified around the big boss feature but only 26 of these havematerial solidified around the small boss feature. This indicates that the large bossfeature is tested for 50 shots and the small feature is tested for 26 shots. It is observedthat a small portion of the big boss feature is chipped off after 46 shots. The chippedmold and the last four parts obtained from this mold are shown in Figure 7.20.

No features failed due to either pullout or flow failure for 50 shots, which ispredicted by the mold life predictor. This validates the solution that one mold issufficient to mold 50 parts with no pullout and flow failure.


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7.5.7 Summary of Physical Validation

The values of all the goals except stress and displacement were experimentallymeasured and validated in this section. The values of the different goals /variablesobtained from physical experimentation are summarized in Table 7.23 along with theirestimated values. From the values in the table, we can see that the variables D, d,Young’s modulus, tensile strength, build time and density have error < 1.5%. This erroris very small and can be considered as experimental or part building error. The surfacefinish values have an error of around 10%. Though not very small, this error is alsoreasonable. It should be noted that the surface finish models are developed for the SLAmolds and are applied for the injection-molded parts. Part fabrication and ejection could

Extra materialon the part

due tochippingfailure

Chipped ofregion of themold feature

Figure 7.20 - Chipped Mold and Parts Obtained from the Mold.

Figure 7.21 - Short (left) and Complete (right) Shots obtained from IJM.


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cause small variations in surface finish values. Weight and part thickness have an errorof 18%. This can be attributed to the flashing because of mold warpage. The error in theaccuracy /tolerance values could be due to the small surface area used for themeasurement.

Table 7.23 - Summary of Physical Validation Results for MPGT of Robot Arm.

Goal Actual value Estimated value ErrorD (mm) 20.71 20.64 -0.3%d (mm) 10.04 9.9 -1.4%t (mm) 3.65 2.99 -18.1%

Young's Modulus (MPa) 3486 3400 -2.5%Tensile Strength (MPa) 32.3 32.8 1.5%

Flatness Tol1 (mils) 0.7 1.7 142.9%Flatness Tol2 (mils) 1.9 1.7 -10.5%

Parallelism Tol (mils) 2.2 1.6 -27.3%Large hole SF (µin) 70 64 -8.7%Small hole SF (µin) 74 65 -12.0%

Build Time (min) 290 293 1.0%Weight (g) 4.09 3.34 -18.3%

Density (g/cc) 1.025 1.04 1.5%

Values of stress and displacement were not measured due to lack of appropriatemeasurement equipment. There was no easy way of measuring the stress anddisplacement of the robot arm under the loading conditions considered in the problem.Hence these values are not experimentally validated. However it should be noticed thatthese values are partially validated by validating the Young’s modulus of tensile bars(Section 7.5.4). The remaining part of validation (obtaining stress and/or displacementfrom Young’s modulus) concerns with the finite element analysis (FEA) of the robot arm.Some of the validating aspects of the FEA model include: verifying if the loadingconditions represent the real problem, verifying if the loading conditions areappropriately applied, verifying if the material properties are entered correctly in correctunits, and verifying if the mesh size is small and the element shape has small aspect ratio.For robot arm problem, these aspects are taken care while preparing the FEA model inPro/Engineer. Apart from these, no other physical validation is performed to explicitlydetermine the stress and displacement at the considered loading conditions.

Based on the experiment results, the solution given by the DFRTS (Section 7.4.3)satisfies the problem requirements given in Section 7.1. Fifty prototypes are producedwith satisfied surface finish and accuracy. The fabrication cost is $570 which is less thanthe budget ($1500). After all the validation work for the robot arm case is presented, anevaluation of the RTMDS and DFRTS based on the robot arm study is given in the nextsection.

7.6 EVALUATION OF ROBOT ARM CASE – POST DFRTS

With the aid of the DFRTS, the material-process geometric tailoring (MPGT) wassuccessfully performed for the robot arm (Section 7.4). To further understand the robotarm case, a comparing experiment is performed and presented here.


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In the experiment, the decisions on design variables and process variables are madesequentially. That is, the MGT problem is first formulated and solved. For the solutionof the MGT problem, the manufacturer then performs the process planning of the RapidTooling. The solutions (values of geometry and RP variables) for the experiment arepresented in Table 7.24 and the corresponding goal achievements are presented in Table7.25. The current solution and goal achievements are also listed in the table.

Table 7.24 - Solutions of Sequential and Concurrent Solution Process.

Variables D(mm)

d(mm)

t(mm)

TA(0/1)

Draft1(o)

draft2(o)

CT(s)

HOC1(mil)

HOC2(mils)

HOC3(mil)

FOC1(mil)

FOC2(mil)

FOC3(mil)

Sequential 20.34 10.17 3.049 0 0.62 0.1 300 0.001 0.002 0.001 0.002 0.012 0.002Concurrent 20.62 9.9 2.987 0 0.61 0 300 0.001 0.002 0.001 0.002 0.012 0.002

Table 7.25 - Goals Achievements for Sequential and Concurrent Solution Processes.

Goals Stress(Mpa)

y-disp(mm)

Weight(g)

SF1(µin)

SF2(µin)

SFP1(µin)

SFP2(µin)

Cost($)

Mold #

Sequential 6.022 0.485 3.404 65 64 6 6 570 1Concurrent 6.05 0.49 3.34 65.3 64.4 6 6 570 1

Target 5.99 0.51 3.4 20 20 10 10 570 1

Comparing the solutions of the sequential and concurrent solution processes, it isnoticed that the sequential process has a better solution of design variables (D, d, t) andrelated goal achievements (Stress, Y-displacement, weight) as their values are closer tothe target values. This is obvious because the design variables are solved first in thesequential solution process. As no process goals are considered in the determinationprocess of the design variables, higher weights are assigned to the design functionalgoals. This results in solutions that are closer to the target values. However, the draftangle and surface finish got from the sequential solution process are slightly worse.Therefore the improvement on design functions is actually compensated by thedeteriorations of process goals. At first it seems that the improvements of designfunctions and variables are larger than the deteriorations of process goals. After furtheranalysis, a possible reason is provided as follows. As stated before, the linear physicalprogramming (Section 2.5) is used in assigning weights for different goals. The feasiblerange of a goal is divided into sub-ranges of ideal, desirable, tolerable, and undesirable.Based on the algorithm for weight calculation (Hernandez and Mistree, 2001), the weightof a goal in desirable sub-range is approximately 10 times bigger than that of a goal inideal sub-range. This is reasonable in making tradeoffs between different goals. InFigure 7.22, the solution of the robot arm (Table 7.12) is positioned in related sub-rangesbased on the specified goals in the MPGT problem formulation (Table 7.7). As draftangle and surface finish are in desirable sub-range, their weights are much higher thanthose of design goals which are in ideal sub-range. Therefore the small changes of draftangle and surface finish will modify the objective function of the whole problem furtherthan those of design variables and goals.

In contrast to the author’s expectation, the solutions of sequential and concurrentsolution processes are almost the same (Table 7.24 and Table 7.25). This is mainly


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because the manufacturing requirements for the robot arm (50 prototypes with $1500 and7 days) are rather easy to achieve. Therefore the process considerations are notsignificant in the solving process of the MPGT problem. As discussed in Table 7.14, thereduction of the cost range by 40% did not change its value at all. As only one mold coreand cavity are needed for 50 prototypes, and all RP and IJM variables have their values inthe lower bounds, the cost of rapid tooling process cannot be reduced further. Thereforefixing design variables in the sequential solution process has little effects on the latersolving process for RP and IJM variables. This also explains why both solutionprocesses generate similar results. However, the process considerations may be muchsignificant for a more complicated part. Consequently the sequential and concurrentsolution processes will result in rather different solutions. A case of camera roller is usedto illustrate it, which is presented in Chapter 8. In the next section, a brief summary isgiven for discussing the relevance of these results with regard to the hypotheses of thedissertation.


In this chapter, the RTMDS and DFRTS were applied to determine the mold designand geometric tailoring for a robot arm design. A problem of producing functionalprototypes of a robot arm was first introduced in Section 7.1. By using the RTMDS, twomold designs were generated for the robot arm (Section 7.2). A standard mold base forMorgan Press injection molding machine was used in the mold piece construction. Theformulation of the geometric tailoring problem for the robot arm was presented in Section7.3. Among the three design functions, two of them were represented by responsesurface equations, and one was represented by an analytical equation. The generation ofthe equations was also presented in Section 7.3. The solution process of the MPGTproblem, which consists of three stages, was described in Section 7.4. The modifiedMPGT problems that are related to eight slicing schemes were solved by OptdesX. The

Ideal

Target

Unacceptable

UndesirableTolerable

Desirable

Desirable

Unacceptable

Undesirable

Tolerable

Ideal

stress

weight

Y-disp

Dd

t

θ1θ2

SF1

SF2PTOL

FTOL1

FTOL2

FTOL3

SFP1

SFP3

SFP2cost

Figure 7.22 – The Ranges of Goals for the Robot Arm Problem.

Lowerweight

Higherweight


288

results of physical experiments for the validation of the problem solution were presentedin Section 7.5. Finally an evaluation of sequential and concurrent solution processes forthe robot arm case was provided in Section 7.6.

The robot arm example highlighted different aspects of mold design and material-process geometric tailoring problem and related solution procedures. In the discussion ofapplication of RTMDS and DFRTS to the robot arm example, the relationships with thehypotheses were identified. The results from testing the hypotheses are summarizedbelow (Figure 7.23):

Hypothesis 1: Two mold designs were generated for the robot arm by using the RapidTooling Mold Design system, which is developed based on the Multi-piece MoldDesign Method. The whole running process only took less than one minute. Thesummary of testing the sub-hypotheses (H1.1 ~ H1.3) presented below will providemore details.

Hypothesis 1.1: 40 concave edges and 66 concave faces are identified. Based on theidentified concave faces, 26 concave regions are generated for the robot arm. Theregion generation process only took seconds for the robot arm.

Hypothesis 1.2: The generated concave regions of the robot arm are combined into 1concave region and 1 convex face for one mold design, and 1 concave region and 25convex faces for another mold design. The region combination process only tookseconds for the robot arm.


Hypothesis 2


Validation


The requirements on molddesign of the robot arm arerepresentative of the mold

design problems. Thesystem can be used to

generate mold design forthe robot arm.


Validation

Mold designs of robotarm are generated by

the RTMDS inacceptable time, andthey can be used to

produce prototpyes inthe Rapid Tooling

process.

Hypothesis 1

The requirements ongeometric tailoring of the

robot arm are representativeof the geometric tailoring

problems. The system canbe used to perform

geometric tailoring for therobot arm.

Solutions of geometrictailoring of robot arm aregenerated by the DFRTSin acceptable time, andtailored part design andprocess parameters can

be used to producefunctional robot arms inRapid Tooling process.

Figure 7.23 – Empirical Structural and Performance Validation for H1 and H2.


289

Hypothesis 1.3: Mold pieces are constructed for the generated mold design for a standardmold base which can be installed in a Morgan Press injection molding machine. Themold piece construction process only took seconds for the robot arm.

Hypothesis 2: Tailored part design with mold design variables, RP process variables andIJM process variables were generated for the robot arm by using the Design for RapidTooling system, which integrates geometric tailoring and process planning by themanufacturer. The satisfied prototypes are produced without any iteration for therobot arm. The summary of testing the sub-hypotheses (H2.1 ~ H2.3) presentedbelow will provide more details.

Hypothesis 2.1: For the robot arm the designer formulated a partial MPGT problembased on the MPGT decision template. The manufacturer formulated a completeMPGT problem which considered the material difference of products and prototypes.

Hypothesis 2.2: The designer’s MPGT problem formulation, RP process planningformulation, and IJM process planning formulation were integrated into a completeMPGT problem formulation for the robot arm.

Hypothesis 2.3: The MPGT problem was solved in a three-stage solution process. Thesolution was reasonable and validated by physical experiments.

In the next chapter RTMDS and DFRTS will be applied to a more complicated case,a camera roller (Figure 7.24). The case study will demonstrate RTMDS’s usefulnesstowards determining mold design for a part with an undercut. A multi-piece mold designis generated for the part (Section 7.2). The difficulties in fabricating the camera roller aremuch more significant (Section 7.3). Therefore in addition to the geometric tailoring dueto the material properties, the camera roller will also illustrate the necessity of geometrictailoring because of the fabrication process requirements (Section 7.4 and 7.5).


Chp 4: RTMDS and its Usage Chp 6: Design for Rapid Tooling

ParameticCAD Model

Relations betweenPart and MoldSize ParametersSurface FinishAccuracy

A. Mold ConfigurationGenerator

Parting directionParting linesMold halvesSide Actions

The RP ProcessCompromise DSPGiven

Mold DesignFind

Process parametersSatisfy

ConstraintsGoals

MinmizeDeviationC. Injection Molding

Process AnalyzerDraft Angle

Rib Height/Width ratioWall Thickness

Boss HeightMold Life

B. RP ProcessGenerator

Surface FinishAccuracy

CostTime

D. HierarchicalCompromise DSP

Processor

The Designer'sCompromise DSPGiven

Part DesignFind

Design parametersSatisfy

ConstraintsGoals

MinmizeDeviation

The IJM ProcessCompromise DSPGiven

Mold DesignFind

Mold parametersSatisfy

ConstraintsGoals

MinmizeDeviation

Tailored Part Design /Mold Design / RPProcess Design

Input and Output

Processor

C-DSP Template

Figure 7.24 – Preview of Chapter 8.

Chapter 8 – Functional Prototypes of a Camera Roller

290

CHAPTER 8

FUNCTIONAL PROTOTYPES OF A CAMERA ROLLER

The RTMDS (Chapter 4) and DFRTS (Chapter 6) are applied to determine the molddesign and geometric tailoring for a camera roller design in this chapter. This example ismore complicated than the robot arm example discussed in the previous chapter.Therefore it tests the applicability of the RTMDS and DFRTS for complex part designs.A problem of producing functional prototypes of a camera roller is first introduced inSection 8.1. By using the RTMDS, a mold design with three mold pieces is generated forthe camera roller (Section 8.2). A standard mold base for Morgan Press injectionmolding machine is used in the mold piece construction. A preliminary study andexperiment are performed which results in a modified camera roller design (Section 8.3).This modified design is considered in the DFRTS. The formulation of the geometrictailoring problem for the modified camera roller is presented in Section 8.4. In theformulation the design functions are represented by response surface models. Thesolution process of the MPGT problem, which consists of three stages, is described inSection 8.5. The modified MPGT problems that are related to 24 slicing schemes aresolved by OptdesX. The results of physical experiments for the validation of the problemsolution are presented in Section 8.6. Finally an evaluation of sequential and concurrentsolution processes for the camera roller case is provided in Section 8.7.

Chp 8: Prototypes of a Camera Roller


291

8.1 A CAMERA ROLLER DESIGN – PROBLEM DESCRIPTION

A more complicated part, a camera roller, is studied and presented in this chapter. Itis used to demonstrate the applicable of the RTMDS and DFRTS to more complicatedparts. This chapter is organized in a similar way as that of Chapter 7, in which a casestudy of a robot arm was presented. Also similar to the case study of the robot arm, theauthor worked with Shiva Sambu, a M.S. student in our lab, in performing the case study.

A camera roller as shown in Figure 8.1 is a component in a disposable camera. It isused as a spool on which a roll of film is wound. The movement of the film in thecamera is achieved by the rotation of the roller (either manually or automatically). Theprimary purpose of the roller is to deliver the film at the desired rate. Although therotation speed while taking a picture is low (manually by fingers), the rotation speed toreel the film onto the roller is pretty high while the camera is produced (automatically bymachines). As thousands of disposable cameras are produced each day, it is desired touse a higher winding speed to reduce the production time for each camera. As ratherhigh velocities and accelerations are involved, the stress of the camera roller in theassembly process is usually rather high. Therefore in the design of the camera roller, thedesigner should verify that the stress generated by the film movement in the high windingspeed is within the material strength. The camera roller should also have small weightto achieve low inertia, low cost and to facilitate easy handling.

Suppose during the film reeling process, loading conditions on the roller part for thecamera assembly process are shown in Figure 8.2. A tangential force of 20 N and a

Undercut

Figure 8.1 – A Production Camera Roller with a Camera.

Ribs Grooves


292

compressive force of 5 N are applied on the cylindrical surface (red color). Thecylindrical surface of the undercut is fixed (blue color). Tangential force is due to theacceleration and tension in winding the film. The compressive force is due to thecompression of wound film on the roller.

Under these loading conditions, the designer finishes a new camera roller designwith a lower weight (2.276g) and an angle of rotation about the axis of the cylinder(0.013 radian). Suppose in production, the camera rollers are to be fabricated in atacticpolystyrene through injection-molding process with steel molds. To test the new design,the designer requires fifty functional prototypes of the camera roller in testing theassembly process. In the testing, the designer is interested in the weight of the roller, andthe angle of rotation of prototypes about the axis of the cylinder. Therefore it is desiredthat the prototypes should be product-representative on the values of weight and therotation angle under the loading conditions.

Considering the assembly requirements, the designer also has surface finish andtolerance requirements for the prototypes, which are shown in Figure 8.3. The cylindricalsurface of the body and the cylindrical surface of the undercut (shown in Figure 8.1)should have a surface finish of less than 450 µin. The surface finish for the cylindricalsurface of the body is used to ensure the reeling of the film. The surface finish for the

Figure 8.2 - Loading Conditions on the Camera Roller.

