topology optimization of compliant mechanisms based...

Topology Optimization of Compliant Mechanisms

Based on the BESO Method

A thesis submitted in fulfillment of the requirements for the degree of

Doctor of Philosophy

Yan Li

BCivEng, MConstMgmt

School of Civil, Environmental and Chemical Engineering

College of Science Engineering and Health

RMIT University

March 2014

I

Declaration

I certify that except where due acknowledgement has been made; the work is that of

the author alone. The work has not been submitted previously, in whole or in part,

to qualify for any other academic award. The content of the thesis is the result of

work which has been carried out since the official commencement date of the

approved research program and ethics procedures and guidelines have been

approved and followed. Any editorial work, paid or unpaid, carried out by a third

party is acknowledged.

Yan Li

March 2014

II

Acknowledgments

The process of conducting research in the pursuit of a doctorate has been a

humbling experience, during which I have been exposed to the lack of depth of my

own knowledge in my field of supposed expertise. It is impossible to imagine

having to go through the process alone, and I would like to recognize the people

who have helped me along the way.

My greatest appreciation goes to my primary supervisor, Dr. Xiaodong Huang, for

his academic guidance and encouragement of my work throughout this process.

This PhD program would not have been completed without his constant direction

and support. He provided terrific guidance and insights, knowing just when to

challenge and when to support in the right balance. His structural optimization

techniques and ideas of designing compliant mechanisms are extremely valuable

for me.

I would also like to express my sincere thanks to my second supervisor, Prof. Mike

Xie, for his supervision and support. He has been always patient, providing fruitful

guidance. His comments during our meetings have also deepened my

understanding of my research.

I would like to thank other members of the Centre for Innovative Structures and

Materials, especially Dr. Shiwei Zhou, Dr. Zhihao Zuo, Dr. Jianhu Shen and Dr.

Annie Yang. I often benefit from their suggestions, experience and ideas. It has

been an honor to share the time with such an intelligent group.

I would like to thank my entire family, especially my parents for their love, support,

and patience. Their love and expectation are a huge encouragement and motivation

III

for me to deal with difficulties from my research. Their constant support and

encouragement have enabled me to face the challenges of this PhD program.

IV

Abstract

This dissertation explores topology optimization techniques for designing

compliant mechanisms actuated by forces. For a compliant mechanism, it has the

potential of reducing part count, mechanical joints, operation noise, and

manufacturing and assembly costs over a traditional rigid-link mechanism. Thus

the application of compliant mechanisms is becoming increasingly prevalent in

medical instruments and mechanical devices. Optimization of compliant

mechanisms has drawn intense attention of many researchers.

However, this design field has been facing many challenges and shortages in

several aspects such as optimization method, optimization algorithm and resulting

topology. For example, convergence problems often lead to vague solutions.

Optimization algorithms are not very suitable to investigate the real physical

significance. Furthermore, designs of compliant mechanisms using topology

optimization techniques naturally lead to the introduction of hinges into final

topologies. In addition, the previous design also focuses mainly on the optimal

design of linear compliant mechanisms. In fact, optimizing nonlinear compliant

mechanisms is proving quite necessary in real applications as the simulation is

more accurate. Therefore, it is important to devote efforts to the modification of

previous optimization techniques for constructing practical compliant mechanism

designs.

This dissertation proposes a modified bi-directional evolutionary structural

optimization (BESO) method for the optimal design of linear and geometrically

nonlinear compliant mechanisms. Numerical algorithms based on the BESO

method are developed through various objectives and constraints in compliant

V

mechanism design.

Firstly, to consider functional behaviors of compliant mechanisms, sets of clear

and suitable structural configurations are produced by quantifying various

performance characteristics and changing the stiffness of attached springs. This

implies that material distribution and hinge formation are demonstrated in this

work. To achieve prescribed structural stiffness for optimized mechanisms, a new

BESO algorithm is established for solving the proposed optimization problem by

gradually updating design variables. The inverter and the gripper optimization

problems serve to demonstrate the practicability and effectiveness of the proposed

method. Besides this, a new formulation is established by considering desirable

deformation and simultaneously precluding the formation of hinges in order to

design hinge-free compliant mechanisms, verified by a large number of numerical

experiments including rare 3D hinge-free designs. Furthermore, compliant

mechanisms often undergo large displacement, in order to provide their

functionality. Therefore, the research also addresses the optimal design of

compliant mechanism with geometrically nonlinear behaviors. With the aid of the

hard-kill BESO method, a new systemic design approach is developed to

overcome the convergence difficulty caused by extreme deformation in the

nonlinear finite element analysis. Large-displacement inverter design with the

desired structural stiffness is provided based on a new evolutionary optimization

technique involved in a developed multi-criteria flexibility-stiffness formulation.

Overall, the modified BESO method has effectively set up new optimizations,

visualizing and analyzing the resulting topologies for 2D and 3D compliant

mechanism designs. The findings shown in this dissertation have also established

appropriate techniques for designing various linear compliant mechanisms. In

addition, an efficient and robust methodology has been provided for the topology

VI

optimization of geometrically nonlinear compliant mechanisms. Furthermore, the

work has provided a solid foundation for creating a practical design tool in the

form of a user-friendly computer program, which is suitable for the conceptual

design of a wide range of compliant mechanisms.

VII

Publication List

The following papers have been produced during the PhD research project.

Li, Y., Huang, X., Xie, Y. M. and Zhou, S. W. (2013). “Bi-directional evolutionary

structural optimization for design of compliant mechanisms", Key Engineering

Material, 535-536, pp.373-376.

Li, Y., Huang, X., Xie, Y. M. and Zhou, S. W. (2014). “Evolutionary topology

optimization of hinge-free compliant mechanisms", International Journal of

Mechanical Science, 86, pp. 69-75.

Huang, X., Li, Y., Zhou, S. W. and Xie, Y. M. (2014). “Topology optimization of

compliant mechanisms with desired structural stiffness”, Engineering Structures,

79, pp. 13-21.

VIII

Table of Contents

Acknowledgments .............................................................................. II

Abstract ............................................................................................. IV

Publication List ................................................................................ VII

Table of Contents ........................................................................... VIII

List of Tables .................................................................................... XII

List of Figures ................................................................................ XIII

Chapter One: Introduction ................................................................ 1

1.1 Definition of Compliant Mechanisms .................................................... 2

1.2 Classification of Compliant Mechanisms .............................................. 3

1.3 Advantages of Compliant Mechanisms .................................................. 5

1.4 Applications of Compliant Mechanisms ................................................ 6

1.5 Motivation of the Dissertation ................................................................ 8

1.6 Aims and Objectives ................................................................................ 9

1.7 Organization of the Dissertation .......................................................... 10

Chapter Two: Literature Review ..................................................... 15

2.1 Kinematic-based Methods for Designing Compliant Mechanisms ... 16

2.2 Optimization-based Methods for Designing Compliant Mechanisms20

2.2.1 Design Criteria of Compliant Mechanisms in Optimization-based

Methods ................................................................................................. 21

2.2.2 Optimal Design of Compliant Mechanisms Using Homogenization

Methods ................................................................................................. 23

2.2.3 Optimal Design of Compliant Mechanisms Using Solid Isotropic

Material with Penalization ..................................................................... 26

IX

2.2.4 Optimal Design of Compliant Mechanisms Using Level-set

Method ................................................................................................... 29

2.2.5 Optimal Design of Compliant Mechanisms Using Other Methods34

2.3 Topology Optimization Methods .......................................................... 35

2.4 Bi-directional Topology Optimization Method ................................... 42

2.4.1 The Early Bi-directional Evolutionary Structural Optimization

Method ................................................................................................... 42

2.4.2 Introduction to the Current Bi-directional Evolutionary Structural

Optimization Method ............................................................................. 45

2.4.3 Hard-kill Based and Soft-kill Based Bi-directional Evolutionary

Structural Optimization Method ............................................................ 48

2.4.4 Applications of the Current Bi-directional Evolutionary Structural


2.4.5 Summary of the Bi-directional Evolutionary Structural


2.5 General Remarks ................................................................................... 53

Chapter Three: Modified Bi-directional Evolutionary Structural

Optimization ...................................................................................... 67

3.1 Problem Statement................................................................................. 68

3.2 Sensitivity Calculation ........................................................................... 69

3.3 Filter Scheme and Stability Process ..................................................... 71

3.4 Volume Constraint and Convergence Criterion ................................. 74

3.5 Evolutionary Procedure of Modified Bi-directional Evolutionary

Structural Optimization Methods .............................................................. 76

3.6 Numerical Implementation of Modified Bi-directional Evolutionary

Structural Optimization Methods .............................................................. 77

3.7 Conclusions ............................................................................................. 80

Chapter Four: Effects of Spring Stiffness on Topology Design of

X

Linear Compliant Mechanisms ........................................................ 82

4.1 Optimization Problem and Structural Analysis .................................. 83

4.2 Sensitivity Number ................................................................................ 89

4.3 Numerical Implementation ................................................................... 92

4.4 Iterative Procedure ................................................................................ 92

4.5 Numerical Examples and Discussion ................................................... 94

4.5.1 2D Numerical Examples ............................................................... 95

4.5.2 3D Numerical Examples ............................................................. 109

4.6 Conclusions ........................................................................................... 111

Chapter Five: Desired Structural Stiffness in Topology Design of

Linear Compliant Mechanisms ...................................................... 114

5.1 Optimization Problem and Structural Analysis ................................ 115

5.2 Sensitivity Number .............................................................................. 118

5.3 Determination of Lagrange Multiplier .............................................. 119

5.4 Numerical Implementation ................................................................. 121

5.5 Iterative Procedure .............................................................................. 121

5.6 Numerical Examples and Discussion ................................................. 124



5.7 Conclusions ........................................................................................... 135

Chapter Six: Evolutionary Topology Optimization of Hinge-free

Compliant Mechanisms .................................................................. 138





6.5 Numerical Examples and Discussion ................................................. 146

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6.6 Conclusions ........................................................................................... 156

Chapter Seven: Topology Design of Nonlinear Compliant

Mechanisms ...................................................................................... 159

7.1 Differences between Structural Linearity and Nonlinearity ........... 160

7.2 Types of Structural Nonlinearity ........................................................ 163

7.3 Geometrical Nonlinearity in Designing Compliant Mechanisms .... 164


7.5 Geometrically Nonlinear Finite Element Analysis ............................ 166


7.7 Determination of Lagrange Multiplier .............................................. 171



7.10 Numerical Examples and Discussion ............................................... 175

7.11 Conclusions ......................................................................................... 179

Chapter Eight: Conclusions ........................................................... 181

Appendix A ....................................................................................... 186

XII

List of Tables

Table 3.1 The evolutionary procedure of the modified BESO method ................. 77

Table 4.1 Resulting topologies of the inverter mechanism using various spring

stiffness values ..................................................................................................... 102

Table 4.2 Resulting topologies of the gripper mechanism using various spring

stiffness values ..................................................................................................... 106

Table 7.1 Comparison of linear and nonlinear structure analysis ........................ 161

XIII

List of Figures

Figure 1.1 A traditional rigid link mechanism ......................................................... 2

Figure 1.2 A compliant mechanism ......................................................................... 3

Figure 1.3 Classification of compliant mechanisms ................................................ 4

Figure 1.4 Compliant devices: a binder clip, scissor and single-piece stapler ........ 7

Figure 2.1 A compliant slider mechanism and its pseudo-rigid-body model ........ 19

Figure 2.2 Optimal configurations of compliant gripper with different total

volume constraints using the homogenization method .......................................... 25

Figure 2.3 Optimal configurations of compliant gripper with one output and two

outputs using the density-based method ................................................................ 26

Figure 2.4 Evolution of the level-set surface ......................................................... 30

Figure 2.5 An optimal configuration of hinge-free compliant using the level-set

method ................................................................................................................... 32

Figure 2.6 Optimal configurations of nonlinear compliant inverter with different

spring stiffness using the level-set method ............................................................ 33

Figure 2.7 Solid continua parameterizations ......................................................... 38

Figure 2.8 An illustration of level-set method ....................................................... 39

Figure 2.9 Design domain and its ESO topology with RR=25% for a Michel-type

structure optimization with von Mises stress criterion .......................................... 41

Figure 2.10 Design area and its ESO topology with the performance index

PI=0.01211 and perimeter limit P*=193L for a Michel-type structure optimization44

Figure 2.11 Typical checkerboard regions in an optimum solution ....................... 46

Figure 2.12 Design domain and its BESO topologies with 32×20 and 160×100

meshes .................................................................................................................... 47

Figure 2.13 The evolutionary history of stiffness optimization for a cantilever ... 48