Parallelism ≤ 20 milsPositional ≤ 30 milsParallelism ≤ 20 milsPositional ≤ 30 mils

Flatness ≤ 10 milsFlatness ≤ 10 mils

Surface Finish ≤ 450 µ inSurface Finish ≤ 450 µ in

Figure 8.3 - Surface Finish and Tolerance Requirements.

Fixed

Loads


293

undercut is for the rotation of the roller. Besides the surface finish, the outer circularfaces of the cylindrical features should have a flatness tolerance of less than 10 mils,parallelism tolerance of less than 20 mils, and positional tolerance of less than 30 mils.These tolerances are required for proper assembly and alignment of the roller in thecamera.

Considering the time and budget constraints, the designer wants to get the prototypeswithin a week for the maximum cost of $2500. To enable the manufacturer to producethe prototypes within such limited time and cost, the designer decides to given muchdesign freedom to the manufacture for geometric tailoring. As the detail features(grooves and ribs) of the part design will not be tested in the functional testing, thedesigner does not care about them in the prototypes. That is, as long as the weight andthe angle of rotation are production-representative, the manufacturer has the freedom tochange them to facilitate the fabrication of the prototypes.

For the problem as stated in this section, the mold design for the camera roller byusing the RTMDS is presented in Section 8.2; the geometric tailoring for the given designrequirements by using the DFRTS is presented in Section 8.3 to 8.5. Finally physicalvalidation of the results is presented in Section 8.6.

8.2 MOLD DESIGN WITH AID OF RTMDS

The Rapid Tooling Mold Design System (RTMDS) was used in the mold design forthe camera roller in the study. Based on the part shown in Figure 8.1, a CAD model ofthe camera roller was created. An additional step was added in the preparation process,that is, face A was used to split all the faces of the part (Figure 8.4). This is because aface is the basic element of the RTMDS. In the region generation and combinationprocess they usually will not be split (Section 3.5). Therefore if face F1 as shown inFigure 8.4 is a single face, it will belong to a region and a corresponding mold piece. If itis desired to form F1 and F2 with two mold pieces, they are required to be two differentfaces in the inputted CAD model. The preparation of the camera roller was performed inSolidWorks@, a 3D CAD system provided by SolidWorks Corporation(www.solidworks.com). The size of the generated CAD file (.sat) is 657 KB.

Figure 8.4 – Dividing Cylindrical Faces for the RTMDS.

F1 A

F2


294

With the aid of the RTMDS, the mold designer can generate a mold design morequickly for the camera roller. The information regarding the part, generated regions,reverse glue operation, and the execution time of each step is listed in Table 8.1. Thereader can refer to the descriptions given in Section 4.4.2 for the meaning of each items.The running time given in the table is based on a personal computer with a 700 MHzIntel-III processor. One thing to be noticed is that the running time of step 7 includes thetime for the user to interactively select the input file name of the mold base. Similarly therunning time of step 8 includes the time for the user to interactively select the output filenames of the mold pieces.

The graphical results given by the RTMDS are also provided in Figure 8.5 to foster abetter understanding of the region generation and combination processes of the system.In the figures, different colors are used to indicate different regions or convex faces(CVX). First in the region generation process, 32 regions are generated based on theconnectivity of concave faces (336 concave faces are identified). These regions areshown in Figure 8.5.a with two different views. In the first cycle of region combinationprocess, 32 regions are combined into 11 regions (Figure 8.5.b). The region combiningcriteria are mainly based on the connectivity of regions and feasible parting directions(Section 3.5.1). In the second cycle of region combination process, the 11 regions arecombined into 6 regions (Figure 8.5.c). Finally 3 regions are generated (Figure 8.5.d)after three cycles. They cannot be further combined due to the undercut shown in Figure8.1. Therefore three mold pieces are needed in producing the camera roller.

Table 8.1 – The Information for Camera Roller.





Reverse glue Info.For R1

1 0 0CPL No. Edge # of CPL1 Edge # of CPLi


Reverse glue Info.For R2

1 0 1Step 1 Step 2 Step 30.14 2.91 0.00

Step 4 Step 5 Step 62.66 33.02 0.04

Step 7 Step 8-1 Step 8-2 Total Time

Running Time(seconds)

9.4* 11.6* 12.5* 63.27*

Note: ‘*” denotes that the time of interactively selecting files is included in the measurement.


295

(a) Generated Regions (32 regions)

(b) Region Combination Result - Cycle 1 (11 regions, running time - 16.65 seconds)

(c) Region Combination Result - Cycle 2 (6 regions, running time - 10.69 seconds)

(d) Region Combination Result – Cycle 3 (3 regions, running time – 5.68 seconds)

Figure 8.5 – Graphical Results of Region Combination Process.

R2 R3

R1


296

Before the regions are used for the mold piece construction, two faces (F1, F2) asshown in Figure 8.6 need to switch from region R3 to R2. The reason of F1 is combinedto R3 was discussed in Section 4.3.2. F2 is a vertical face and is neighboring to both R2

and R3. Therefore it is combinable to both regions according to our combining criteria.In RTMDS, interactive tools are provided to select faces and switch the region numbersof the selected faces.

The mold base used for the camera roller is shown in Figure 8.7.a. The generatedmold pieces for R1 is shown in Figure 8.7.c, and the mold piece for R2 and R3 is shown inFigure 8.7.d. The generated glue face (GFps) is also shown in the mold pieces.

Figure 8.6 – Changing Region Number of Selected Faces in the RTMDS.

F1

F2R2

R3


297

After loading M2 as the new mold base for regions R2 and R3, the mold piececonstruction algorithm (Section 3.6.3) is executed again. The parting surface used in theconstruction is shown in Figure 8.8.b. The two generated mold pieces are shown inFigure 8.8.c. The glue faces (one GFps and one GFinner) used for the mold piececonstruction are also shown in the figure.

(a) A Mold Base for Morgan Press (b) Mold Base after Boolean Operation

(c) Mold Piece M1 for R1 (d) Mold Piece M2 for R2 and R3

Figure 8.7 – Graphical Results of a Mold Design for the Camera Roller –Phase 1.

GFps


298

The generation of CAD moldes of mold pieces is an important task in the molddesign process. After it is finished, other tasks can be started. In IronCAD system(www.ironcad.com), the author added ejector-pin holes, gate and runner in the moldpieces. A complete mold design for the camera roller is shown in Figure 8.9. The molddesign is further verified by physical validations. A photo of the built mold pieces andprototype parts is given in Figure 8.10.

(a) A Mold Base for Morgan Press (b) Part with the Parting Surface

(c) Two Generated Mold Pieces

Figure 8.8 – Graphical Results of a Mold Design for the Camera Roller – Phase 2.

GFinner

GFps


299

8.3 GEOMETRIC TAILORING WITH AID OF DFRTS – PROBLEM ANALYSIS

Suppose the direct AIM tooling (SLA 3500 – SL7510 molds) for Atactic polystyreneis chosen for the prototypes. Because of the complicated shape of the part, themanufacturer cannot determine if satisfactory prototypes can be produced in such a shorttime and limited budget. Therefore a preliminary analysis and experiment are performed

Figure 8.9 – A Completer Mold Design for the Camera Roller.

Figure 8.10 – Physical Validation of the Mold Design for the Camera Roller.

Gate

Ejector Pin Holes

Runner


300

to identify the challenges in injection molding the camera roller parts without anygeometric tailoring.

From Figure 8.9, one can see that the mold pieces have several thin and tall ribfeatures. Injection-molding parts with such features in SLA molds could be difficult. Asa preliminary study, the mold life predictor (Section 6.4.2) is used to determine the life ofall the protrusion features in the mold pieces. The calculations are based on the modelsfor the mold pieces fabricated on SLA 3500 with SL 7510 resin with no thermal aging. Acooling time of 300 seconds, injection pressure of 3 MPa, layer thickness of 2 mils and adraft angle of 10 are used in the calculation. The mold life of different features in moldcore, mold cavity and third mold piece are shown in Figure 8.11. All the mold featureshave very low life (≤ 30 parts).

As the mold life of a mold piece is determined by the lowest life of all its features,the required mold piece number for producing 50 prototypes is estimated in Table 8.2.

Table 8.2 – Estimated Mold Life and Required Number of Each Mold Piece.

Mold Piece Mold life of Mold Piece Required Number of Mold PieceMold Core 9 6

Mold Cavity 4 13Third Piece 16 4

Mold Core

9

23

24

26

30 26

Mold Cavity

4

21

9

610 21 23 10 29

22

119

10

Third Piece

16

Figure 8.11 - Mold Life of Different Features in Mold Pieces.


301

A physical experiment was also performed for the camera roller. The experimentalresult verified that the mold pieces, especially the mold cavity, have very low mold life.In the experiment, two thin ribs in the mold cavity are deformed only after one shot,which resulted in incomplete fill of all later shots (Figure 8.12). These two ribscorrespond to the features of mold cavity with mold life 4 and 6 in Figure 8.5. Thiscoincidence also verified that the mold life calculation presented in Figure 8.5 isreasonable.

Based on his/her experience, the manufacturer determines that it is infeasible toproduce the required prototypes in one week with only $2500. In the current usage ofRapid Tooling, the design will be sent back to the designer with a note that it cannot beproduced within the budget. However, the designer may not know what should bechanged in order to make it fabricatable, as he/she may have no idea about why the partcannot be made. Therefore it may take a long time with several iterations before therequired prototypes are produced. Sometimes it may also lead to the designer choosinganother fabrication process, or just increasing the budget to save the trouble.

However, as stated in Section 8.1, suppose the designer has already formulated thedesign intension (weight and angle of rotation) and other requirements (surface finish andtolerance) of the camera roller into a formulation which is then transferred to themanufacturer. The manufacturer can try to tailor the part design based on the design andmanufacturing requirements. Within the given design freedom, the tailored design maybe fabricatable. Therefore the iteration between the designer and manufacturer can beavoided.

In the case of camera roller, as the designer gave the design freedom of changingdetail features (grooves and ribs) of the part design (Section 8.1), the manufacturerdetermines to tailor their topology before performing parametric tailoring. Thesetopology tailoring and the related reasoning are provided as follows.

The camera roller is designed with large number of ribs to obtain good strength withlow weight. Most of the rib features have no specific purpose apart from providing

Figure 8.12 – Problem Identified in the Preliminary Experiment.

Incomplete Fills

Related Ribs in Mold Piece


302

strength to the part. To enable injection molding of the parts through SL molds, some ofthe slots in the part are filled with material to avoid the corresponding mold featurefailure. In Figure 8.13, the ‘Rib’ feature is required to hold the film in position and henceis an important feature. A ‘Support Rib’ is added to provide strength to the rib. Additionof support rib results in a small ‘Slot A’. This slot corresponds to the rib feature withmold life of 4 parts in Figure 8.11. Fabricating this slot feature with SL molds would bedifficult and hence the whole feature is removed by filling ‘Slot A’ with material.

In Figure 8.13, ‘Reel slot’ is required to hold one end of the film in the roller andhence is an important feature. To avoid the failure of the corresponding mold feature,slots B and C are filled with material. Filling slots B and C provides freedom to increasethe width of the ‘reel slot’ feature that would result in increased mold life. Also, fillingslots B and C avoids the risk of the corresponding mold feature failure. The slots locatedon the opposite side of slots B and C are also filled for the same reason.

Another feature in the part that should be modified to facilitate injection molding is‘Thin Wall’ shown in Figure 8.13. This thin wall has 0.5 mm wall thickness, 23 mmlength and 4 mm height. Filling such thin and long wall features with plastic requireshigh injection pressures. But using high pressures would cause flow failure of the moldfeatures. Hence, based on experience, a guideline on thin wall is that all the features inthe part should have a wall thickness of larger than 0.75 mm. Accordingly the thickness

Rib

Support Rib

Slot A

Slot B

Slot C

Thin Wall

Reel slot

Figure 8.13 - Different Features in Camera Roller.

Figure 8.14 - Modified Camera Roller Part.


303

of the thin wall in camera roller is increased to 0.75 mm. The part obtained after thesemodifications is shown in Figure 8.14.

The topology and parametric modifications presented so far are performed beforeformulating the MPGT problem. These modifications are required to ensure themanufacturability of the part. In MPGT problem, the functional property requirementand mold life concern are further considered to ensure that satisfactory prototypes areproduced. The formulation and solving of the MPGT problem are presented in Section8.4 and 8.5 respectively.

8.4 GEOMETRIC TAILORING WITH AID OF DFRTS – MODELING

To formulate the MPGT problem, the design functions and the fabrication processesshould be modeled first.

8.4.1 Modeling Design Functions

The design requirements for the camera roller are the weight (W) and the Rotationalong z-axis (ZR). In this study, the equations of W and ZR are represented by responsesurface models because the camera roller has rather complicated geometry. The steps ofgetting the equations are described as follows.

• Generating Z-Rotation Response SurfacesAs shown in Figure 6.10, the approach used in this study is similar to the RCEM

(Section 2.5). Five steps are used to generate the response surface equations of ZR.

(1) Identify factors and ranges

Based on the available design freedom for the modified camera roller (Figure 8.14),the geometry variables that are considered in the geometric tailoring are shown in Figure8.15. In the figure, D is diameter of the cylindrical hole, Wi is width of the ‘Reel Slot’, tis thickness of the center plane, NC is number of columns of slots in the camera rollerarray, and NR is number of rows. These variables are the key factors that affect thefunctional properties and mold life of the features. D, Wi and t are continuous variables.NC and NR are discrete variables. The ranges of the geometry variables are presented in

D

Wi

T

NC

NR

Figure 8.15 - Geometry Variables in the Modified Camera Roller.

t


304

Table 8.3.

(2) Design of Experiments

A three factor – three level central composite response surface design with α = 1.0(for rotatability) is used as the design for the experiments to generate Z-Rotation responsesurface models. In this design, only one replication is considered because theexperiments are performed on a software system (ANSYS), which would result in thesame solution for all the replications. Two discrete variables are NC and NR. They canhave 4 combinations as (2, 2), (2, 3), (3, 2), and (3, 3). Three continuous variables are D,Wi, and t. Table 8.4 lists the values of the continuous design variables for each of theexperiments.

Table 8.4 - List of Experiments for Response Surface Generation.

Expt. No. D (mm) Wi (mm) t (mm)1 4.75 4 32 4.75 4 13 4.75 2 34 4.75 2 15 2.75 4 36 2.75 4 17 2.75 2 38 2.75 2 19 4.75 3 210 2.75 3 211 3.75 4 212 3.75 2 213 3.75 3 314 3.75 3 115 3.75 3 2

(3) Simulation

Considering both discrete variables (NC and NR) and continuous variables (D, Wi,and t), a central composite design is used for three level variables and a full factorialdesign is used for the two level variables. Therefore there are totally 4x15 = 60experiments. For each experiment, the analysis of the robot arm is performed by a FEAsoftware system (ANSYS). First the parametric CAD models are generated in a CADsoftware system, SolidWorks (www.solidworks.com). Then FEA modeling is performed

Table 8.3 – Design Factors and Their Ranges of Camera Roller.

Design Factor Low Band (-1) Middle (0) High Band (1)D (mm) 2.75 3.75 4.75

Wi (mm) 2.0 3.0 4.0t (mm) 1.0 2.0 3.0

NC 2 - 3NR 2 - 3


305

in another CAD system, Pro/Engineer (www.ptc.com). Finally finite element analysis isperformed in ANSYS. The loading conditions shown in Figure 8.2 are applied on thecamera roller. A cylindrical coordinate system is used in adding the loads. The variationof Z-Rotation in the camera roller for experiment 1 for NC = NR = 3 is shown in Figure8.16. The Z-Rotation has a value of zero near the cylindrical hole (where it is fixed) andit has a maximum value of 6.517x10-3 radians at the other end. The variation of Z-Rotation within the camera roller is similar for all the experiments. In the plot, blue color(right end of the part) indicates small rotations while red color (left end) indicates bigrotations.

Before performing the experiments listed in Table 8.4, a set of experiments areperformed to determine a good mesh size considering the trade-off between the accuracyof the solution and running time. The experiments performed on the modified cameraroller part and the results obtained are presented in Table 8.5.

Table 8.5 - Experiments to Study the Effect of Mesh Size.

Expt No.Max size

(mm)Min size

(mm)Z-Rotation

(radian) # Elements Smart mesh1 3 0.5 0.007437 8868 No2 2 0.5 0.007679 13967 No3 1.6 0.5 0.007794 10045 No4 1 0.5 0.007870 10445 No5 1 0.3 0.007870 27728 No6 1 0.5 0.007957 30679 Cylinders (max mesh size = 0.5)

7 1.5 0.5 0.008013 28688Cylinders and surfaces near it

(max mesh size = 0.5)8 0.75 0.5 0.008034 54810 No

9 1 0.5 0.008178 70061Cylinders and surfaces near it

(max mesh size = 0.5)

Figure 8.16 - ANSYS Output of Z-Rotation for Camera Roller (Exp.1: NC=NR = 3).