Figure 2.14 The BESO topology for a MBB beam using multiple materials ........ 50

Figure 3.1 The filter scheme in the BESO method ................................................ 72

XIV

Figure 3.2 The filter scheme in the modified BESO method ................................ 73

Figure 3.3 The evolutionary history of the mean compliance using the modified

BESO method ........................................................................................................ 78

Figure 3.4 Resulting topologies with 80×50 and 160×100 meshes ....................... 79

Figure 3.5 Dimensions of the design domain and boundary conditions for a beam79

Figure 3.6 Evolution histories of mean compliance and volume fraction using the

modified BESO method ......................................................................................... 80

Figure 4.1 Model A for the optimal design of compliant mechanisms .................. 85

Figure 4.2 Model B for the optimal design of compliant mechanisms .................. 89

Figure 4.3 Flow chart of the modified BESO procedure for the optimal design of

linear compliant mechanisms ................................................................................. 94

Figure 4.4 Design domain and boundary conditions of the inverter mechanism .. 95

Figure 4.5 Optimum topologies for the inverter mechanism ................................. 97

Figure 4.6 Evolution histories of the inverter mechanism using different design

problems ................................................................................................................. 99

Figure 4.7 Output spring‟s effects on objective functions ................................... 101

Figure 4.8 Design domain and boundary conditions of the gripper mechanism . 103

Figure 4.9 Optimum topologies for the gripper mechanism ................................ 104

Figure 4.10 Evolution histories of the gripper mechanism using different design

problems ............................................................................................................... 105

Figure 4.11 Resulting topologies for the gripper mechanism by using Function C107

Figure 4.12 Design domain and boundary conditions of the compliant force

inverter with two output ports .............................................................................. 108

Figure 4.13 Optimum topologies for the force inverter with two output ports ... 109

Figure 4.14 Design domain and boundary conditions of 3D gripper mechanism110

Figure 4.15 Optimized topologies and CAD models of 3D gripper mechanisms

with using different design criteria ...................................................................... 111

Figure 5.1 Flow chart of the proposed BESO procedure with the compliance

XV

constraint for the optimal design of compliant mechanisms ............................... 123

Figure 5.2 Evolution history of the topology for the compliant inverter ............. 125

Figure 5.3 Evolution histories of the objective function and constraints ............ 126

Figure 5.4 Optimized topologies and output displacements of the compliant

inverters for various compliance constraints ....................................................... 127

Figure 5.5 Relationship between the output displacement and mean compliance of

optimized compliant mechanisms. ....................................................................... 128

Figure 5.6 Design domain and boundary conditions of the compliant gripper ... 129


grippers for various compliance constraints ........................................................ 130

Figure 5.8 Optimized topologies of the compliant grippers for various stiffness of

the spring .............................................................................................................. 132

Figure 5.9 Optimized designs with hinges or without hinges under various

compliance constraints and stiffness of the spring .............................................. 132

Figure 5.10 Optimized topologies of the compliant gripper with soft materials . 133

Figure 5.11 Design domain and boundary conditions of a 3D compliant gripper134

Figure 5.12 Optimized topologies and CAD models of 3D compliant grippers

with various compliance constraints .................................................................... 135

Figure 6.1 A general design model for hinge-free compliant mechanism ........... 141

Figure 6.2 Flow chart of the BESO procedure for the optimal design of compliant

mechanisms .......................................................................................................... 146

Figure 6.3 Design domain and boundary conditions of designing hinge-free

inverter mechanism .............................................................................................. 147

Figure 6.4 Optimum topologies for the hinge-free inverter mechanism ............. 148

Figure 6.5 Evolution histories of the hinge-free inverter mechanism ................. 149

Figure 6.6 Topology optimization for the hinge-free gripper mechanism ........... 150

Figure 6.7 Topology optimization for the 3D hinge-free gripper mechanism ..... 152

Figure 6.8 Topology optimization for the 3D hinge-free elevation mechanism .. 154

XVI

Figure 6.9 Topology optimization for the 3D hinge-free contraction mechanism155

Figure 7.1 Tangent and secant stiffness ............................................................... 162

Figure 7.2 Newton-Raphson iterative method ..................................................... 168

Figure 7.3 Flow chart of the BESO procedure for the optimal design of

geometrical nonlinear compliant mechanisms ..................................................... 174

Figure 7.4 Evolution history of the topology of designing the compliant inverter

with geometrical nonlinearity .............................................................................. 176

Figure 7.5 Evolution histories of the objective function and constraints for the

compliant inverter with geometrical nonlinearity ................................................ 176

Figure 7.6 Performance of the topology after loading ......................................... 177

Figure 7.7 Optimized topologies and output displacements of the nonlinear

compliant inverter for various compliance constraints ........................................ 178

1

Chapter 1

Introduction

2

Chapter Overview

Compliant mechanisms represent a relatively new breed of lightweight structures.

To address the topology optimization of compliant mechanism, it is necessary to

introduce its fundamental aspects. This chapter first introduces the definition and

classification of compliant mechanisms, and describes their merits and applications

in medical instruments and mechanical devices. Then the motivation and objectives

of this dissertation are presented. Finally, the organization of this dissertation is

presented in this chapter.

1.1 Definition of Compliant Mechanisms

A mechanism is defined as a mechanical device used to transmit motion, force or

energy. The family of mechanisms can be classified into two main groups based on

the type of construction, namely rigid-body mechanism and compliant mechanism.

The traditional rigid-body mechanisms, such as the slider-crank mechanisms

shown in Figure 1.1, consist of rigid links connected at movable joints. They are

designed to obtain mobility exclusively by using rigid links (large stiffness) and

kinematic joints (large compliance) so that an input force can be transformed to an

output torque.

Figure 1.1 A traditional rigid link mechanism

Figure 1.1 A traditional rigid link mechanism

T

θ1 Fi

-θ2

3

However, compliant mechanisms are single-piece flexible structures that transfer

an input force or displacement to another point through elastic deformation (Kota et

al., 2001). They can be designed in different shapes and sizes for various purposes.

As shown in Figure 1.2, compliant mechanisms are different from traditional

mechanisms consisting of rigid links connected with joints. Their mobility is

derived from structural deformation rather than the relative rigid-body motion. An

efficient compliant mechanism should be flexible enough to produce expected

kinematic motion under the action of applied loads for satisfying the flexibility

requirement. Meanwhile, for compliant mechanisms, a certain stiffness is

necessary to resist the applied input force. (Kikuchi et al., 1998; Joo et al., 2000;

Joo and Kota, 2004). Therefore, a compliant mechanism is considered as a

combination of a structure and a mechanism.