306

In Table 8.5, the experiments are arranged in ascending order of the accuracy of Z-Rotation value. In FEA, using large number of elements results in more accurate value ofstress or strain /displacement. Using lower number of elements usually results in smallervalues of stress or strain/displacement. This is also evident from Table 8.5. In this table,max size and min size are the maximum and minimum allowable element sizes. Smartmesh column indicates if any smart meshing is used for the part. Smart meshing involvesidentifying small or crucial features and locally reducing the element size in that region.Number of elements in the part after meshing is presented in # elements column. Z-Rotation is the value of the rotation of the part along its axis obtained after FE analysis.

Experiments 1 ~ 4 and 8 have same values of minimum element size but differentmaximum size. The number of elements and z-rotation increase with reducing the maxsize. Experiment 3 is an exception since the number of elements reduces by reducing themax size. Though, the value of z-rotation increases. This could be due to more uniformelement size throughout the part. Experiments 4 and 5 have same values of max size butdifferent min size. The number of elements in the part increases by reducing the value ofmin size for experiment 5 but the z-rotation value did not change indicating that reducingthe value of min size is not helpful in obtaining a better solution. Comparing experiments4 and 6, the values of min size and max size are same but in experiment 6, smart mesh isused. The small cylindrical features in the part are meshed with a max element size of0.5 mm. This increased the number of elements and also the accuracy of z-rotation. Thecomparison between experiments 3 and 7 is similar. In experiment 7, smart mesh is usedfor the cylindrical features and the surfaces around them. Experiments 6 and 9 aresimilar except that surfaces around the cylindrical feature have smaller mesh size forexperiment 9.

Comparing the values of z-rotation for the first and last experiment, we can see thatexperiment 1 has a 9% lower value. This means that using a max size of 3 mm with nosmart meshing results in a 9% error in z-rotation value when compared to using a maxsize of 1 mm with smart meshing. However as the computation time for an experiment isproportional to the number of elements used in the experiment, it is noticed thatexperiment 9 requires approximately 25 minutes running time (on a PC with a 700MHzIntel-III processor). This time is unacceptable when running 60 experiments to fitresponse surface models. Comparing the accuracy of the solution and number ofelements in the part, experiment 7 (using max element size of 1.5 mm, min element sizeof 0.5 mm and max element size of 0.5 mm around the cylindrical features) is chosen asthe standard for the experiments in this study. This experiment has a z-rotation error of2% (compared to experiment 9) and takes approximately 10 minutes to run.

The z-rotation given by ANSYS for each experiment is shown in Table 8.6.

Table 8.6 – Results of the Z-Rotation Experiments for Camera Roller.

Z-Rotation (Radian)Expt. No. D (mm) Wi (mm) t (mm) NC=2,

NR=2NC=2,NR=3

NC=3,NR=2

NC=3,NR=3

1 4.75 4 3 0.007229 0.006919 0.006819 0.0065172 4.75 4 1 0.012065 0.010893 0.010386 0.0091233 4.75 2 3 0.00588 0.005612 0.005539 0.005187


307

4 4.75 2 1 0.009963 0.008994 0.008474 0.0073875 2.75 4 3 0.007864 0.007501 0.007381 0.0071786 2.75 4 1 0.012597 0.011494 0.010961 0.0098777 2.75 2 3 0.006401 0.006137 0.006073 0.0057928 2.75 2 1 0.010419 0.009629 0.00897 0.0080139 4.75 3 2 0.008588 0.007962 0.007741 0.00702910 2.75 3 2 0.009109 0.008594 0.008441 0.00763111 3.75 4 2 0.009894 0.009249 0.008854 0.00813212 3.75 2 2 0.008018 0.007537 0.007172 0.00658813 3.75 3 3 0.006667 0.006325 0.006264 0.00592314 3.75 3 1 0.010888 0.009929 0.009409 0.00834815 3.75 3 2 0.008752 0.008202 0.007809 0.007242

(4) Build Response Surface Models

A statistical software system, MINITAB, was used to perform regression analysisand ANOVA of the obtained data. The response surfaces generated by the system arepresented in equations 8.1 ~ 8.4 along with their R2, R2 (adj), maximum deviation andaverage deviation values. In the models, the units of geometry variables (D, Wi and t) aremm and of z-rotation (ZR) are radians.

NC =2, NR=2:

ZR2,2 = (12.1910 - 0.7585•D + 0.1764•Wi - 1.5277• t + 0.0819•D•D + 0.1828•Wi•Wi +0.0149• t• t - 0.0223•D•Wi - 0.0197•D• t - 0.1727•Wi• t) •10-3 [8.1]


NC =2, NR=3:

ZR2,3 = (11.6270 - 0.8488•D - 0.0506•Wi - 1.1300• t + 0.0729•D•D + 0.1808•Wi•Wi -0.0693• t• t - 0.0028•D•Wi + 0.0153•D• t - 0.1287•Wi• t) •10-3 [8.2]


NC =3, NR=2:

ZR3,2 = (11.2048 - 1.5515•D + 0.5053•Wi - 0.7899• t + 0.1768•D•D + 0.1035•Wi•Wi -0.0629• t• t - 0.0126•D•Wi - 0.0031•D• t - 0.1548•Wi• t) •10-3 [8.3]


NC =3, NR=3:

ZR3,3 = (9.43317 - 1.0040•D + 0.2710•Wi - 0.5420• t + 0.09824•D•D + 0.12634•Wi•Wi -0.08486• t• t - 0.02167•D•Wi + 0.01327•D• t - 0.10400•Wi• t) •10-3 [8.4]



308

(5) Validation of response surfaces

The response surfaces of z-rotation (equations 8.1 ~ 8.4) have very high R2 and R2

(adj) values (≥ 99.8 %). This indicates that the response surface fits very well throughactual data. The values of maximum and average deviation for Z-Rotation responsesurfaces are ≤ 1.4% and 0.5% respectively. These values are very small which reflectsthe accuracy of the response surface models.

To gain a better understanding between the responses and design variables, theresponse surface Z2,2 is plotted in Figure 8.17. The response surfaces for other values ofNC and NR are very similar.

In Figure 8.17, Z-Rotation is higher for higher values of width and for lower valuesof thickness and diameter. The variation of Z-Rotation with diameter is very smallthough. Increasing width removes more material in camera roller resulting in lowerinertia and hence higher values of Z-Rotation. On the other hand, increasing thicknessadds material in camera roller in the center plane resulting in increased inertia and hencelower values of Z-Rotation. Increasing diameter changes the location of the fixedcylindrical surface in the camera roller. Apparently, this did not have noticeable effect onZ-Rotation. The behavior of the response surfaces is intuitive and reasonable.

To validate the quantitative models more completely, additional experiments areperformed to determine the error between actual and approximated design space. Fourvalidation experiments are performed (one for each response surface). The experimentsare performed in the same way as the original set of experiments. The values ofgeometry variables used in these experiments along with the obtained results arepresented in Table 8.7. The maximum and average errors of Z-Rotation response surfaceare 2.1% and 1.1% respectively. The error values for Z-Rotation are very small and arecomparable to the values obtained from the original set of experiments. This indicatesthat the response surfaces are in fact an approximation of the actual design space and arenot merely the surfaces fitted through the data points used in the DOE.

Table 8.7 - Results of Validation Experiments of Z-rotation.

Z-Rotation (* 10-3) (rad.)Expt. No. D Wi t NC NR

Actual Estimated % Error1 4.25 3.5 2.5 3 3 6.86 6.85 0.082 4.25 2.5 1.5 3 2 8.22 8.14 0.96

Figure 8.17 – Graphical Relations of z-rotation and Variables.


309

3 3.25 3.5 1.5 2 3 9.97 9.76 2.114 3.25 2.5 2.5 2 2 7.55 7.46 1.23

• Generating Weight Response SurfacesUnlike the robot arm example where the weight is computed using analytical

equation, in this case weight is computed using response surfaces. This is due to thecomplicated part geometry of the camera roller. The same 60 experiments used for Z-Rotation are also used in generating the weight response surfaces. The volume of theCAD models is obtained from SolidWorks, in which a tool of mass properties function isprovided to calculate the volume of a part.

Since the first three steps are similar to those of generating z-rotation responsesurface, they are not repeated here. The volume given by SolidWorks for eachexperiment is shown in Table 8.8.

Table 8.8 – Results of the Volume Experiments for Camera Roller.

Volume (mm3)Expt. No. D (mm) Wi (mm) t (mm) NC=2,

NR=2NC=2,NR=3

NC=3,NR=2

NC=3,NR=3

1 4.75 4 3 2407.16 2454.03 2434.44 2477.952 4.75 4 1 2151.64 2214.25 2187.44 2245.63 4.75 2 3 2580.34 2627.21 2607.63 2651.134 4.75 2 1 2336.22 2398.84 2372.03 2430.185 2.75 4 3 2452.92 2499.79 2480.2 2523.716 2.75 4 1 2197.4 2260.01 2233.2 2291.367 2.75 2 3 2626.11 2672.97 2653.39 2696.898 2.75 2 1 2381.98 2444.6 2417.79 2475.949 4.75 3 2 2367.48 2422.22 2399.02 2449.8610 2.75 3 2 2413.24 2467.98 2444.79 2495.6211 3.75 4 2 2306.35 2361.09 2337.89 2388.7212 3.75 2 2 2485.23 2539.97 2516.77 2567.613 3.75 3 3 2519.34 2566.2 2546.62 2590.1314 3.75 3 1 2269.51 2329.99 2303.18 2361.3415 3.75 3 2 2394.43 2449.17 2425.97 2476.8

Similarly MINITAB was used to perform regression analysis and ANOVA of theobtained data. The response surfaces generated by the system are presented in equations8.5 ~ 8.9 along with their R2, R2 (adj), maximum deviation and average deviation values.In the models, the units of geometry variables (D, Wi and t) are mm, of volume (V) ismm3, and of weight (W) is g.

NC =2, NR=2:

V2,2 = 2.47088 + 0.00763•D - 0.10332• Wi + 0.11638• t - 0.00407•D•D +0.00136•Wi•Wi + 0.00285•Wi • t [8.5]



310

NC =2, NR=3:

V2,3 = 2.54752 + 0.00457•D - 0.10577•Wi + 0.11135• t - 0.00366•D•D + 0.00177•Wi•Wi- 0.00066• t • t + 0.00285•Wi• t [8.6]


NC =3, NR=2:



NC =3, NR=3:



W = 1.04 * V[8.9]

The response surfaces of volume (equations 8.4 ~ 8.8) have very high R2 and R2

(adj) values (100 %). This indicates that the response surface fits very well throughactual data. The values of maximum and average deviation for volume response surfacesare ≤ 0.04% and 0.01% respectively. These values are very small which reflects theaccuracy of the response surface models.

To gain a better understanding between the responses and design variables, theresponse surface V2,2 is plotted in Figure 8.18. The response surfaces for other values ofNC and NR are very similar.

Figure 8.18 – Graphical Relations of Volume and Variables.


311

Variation of volume with geometry variables, as shown in Figure 8.18, is opposite tothat of Z-Rotation. This is intuitive, as Z-Rotation is inversely proportional to inertiawhile volume is directly proportional to inertia. Volume is higher for lower values ofdiameter and width and for higher values of thickness. Using lower values of diameterand width removes smaller amounts of material and hence the part has higher volume.Using higher value of thickness adds more material in the center plane and henceincreases the volume of the part. This indicates that the behavior of the response surfacesis intuitive and reasonable.

To validate the quantitative models more completely, additional experiments areperformed to determine the error between actual and approximated design space. Fourvalidation experiments are performed (one for each response surface). The experimentsare performed in the same way as the original set of experiments. The values ofgeometry variables used in these experiments along with the obtained results arepresented in Table 8.9. The maximum and average errors of volume response surface are0.018% and 0.008% respectively. The error values for volume are quite insignificant andit can be very well assumed that the estimated values of volume from response surfaceare equal to the actual values.

Table 8.9 - Results of Validation Experiments of Volume.

Volume (mm3)Expt. No. D Wi t NC NR

Actual Estimated % Error1 4.25 3.5 2.5 3 3 2.4774 2.4772 0.0082 4.25 2.5 1.5 3 2 2.3991 2.3987 0.0183 3.25 3.5 1.5 2 3 2.3559 2.3557 0.0064 3.25 2.5 2.5 2 2 2.5117 2.5117 0.000

As a summary, the Z-rotation and weight response surfaces are validated by:

a) The response surfaces have a reasonably good fit through the actual data points.This is tested by checking the R2, R2 (adj), maximum deviation and averagedeviation values for these response surfaces.

b) It is qualitatively verified (from the response surface plots) that the behavior ofthe response surfaces is in accordance with the expected behavior.

c) The low values of maximum deviation and average deviation for the validationexperiments indicate that the response surface models are in fact a reasonablygood approximation of the original design space.

8.4.2 Modeling Fabrication Processes

For the fabrication process of prototypes (direct AIM tooling), suppose SLA 3500 –SL 7510 molds are used in the injection molding process. As discussed in Section 6.4.2,the models of surface finish, accuracy, mold life, build time and cost for the fabricationprocess were formulated based on other research work. These models are the same fordifferent parts. Therefore they were used in the geometric tailoring for the camera rollerdirectly. The models of process that are related to the robot arm case are listed asfollows.


312

Surface finish SF = f (PO, LT, θ1, θ2)Surface finish of parting surface SFP = f (PO, LT)Parallelism tolerance PTOL = f (PO, LT, HOC, FOC)Flat tolerance FTOL = f (PO, LT, HOC, FOC)Mold life ML = f (LT, CT, TA, D, d, t, θ1, θ2)

Mold number NN

MLmp=

−+L

NMOQP

11

SLA build time BT = f (PO, LT, HOC, FOC, Nm)Mold cost Cm = f (BT, TA)Part cost Cp = f (CT)Total cost C = Cm + Cp

Mold time Tm = f (BT, TA)Part time Tp = f (CT)Total time T = Tm + Tp

For the direct AIM tooling, the RP process variables are mold orientation (PO),slicing scheme (LTp), hatch overcure (HOCp), fill overcure (FOCp), and thermal aging(TA). In the variables, the sub-script ‘p’ corresponds to the block number of differentlayers. TA is a boolean variable, which related to if the mold pieces are thermal curedbefore they are used in the IJM process. The IJM process variable is cooling time (CT).The bounds on the cooling time depend on part geometry and are different for differentparts. Experimentally it is verified that a cooling time of 5 – 7 minutes is required toinjection-mold the robot arm with SLA molds on Morgan press injection-moldingmachine. Attempts to mold parts with lower cooling times resulted in deformed parts.Beside the process variables, the manufacturer may add three draft angle variables θ1, θ2,and θ3 (Figure 8.19) for mold life consideration. In this study a uniform draft angle isused for a mold piece. That is, draft angles θ1, θ2 and θ3 are considered for all protrusionfeatures of the mold core, mold cavity, and the third mold piece respectively. Alltogether, there are a total of 15 protrusion features in the mold pieces. It is possible tofabricate each of them with different draft angles. However considering 15 different draftangle variables significantly increases the computational time and hence only threedifferent variables (one for each mold piece) are considered in this study.

Obtaining low values of draft angles is considered as a goal in the MPGT problem.Using higher draft angles increases the volume (and hence weight) of the part and alsodistorts the whole part geometry and hence is undesirable. Other goals of the cameraroller given by the designer can be transferred to goals of mold pieces directly based onthe accuracy and surface finish of the part surfaces are very close to those of the moldsurfaces (Cedorge, 1999). For a mold design given by RTMDS (Figure 8.9), someprocess-related goals are shown in Figure 8.19. The manufacturer may add an additionalsurface finish requirement (SFP ≤ 200 µin) for the parting surfaces of the mold pieces.The manufacturer may also add some other requirements in the mold pieces forproducing the required prototypes parts. These requirements include positional tolerance(PoTOL1, PoTOL2), flatness tolerance of parting surfaces (FTOLP1, FTOLP2, FTOLP3,FTOLP4, FTOLP5), and perpendicularity tolerance (PeTOLP1, PeTOLP2) (refer to Figure8.19).


313

It is noticed that in the third mold piece, two of the surface finish goals are coupledwith draft angle variables (θ3). These surface finish goals correspond to the cylindricalundercut in the camera roller and the secondary parting surface. The drafts in the othermold pieces affect the rib features in the mold but not the cylindrical surface in the mold(that has surface finish requirement). Hence the other surface finish requirements are notcoupled.


Based on the information given by the designer, and the models of design functionsand fabrication processes, the manufacturer can formulate a complete MPGT problem asshown in Table 8.10. In the formulation the injection pressure (IP) used in fabricating theparts is also added because high injection pressure may lead to flow failure of the molds.Based on the preliminary experiment, it is determined that an injection pressure of 2.0Mpsi is sufficient for molding camera roller parts.

Primary PartingSurface

SecondaryParting Surface

Surface Finish≤ 200 µin


Flatness ≤ 20 milsPerpendicularity ≤ 10 mils

Flatness ≤ 10 milsParallelism ≤ 20 milsPositional ≤ 30 mils


Flatness ≤ 10 milsParallelism ≤ 20 milsPositional ≤ 30 mils

Flatness ≤ 20 milsPerpendicularity ≤ 10 mils



Primary PartingSurface




Flatness ≤ 20 mils

Figure 8.19 – Process Planning Goals for the Mold Pieces.

(PoTOL1) (PoTOL2)

(FTOLP1)

(FTOLP2)

(SFP3)

(FTOLP4)

(SFP5)

(PeTOLP1)

(PeTOLP2)

(FTOLP3)

(FTOLP2)

(FTOLP5)

Draft angle

θ1

Draft angle

θ2

Draft angle

θ3


314

Table 8.10 – MPGT Camera Roller Problem Formulation.

PROBLEM STATEMENT:Find a tailored part design and related process parameters for producing functional

prototypes of the camera roller as shown in Figure 8.1. In the problem the mold design of thecamera roller is already determined which is shown in Figure 8.19.GIVEN:! Parametric CAD model of modified camera roller! Parametric CAD models of the mold pieces for the camera roller! Geometry variables that affect part functionality: D, Wi, t, NC, NR! Tangential force = 20N, compressive force = 5N! Production parts injection molded in Atactic polystyrene with steel molds

• Young’s modulus YMp = 3200 Mpa• Tensile strength TSp = 37.4 Mpa• Density Denp = 1.04 g/cc

! Prototype parts injection molded in Atactic polystyrene with SLA 3500 – SL 7510 molds• Young’s modulus YMm = 3400 MPa• Tensile strength TSm = 32.8 MPa• Density Denm = 1.04 g/cc

! Required number of prototype parts Np = 50! Injection pressure (IP) = 2.0 Mpsi! Goals of prototypes under loads:

• Z-Rotation requirement (ZR),• Weight requirement (W)• Geometry variables requirements (D, Wi, t)

! Goals of produced prototypes:• Parallelism tolerance between cylindrical surfaces (PTOL1, PTOL2)• Flatness tolerance of mold pieces (FTOL1, FTOL2, FTOL3, FTOL4)• Positional tolerance of mold pieces (PoTOL1, PoTOL2)• Flatness tolerance of parting surfaces (FTOLP1, FTOLP2, FTOLP3, FTOLP4, FTOLP5)• Perpendicularity tolerance of mold pieces (PeTOLP1, PeTOLP2)• Surface finish of mold pieces (SF1, SF2, SF3, SF4)• Surface finish of parting surfaces (SFP1, SFP2, SFP3, SFP4, SFP5)• Total cost (C)• Total time (T)• Draft Angle (θ1– Features in mold core, θ2– mold cavity, θ3– third mold piece)

! Targets for the goals: (Ideal, desirable, tolerable, undesirable, unacceptable) (35 goals)• ZR (1S)# 0.01308, 0.01439, 0.01570, 0.01766, 0.01962 (radians)• ZR (2S)# 0.01308, 0.01177, 0.01046, 0.00850, 0.00654 (radians)• W (1S) # 2.276, 2.390, 2.504, 2.663, 2.845 (g)• W (2S) # 2.276, 2.162, 2.049, 1.889, 1.707 (g)• D # 2.75, 3.15, 3.55, 4.15, 4.75 (mm)• Wi # 2.0, 2.4, 2.8, 3.4, 4.0 (mm)• t # 1.0, 1.4, 1.8, 2.4, 3.0 (mm)• θ1 # 0.0, 1.0, 2.0, 3.5, 5.0 (degrees)• θ2 # 0.0, 1.0, 2.0, 3.5, 5.0 (degrees)• θ3 # 0.0, 1.0, 2.0, 3.5, 5.0 (degrees)• PTOL1, PTOL2 # 0.002, 0.004, 0.008, 0.013, 0.020 (in)• FTOL1, FTOL2, FTOL3, FTOL4 # 0.001, 0.002, 0.004, 0.007, 0.010 (in)


315

• PoTOL1, PoTOL2 # 0.003, 0.005, 0.010, 0.017, 0.030 (in)• FTOLP1, FTOLP2, FTOLP3, FTOLP4, FTOLP5# 0.002, 0.004, 0.008, 0.013, 0.020 (in)• PeTOLP1, PeTOLP2 # 0.001, 0.002, 0.004, 0.007, 0.010 (in)• SF1, SF2, SF3, SF4 # 75, 125, 200, 300, 450 (µin)• SFP1, SFP2, SFP3, SFP4, SFP5 # 20, 40, 80, 130, 200 (µin)• C # 500, 900, 1300, 1900, 2500 ($)• T # 3, 4, 5, 6, 7 (days)

! Allowable layer thickness (LT): 2, 4 and 8 milsFIND:! The system variables:

• D, Wi, t, NC, NR• Draft angle variables: θ1, θ2, θ3

• RP process variables: PO, LTP, HOCp, FOCp, TA. Where p corresponds to block number.• IJM process variables: CT

! The requirements: ZR, W! Deviation variables

• d din in+ −, i = 1,…,35, n = 1,…,4

SATISFY:! Goals:

• C, T, PTOL, FTOL, PoTOL, FTOLP, PeTOL, SF, SFP, D, Wi, t, θ1, θ2, θ3 areminimization goals (class 1S)

• ZR and W are target-matching goals (class 3S) and each of these goals is split into twoindependent goals of class 1S and class 2S.

• Class 1S goals formulation:

A x max A x t

td d

q q q n

q nq n q n

( ) ( ) ,,

,, ,

− −+ − =+ − +1 0

1c h

q = 1,…,33


td dr r r n

r nr n r n

( ) ( ) ,,

,, ,

+ −+ − =+ − +1 0

1c h

r = 1, 2


• ZR (radian) = f (D, Wi, t, NC, NR)• W (g) = f (D, Wi, t, NC, NR)• SFi = f (PO, LT), i =1, 2• SFi = f (PO, LT, θ3), i =3, 4• SFPi = f (PO, LT), i =1, …, 5• PTOLi = f (PO, LT, HOC, FOC), i =1, 2• FTOLi = f (PO, LT, HOC, FOC), i =1, …, 4• PoTOLi = f (PO, LT, HOC, FOC), i =1, 2• FTOLPi = f (PO, LT, HOC, FOC), i =1, …, 5• PeTOLPi = f (PO, LT, HOC, FOC), i =1, 2• ML1 = f (LT, CT, IP, TA, Wi, t, NC, NR, θ1) [8.10]• ML2 = f (LT, CT, IP, TA, Wi, t, NC, NR, θ2) [8.11]• ML3 = f (LT, CT, IP, TA, D, θ3) [8.12]

• NN

MLmip

i

=−

+LNM

OQP

11 i= 1, 2, 3


316

• BT = f (PO, LT, HOC, FOC, Nmi)• Cm = f (BT, TA)• Cp = f (CT)• C = Cm + Cp


! Constraints:• 0.00654 radian ≤ ZR ≤ 0.01962 radian• 1.707 g ≤ W ≤ 2.845g• C ≤ $2500• T ≤ 7 days• PTOLj ≤ 0.020 inch, j = 1, 2• FTOLj ≤ 0.010 inch, j = 1, …, 4• PoTOLj ≤ 0.030 inch, j = 1, 2• FTOLPj ≤ 0.010 inch, j = 1, …, 5• PeTOLPj ≤ 0.010 inch, j = 1, 2• SFj ≤ 450 µin j = 1, … ,4• SFPj ≤ 200 µin j = 1, …, 5

• d din in+ −• = 0

• d din in+ − ≥, 0

! Bounds:• 2.75 mm ≤ D ≤ 4.75 mm• 2.0 mm ≤ Wi ≤ 4.0 mm• 1.0 mm ≤ t ≤ 3.0 mm• 2 ≤ NC ≤ 3 – Discrete variable• 2 ≤ NR ≤ 3 – Discrete variable• 0 degree ≤ θ1 ≤ 5 degree• 0 degree ≤ θ2 ≤ 5 degree• 0 degree ≤ θ3 ≤ 5 degree• PO (θx, θy, θz) – Discrete variable• TA (0 or 1) – Discrete variable• 2 ≤ LT ≤ 8 (mils) – Discrete variable• LT=2, 0.002 (mils) ≤ HOC ≤ 0.006 (mils), 0.012 (mils) ≤ FOC ≤ 0.016 (mils)• LT=4, 0.003 (mils) ≤ HOC ≤ 0.007 (mils), 0.004 (mils) ≤ FOC ≤ 0.008 (mils)• LT=8, 0.001 (mils) ≤ HOC ≤ 0.005 (mils), 0.002 (mils) ≤ FOC ≤ 0.006 (mils)• 300 second ≤ CT ≤ 420 second


Z w d di nin

i n i n= ++==

+ −∑∑ , , ,11

35

1

4

c h

In the problem formulation (Table 8.10), the targets for the goals are formulated inLinear Physical Programming formulation (Section 2.6.2). The meaning of target valuesis also explained in the robot arm study (7.3.1). The mold life (ML1, ML2, ML3, Equation8.10 ~ 8.12) is calculated for all the protrusion features. There are a total of 4 rib features


317

in mold core (ML1), 10 rib features in mold cavity (ML2) and 1 boss feature in third moldpiece (ML3). The dimensions of these mold features are presented in Table 8.11. Theseare the dimensions of the features in modified camera roller part shown in Figure 8.15.The system variables NC and NR affect the quantity and dimensions of these features.Therefore based on the values of NC and NR, the dimension of each feature should becalculated and used in the evaluation of the mold life of the mold pieces. As statedbefore, life of a mold piece is equal to the smallest value of mold life of the features inthe corresponding mold piece. Mold life affects the number of different mold piecesrequired to fabricate desired number of functional prototypes. Number of molds in turnaffects build time and fabrication cost.

Table 8.11 - Dimensions of the Different Mold Features.

Feature No Type Mold Piece Height(mm)

Width /Dia(mm)

Thick(mm)

Part Thick(mm)

1 Rib Core 4.90 12.20 2.00 1.12 Rib Core 4.45 5.80 2.25 1.13 Rib Core 4.45 5.80 2.25 1.14 Rib Core 4.45 7.40 2.25 1.15 Rib Cavity 4.90 6.50 2.00 1.16 Rib Cavity 4.45 2.90 1.25 1.17 Rib Cavity 4.45 5.80 1.25 1.18 Rib Cavity 4.45 5.80 1.25 1.19 Rib Cavity 4.90 7.40 2.00 1.110 Rib Cavity 4.90 7.40 2.00 1.111 Rib Cavity 4.90 7.40 2.00 1.112 Rib Cavity 4.45 5.80 1.25 1.113 Rib Cavity 4.45 5.80 1.25 1.114 Rib Cavity 4.45 5.80 1.25 1.115 Boss Third Piece 3.50 2.75 *** 3.0

After the MPGT problem for the camera roller is formulated, the solution process forsystem variables are presented in the next section.

8.5 GEOMETRIC TAILORING WITH AID OF DFRTS – SOLVING

In the DFRTS, the solution process of the MPGT problem has three stages, solvingdiscrete variables, solving continuous variables, and selecting a solution (Figure 6.8). Forthe camera roller, these steps are introduced in Section 8.5.1~8.5.3 respectively.

8.5.1 Solving Discrete Variables

The discrete variables in the DFRTS are mold design variables, part orientations andlayer thickness. They are determined with the aid of RTMDS and refined RP processplanner.

• Mold Design VariablesAs presented in Section 8.2, the RTMDS can aid the manufacturer to generate

feasible mold designs for the camera roller. The camera roller is a complicated part withundercuts and intricate geometry. Consequently there is little freedom in choosing the


318

mold design variables. It should be noted that more mold designs are possible with largernumber of mold pieces. However since it is desirable to use the least possible number ofmold pieces (Section 3.2), only one mold design, which is shown in Figure 8.9, is furtherconsidered.

• Part Orientation in SLA BuildingFor the mold design, the mold pieces are inputted to the refined RP process planner

to generate a set of part orientations (PO) and layer thickness (LT). As describe inSection 6.4.2, only the discrete variables (PO and LT) are considered in the modified RPprocess planner. Hatch overcure (HOC) and fill overcure (FOC) are set as default values.Its problem formulation is similar to RP-PP problem (Table 6.2). Sambu (2001) gave adetail description of the refined RP process planner. Solving the modified RT-PPproblem results in a set of promising mold orientations and slicing schemes for each ofthe mold pieces.

Two orientations (one favorable for build time and the other favorable for surfacefinish) are obtained for mold core and mold cavity. One orientation is obtained for thirdmold piece. These orientations result in 4 possible combinations as shown in Figure8.20.

The mold pieces are combined together in IronCAD to create a single ACIS file.Slicing module is run for all the four combinations or orientations. Six promising slicingschemes are generated for each of the combinations. The slicing schemes obtained forcombination 1 are shown in Figure 8.21. In this figure, dark shade corresponds to 2-millayer thickness, light shade corresponds to 8-mil layer thickness and the medium levelshade corresponds to 4-mil layer thickness.


319

(a) PO1

(b) PO2

(c) PO3

(d) PO4

Figure 8.20 – Four Part Orientations for the Mold Design (Sambu, 2001).


320

Slicing schemes 1 ~ 4 correspond to the solutions that result in the smallest values ofthe objective function. They have small layer thickness and hence achieve best possibleaccuracy and surface finish. Slicing schemes 5 and 6 have best possible build timeachievement. However the surface finish requirements are barely met. It is observed thatthe primary trade-off in the slicing scheme is between surface finish and build time goals.A comparison of surface finish and build time goal achievements for these slicingschemes is presented in Table 8.12. The slicing schemes for the other combinations aresimilar.

Table 8.12 - Surface Finish and Build Time Achievements for Slicing Schemes.

SlicingScheme SF_mold1 SF_mold2 SF_mold3 SF_part_

mold1SF_part_

mold2 Build Time

1 156 156 64 64 64 13:592 156 156 64 68 68 13:393 156 156 131 74 72 13:134 156 156 64 77 75 12:525 390 390 400 187 177 6:286 382 390 130 187 177 6:29




Figure 8.21 - Promising Slicing Schemes for PO1 (Sambu, 2001).


321

There are totally 4x6 = 24 slicing schemes. For each slicing scheme, a modifiedMPGT problem can be formulated (Section 6.4.1) and solve with the aid of OpdesX.

8.5.2 Solving Continuous Variables

The values of other variables of the MPGT problem (including three discretevariables: NC, NR, TA) are determined by solving a modified MPGT problem. Theformulation of the modified MPGT problem is identical to the MPGT problem (Table8.10) except for one difference. The set of promising mold orientations and the slicingschemes are available in the modified MPGT problem. As these values are alreadydetermined, PO and LT are dropped from the system variables. Similar to the robot armstudy (Section 7.4), the modified MPGT formulations for camera roller problem aredeveloped from the MPGT problem (Table 8.10).

The formulated modified MPGT problems are then solved by an engineeringoptimization software system to determine a solution for the related slicing scheme. Inthe camera roller study, OptdesX (with SAN algorithm) was used to solve the modifiedMPGT problem for all the 24 slicing schemes obtained from the modified RP PPproblem. The solving process is similar to that of the robot arm study (Section 7.4.2).Therefore only the differences are explained and presented here.

Three starting points, low, medium and high, are investigated to determine the effectof starting point on the obtained solution. The starting points used in the study arepresented in Table 8.13. The available models indicate that using the smallest possiblecooling time, HOC and FOC values results in the best solution for molds built on SLA3500 with SL 7510 resin and hence different values for these variables are notinvestigated in starting points.

Table 8.13 - Starting Points Investigated for Each Slicing Scheme.

Variable Low Medium HighD 2.75 3.75 4.75Wi 2 3 4t 1 2 3

NR 2 2 3NC 2 3 3θθθθ1 0 2.5 5θθθθ2 0 2.5 5θθθθ3 0 2.5 5TC 0 0 1CT 300 300 300

HOC1-6 0.002 0.002 0.002FOC1-6 0.012 0.012 0.012

The obtained solutions for slicing scheme 1 are presented in Table 8.14. The valuesof the system variables for the solutions obtained for all the starting points are very closeindicating a good convergence. Also, the values of objective function are very close too.This indicates that the solution of modified MPGT/RT problem is independent of startingpoint.


322

Table 8.14 - Solutions Obtained for Slicing Scheme 1 for Different Starting Points.

Variable Low Medium HighD 2.76 2.78 2.76Wi 2.80 2.80 2.78t 1.00 1.00 1.00

NR 2 2 2NC 2 2 2θθθθ1 1.3 1.3 1.3θθθθ2 1.2 1.2 1.2θθθθ3 1.0 1.0 1.0TC 0 0 0CT 300 300 300

HOC1-6 0.002 0.002 0.002FOC1-6 0.012 0.012 0.012

Z 0.0152 0.0152 0.0154

The variation of the objective function with iteration (cycle) number for mediumstarting point is shown in Figure 8.22. The plot is for 5000 cycles and the total run timeis 2 minutes and 30 seconds. This plot shows an expected behavior of the SAN algorithm(large fluctuations in the beginning and small fluctuations at the end) indicating its properfunctioning for this problem.

The objective function values for the solution obtained for the 24 slicing schemes arepresented in Table 8.15. The complete listing of the obtained solutions is presented inAppendix B. The analysis and validation of the solutions are presented in section 8.5.4.Among the 24 solutions, a solution is chosen as the solution of the MPGT problem,which is presented in the next section.

Figure 8.22 - Objective Function vs. Iteration Number for MPGT Problem.


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Table 8.15 - Objective Function Values Obtained for Different Slicing Schemes.

Z SS1 SS2 SS3 SS4 SS5 SS6PO1 0.0152 0.0157 0.0193 0.0168 4.4450 3.4922PO2 0.0348 0.0339 0.0327 0.0384 1.1760 1.0641PO3 0.0371 0.0336 0.0361 0.0357 3.9700 4.0338PO4 0.0363 0.0324 0.0352 0.0344 4.1710 3.8244

8.5.3 Selecting A Solution

From the 24 values in Table 8.15, it can be seen that slicing scheme 1 for PO1(Figure 8.20.a) has the smallest value of objective function. Therefore, it is determinedthat the solution corresponding to PO1-SS1 is regarded as the solution of the MPGTproblem.