Figure 1.2 A compliant mechanism

Figure 1.2 A compliant mechanism

1.2 Classification of Compliant Mechanisms

As noted earlier, mechanisms can be divided into two groups, namely rigid-body

mechanisms and compliant mechanisms. Figure 1.3 illustrates the classification of

compliant mechanisms. Compliant mechanism can be classified into partially

compliant mechanisms and fully compliant mechanisms according to the various

Workpiece

Deformed configuration

Initial configuration

4

rigid and flexural members which make up the devices (Midha et al., 1994;

Cardoso and Fonseca, 2004). Partially compliant mechanisms consist of some rigid

members, traditional joints and compliant members. On the other hand, fully

compliant mechanisms do not contain mechanical joint, and the mobility is

obtained from the elastic deformation of compliant members (Howell, 2001). In

addition, compliant mechanisms can also be categorized as distributed and lumped

compliant mechanisms in terms of the distribution of compliance. More specifically,

the flexibility is concentrated in flexural hinges (or flexural pivots) for lumped

compliant mechanisms, but it is distributed throughout most compliant members

for distributed compliant mechanisms (Sigmund, 1997). This implies that

distributed compliant mechanisms can generally avoid high stress concentrations

as they can utilize most compliant members to store elastic energy.

Figure 1.3 Classification of compliant mechanisms

Figure 1.3 Classification of compliant mechanisms

Compliant Mechanisms

Partially Compliant

Mechanisms

Partially Compliant

Mechanisms Lumped

Compliance

Partially Compliant

Mechanisms Distributed Compliance

Fully Compliant

Mechanisms

Fully Compliant

Mechanisms Lumped

Compliance

Fully Compliant

Mechanisms Distributed Compliance

5

1.3 Advantages of Compliant Mechanisms

Compliant mechanisms can offer a number of benefits over traditional rigid-link

mechanisms due to the elimination or significant reduction of mechanical joints

and the existence of compliant members (Howell, 2001; Bendsøe and Sigmund,

2003). Some advantages are listed as follows:

Assembly procedures: Compliant mechanisms can be designed to be

monolithic or built using fewer parts, which means the need for assembly

procedures can be cut to a minimum (Nishiwaki et al., 1998).

No operation noise: Friction exists in rigid-body joints. Due to the absence or

reduction of mechanical joints, compliant mechanisms could produce less

operation noise when operating (Kobayashi et al., 2009).

Zero backlash: Due to the absence or reduction of mechanical joints,

compliant mechanisms suffer from less backlash. Consequently, high

precision and highly repeatable output motions can be achieved (Shuib et al.,

2007).

Light weight: Compliant mechanisms can be much lighter in weight than

rigid-link mechanisms since a flexible member contains less material than a

counterpart rigid member.

Miniaturization: One of the advantages of compliant mechanisms is the

easiness for miniature of mechanical structural devices, enabling

Micro/Nano-Mechanical Systems (MEMS/NEMS) (Ananthasuresh and

6

Howell, 2005).

Energy savings: Elastic energy is stored in compliant mechanisms after

deformation. The energy can be used to assist in application requiring a return

stage, which allows the controlled release of this energy for springs and

possible actuation.

Cost reduction: The number of parts of compliant mechanisms is largely

reduced in construction. The simplified manufacture reduces assemble costs.

In addition, there is no need for lubrication in operation. As a result,

compliant mechanisms‟ maintenance cost is much lower than their rigid-link

counterparts.

1.4 Applications of Compliant Mechanisms

Unlike rigid-link mechanisms, compliant mechanisms take advantage of the elastic

deformation of their flexible members to produce force, motion or energy

transmission. Numerous favorable features are achieved. Applications of compliant

mechanisms appeared thousands of years ago such as bows and catapults.

Nowadays, compliant mechanisms can be found everywhere in everyday life, such

as in binder clips, nail clippers, staplers and so on (refer to Figure 1.4). In addition,

the application of compliant mechanisms has become increasingly prevalent in

medical instruments and mechanical devices (Kobayashi et al., 2009).

7

Figure 1.4 Compliant devices: a binder clip, scissor and single-piece stapler

Figure 1.4 Compliant devices: a binder clip, scissor and single-piece stapler

Mechanical applications: Some current and potential practical applications

have been presented in aeromechanics, robotics and automation. Shape

morphing aircraft structures based on compliant mechanism technologies can

significantly enhance air vehicle performance (Kota et al., 2003). In addition,

compliant mechanisms are quite suitable for micro electro-mechanical

(MEMS) devices like micro grippers (Lee et al., 2003) and crimping devices

(Kota et al., 2001) owing to minimal or no assembly. Additionally, compliant

mechanisms have been applied for hand-held tools based on single-piece

compliant mechanisms like pliers and shears, manufacturing devices like

grippers, flexible fixtures and flexible robotic manipulators, optical

instruments like mirror adjustment devices and balance devices like

automotive suspension (Allred, 2003).

Medical instruments: Biomedical and biomechanical engineering are quickly

emerging areas of research. Applications of compliant mechanisms have been

extended to these fields such as surgical forceps (Howell, 2001) and medical

knee brace (Erdman et al., 2001). Due to the absence of mechanical joints,

compliant instruments can be cleaned and sterilized more easily. In addition,

the benefits of flex-feet with compliant members over traditional prosthetic

feet are their light weight and compliance.

8

1.5 Motivation of the Dissertation

The main motivation of this dissertation is to design compliant mechanisms using

the bi-directional evolutionary structural optimization (BESO) method. At present,

a large amount of effort has also gone into establishing and developing topology

optimization of compliant mechanisms (Luo and Tong, 2008). However, previous

research has been facing many challenges and shortages in several aspects such as

optimization method, optimization algorithm and resulting topology.

Firstly, to achieve structural design of compliant mechanisms, various topology

optimization methods have been widely exploited and used in this design field.

Nevertheless, previous optimization methods often produce vague topologies of

compliant mechanisms, which can be found in previous literature (Sigmund, 1997;

Nishiwaki et al., 1998; Lau et al., 2001; Bendsøe and Sigmund, 2003). The

modified BESO method proposed in this research can effectively update discrete

design variables to generate clear solutions. In addition, the previous optimization

process often experiences a long convergence history, like the level-set method,

so an efficient optimization tool with high computational efficiency is considered

for the optimal design of compliant mechanisms in this study.