With the tailored camera roller design, the values of mold design variables, RPprocess variables, and IJM process variables are also determined. They are summarizedas shown in Table 8.16. The values of goals, significant intermediate responses andobjective function (along with the ideal target values) for this solution are presented inTable 8.17.

Table 8.16 – Solutions of the MPGT Problem for Camera Roller.

D(mm)

Wi(mm)

t(mm)

NR NC θθθθ1

(o)θθθθ2

(o)θθθθ3

(o)TA CT

(sec)HOC1-6

(mil)FOC1-6

(mil)2.78 2.80 1.00 2 2 1.3 1.2 1.0 0 300 0.002 0.012

Table 8.17 - Values of Goals and Intermediate Responses for the Solution.

ZR(rad.)

W(g)

SF1

(µin)SF2

(µin)SF3

(µin)SFP1

(µin)SFP2

(µin)SFP3

(µin)Achieved 0.0104 2.40 156 156 65 6 64 6

Target 0.0131 2.28 75 75 75 20 20 20SFP4

(µin)SFP5

(µin)FTOL(inch)

PTOL(inch)

PoTOL(inch)

FTOLP(inch)

PeTOLP(inch)

Cost($)

Achieved 64 6 0.0018 0.0016 0.0064 0.0018 0.0020 1291Target 20 20 0.0010 0.0020 0.0030 0.0020 0.0010 500

D(mm)

Wi(mm)

t(mm)

Z # molds1 # molds2 # molds3

Achieved 2.78 2.80 1.00 0.0152 1 1 3Target 2.75 2.00 1.00 0 - - -

8.5.4 Post-Solution Analysis

From Table 8.16, one can see that the solution has NR = NC = 2 indicating that thiscombination results in a better solution than the original values used in the productionpart (NR = NC = 3). Considering the mold life concern for thin and tall features in SLmolds, this is reasonable. Using lower number of columns and rows increases the width


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of the rib features in the mold and hence increases their life. The solution has the draftangle values > 10 indicating that drafts are necessary in the mold pieces for the cameraroller. As in the robot arm example (Section 7.4.3), no thermal aging is preferred.Cooling time, HOC and FOC converge to their lower bounds as expected.

From the values of goal achievements for the solution presented in Table 8.17, onecan see that z-rotation (ZR) is lower than its target value and weight (W) is higher than itstarget. The difference between the achieved and target values is 21% for z-rotation and5.3% for weight. We also run ANSYS for the initially modified part (Figure 8.14). Thevalues of ZR and W for the part are 0.0080 (radian) and 2.57 (g) respectively. Thedifference between them and the target values is 40% for z-rotation and 13% for weight.Therefore by performing geometric tailoring, the difference of these values is reduced.

The cylindrical surfaces on mold core (SF1) and mold cavity (SF2) have a surfacefinish of 156 µin and the cylindrical surface on third mold piece (SF3) has a surface finishof 65 µin. The cylindrical surfaces in core and cavity are horizontal and hence haveworse surface finish than the cylindrical surface in third mold piece, which is vertical.The primary parting surface for all the mold pieces are horizontal and hence have a goodsurface finish of 6 µin. The secondary parting surfaces for mold core and mold cavity arevertical and have a surface finish of 64 µin. The flatness tolerance on parting surface andparallelism tolerance between the cylindrical faces are lower than the desired value(achieved better than what is desired) while the other tolerances are more than their idealvalues. The ideal value for cost is $500 and maximum allowable value is $2500. Thecost for the obtained solution is $1291.

Diameter (D), width (Wi) and thickness (t) are all system variables and goals in theMPGT problem. For the obtained solution, the difference between ideal and achievedvalue for D, Wi and t are 1.1%, 40% and 0% respectively. This indicates that thediameter and thickness need not be varied much to satisfy the functional property andmold life requirements, though width needs to be modified significantly. The goals arewell satisfied by modifying width (Wi), number of rows (NR) and number of columns(NR) in the part.

Table 8.17 also contains the values of objective function (Z) and the number ofdifferent mold pieces required to fabricate the desired number of parts. According to theobtained solution, one of mold core and cavity and three of third mold pieces are requiredto fabricate the desired number of functional prototypes. Compared to the requirednumber of each mold pieces (6 for mold core, 13 for mold cavity, 4 for the third piece asshown in Table 8.2), this is significantly smaller. It is also possible to mold 50 parts withonly one of third mold piece. However higher draft angle (θ3) is required. Using threemold pieces results in a better trade-off between draft angle and cost goals. Therefore theobtained solution shows a reasonable trade-off between different goals and also hasreasonable values for the individual goal achievements.

Further validation of the solution is performed by experiments of reducing the targetranges of different goals. For the camera roller, as surface finish and tolerance goals areaffected by only mold orientation and slicing scheme, the effect of target range on thesegoals is not studied. The experiments by modifying target ranges of each other goal arepresented in Table 8.18. In each of these experiments, the target range for the goal in the


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corresponding category is reduced by 40%. By running OptdesX for the new values ofthe goals, the results for the experiments are presented in Table 8.19. The improvementand deterioration of the goals due to this modifying in target ranges are shown in Table8.20. Positive values indicate improvement and negative values indicate deterioration ofthe corresponding goal.

Table 8.18 - Target Modification Experiments.


ZR (rad.) 0.00654 0.01308 0.01962 0.009156 0.01308 0.017004W (g) 1.707 2.276 2.845 1.9346 2.276 2.6174

D (mm) 2.75 2.75 4.75 2.75 2.75 3.95Wi (mm) 2.0 2.0 4.0 2.0 2.0 3.2t (mm) 1.0 1.0 3.0 1.0 1.0 2.2

cost ($) 500 500 2500 500 500 1700

Table 8.19 – Results of Target Modification Experiments.

Target whose range is reduced

GoalsTargetValues

CurrentSolution ZR exp. W exp. D exp. Wi exp. t exp. Cost exp.

ZR (rad.) 0.01308 0.01041 0.01041 0.01041 0.01042 0.01012 0.01042 0.01029W (g) 2.276 2.40 2.40 2.40 2.40 2.43 2.40 2.39

D (mm) 2.75 2.78 2.76 2.77 2.76 2.78 2.75 3.14Wi (mm) 2.00 2.80 2.80 2.80 2.80 2.48 2.80 2.81t (mm) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00cost ($) 500 1291 1291 1291 1291 1291 1291 1153

Table 8.20 – Percent Change of the Goals for Target Modification Experiments.

Target whose range is reducedGoals ZR exp. W exp. D exp. Wi exp. t exp. Cost exp.ZR (%) 0.0 0.0 0.1 -2.2 0.1 -0.9W (%) 0.0 0.0 0.0 -1.4 0.0 0.3D (%) 0.6 0.3 0.7 0.3 1.2 -12.9Wi (%) 0.2 0.1 0.0 16.0 0.0 -0.6t (%) -0.1 0.1 0.0 0.0 0.0 -0.1

Cost (%) 0.0 0.0 0.0 0.0 0.0 27.7

Z-rotation, weight and thickness have no improvement by reducing their targetrange. Thickness is already at the ideal value for nominal targets and hence cannotimprove further. Apparently z-rotation and weight have trade-off with other goals that donot let them improve. This is evident by observing the corresponding rows. The valuesof ZR and W never improved in all the experiments. Even the deterioration of thesevariables is not significant. The improvement in diameter goal with reducing its target isvery small (0.7%). Width and cost have an improvement of 16% and 28% respectively.These are high values indicating positive effect on these goals by reducing their target


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range. Width is improved by worsening z-rotation and weight goals. Cost is improvedby worsening diameter and draft angle (shown in Table 8.21) goals. In fact, cost isreduced by reducing the required number of third mold pieces from 3 to 1. This isachieved by increasing the draft angle of the corresponding boss feature and reducing thepart thickness (by increasing diameter).

The variation in the values of the other system variables (NR, NC, θ1, θ2, θ3, CT, TA,HOC and FOC) is not significant. Their values for different experiments are presented inTable 8.21. One thing to be noticed for different experiments is the value of θ3, which isthe draft angle that corresponds to the boss feature in the third mold piece. It increasesfrom 1.0o to 1.7o when the target range for cost goal is reduced. As explained earlier,increasing draft angle results in fewer mold pieces and hence lower cost.

Table 8.21 - Values of System Variables for Target Modification Experiments.

Target whose range is reduced

GoalsCurrentSolution ZR exp. W exp. D exp. Wi exp. t exp. Cost exp.

D (mm) 2.78 2.76 2.77 2.76 2.78 2.75 3.14Wi (mm) 2.80 2.80 2.80 2.80 2.48 2.80 2.81t (mm) 1.00 1.00 1.00 1.00 1.00 1.00 1.00

NR 2 2 2 2 2 2 2NC 2 2 2 2 2 2 2

θθθθ1 (o) 1.3 1.3 1.3 1.3 1.3 1.3 1.3θθθθ2 (o) 1.2 1.2 1.2 1.2 1.4 1.2 1.2θθθθ3 (o) 1.0 1.0 1.0 1.0 1.0 1.0 1.7TA 0 0 0 0 0 0 0

CT (sec.) 300 300 300 300 300 300 300HOC1-6 (inch) 0.002 0.002 0.002 0.002 0.002 0.002 0.002FOC1-6 (inch) 0.012 0.012 0.012 0.012 0.012 0.012 0.012

The solutions of the target range reduction experiments indicate that the goals do notworsen (if not improve) by reducing their target range. This is an expected behavior andhence provides more faith in the correctness of the algorithm and analysis functions (usedin OptdesX) as well as the actual values of the obtained solution.

In the next section, physical experiments are presented which further validate thesolution of the problem.

8.6 PHYSICAL VALIDATION

The RTMDS was used to generate the mold pieces for the tailored part design(Figure 8.23.a). The running process is the same as the mold design process for theoriginal camera roller design (Section 8.2). Therefore it is not repeated here. Thegenerated mold design is shown in Figure 8.23.b. Compared them with the mold piecesin Figure 8.9, it is expected that the mold pieces for the tailor part will have a much bettermold life than the mold pieces for the original part. The designed mold pieces werefabricated on SLA 3500 with SL 7510 resin using the mold orientations, slicing schemesand the process parameters obtained by solving the MPGT problem. The molds are post


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processed in the usual manner (cleaning with TPM, water and alcohol, drying with air,post-curing in UV oven for an hour and thermal curing for 2 hours).

Placing the mold pieces in the mold base of the Morgan Press machine waschallenging because the three mold pieces are assembled and dissembled manually. Themold core and mold cavity were fixed in the corresponding mold halves and the thirdmold piece was placed in the bottom half without fixing it. This piece was fixed afterclamping the mold halves. The injection molding parameters used to mold the parts arepresented in Table 8.22. The ejection of the parts from the mold is in the reverse order.The fixture of the third piece is loosed before the molds are opened. The ejection of the

(a) Tailored Camera Roller (Two Views)

(b) Mold Piece for the tailored Camera Roller

Figure 8.23 – Mold Design for the Tailored Camera Roller Part.

Third Mold Piece

Mold Cavity Mold Core


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parts from the mold core is also performed manually. After removing the mold core withthe part from the mold base, the third piece is ejected first and the part is ejected from themold core with the aid of four ejector pins. A photo of the physical mold pieces and theparts fabricated from them are shown in Figure 8.24.

Table 8.22 - Injection Molding Parameters Used for the Camera Roller.

Parameter Value UnitsClamping Force 11 TonsInjection Pressure 2 MpsiPilot Valve Pressure 50 psiPlastic Temp. in barrel 430 0FPlastic Temp. in nozzle 450 0FInjection Time 10 SecondCooling Time 300 SecondRelease Agent Used No units

In the experiment, it is observed that the mold pieces for the tailored part design aremuch stronger than the mold pieces shown in Figure 8.9. All the features in the moldpieces are still intact after three shots (it was ruined by a mistake made by the author).This also verifies the correctness of the solutions obtained from the MPGT problem.

Figure 8.24 – Physical Mold Pieces and Injection Molded Camera Roller.


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8.7 EVALUATION OF CAMERA ROLLER CASE – POST DFRTS

With the aid of the DFRTS, the material-process geometric tailoring (MPGT) wassuccessfully performed for the robot arm (Section 8.5). To further understand the cameraroller case, some comparing experiments are performed and presented here.

When putting the original design and the tailored design together (Figure 8.25), onecan obviously see that the geometric tailoring tremendously reduced the fabricationdifficulties in the Rapid Tooling process for the camera roller. A simple calculation isprovided here to illustrate the necessity of geometric tailoring for the case.

As estimated by the mold life predictor, the mold life of mold core, mold cavity andthe third piece is 9, 4, and 16 for the original design (Section 8.3). Therefore the requirednumber of mold core, mold cavity and the third piece is 6, 13, and 4 respectively. In thesolution of the MPGT problem, the cost of rapid tooling process is $1291 ($1091 for SLAfabrication process and $200 for injection molding process). It should be noticed that thiscost is for one set of mold pieces (1 of mold core, 1 of more cavity and 3 of the thirdpiece) for the tailored part. Based on our experience, the cost of the SLA fabricationprocess for 6 of mold core, 13 of mold cavity and 4 of the third piece for the original partwould be 5 ~ 7 times of $1091, which is $5455~7637. After a mold piece is ruined, anew mold piece needs to replace the old one, which leads to a cost for assembling anddisassembling the mold piece. More frequent replacements leads to higher cost andlonger time. Therefore it is estimated that the cost of Rapid Tooling process for theoriginal design would be around $6500. Compared to the cost of $1291 for the tailoreddesign, it is significantly decreased. Suppose the designer’s budget is $2500, themanufacturer will have a profit of $1209 instead of a loss of $4000.

Another comparing experiments are performed for the modified camera roller part(Figure 8.14). Similar to the experiment given in Section 7.6, the sequential and

(a) Before Tailoring

(b) After Tailoring

Figure 8.25 – Comparison the Camera Roller before and after Geometric Tailoring.


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concurrent solution processes are tested in the experiments. First in the sequentialexperiment, the MGT problem is formulated and solved. For the solution of the MGTproblem, the manufacturer then performs the process planning of the Rapid Tooling. Inthe concurrent experiment, the MPGT problem is formulated and solved. Since the costfactor is important here, the cost experiment (reducing the target range of cost by 40%) isadopted (Table 8.18). The solutions (values of geometry) for the experiments arepresented in Table 8.23 and the corresponding goal achievements are presented in Table8.24. The target value of each goal is also listed in the table.

Table 8.23 - Solutions of Sequential and Concurrent Solution Process.

Variables D(mm)

Wi(mm)

t(mm)

NR NC θθθθ1 (o) θθθθ2 (o) θθθθ3 (o)

Sequential 2.75 2.8 1.00 2 2 1.3 1.2 1.4Concurrent 3.14 2.81 1.00 2 2 1.3 1.2 1.7

Table 8.24 - Goals Achievements for Sequential and Concurrent Solution Processes.Goals ZR (rad.) W (g) Cost ($) # Mold1 # Mold2 # Mold3 Obj. Function

Sequential 0.01043 2.399 1221.66 1 1 2 0.0365Concurrent 0.01029 2.394 1152.5 1 1 1 0.0408

Target 0.01310 2.28 500 - - - 0.0

Comparing the results of the sequential and concurrent solution processes, it is asexpected that the sequential process has a better solution of design variables (D) anddesign goal achievements (Z-Rotation). However, its cost and the total objective functionare higher than those of the solutions obtained from the concurrent process. After furtheranalysis, we can see the increase of cost is mainly because the required number of thethird mold piece is increased from 1 to 2. The main dimensions of the boss in the thirdmold piece are D and θ3 as shown in Figure 8.26. The mold life of the mold piece isdetermined by their values. In the concurrent solution process, D and θ3 are formulatedand solved together. Therefore the values of D and θ3 can all be increased to reduce the

Figure 8.26 – The Dimensions of the Third Mold Piece.

D

θ3


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required mold number. However, in the sequential solution process, D has already beenfixed in the former process. The manufacturer can only change the value of θ3. If thedesign freedom of θ3 is not sufficient to make the required mold number as 1, anadditional mold piece will be required. Correspondingly the value of θ3 can be reducedfor the required mold number (2). As the objective function value of the concurrentsolution process is smaller that of the sequential solution process, it verifies that theconcurrent formulation and solving processes utilized in the DFRTS is better than thecurrent sequential solving process.

The experiments also verify that the mapping from design space to themanufacturing space may have discontinuities. That is, for the continuous variables Dand θ3, some design spaces correspond to required mold piece number 1. However otherdesign spaces may require two mold pieces. Consequently the fabrication cost will berather different for them. Considering the discontinuity of the design-for-manufacture(DFM), it is generally desired to transfer more design freedom to the manufacturer.

In the next section, a brief summary is given for discussing the relevance of theseresults with regard to the hypotheses of the dissertation.


In this chapter, the RTMDS and DFRTS were applied to determine the mold designand geometric tailoring for a camera roller design. A problem of producing functionalprototypes of a camera roller was first introduced in Section 8.1. By using the RTMDS, amold design with three mold pieces was generated for the camera roller (Section 8.2). Astandard mold base for Morgan Press injection molding machine was used in the moldpiece construction. A preliminary study and experiment were performed which results ina modified camera roller design (Section 8.3). This modified design was considered inthe DFRTS. The formulation of the geometric tailoring problem for the modified cameraroller was presented in Section 8.4. In the formulation the design functions wererepresented by response surface models. The solution process of the MPGT problem,which consists of three stages, was described in Section 8.5. The modified MPGTproblems that are related to 24 slicing schemes were solved by OptdesX. The results ofphysical experiments for the validation of the problem solution were presented in Section8.6. Finally an evaluation of sequential and concurrent solution processes for the cameraroller case was provided in Section 8.7.

Similar steps are utilized in the camera roller example as those in the robot armexample. This indicates the generality of the approaches developed in the RTMDS andDFRTS. Hence, this example extends the faith in the developed approaches from theparticular examples presented in this thesis to other problems that fall under this category(theoretical performance validation). In the discussion of application of RTMDS andDFRTS to the camera roller example, the relationships with the hypotheses wereidentified. The results from testing the hypotheses are summarized below (Figure 8.27):

Hypothesis 1: A mold design with three mold pieces was generated for the camera rollerby using the Rapid Tooling Mold Design system, which is developed based on theMulti-piece Mold Design Method. The whole running process only took severalminutes. The summary of testing the sub-hypotheses (H1.1 ~ H1.3) presented belowwill provide more details.


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Hypothesis 1.1: 446 concave edges and 336 concave faces are identified. Based on theidentified concave faces, 32 concave regions are generated for the camera roller. Theregion generation process only took seconds for the camera roller.

Hypothesis 1.2: The generated concave regions of the camera roller are combined into 3concave regions. The region combination process only took less than one minute forthe camera roller.

Hypothesis 1.3: Mold pieces are constructed for a standard mold base which can beinstalled in a Morgan Press injection molding machine. The mold piece constructionprocess only took less than one minute for the camera roller.

Hypothesis 2: Tailored part design with mold design variables, RP process variables andIJM process variables were generated for the camera roller by using the Design forRapid Tooling system, which integrates geometric tailoring and process planning bythe manufacturer. The satisfactory prototypes are produced without any iteration forthe camera roller. The summary of testing the sub-hypotheses (H2.1 ~ H2.3)presented below will provide more details.

Hypothesis 2.1: For the camera roller the designer formulated a partial MPGT problembased on the MPGT decision template. The manufacturer formulated a completeMPGT problem which considered the material difference of products and prototypes.

Hypothesis 2.2: The designer’s MPGT problem formulation, RP process planningformulation, and IJM process planning formulation were integrated into a completerMPGT problem formulation for the camera roller.

Hypothesis 2.3: The MPGT problem was solved in a three-stage solution process. Thesolution was reasonable and validated by physical experiments.


Hypothesis 2


Validation


The requirements on molddesign of the camera rollerare representative of themold design problems.

The system can be used togenerate mold design for

the camera roller.


Validation

Mold designs ofcamera roller aregenerated by the

RTMDS in acceptabletime, and they can be

used to produceprototpyes in the Rapid

Tooling process.

Hypothesis 1

The requirements ongeometric tailoring of the

camera roller arerepresentative of the

geometric tailoring problems.The system can be used toperform geometric tailoring

for the camera roller.

Solutions of geometrictailoring of camera roller

are generated by theDFRTS in acceptable time,

and tailored part designand process parameterscan be used to produce

functional camera roller inRapid Tooling process.

Figure 8.27 – Empirical Structural and Performance Validation for H1 and H2.


333

In the next and final chapter, contributions form this work will be presented alongwith a critical review of the research and future work (Figure 8.28).

Chapter 9: Achievements and Recommendations



Figure 8.28 – Preview of Chapter 9.

Chapter 9 – Achievements and Recommendations

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CHAPTER 9

ACHIEVEMENTS AND RECOMMENDATIONS

In this dissertation, a mold design system and a design-for-manufacture system havebeen developed, presented, and tested to facilitate producing function prototypes for apart design. The development and presentation of these systems is brought to a close inthis chapter. In Section 9.1, closure is sought by returning to the research questionsposed in Chapter 1 and reviewing the answers that have been offered. The resultingcontributions are then summarized in Section 9.2. Limitations of the research arediscussed in Section 9.3, and possible avenues of future work are described in Section9.4.


335

9.1 ANSWERING THE RESEARCH QUESTIONS

In this section focus is returned to the research questions justified through thediscussions of Chapter 1 and formulated specifically in Section 1.2.1. In essence, thissection is a review of hypothesis testing presented throughout the dissertation, as thehypotheses offered in direct support of the research questions. The relationship betweenthe research questions, and the hypotheses formulated are shown in Figure 9.1. Eachhypothesis has been supported through the investigations of the previous chapters.

9.1.1 Research Question Overview

As stated in the introduction to Chapter 1, the main objective of this research is toreduce the time for functional prototypes in the product realization process. The keyresearch question being investigated is:

How to reduce the lead-time in the usage of Rapid Tooling to produce functionalprototypes for a part design?

The key question directly ties to the principal technology development goal for thisresearch project (RTTB): to enable “customer” of the rapid tooling testbed receivenearly production-representative components in a variety of polymer, ceramic, and metalmaterials within 3-4 days of submitting a product model. In this research, the focus is on

Primary Research QuestionHow to reduce the lead-time in the usage of RapidTooling to produce functional prototypes for a part

design?

Q1. How to aid the molddesigner to reduce the mold

design time ?

Q2. How to reduce the time ofiteration between the designer and

manufacturer ?

Q1.1 What areappropriate

basic elementsto automate

mold design?

Q1.2 How togenerate moldconfigurationsbased on the

basic elements?

H1. Multi-Piece Mold Design Method provides a methodto automate several key steps of the mold design.

H2. Geometric tailoring can be integrated withprocess planning and solved by the manufacturer.

Q1.3 How togenerate moldpieces from a

mold configurationdesign?

H1.1 Concaveregion and

convex face.

H1.2 The regionbased combining

process andalgorithms.

H1.3 TheReverse Glue

Mold ConstructionMethod andalgorithms.

Q2.1 How toreduce theiterations

because ofdifferent material

properties?

Q2.2 How toformulate the

design for RapidTooling

problem?

Q2.3 How tosolve thedesign for

Rapid Toolingproblem?

H2.1 MaterialGeometricTailoringDecision

Template.

H2.2Formulation of

design for RapidTooling system

H2.2 A three-stage solutionprocess for a

satisficingsolution.

Figure 9.1 – Context for Research Questions and Hypotheses.


336

the mold design and design-for-manufacture for Rapid Tooling. The idea is to reduce thetime of several time-consuming mold design steps and the number of iterations betweenthe designer and manufacturer in Rapid Tooling. Towards achieving this objective, aRapid Tooling Mold Design System (RTMDS) and Design for Rapid Tooling System(DFRTS) are developed.

The key question has two related questions associated with it:

Q1. How to aid the mold designer to reduce the mold design time for a wide variety ofpart geometries in the design for a Rapid Tooling process?

Q2. How to reduce the time of iteration between the designer and manufacturer in theusage of Rapid Tooling for a wide variation of design requirements?

To address these questions, research hypotheses are proposed in support of achievingthe principal objective for the dissertation.

H1 Multi-Piece Mold Design Method provides a method to automate several keysteps of the mold design process, which can greatly reduce the mold design timefor a wide variety of part geometries.

H2 Geometric tailoring for Rapid Tooling can be integrated with process planningbased on decision templates and solved by the manufacturer, which can reducethe time of iteration between the designer and manufacturer.

Their elaboration and verification have provided the context in which the researchwork has proceeded. This work has been based in computational geometry and decision-based design. The sub-questions associated with these two questions are:

Q1.1 What are appropriate basic elements to automate several mold design steps for awide variety of part geometries?

Q1.2 How to generate mold configurations by a systematic design process based on thebasic elements?

Q1.3 How to generate mold pieces from a given mold base effectively and efficientlyaccording to a mold configuration design?

Q2.1 How to reduce the iterations between the designer and manufacturer in producingfunctional prototypes that have different material properties from products?

Q2.2 How to formulate the design for Rapid Tooling problem which integratesdecisions on design and manufacturing variables and other design andmanufacturing requirements including goals, constraints, and preferences?

Q2.3 How to solve the design for Rapid Tooling problem effectively and efficiently?

To facilitate answering these questions, each has hypothesis associated with them:

H1.1 Concave region and convex face are two kinds of basic elements that provide anefficient and effective approach for exploring and developing molds designmethod for Rapid Tooling.


337

H1.2 The Region based combining process and related algorithms provide a systematicmethod to generate mold configurations of multi-piece molds design.

H1.3 The Reverse Glue Mold Construction Method and related algorithms provide anefficient and effective method for constructing multi-piece mold designs forRapid Tooling.

H2.1 The designer can initiate a material geometric tailoring (MGT) formulation basedon a MGT decision template; therefore the manufacturer, who completes andsolves the MGT problem, can produce production-representative prototypes morequickly.

H2.2 The design for Rapid Tooling problem can be formulated by several compromiseDSPs and tasks, which can then be integrated into a design for Rapid Toolingsystem (DFRTS).

H2.3 A three-stage solution process can be utilized to get a satisficing solutioneffectively and efficiently based on design and manufacturing models andcontinual/discrete variables.

With an overview of the research questions and hypotheses presented, answers tothese research questions are summarized in the next section. As mentioned the researchquestions are answered through testing the hypotheses.

9.1.2 Answering Research Questions

In general a research question is answered when the corresponding hypothesis isvalidated. Hence, in order to answer the fundamental question, we validate each of thesupporting hypotheses. This is done according to the process given in Section 1.3.2,where each of the ‘validities’ in the Validity Square (Figure 9.2) is used to test eachsupporting hypotheses separately. Then, by adding necessary additional information, thevalidity of the fundamental hypothesis – i.e., the validity of the RTMDS and DFRTS – isasserted through induction.

Answering the research questions is performed following the hierarchy of thequestions presented in Figure 9.1. First the two questions are answered using the answers

THEORETICALSTRUCTURAL

VALIDITY

EMPIRICALSTRUCTURAL

VALIDITY

EMPIRICALPERFORMANCE

VALIDITY

THEORETICALPERFORMANCE

VALIDITY

Figure 9.2 – Validation Square.


338

of the supporting questions. These two questions are then used to answer the keyquestion.

• Answer to Question 1:This research question has three supporting questions, answer to these questions are

summarized as follows:

Answer to Basic Element Question (1.1)

By identifying concave region, combined region and convex face, this work extendsthe basic elements used in other works, which are pockets (Chen, et al., 1993; Weinsteinand Manoochehri, 1996) and features (Fu, et al., 1999; Gu, et al., 1999). The basicelements of regions are more general for different geometries. Therefore more parts canbe handled by the developed mold design method. The theoretical structural validity wasdiscussed in Section 3.3 through mathematical definition and proofs related with it.Empirical structural validity and performance validity was presented using the illustrativeexamples (Section 4.4 and 4.5) and case studies (Chapter 8 and 9). In all cases theconcave regions and convex faces provided basic elements for mold configuration design.

Answer to Region Combination Question (1.2)

In this research, the mold configuration design is transferred to a region-growingprocess, which captures the connectivity of faces and the demoldability of regions. Thistransferring enables the automation of the generation of parting directions and partinglines for mold configuration design. The mold design knowledge can also be added tothe region combination process easily. The theoretical structural validity was discussedin Section 3.4 through mathematical definition and relations related with it. Empiricalstructural validity and performance validity was presented using the illustrative examples(Section 4.4 and 4.5) and case studies (Chapter 7 and 8). In all cases combined regionswere generated which provided mold configuration design for mold piece construction.

Answer to Mold Piece Construction Question (1.3)

Gluing operation is a standard high-level Euler operation for half-edge data structure.A novel approach based on reversing the gluing operation was developed in this researchand used in the mold piece construction. The mold piece construction problem is thentransferred to a glue face generation problem. It is illustrated in this dissertation that theglue face generation is actually a geometric reconstruction problem. The theoreticalstructural validity was discussed in Section 3.5 through mathematical definition andrelations related with it. Empirical structural validity and performance validity waspresented using the illustrative examples (Section 4.4 and 4.5) and case studies (Chapter7 and 8). In all cases mold pieces were generated for mold configuration designs withinsatisfied time.

Answer to Mold Design Question

Answering the research sub-questions related to Hypothesis 1 provides a foundationfor answering the high-level question. In this research computation geometry and solidmodeling have been utilized for multi-piece mold design for Rapid Tooling. By using theRTMDS to automate several steps of the mold design, mold pieces are generated in ashort time. Other mold design components (e.g. gate, runner, ejector pin holes) can be


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added to the generated mold pieces. Consequently the total mold design time isdramatically decreased.

In Section 3.3 the relationships with the different steps of the mold design methodwere discussed which validated the structural (theoretical and empirical) validity of themethod. The application of the RTMDS was demonstrated in illustrative examples(Section 4.4 and 4.5) and case studies (Chapter 7 and 8). The mold design results verifythe usefulness of the mold design method and the related system in designing molds for aset of products. The mold designs are further verified by producing physical prototypeswith the molds. Although used for a limited number of examples, the obtained resultsindicate that the mold design method presented in this dissertation is applicable for otherproblems.

• Answer to Question 2The second research question also has three related questions associated with it,

answers to these three questions are first provided because they will help answerQuestion 2, which is related to geometric tailoring problem.

Answer to Material Difference Question (2.1)

In different Rapid Tooling process, the material difference between production andprototypes are normal. The geometric tailoring, which is based on Buckingham πtheorem, was utilized to transfer the material difference problem to a process of makingdecisions on unimportant system variables (Section 5.2). Then a decision template in thecompromise DSP formulation is proposed for integrating the design and manufacturingrequirements (Section 5.3). The Buckingham Π theorem and the compromise DSPprovide the theoretical structural validity of material geometric tailoring and its decisiontemplate. Empirical structural validity and performance validity was presented usingthree illustrative examples (Section 5.5) and two case studies (Chapter 7 and 8). In all thecases both design and manufacturing requirements were satisfied without iterationsbetween the designer and manufacturer.

Answer to Problem Formulation Question (2.2)

In this dissertation decisions on design and process variables are used to model thedesign-for-manufacture problem for different viewpoints. The decisions in the threesteps of the Rapid Tooling process (material-process geometric tailoring, RP processplanning and IJM process planning) can be formulated as compromise-DSPs, and molddesign can be considered as a task. Therefore a design for rapid tooling system waspresented in which the decisions on different steps are integrated into a singlecompromise-DSP (Section 6.3). The theoretical structural validity was discussed inSection 6.1 through different modules and their relations. Empirical structural validityand performance validity was presented using a robot arm and a camera roller (Chapter 7and 8). In the case studies formulations that were considered in the DFRTS werepresented, which provided a unified decision framework for solving the geometrictailoring problem.

Answer to Problem Solving Question (2.2)

By identifying coupling between goals and variables, this work presented a three-stage solution process for solving the material-process geometric tailoring problem. A


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satisficing solution is generated instead of an optimal solution because of the complexityof the problem. The theoretical structural validity was discussed in Section 6.4 throughdifferent modules and their relations. Empirical structural validity and performancevalidity was presented using two case studies (Chapter 7 and 8). In all cases followingthe solution process of the DFRTS generated satisfied solutions of the geometric tailoringproblem, which were further validated by physical experiments.

Answer to Geometric Tailoring Question

Answering the research sub-questions, associated with Hypothesis 2, provide afoundation that can be used to answer the overall geometric tailoring question. Theiterations between the designer and manufacturer are actually a process of integratingdesign and manufacturing requirements. Geometric tailoring is also a method to integratedesign and manufacturing requirements. However much less time is needed becausecomputers instead of negotiations between the designer and manufacturer solve thetradeoffs between the requirements. Therefore a satisficing solution can be found morequickly.

The application of the DFRTS was demonstrated in two case studies (Chapter 7 and8). Solutions were generated by the system based on the design and manufacturingrequirements. Physical prototypes were produced for the solutions. These results verifythe usefulness of the geometric tailoring and the Design for Rapid Tooling System inproducing functional prototypes for a set of products.

• Answering Key QuestionAnswer to the key question proposed for this work is embodied by the Rapid Tooling

Mold Design System and Design for Rapid Tooling System, which reduce the time ofmold design and iterations between the designer and manufacturer respectively. The twosystems are important components of the Rapid Tooling Testbed, which uses RapidPrototyping technology in the product tooling design and manufacture. Several casestudies (including the robot arm and camera roller in Chapter 7 and 8) were tested in theRTTB. The results affirmed that the goal of producing production-representativecomponents in a variety of polymer, ceramic, and metal materials within 3-4 days ofsubmitting a product model is feasible. The experiments also verify the importance ofthe research questions addressed in this dissertation, that is, how to reduce the time ofmold design and how to avoid iterations between designer and manufacturer.

Producing functional prototypes for a part design within days is always desired tocompress the product development time and cost. Today’s competitive and highlyvolatile market makes this desire much stronger. This research is an effort towardachieving this goal. With this in mind, a summary of the research contributions is offeredin the next section.