Secondly, input displacement is often considered as a constraint and cited in

previous optimization algorithms. However, its role is not to ensure a certain

amount of flexibility exists in a mechanism, which is used to cause elastic

deformation according to the definition of compliant mechanism. This is not very

suitable for compliant mechanism design to investigate the real physical

significance. Reasonable constraints are quite indispensable to balance flexibility

and rigidity effectively for optimizing compliant mechanisms with linear or

geometrically nonlinear behavior.

9

Thirdly, designs of compliant mechanisms using topology optimization techniques

naturally lead to the introduction of hinges into final topologies, making them

function essentially as rigid-body mechanisms (Rahmatalla and Swan, 2005). The

material around flexural hinges is easily subject to overstress and overstrains.

Therefore, current formulations should be promoted for controlling and

precluding de facto hinge regions to generate 2D and 3D hinge-free numerical

examples.

Fourthly, previous research focuses mainly on the optimal design of linear

compliant mechanisms. In fact, optimal design of compliant mechanisms using

linear analysis is not very accurate as force and motion transmission is often

achieved through large deformation. Geometrically large deformation continuum

models should be incorporated in the optimal design to appropriately capture the

real behavior of large-displacement compliant mechanisms. Furthermore, the

convergence will experience difficulties due to the extreme deformation of

low-density elements (Luo and Tong, 2008). With the help of the hard-kill BESO

method, this systemic design approach will be especially suitable for solving this

problem.

Overall, it is quite necessary to devote efforts to the modification and promotion

of previous topology optimization techniques for constructing practical compliant

mechanism designs. It would also be desirable to use the models exhibited in this

dissertation into practical applications.

1.6 Aims and Objectives

10

This dissertation aims to propose a new BESO method for the topology

optimization of compliant mechanisms. The specific objectives are:

(1) To introduce a new BESO method into the systematic design of compliant

mechanisms.

(2) To establish new computational algorithms for the optimal design of

compliant mechanisms

(3) To incorporate nonlinear analysis procedures of compliant mechanisms into

the topology synthesis method.

(4) To exhibit a number of 2D and 3D examples for verifying proposed

optimization techniques.

(5) To find a suitable technique for industrial applications.

1.7 Organization of the Dissertation

This dissertation consists of eight chapters:

Chapter 1 of the dissertation presents the definition, classification, advantages

merits and application fields of compliant mechanisms. In addition, this

dissertation proposes the motivation to the research. Next, the purpose and

objectives of this study are also introduced. Finally, the outline of this dissertation

is proposed.

11

Chapter 2 first surveys past and present research of designing compliant

mechanisms using kinematic-based approaches and more recent structural

optimization-based approaches such as Solid Isotropic Material with Penalization

(SIMP), the level-set method and other methods. In addition, the chapter

introduces the evolutionary structural optimization (ESO) method. Next, the

chapter reviews the later version of ESO, the BESO method, including its basic

concepts, related theory and its applications in structural optimization.

Chapter 3 aims to propose a revised version of the BESO method for topology

optimization of compliant mechanism because the traditional BESO method is not

suitable to being directly applied to this design field. In this chapter, related

techniques are discussed, such as material distribution theory, the filter scheme,

stability process, element density transformation and convergence criterion.

Numerical examples are shown of its effectiveness and practicability.

Chapter 4 adopts a modified BESO method to produce sets of resulting topologies

of compliant mechanisms, and different objectives are considered to achieve

various functional behaviors. A sensitivity analysis is conducted by applying an

adjoint method. Next, numerical applications are performed to demonstrate the

capability and effectiveness of the developed method for the optimal design of

compliant mechanisms. Finally, this chapter discusses and analyses effects of the

attached springs on the topology optimization of compliant mechanisms.

Chapter 5 uses a new optimization formulation with compliance and volume

constraints, establishing the topology synthesis approach based on a modified

BESO method for designing compliant mechanisms. The new algorithm is

beneficial to control effectively the energy stored in compliant mechanisms after

deformation. Then the research analyzes the flexibility and hinge-related

12

properties through the effective control for the desired structural stiffness.

Numerical applications including 2D and 3D compliant mechanisms demonstrate

the effectiveness of the developed method and reliability of the proposed theory.

Chapter 6 defines a new optimization formulation and establishes the topology

synthesis approach based on the traditional BESO method for designing hinge-free

compliant mechanisms. The input-restrained compliance is addressed to preclude

de facto hinge regions. The sensitivity analysis is conducted by applying the

adjoint method to both the kinematical function and the structural function.

Research provides various numerical applications to demonstrate the practicability

of the developed method and the feasibility of the proposed theory.

Chapter 7 provides solutions for nonlinear compliant mechanism topologies.

Firstly, the chapter introduces structural nonlinearities and reviews solutions of

geometrically nonlinear finite element equations including incremental and

iterative methods. Furthermore, the research extends the idea of designing linear

compliant mechanisms to the optimal design of compliant mechanisms with

geometrically nonlinear behaviors. The work will define a new optimization

formulation and establish a topology synthesis approach using an end-compliance

constraint. Synthesis example is proposed and used to verify the reliability and

effectiveness of the established algorithm. Finally, insights to the topology shape

and performance characteristics are discussed following the numerical experiment.

Chapter 8 summarizes the conclusions and achievements from this research and

presents some suggestions for future research.

References

13

Allred, T. M., 2003. Compliant mechanism Suspensions, Department of Mechanical Engineering,

Brigham Young University.

Ananthasuresh, G. K. and Howell, L. L., 2005, "Mechanical design of compliant microsystems - a

perspective and prospects", Journal of Mechanical Design 127(4), 736-738.

Bendsøe, M. P. and Sigmund, O., 2003. Topology Optimization. Theory, Methods and Applications,

Springer, Berlin.

Cardoso, E. L. and Fonseca, J. S. O., 2004, "Strain energy maximization approach to the design of

fully compliant mechanisms using topology optimization", Latin American Journal of

Solids and Structures 1(3), 263-275.

Erdman, A. G., Sandor, G. N. and Kota, S., 2001. Mechanism Design: Analysis and Synthesis,

Prentice Hall, New Jersey.

Howell, L. L., 2001. Compliant Mechanisms, John Wiley & Sons, New York.

Joo, J. and Kota, S., 2004, "Topological synthesis of compliant mechanisms using nonlinear beam

elements", Mechanics Based Design of Structures and Machines 39(1), 17-38.