9.2 ACHIEVEMENTS: REVIEW OF RESEARCH CONTRIBUTIONS

The contributions offered in this dissertation are introduced in Section 1.3.3 andrealized throughout the dissertation. The list provided in Chapter 1 was partial andidentified only the high level contributions. In this section contributions from this workwill be revisited to highlight the significance of this work.


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The primary contribution from this work is a systematic method and a set ofalgorithms for multi-piece mold design and a systematic way (formulations and asolution process) to integrate design and manufacturing requirements. They areembodied in the Rapid Tooling Mold Design System and the Design for Rapid ToolingSystem respectively. The methods are tested by several industrial cases via thedeveloped systems. The contributions can be categorized in three areas.

Contributions related to Computer-aided Mold Design (H1 and H1.1~1.3):

• A problem definition for multi-piece mold design is proposed as the foundation ofa computer-aided mold design method. Compared to other representative problemdefinitions, the problem definition presented in this research added the faceconnectivity as an important factor in the mold design (Section 3.2).

• A solution process for the formulated problem definition is proposed (Section3.3). As an additional requirement, face connectivity, is considered, the molddesign process is more complicated. Another important contribution of thisresearch is identifying a systematic process to generate mold design for theproblem definition.

• The notions of concave region and convex face in mold configuration design anda means of generating them for a given part. As an extension of “pocket”, whichis a common notion in the computer-aided mold design, concave region canhandle more general cases (Section 3.4). An algorithm, which is based on severallemmas, is also presented to generate concave regions for a part effectively.

• A combining approach to generate regions from the basic elements based onlinear programming and mold design knowledge (Section 3.5). In judgingfeasible parting direction of several faces, a simple yet efficient approach basedon linear programming is proposed. It enables more combinations of regions andfaces are explored because of its fast solution process. An algorithm is developedwhich allows more mold design knowledge considered.

• A mold construction method based on the reverse glue operation and a geometricreconstruction problem for any mold base. A novel mold construction approachbased on reverse gluing operation is proposed in this research, which is muchmore efficient than the current industrial approaches (Section 3.6). Also it isillustrated in this research that the mold construction problem is actually ageometric reconstruction problem. Therefore computer-aided mold designmethods may find theoretical basis in the area of computational geometry.

• A set of examples and industrial cases are valuable for the future research incomputer-aided mold design area. In this research the author constructs severalexample parts to test different aspects of a mold design method (Section 4.4).Several representative industrial cases are also utilized in the dissertation (Section4.5, 7.2 and 8.2). They provide a basis to compare different mold designapproaches.


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Contributions related to Design-for-Manufacture (H2 and H2.1 ~2.3):

• A demonstration of transferring part of DFM responsibility (geometric tailoring)from the designer to the manufacturer is feasible, at least for Rapid Tooling.Traditionally the designer takes all the responsibility of DFM. This was taken forgranted by lots of researchers for a long time. However from the perspective ofinformation integration, design-for-manufacture is actually a process in whichdesign and manufacturing requirements are integrated and tradeoffs are made onthe design and manufacturing goals and variables. Therefore it is not necessarilyto let the designer have all the DFM responsibility. This research demonstratesthat by given some design freedom to the manufacturer, the cost/time offabrication process may be reduced (Section 5.5 and Chapter 8).

• Geometric tailoring is identified as part of DFM tasks that may be transferredfrom the designer to the manufacturer in the process of producing functionalprototypes. The definition and category of geometric tailoring are presented withits fundamentals (Section 5.2).

• Decision templates based on the compromise DSP are proposed as the digitalinterface to transform design information to the manufacturer for geometrictailoring problem. Their methodology is identified based on design freedom(Section 5.3). Two kinds of decision templates are formulated in this research andare applied to a set of examples (Section 5.5 and 7.3).

• An integrated design for Rapid Tooling system is developed based on Decision-Based Design. The DFRTS is a collaborative system which can handle conflictsbetween designers and manufacturers. A problem formulation which considersboth the design and manufacturing requirements is presented. This researchdemonstrates that a concurrent approach to formulate and solve problem mayresult in a better solution than a sequential approach (Section 6.5 and 8.6).

• A solution strategy and related solution process are proposed for the integratedproblem formulation. The solution process is systematic and the same threestages can be followed to obtain solutions for different parts (Section 7.4 and 8.5).

Contributions related to Rapid Tooling (H1 and H2):

• The Rapid Tooling Mold Design System is a useful tool in designing molds for apart design. The implementation of the mold design method into a convenienttool is another kind of contributions of this research.

• The Design for Rapid Tooling System, which integrates several softwarepackages, is developed to aid the manufacturer to produce functional prototypesby the direct AIM tooling. In this research, different software systems are used insolving the geometric tailoring problem, which reduces the time in achievingsatisfied prototypes.

• The developed systems are important components of the Rapid Tooling TestBed.The RTTB is an experimental testbed that supports exploration of design anddesign-for-manufacture issues related to rapid prototyping and rapid tooling forinjection molding (Section 1.2.2). The efforts in producing prototypes within 3-4


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days of submitting a product model will significantly reduce the productrealization time and cost.

• Several industrial cases are investigated, which demonstrates the advantage anddisadvantage of the direct AIM tooling. As is a new technique, the direction AIMtooling has only limited case studies published in literature. In this researchprototypes of a ring gear (Section 5.5.3) and a camera roller (Section 8.5) wereproduced. These studies illustrated that the direction AIM tooling can be used forrather complicated part. However its limited mold life should be considered inselecting a fabrication process.

Anticipated Impact:

In summary, the author expects this research will have two impacts.

(1) It is expected that this research will enhance our formal understanding of thebasic computational issues that lie behind the computer-aided mold designproblem. In this way, improved and more rigorous mold design systems can bedeveloped and used in the injection molding industry.

(2) Design-for-Manufacture (DFM) is actually a process to integrate both design andmanufacturing requirements. From this perspective, there is no particular reasonfor the designer to play a dominant role in performing DFM. A differentapproach is presented in this research, which, in the author’s hope, will enhanceour understanding of the basic strategies of DFM. It is believed that furtherresearch in this direction can enhance our understanding of the advantages anddisadvantages of different DFM approaches.

After the contributions of this research were highlighted in this section, limitations ofthe research will be identified and discussed in the next section.

9.3 CRITICAL ANALYSIS: LIMITATIONS OF THE RESEARCH

With the research questions answered and the contributions of this work presented,in this section a critical evaluation of this work is performed, with the focus on thelimitations of RTMDS and DFRTS and their different elements.

RTMDS

The mold design method developed in this research only considers planar surfaces.Although quadric and parametric surfaces can be approximated by a series of planarsurfaces, the face number and file size may increase dramatically. More importantly, itmay bring problems in the integration of the RTMDS with other software systems. Forexample, the RP process planner developed by West (1999) checks all directions ofplanar surfaces of a part in selecting a building orientation in a SLA machine. Thereforeit may take a long time for a mold piece generated by the RTMDS because too manyplanar faces are generated in the approximation of a quadric surface.

The RTMDS can only handle a SAT file as the input. This may add a burden to themold designer in preparing the input file of a part. Also the transfer of a STL file to aSAT file may bring in some errors in the CAD model. In the transferring, all planar facesthat are in the same surface are combined into one face in the SAT file. This may not be


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desired for some parts (e.g. the camera roller in Section 8.2). Therefore the mold designmethod needs to be extended to handle a part represented in STL file.

Some implementation limitations of the modules of the RTMDS are listed asfollows. The limitations of the region generation module include (1) the face splittingalgorithm is not general; (2) data structure of the system is not efficient which results inlarge memory requirements. The limitations of the region combination module include(1) limited more mold design knowledge is considered in the system; (2) the robustnessand accuracy of the combination process are different for parts; (3) the combining ordersof regions and faces are not changeable by users. The limitations of the mold piecemodule include (1) the algorithm of judging parting surface for a region needs to berefined; (2) the generation of inner glue faces can only handle regular cases; (3) the partcan not be positioned freely within the mold base.

DFRTS

In the DFRTS, much effort is required to model the manufacturing process before ageometric tailoring problem can be solved. Currently some manufacturing requirementsfor the direction AIM tooling have been formulated in the DFRTS (Section 6.1). Theycan be reused for different parts. However, to add more requirements or processvariables of the direction AIM tooling (e.g. injection molding process variables besidesthe cooling time), more process models are needed. To extend the DFRTS to otherprocesses and materials, their manufacturing models should also be built by experimentsor simulation. Related to this limitation is that as only quantitative equations can beconsidered in the compromise DSP, some properties of a fabrication process or a designfunction may not be able to be considered in the DFRTS if they cannot be formulated asequations.

Geometric tailoring is mainly due to the material difference of products andprototypes (Section 5.5), or the fabrication difficulties for a part design (Section 8.3).Not surprisingly, if the material properties of products and prototypes are similar, thematerial geometric tailoring will generate similar results as the original design. Similarlyif there is only little difficulty in producing a part design, the material-process geometrictailoring will generate similar results as those of material geometric tailoring (Section7.6). Since it takes much time and effort to couple the decisions in the geometrictailoring problem, this effort may not be needed if the two results will be similar.Therefore a criteria and a related judging method may be needed in the DFRTS todetermine if the geometric tailoring is necessary for a part design.

The dimensions in the decisions templates are related to a feature in CAD models.Currently their relations are not recorded. In this research the author acts as the designerand the manufacturer at the same time, hence this is not a concern in the current DFRTS.However, in real applications, a mechanism to build and record the relation ofdimensions in decision templates and features in CAD models is necessary. The researchin feature representation and recognition may be incorporated in this development.

In the camera roller example (Chapter 8), the topology of the part is changed becausesmall dimension changes cannot satisfy the manufacturing requirements for the part. Inthe DFRTS a systematical approach was developed for the dimensional changes.However the topological changes are still based on the experience and knowledge of the


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manufacturer. Therefore a systematic approach is desired for judging if the topology of apart needs to be changed, and how is should be changed if it is necessary.

With some of the limitation of the RTMDS and DFRTS highlighted in this section,recommendations for future work will be presented in the next section.

9.4 FUTURE WORK

The limitations of the RTMDS and DFRTS described in the last section can be usedto guide future research related to the systems. Some of the future research avenues aredescribed as follows.

Future Work in Computer-Aided Mold Design

Refining the RTMDS

As mentioned in Section 9.3, several refinements can be added in the currentimplementation of the RTMDS. Some tasks are further described as follows.

! The mold design method can be extended to handle a CAD model represented in STLformat. As the concave regions are generated based on the convex and concaveproperty of two neighboring faces, a part represented in triangles can also be handledby the region-based mold design method. The only task here is to generate topologyrelation from a STL file. Research on the topology generation for STL file is rathermature in computer graphics area.

! In the later mold design process, it is desired to move the part position in the moldbase based on the ejector pattern of the injection molding machine. Therefore in theRTMDS, it is important to enable the user to position the part freely within the moldbase.

! Parting surface is a important mold design variable. The current algorithm of judgingparting surface for a region is rather simple, and can be improved to handle morecases.

! The current data structure of the RTMDS is not efficient. By using pointer instead ofarray, the memory requirements for large CAD models will be reduced.

Extending the RTMDS

! Shrinkage compensation is also an important consideration in the mold design,especially for low-pressure powder injection molding (shrinkage can be as high as14%). Based on the models developed in (Judson, 1999), it is possible to predict theshrinkages and change the shape of the mold correspondingly. In ACIS, theintroduction of laws (mathematical functions) provides a powerful tool to modelcomplex geometries. By using different law classes, complex functions can be usedfor creating curve and surface geometry.

! More design knowledge can be added to the multi-piece mold design. One possiblecriterion is the ejection property of a mold piece. Considering the fragility of theinjection molded part in the low pressure power injection molding, more mold piecesmay be necessary for a complete part.


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New Directions

! A geometric reconstruction problem, Problem GFG (Section 3.6.2), is defined in thisresearch. No approaches for this problem are found in literatures of computationalgeometry. However it is an important problem in computer-aided mold designresearch in order to construct mold pieces for a part (refer to Figure 4.12). InRTMDS an algorithm is proposed for simple cases. Further research is necessary todevelop a theoretical basis and more advanced approach.Future Work in Design-for-Manufacture

Refining the DFRTS

! It is desired to formulate the updated research results on the direct AIM tooling andincorporate them into the DFRTS.

! To further testing the idea of geometric tailoring, it is desired to test the DFRTS withdifferent fabrication process (e.g. low-pressure powder injection molding) anddifferent materials.

! The automation levels in utilizing different software systems of the DFRTS need tobe increased in order to handle different parts more efficiently.Extending the DFRTS

! A mechanism to build and record the relation of dimensions in decision templates andfeatures in CAD models needs to be developed. As mentioned before, a decisiontemplate can be linked with a parametric feature-based CAD model which is similarto behavioral modeling (Section 2.4). Therefore the intent and performance of thedesign are an integral part of the product model. Their values can also referenceexternal applications as part of the product model definition.

! A systematic approach is desired for judging if the topology of a part needs to bechanged, and how is should be changed if it is necessary. The task would be ratherchallenging considering the representation and manipulation of the topology are allmuch more difficult than those of dimensions. However research on graph theorymay be useful.

! In the DFRTS, a criteria and a related judging method can be developed to determineif the geometric tailoring is necessary for a part design. It would save the efforts ofthe designer and manufacturer in formulating and solving the geometric tailoringproblem.

New Directions

! Another significance of transferring the responsibility of geometric tailoring from thedesigner to the manufacturer is that it enables the designer to consider severalfabrication processes at the same time. It is well known that different prototypingprocesses have their advantages and disadvantages. Typically no single processmeets all the requirements of a part. Therefore it is difficult for the designer toconsider several fabrication processes and perform design-for-manufacture tasks forthem. Instead it is feasible to develop a distributed system (Figure 9.3). With the aidof decision templates, the designer formulates the design requirements and sendsthem to a process broker. The process broker can then select several candidateprocesses and send the manufacturers the design requirements. Based on the design


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requirements and the property of a fabrication processes, the manufacturer performsthe geometric tailoring and sends the results back to the process broker. The processbroker can aid the designer in selecting a process based on the tailored part design.Therefore more fabrication processes are considered for a par design.

1. CAD Model of part &Requirements 2. CAD Model of part &requirements

Designer

ProcessBroker

RapidToolingServices

5. Recomendation

4.Tailored Part Design& Goal Achievements

Internet

Internet

Manufacturer

Manufacturer

Manufacturer

3. GeometricTailoring



Figure 9.3 – An Ideal System Working Between the Designer and Manufacturer.

Appendix A

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APPENDIX A

RTMDS IMPLEMENTATION

In this appendix the files and classes are summarized for the RTMDS implementation.The RTMDS was implemented in Microsoft Visual C++ for Windows NT on personalcomputers. ACIS geometric modeler and LINGO are also used in the systemimplementation.

• Appendix A.1 contains the code files and simple descriptions of theirfunctions.

• Appendix A.2 contains the main classes of the RTMDS and relateddescriptions of their functions.

• Appendix A.3 contains the implementation information based on ACISgeometric modeler.

• Appendix A.4 contains the implementation information based on LINGOsoftware system.

Appendix A

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A.1 Code Files of RTMDSThe RTMDS was written in C++ using Microsoft Wizard and the Microsoft FoundationClasses (MFC). The codes include C++ source (.CPP) files, header (.H) files, resource(.RC) files, and a project (.DSP) file. They are described with their source code size asfollows:

• Source (.CPP) Files:RTMold.cpp – The main program source file. It also contains the initialization and

deletion of ACIS and LINGO libraries. (9 KBytes)MainFrm.cpp – Derive the CmainFrame class, which handles the creation of toolbar

buttons and the status bar. (5 KBytes)RTMoldDoc.cpp – Derive and implement the document class, which initialize and

serialize a document. This is also the main file which contains the functions ofmold design modules that may change the context of a CAD model. (52 KBytes)

RTMoldView.cpp – Derive and implement the view class, which display and print thedocument data. (13 KBytes)

Attrib_Connect_Region.cpp – Implement of an attribute (connected regionproperties) attached to an entity of an ACIS model. (3 KBytes)

Attrib_PEdge.cpp – Implement of an attribute (parting edges) attached to an entity ofan ACIS model. (3 KBytes)

Attrib_Region.cpp – Implement of an attribute (region properties) attached to anentity of a ACIS model. (3 KBytes)

Attrib_Rtmold.cpp – Implement of a basic attribute used for deriving other attributes.(1 KBytes)

Concave_Edge.cpp – Code to generate and record concave edges. (1 KBytes)Concave_Face.cpp – Code to generate and record concave faces from generated

concave edges. (2 KBytes)Concave_Region.cpp – Code to generate and record concave regions from generated

concave faces. (50 KBytes)Connect_Edge.cpp – Code to generate and record connect edges of a region. (1

KBytes)Geometry.cpp – Code to judge geometry properties. It also contains common

functions to call ACIS functions. (41 KBytes)LP_Parting_Direct.cpp – Code to generate parting direction of a region by calling

LINGO software system. (7 KBytes)Region_Split.cpp – Code to divide a region or a face by using a given plane. (31

KBytes)Rev_Glue.cpp – Code to generate glue faces and generate mold piece bodies based on

the glue faces and the mold base. (48 KBytes)RTMold_Dialog.cpp – Code to generate an error dialog box. (3 KBytes)RTMSelectTool.cpp – Code to select entities interactively by using a mouse. (13

KBytes)SetTolDlg.cpp – Code to generate a dialog box for setting combination tolerances of

faces and regions. (2 KBytes)ChangeRegDlg.cpp – Code to generate a dialog box for changing region numbers. (2

KBytes)

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Set_CurR.cpp – Code to generate a dialog box to set the current region number forjudgment. (2 KBytes)

Set_MP_CurR_Dlg.cpp – Code to generate a dialog box to set the current regionnumber for mold piece construction. (2 KBytes)

StartLoopDlg.cpp – Code to generate a dialog box to set the iteration number before aregion which is not in main parting direction is considered in the combinationprocess. (2 KBytes)

• Header (.H) Files:For each source file (.cpp) , there is a header file in the RTMDS. The header file (.h)contains the definitions of parameters that are used in the source files, and also thedefinitions of functions that are implemented in the source files. The functiondescripoint of each header file is omitted here because the header file has the samefunction as the source file in the same name. Therefore only the source code size isprovided.RTMold.h (2 KBytes) MainFrm.h (2 KBytes)RTMoldDoc.h (5 KBytes) RTMoldView.h (4 KBytes)Attrib_Connect_Region.h (1 KBytes) Attrib_PEdge.h (1 KBytes)Attrib_Region.h (1 KBytes) Attrib_Rtmold.h (1 KBytes)Concave_Edge.h (1 KBytes) Concave_Face.h (1 KBytes)Concave_Region.h (3 KBytes) Connect_Edge.h (1 KBytes)Geometry.h (3 KBytes) LP_Parting_Direct.h (2 KBytes)Region_Split.h (2 KBytes) Rev_Glue.h (4 KBytes)RTMSelectTool.h (3 KBytes) SetTolDlg.h (1 KBytes)ChangeRegDlg.h (2 KBytes) Set_CurR.h (2 KBytes)Set_MP_CurR_Dlg.h (2 KBytes) StartLoopDlg.h (2 KBytes)

• Resource (.RC) Files:Resource.h – Definitions of parameters in RTMold.rc. (4 Kbytes)RTMold.rc – code for the definitions of menus, accelerators, dialogs, icons, and

toolbars. (20 KBytes)

• Project (.DSP) File:RTMold.dsp – code used within the development environment. It stores the

information specific to the project. (13 KBytes)

Total size of source code: 379 KBytesExecutable Size: 913 KBytes, compiled optimized.