Joo, J., Kota, S. and Kikuchi, N., 2000, "Topological synthesis of compliant mechanisms using

linear beam elements", Mechanics of Structures and Machines 28(4), 245-280.

Kikuchi, N., Nishiwaki, S., Fonseca, J. S. O. and Silva, E. C. N., 1998, "Design optimization method

for compliant mechanisms and material microstructure ", Computer Methods in Applied

Mechanics and Engineering 151(3-4), 401-417.

Kobayashi, M., Nishiwaki, S., Izui, K. and Yoshimura, M., 2009, "An innovative design method for

compliant mechanisms combining structural optimisations and designer creativity",

Journal of Engineering Design 20(2), 125-154.

Kota, S., Hetrick, J., Osborn, R., Paul, D., Pendleton, E., Flick, P. and Tilmann, C., 2003, Design and

application of compliant mechanisms for morphing aircraft structures, Smart Structures and

Materials 2003: Industrial and Commercial Applications of Smart Structures Technologies,

SPIE, San Diego.

Kota, S., Joo, J., Li, Z., Rodgers, S. M. and Sniegowski, J., 2001, "Design of compliant mechanisms:

applications to MEMS ", Analog Integrated Circuits and Signal Processing 29(1-2), 7-15.

14

Lau, G. K., Du, H. and Lim, M. K., 2001, "Use of functional specifications as objective functions in

topological optimization of compliant mechanism", Computer Methods in Applied

Mechanics and Engineering 190(34), 4421-4433.

Lee, W. H., Kang, B. H., Oh, Y. S., Staphanou, H., Sanderson, A. C., Skidmore, G. and Ellis, M.,

2003, Micropeg manipulation with a compliant microgripper, IEEE International

Conference on Robotics and Automation, Taiwan.

Luo, Z. and Tong, L., 2008, "A level set method for shape and topology optimization of

large-displacement compliant mechanisms", International Journal for Numerical Methods

in Engineering 76(6), 862-892.

Midha, A., Norton, T. W. and Howell, L. L., 1994, "On the nomenclature, classification, and

abstractions of compliant mechanisms", Journal of Mechanical Design 116(1), 270-279.

Nishiwaki, S., Frecker, M. I., Min, S. and Kikuchi, N., 1998, "Topology optimisation of compliant

mechanisms using the homogenization method", International Journal for Numerical

Methods in Engineering 42(3), 535-559.

Rahmatalla, S. and Swan, C. C., 2005, "Sparse monolithic compliant mechanisms using continuum

structural topology optimization", International Journal for Numerical Methods in

Engineering 62(12), 1579-1605.

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and its current developments in applications: a review", American Journal of Applied

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Mechanics of Structures and Machines 25(4), 493-524.

15

Chapter 2

Literature Review

16

Chapter Overview

In order to design ideal compliant mechanisms, many approaches have been

adopted by engineering researchers. This chapter introduces two main approaches

provided by researchers for the systematic design of compliant mechanisms. One

of the two, the structural optimization-based approach, is more prevalent in the

field of compliant mechanism design. Thus the chapter first describes most design

criteria used in topology optimization of compliant mechanisms, reviewing and

discussing previous research related to the optimal design of compliant

mechanisms using continuum-based methods. Then the chapter describes present

structural optimization approaches, such as homogenization method, the level-set

method and the ESO method. Next, this chapter focuses on the review of the

BESO method, which is the topology optimization method closely related to this

study. Finally, general remarks are presented in this chapter.

2.1 Kinematic-based Methods for Designing Compliant Mechanisms

Many structural design approaches have been applied to the synthesis of compliant

mechanisms over the years. There are two main approaches provided by

researchers in the systematic design of compliant mechanisms, namely the

kinematics-based approach and the structural optimization-based approach

(Howell, 2001). The former is built on traditional rigid-link design techniques, and

the latter seeks optimum compliant structure.

One of the two mentioned approaches, the kinematics-based compliant

mechanism technique, was developed and measured the effects from compliant

members by appropriately simulating mechanism behaviors. The approach has

been successfully applied to the design of lumped compliant mechanisms and

17

mechanisms with cantilever-like flexible links. Related design techniques, like

kinematic chains, Burmester theory and graphical synthesis methods have also

been proposed and modified for designing compliant mechanisms (Hill and Midha,

1990; Howell and Midha, 1994; Mettlach and Midha, 1996).

Specifically, engineering researchers considered lumped compliant segments to be

appropriate substitutes for revolute joints, taking advantage of trial and error

methods to design the dimensions of these compliant hinges. In 1965, Paros and

Weisbord analyzed the behavior of flexural pivots, providing equations for

calculating linear deflections produced by loads on hinge axes of flexure hinges

(Paros and Weisbord, 1965). Then, Burns and Crossley presented type synthesis

methods for flexible link mechanisms (Burns and Crossley, 1966), and proposed an

overlay technique for the design of a four-bar linkage (Burns and Crossley, 1968).

In 1987, Midha and Howell (1987) first explored the research in the field of

compliant mechanism by taking advantage of the kinematics-based synthesis

method. The research defined concepts of compliant mechanism, link compliance

and the degree of compliance concept based on a modified Grubler‟s equation

(Her and Midha, 1987). In addition, Midha et al. presented the classification of

compliant mechanisms based on link and segment characteristics and analyzed

kinematic properties of complaint mechanisms. Researchers then derived several

compliant mechanisms where rigid links were replaced by flexible links, and

discrete compliances were imposed at the joints of rigid-link kinematic chain

(Midha et al., 1992a; Midha et al., 1992b; Midha et al., 1994).

A computationally more efficient procedure, called the chain algorithm, was

presented and formed for the nonlinear deformation analysis of compliant

mechanisms (Hill and Midha, 1990; Midha et al., 1992). Furthermore, to address

18

the function synthesis of designing compliant mechanisms, Her et al. (1992)

promoted the approach by combining the trial and error methods with the chain

algorithm.