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A.2 Classes of RTMDSThe major implementation of the RTMDS is in classes, which a format of Object-Oriented Programming. The classes in the code are presented as follows with a briefdescription of the purpose of each class:

class CRTMoldApp : public CwinApp – This class initializes the main application. Italso contains the initialization and deletion of ACIS and LINGO libraries.

class CMainFrame : public CframeWnd – This class creates the toolbar buttons andthe status bar. It also set their styles and positions.

class CRTMoldDoc : public CDocument – This class creates and modifies adocument. The main functions of three mold design modules are also in this classbecause they change the context of a document.

class CRTMoldView : public CView – This class displays and prints the documentdata. All the functions related to showing a CAD model are in this class.

class ATTRIB_RTMOLD : public ATTRIB – This class defines the commonattribute class used in the RTMDS. All other attribute classes are derived from it.

class ATTRIB_PEDGE: public ATTRIB_RTMOLD – This class defines an attribute(parting edges) which is attached to an entity of an ACIS model.

class ATTRIB_CONNECT_REGION: public ATTRIB_RTMOLD – This classdefines an attribute (connected region properties) which is attached to an entity ofan ACIS model.

class ATTRIB_REGION: public ATTRIB_RTMOLD – This class defines anattribute (region properties) which is attached to an entity of an ACIS model.

class Concave_Edge – This class generates and records concave edges in a CADmodel.

class Concave_Face – This class generates and records concave faces from generatedconcave edges.

class Concave_Region – This class generates and records concave regions fromgenerated concave faces.

class Connect_Edge – This class generates and records connect edges of a region.class RTMSelectTool : public MouseTool – This class provides functions to enable

users to interactively select entities in a CAD model by using a mouse.class CSetTolDlg : public Cdialog – This class creates a dialog box for setting

combination tolerances of faces and regions.class CStartLoopDlg : public Cdialog – This class creates a dialog box to set the

iteration number before a region which is not in main parting direction isconsidered in the combination process

class CChangeRegDlg : public Cdialog – This class create a dialog box for changingthe number of a region.

class CSet_CurR : public Cdialog – This class creates a dialog box to set the currentregion number for judgment.

class CSet_MP_CurR_Dlg : public CDialog – This class creates a dialog box to setthe current region number for mold piece construction.

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A.3 Implementation on ACISThe RTMDS is developed based on ACIS 6.2 and Microsoft Visual Studio 6.0. Beforeusing ACIS Microsoft Foundation Class Component (AMFC), the environment andVC++ project should be set.• Environment Variables

1. Set system variables: A3DT = [acis install directory]; ARCH = NT_DLLD;A3DT_MAJOR = 6; A3DT_MINOR = 2.

2. A path to the libraries associated with the ARCH should be set up by adding thestring “;%A3DT%\lib\%ARCH%” in path environment variable.

• Microsoft Visual C++ Project SettingsProject settings are assessed under the “Project->Setting” menu command.

1. “Use run-time library” Setting: Select the “C/C++” tab and “Code Generation” inthe category pop-up. Select “Debug Multithreaded DLL” under the “Use run-timelibrary” pop-up.

2. Precompiled Header Option: In the “C/C++” tab, select “Precompiled Header”from the category pop-up. Select the “Not using precompiled headers” radiobutton.

3. Preprocessor definitions: In the “C/C++” tab, select “Preprocessor” from thecategory pop-up. Click into the field for “Preprocessor definitions:” at the end ofthe existing text. Add the string “,ACIS_DLL,NT,STRICT” into the projectsetting field.

4. Additional Include Directories: In the “Preprocessor” category of the “C/C++”tab, move the insertion cursor to the “Additional include directories:” field. Addthe string “., $(A3DT)\abl, $(A3DT)\aec, $(A3DT)\ag, $(A3DT)\ar,$(A3DT)\amfc, $(A3DT)\amfc\acismfc, $(A3DT)\base, $(A3DT)\blnd,$(A3DT)\bool, $(A3DT)\br, $(A3DT)\clr, $(A3DT)\covr, $(A3DT)\cstr,$(A3DT)\ct, $(A3DT)\ds, $(A3DT)\eulr, $(A3DT)\fct, $(A3DT)\ga, $(A3DT)\gi,$(A3DT)\gl, $(A3DT)\igl, $(A3DT)\ihl, $(A3DT)\intr, $(A3DT)\iges,$(A3DT)\kern, (A3DT)\kern\kernel, $(A3DT)\law, $(A3DT)\lop, $(A3DT)\lopt,$(A3DT)\mesh, $(A3DT)\ofst, $(A3DT)\oper, $(A3DT)\part, $(A3DT)\phl,$(A3DT)\pid, $(A3DT)\rbase, $(A3DT)\rbi, $(A3DT)\rem, $(A3DT)\sbool,$(A3DT)\scm, $(A3DT)\shl, $(A3DT)\skin, $(A3DT)\swp, $(A3DT)\vda,$(A3DT)\vm, $(A3DT)\warp, $(A3DT)\xgeom” into the field.

5. Additional Library Path: Select the “Link” tab and “Input” from the category pop-up menu. Insert the string “$(A3DT)\lib\$(ARCH)” into the “Additional librarypath” field.

After the environment is set, a new project can be created to implement intendedfunctions.• ACIS AppWizard

1. Copy the file “AcisAW.awx” from “$(A3DT)\amfc\aw-i386” to “Microsoft VisualStudio\Common\MSDev98\Template” folder.

2. Initialize a new project by using ACIS AppWizard in VC++ environment.• Function Implementations

Appendix A

353

1. Add all the source files (.cpp) and header files (.h) in folders“$(A3DT)\amfc\acismfc” and “$(A3DT)\amfc\acismfc\tools” to the project.

2. Finally, ACIS’s API functions can be called in the source code to implement theintended functions. Spatial Technology provides an online help for all thefunctions and classes. By checking the description of functions, the developer candetermine which function should be used and how it should be used.

Appendix A

354

A.4 Implementation on LINGOLINGO provides the Dynamic Link Library (DLL) standard for interfacing with otherapplications. In C++ code, the steps to use LINGO to solve a linear problem are listed asfollows:

(1) First loading the LINGO DLL into the system by:// Load the LINGO DLLHINSTANCE m_hInstLINGO = LoadLibrary( "lingodll.dll");// Get a pointer to the LINGO script processor – LGCSCRIPTLINGOCALL* m_pLINGOCall = (LINGOCALL*) ::GetProcAddress((HMODULE)

m_hInstLINGO, "LGCSCRIPT");(2) Generate a script file (.LNG);(3) Use the C/C++ wrapper function LGCSCRIPT to call the LINGO script processor

to solve the linear problem. Some sample codes are given as:pArgs[0] = (void*) "SET ECHOIN 1\nSET TERSEO 1\nTAKE

D:\\cv_Parting_dir.LNG\nGO\nQUIT\n";// Capture all standard output in a filepArgs[1] = (void*) "D:\\cv_Parting_Dir.LOG";// Let LINGO allocate working memorypArgs[2] = NULL;double* pTransfer[8] = &x1, &x2, &y1, &y2, &z1, &z2, &dStatus, NULL;pArgs[3] = pTransfer;int nArgs = 4;// Pass the command script to LINGOint nErrorCode;(*m_pLINGOCall)( pArgs, &nArgs, &nErrorCode);

(4) The results given by LINGO are stored in the arguments pArgs[3], which can bemanipulated by C++ code;

(5) Finally after solving all linear problems, we should release the LINGO DLL fromthe system by:// Release LINGO libraryFreeLibrary( m_hInstLINGO);

A sample of the generated LNG problem are given as follows:

MODEL:MAX= (-0.004378*(x1-x2))+(0.004656*(y1-y2))+(1.001311 *(z1-z2));

!The surface constraints;(-0.341977*(x1-x2))+(0.939683*(y1-y2))+(0.006876*(z1-z2))>=(-0.174000);(-0.642831*(x1-x2))+(0.765977*(y1-y2))+(0.006878*(z1-z2))>=(-0.174000);…(0.980711*(x1-x2))+(-0.195275*(y1-y2))+(0.008613*(z1-z2))>=(-0.174000);(0.980711*(x1-x2))+(0.195275*(y1-y2))+(0.008613*(z1-z2))>=(-0.174000);

!The sphere constraints;(-0.130446*(x1-x2))+(0.034953*(y1-y2))+(-0.990839*(z1-z2))>=-0.990839;(-0.130446*(x1-x2))+(-0.034953*(y1-y2))+(-0.990839*(z1-z2))>=-0.990839;…

Appendix A

355

(-0.095493*(x1-x2))+(0.095493*(y1-y2))+(0.990839*(z1-z2))>=-0.990839;(-0.130446*(x1-x2))+(0.034953*(y1-y2))+(0.990839*(z1-z2))>=-0.990839;

!Here is the data;DATA:

@POINTER (1) = x1;@POINTER (2)= x2;@POINTER (3)= y1;@POINTER (4)= y2;@POINTER (5)= z1;@POINTER (6)= z2;@POINTER (7) = @STATUS();

ENDDATAEND

Appendix B

356

APPENDIX B

CAMERA ROLLER CASE STUDY:COMPLETE SOLUTIONS OF THE

MODIFIED MPGT PROBLMES

In this appendix the results of solving the 24 modified MPGT problems in Section 8.5.2are provided.

• Table B.1 contains the results of variables and goals given by OptdesX for theslicing schemes 1 to 6 for PO1 (Figure 8.20.a).

• Table B.2 contains the results of variables and goals given by OptdesX for theslicing schemes 1 to 6 for PO2 (Figure 8.20.b).

• Table B.3 contains the results of variables and goals given by OptdesX for theslicing schemes 1 to 6 for PO3 (Figure 8.20.c).

• Table B.4 contains the results of variables and goals given by OptdesX for theslicing schemes 1 to 6 for PO4 (Figure 8.20.d).

Appendix B

357

Table B.1 – Solutions Obtained from OptdesX for Slicing Schemes 1~6 of PO1.

PO 1SS1 SS2 SS3 SS4 SS5 SS6

Dia (mm) 2.782406 2.75355 3.280292 2.762365 3.631252 2.833394Width (mm) 2.800527 2.796417 2.801306 2.799082 2.802805 2.80029Thick (mm) 1.00125 1.000293 1.000419 1.001088 1.03211 1.000053

Rows 2 2 2 2 2 2Cols 2 2 2 2 2 2

Draft1 (o) 1.292404 1.286405 1.289208 1.302345 1.455682 0.913498Draft2 (o) 1.17641 1.182111 1.183049 1.179806 1.823202 0.962735Draft3 (o) 0.992764 1.017975 1.167627 1.008618 0.001585 1.002316

TA 0 0 0 0 0 0CT (sec.) 300 300 300 300 300 300

HOC1-6 (mil) 0.002 0.002 0.002 0.002 0.002 0.002FOC1-6 (mil) 0.012 0.012 0.012 0.012 0.012 0.012

Z-Rot (rad.) 0.010408 0.010417 0.010239 0.010415 0.010078 0.010391Weight (g) 2.398691 2.399411 2.389682 2.399121 2.386154 2.397752

mold1_SF (µin) 156 156 156 156 390 382mold2_SF (µin) 156 156 156 156 390 390mold3_SF (µin) 64.91126 64.92935 132.7919 64.92262 407.0193 132.5313mold1_PP (µin) 6 6 6 6 6 6mold1_SP (µin) 64 68 74 77 187 187mold2_PP (µin) 6 6 6 6 6 6mold2_SP (µin) 64 68 72 75 177 177mold3_P (µin) 6 6 6 6 6 6Flat_Acc (inch) 0.001771 0.001771 0.001771 0.001771 0.001771 0.001771Par_Acc (inch) 0.001616 0.001616 0.001616 0.001616 0.001616 0.001616Per_Acc (inch) 0.002002 0.002002 0.002002 0.002002 0.002002 0.002002Pos_Acc (inch) 0.006434 0.006434 0.006434 0.006434 0.006434 0.006434

Cost ($) 1290.801 1268.699 1238.415 1215.334 1381.207 1293.074# molds1 1 1 1 1 2 3# molds2 1 1 1 1 1 3# molds3 3 3 3 3 17 6

Z 0.015166 0.015664 0.019315 0.016753 4.445024 3.492163

Appendix B

358

Table B.2 - Solutions Obtained from OptdesX for Slicing Schemes 1~6 of PO2.

Assembly 2SS1 SS2 SS3 SS4 SS5 SS6


Rows 2 2 2 2 2 2Cols 2 2 2 2 2 2


TA 0 0 0 0 0 0CT (sec.) 300 300 300 300 300 300

HOC1-6 (mil) 0.002 0.002 0.002 0.002 0.002 0.002FOC1-6 (mil) 0.012 0.012 0.012 0.012 0.012 0.012

Z-Rot (rad.) 0.010233 0.010242 0.010275 0.01016 0.010341 0.010358Weight (g) 2.389997 2.390428 2.391826 2.384321 2.39644 2.396307



Z 0.034796 0.033874 0.032748 0.038377 1.175995 1.064109

Appendix B

359




Rows 2 2 2 2 2 2Cols 2 2 2 2 2 2


TA 0 0 0 0 0 0CT (sec.) 300 300 300 300 300 300

HOC1-6 (mil) 0.002 0.002 0.002 0.002 0.002 0.002FOC1-6 (mil) 0.012 0.012 0.012 0.012 0.012 0.012

Z-Rot (rad.) 0.01022 0.01026 0.010225 0.010216 0.010142 0.010034Weight (g) 2.388818 2.392824 2.38971 2.389256 2.3862 2.391746



Z 0.037134 0.033593 0.036098 0.035687 3.969953 4.033753

Appendix B

360




Rows 2 2 2 2 2 2Cols 2 2 2 2 2 2


TA 0 0 0 0 0 0CT (sec.) 300 300 300 300 300 300

HOC1-6 (mil) 0.002 0.002 0.002 0.002 0.002 0.002FOC1-6 (mil) 0.012 0.012 0.012 0.012 0.012 0.012

Z-Rot (rad.) 0.010245 0.010262 0.010268 0.010245 0.010077 0.009911Weight (g) 2.391459 2.391151 2.39185 2.390661 2.390754 2.399862



Z 0.036288 0.032353 0.035243 0.034411 4.171018 3.824423

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Vita

375

VITA

Yong Chen was born in Guiyang, Guizhou, China on October 20, 1972. He grew up in

Zunyi, Guizhou and attended Zuiyi No. 4 High School. He earned his Bachelor of

Science degree in Mechanical Engineering from Zhejiang University (Hangzhou,

Zhejian, China) in 1993. He earned his Master of Science degree in Mechanical

Engineering from Huazhong University of Science and Technology (Wuhan, Hubei,

China) in 1996. After that he worked as a lecturer at school of Mechanical Engineering

and a research scientist at Graphical Software Center at Huazhong University of Science

and Technology before attending Georgia Institute of Technology in March 1998. His

graduate work was funded by National Science Foundation Grant DMI-96-18039. Upon

graduation, Yong has accepted a position as a Senior Software Engineer with the 3D

Systems Inc. in Valencia, California.

computer-aided design for rapid tooling: methods for mold design

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