Thereafter Salamon and Midha (1992) introduced and analyzed the mechanical

advantage in compliant mechanisms, and demonstrated the resulting mechanical

advantage. These researchers concluded that the change of the mechanical

advantage relies on the spring-rate approximation of a flexible workpiece like a

compliant crimper. Next, in order to get the minimum and maximum number of

inputs for compliant mechanisms, researchers proposed a generalized Grubler‟s

formula (Murphy et al., 1994)

In 1996, Murphy et al. employed the graph theory in the topological synthesis of

compliant mechanisms (Murphy et al., 1996). They permutated non-isomorphic

topologies by introducing flexure pivots and compliant segments into rigid-link

mechanisms. In addition, Ananthasuresh and Howell (1996) estimated degrees of

freedom of compliant mechanisms by creating virtual rigid segments. In order to

provide a visual representation of all possible resulting solutions, Mettlach and

Midha (1996) further developed the graphical technique for the shape and size

design of compliant mechanisms

To design compliant mechanisms, Howell and Midha (1994) proposed an analysis

and synthesis method using a pseudo-rigid-body model. The pseudo-rigid-body

model has made an important progress in simplifying analysis of compliant

mechanisms. It is based on the structure and kinematics of rigid-body mechanism.

Its main idea is to model behaviors of flexible elements by combining rigid-body

components to give equivalent force-deflection characteristics. Specifically, this

kind of kinematics-based approach utilizes the kinematic synthesis of a rigid-link

19

mechanism which consists of rigid parts and joints (Howell and Midha, 1996a). It

can be observed from Figure 2.1 that the topologies of compliant mechanisms are

derived directly from rigid-link mechanisms with torsional springs at joints. In

other words, researchers first turned the rigid-link mechanism into a compliant

mechanism when adding torsional springs and transforming their stiffness into

equivalent flexural hinges. Through a modified kinematic approach, researchers

can then identify the lengths and orientations of these rigid links for the desired

motion features. After that, a number of basic compliant configurations using the

pseudo-rigid-body model have been developed, such as small-length flexural

pivots (Howell and Midha, 1994), initially straight slender beams with tip forces

(Howell and Midha, 1995) and functionally binary pinned-pinned segments in

compliant mechanisms (Edwards et al., 1999).

Figure 2.1 A compliant slider mechanism and its pseudo-rigid-body model

Figure 2.1 A compliant slider mechanism and its pseudo-rigid-body model

Besides, the pseudo-rigid-body model can simplify the large-deflection nonlinear

analysis in compliant mechanism design by using the theory of rigid-body

mechanisms (Yu et al., 2005). Thus the model has been successfully applied to the

design of nonlinear compliant mechanisms. In 1996, large deformation based on

the pseudo-rigid-body model was presented by Howell and Midha (1996b). Based

on the kinematics of rigid-body mechanisms, a developed approach, a

pseudo-rigid-body model of two degrees of freedom, was presented to simulate

Flexible segments

Rigid segments

Torsional springs

Pseudo-rigid-body link

20

both the tip locus and tip deflection angle of large-deflection links for compliant

mechanisms (Feng et al., 2010).

Overall, the kinematics-based design approach is considered as a simplified

method for the analysis and synthesis of compliant mechanism. The

well-established rigid-link synthesis theory has also been developed in the

approach. However, this approach is limited to special simple applications, and

resulting solutions are often simple compliant mechanism with lumped compliance.

In other words, the weakness of the method is that kinematic approximation cannot

adequately model the structure under all conditions. For example, the resulting

designs can be unable to fully reproduce the motion of its rigid-body counterpart

(Luo et al., 2007). In practice, flexural hinges located in stress concentration

regions limit the load-bearing capability and reduce the fatigue life of compliant

mechanisms. In addition, as opposed to resulting solutions based on the

kinematics-based design approach, the continuum model reflects the real artifact

more closely. Therefore, this research cannot transplant and utilize the

kinematics-based design approach for the optimal design of compliant mechanisms.

The structural optimization-based approach as a more prevalent method of

designing compliant mechanisms has been widely advocated by engineering

researchers. Thus, several popular optimization-based methods for designing

compliant mechanisms will be reviewed in the next section.

2.2 Optimization-based Methods for Designing Compliant Mechanisms

At present, the study on compliant mechanism has mainly concentrated on

developing topology optimization techniques to automate topology synthesis of

compliant mechanisms. Unlike the kinematics-based approach starting with an

assumed rigid-link mechanism, the structural optimization-based approach

21

formulates the problem with topology optimization formulations. Therefore,

structural optimization techniques can allow for a more fundamental approach for

the optimal design of compliant mechanisms by integrating principles of

kinematics and structural mechanics. Their application in the compliant mechanism

synthesis will be reviewed and discussed in the following sections.

2.2.1 Design Criteria of Compliant Mechanisms in Optimization-based Methods

It is necessary to first give a brief review of some related concepts, theories and

design criteria applied in topology optimization of compliant mechanisms. To

design compliant mechanisms with distributed compliance in topology

optimization, the primary objective is to consider functional behavior. That means

that the synthesis approach should be able to produce a suitable structural

configuration so that resulting solutions are able to achieve their functional

behaviors (Ananthasuresh, 1994). The following design criteria are usually

addressed and applied in compliant mechanism designs using structural

optimization-based methods at both macro and micro levels: (a) Geometric

advantage (the ratio of output displacement to input displacement); (b) Mechanical

advantage (the ratio of output force to input force); (c) Mechanical efficiency (the

product of geometric advantage and mechanical advantage); (d) Volume and

weight of material; (e) Compliance (work done by external forces); (f) Size and

space constraints; (g) Industrial manufacturability; (h) Acceptable strain levels; (i)

No stress concentration regions; (j) Acceptable fatigue strength; (k) Ergonomics

and aesthetics.

Certainly, it is obvious that it is not possible to quantify all design criteria of the

above performance characteristics for topology optimization of compliant

mechanisms. Thus it is not easy to incorporate all performances into the

22

optimization procedure. Engineering researchers usually consider several criteria

at the stage of topology optimization. Other criteria can be addressed in the

post-synthesis phase and used to refine and compare resulting topologies.

More specifically, constraint on the volume serves as a resource cost in the optimal

synthesis problem. Reducing volume is to economize on the amount of material

and to obtain low cost structures for easy handling. In the optimal design of

compliant mechanisms, it is preferable to have less volume designs for better

functionality with improved flexibility and dynamic characteristics.

For structural compliance, it is relatively easy to deal with its global measures.

This measure index describes the behavior of the overall system unlike local

measures such as stresses and deflections. Compliance can be qualified as mean

compliance and defined as the work done by the external loads. It is an idea

performance to measure the stiffness characteristics in a structural system. It is

universally understood and acknowledged that lower compliance denotes a stiffer

structure. Compliant mechanisms are supposed to be stiff to resist the external

loads by the workpiece, even though the mechanisms should be flexible to generate

motion. Thus, compliance is often considered as an objective in the optimization

algorithm of compliant mechanism design.

For topology optimization of compliant mechanisms, previous design often

considers a relationship of force and displacement at the output port. It is supposed

that a spring with spring stiffness kout is introduced to express the applied force at

the output port by 𝐹𝑜𝑢𝑡 = 𝑘𝑜𝑢𝑡∆𝑜𝑢𝑡 . In designing mechanisms, an important

criterion is the mechanical advantage (MA), which is defined as the ratio of the

output force to the input force. Energy is absorbed through deformation owing to

the elastic deformation in the structure of a compliant mechanism. Unlike

23

rigid-body mechanisms, the general relations for the mechanical advantage of

single-input and single-output port compliant mechanisms are more involved and

can be formulated as 𝑀𝐴 = 𝐹𝑜𝑢𝑡 𝐹𝑖𝑛 = 𝛿∆𝑖𝑛 𝛿∆𝑜𝑢𝑡 − 𝛿𝐸 (𝛿∆𝑜𝑢𝑡𝐹𝑖𝑛)⁄⁄⁄ =

= 𝑀𝑟 −𝑀𝑐 (Salamon and Midha, 1998). 𝛿𝐸 represents the incremental change in

the total strain energy of the compliance mechanism. The term 𝑀𝑟 takes the form

of the mechanical advantage of a rigid-like mechanism. The term 𝑀𝑐 accounts for

the strain energy stored in the mechanism (Wang, 2009). In addition, Larsen et al.

(1997) proposed the concept of geometrical advantage (GA) since the measure of

motion transfer is no longer implied in the concept of mechanical advantage. It can

be defined as the ratio of output displacement to input displacement. It is given by

𝐺𝐴 = ∆𝑜𝑢𝑡 ∆𝑖𝑛⁄ . On the other hand, work ratio, which measures the efficiency of

work transfer, is the ratio of output work to input work, 𝑀𝐸 = 𝑀𝐴 × 𝐺𝐴 (Lau et

al., 2001).

Overall, the above mentioned design criterion are critical to topology optimization

of compliant mechanism. In particular, structural compliance and the relationship

of force and displacement are often applied to optimization formulations in current

structural optimization-based methods like the density-based method and level-set

method. Some other design criteria are also addressed by engineering researchers

to refine and compare resulting topologies (Ananthasuresh et al., 1994).

2.2.2 Optimal Design of Compliant Mechanisms Using Homogenization

Methods

With establishing mentioned design criteria for continuum-based models of

compliant mechanisms, various topology optimization methods have been applied

in this design field. The homogenization method is viewed as the first

continuum-based method used to optimize the distributed compliance of compliant

24

mechanisms.

Anathasuresh et al. originally developed the homogenization method as a structural

optimization technique for the synthesis of compliant mechanisms (Ananthasuresh

et al., 1993; Ananthasuresh et al., 1994). Thereafter Frecker et al. (1997) adopted

the homogenization method for solving topology optimization problems of

compliant mechanisms by introducing a mutual energy concept. In their study, the

formulation was based on a multi-criteria optimization procedure for single output

cases to analyze compliant mechanisms‟ performances and satisfy their different

design requirements. Specifically, these researchers proposed objective functions

according to the different types of combinations of mutual potential energy and

strain energy needed to accomplish the design objectives of maximizing the

former and minimizing the latter simultaneously, namely a weighted linear

combination and a ratio equation. Kikuchi et al. (1998) displayed the design of the

optimum layout of compliant mechanisms by minimizing weight subject to a

prescribed elasticity tensor constraint.

In 1998, Nishiwaki et al. performed an investigation on the Pareto optimality

conditions of the multi-criteria compliant mechanism formulation (Nishiwaki et al.,

1998). The advantage of the proposed approach is that it can determine all Pareto

optimal designs. The optimization algorithm formulates the structural flexibility

based on the previous mutual energy concept. Specifically, the mutual mean

compliances of a structure were interpreted according to an elastic body subjected

to two different tractions. Then the formulation is to maximize the mutual mean

compliance for sufficient flexibility of the structure. In addition, the researchers

considered that both kinematic function and structural function were required in

the compliant mechanism design. In order to resist reaction forces from a

workpiece and maintain structural shape, the sufficient rigidity should be obtained

25

by considering minimizing the mean compliance from the design domain, the

input port of which is fixed.

Typical examples can be found in Figure 2.2, where these researchers proposed

the optimal topology configurations for a compliant gripper with different

material volumes. These bear significance to the optimal design compliant

mechanisms, but some shortcomings of the resulting solutions are immediately

apparent. For examples, the optimization technique generates vague topologies

since the intermediate densities of elements are not polarized very well. In

addition, the total volume constraint significantly affects the optimal shape. It can

also be observed from Figure 2.2 that resulting topologies are not really similar

for using different volume constraints and the problem with respect to structural

connectivity exists in the resulting topology with lower volume material.

Figure 2.2 Optimal configurations of compliant gripper with different total volume constraints using the homogenization method

Figure 2.2 Optimal configurations of compliant gripper with different total

volume constraints using the homogenization method (Nishiwaki et al., 1998)

Although the homogenization method first exploited topology optimization

compliant mechanisms and it has a great significance in optimizing compliant

mechanisms using continuum-based models, this dissertation will not apply the

homogenization method to the optimal design of compliant mechanisms. The

research should seek and develop an optimization approach, which can produce

clear and stable final designs of compliant mechanisms.

26

2.2.3 Optimal Design of Compliant Mechanisms Using Solid Isotropic Material

with Penalization

The Solid Isotropic Material with Penalization (SIMP) method has been

developed for solving compliant mechanism design problem. Sigmund (1997)

described this density-based method for the optimal design of compliant

mechanism. In Sigmund‟s work, a sequential linear programming approach is

introduced in the optimization algorithm. The objective is to maximize the

mechanical advantage of a compliant mechanism subject to a limited material

volume and total input constraint. There are different numerical examples

optimized and exhibited by Sigmund to demonstrate the influence of workpiece

stiffness and size on the optimal mechanism topology. Figure 2.3 displays the

optimal configurations of compliant grippers with one output and two outputs

using this density-based method. When compared with resulting topologies

produced by the homogenization method, solutions are more complex and reflect

the real artifact more closely. However, some features, such as vague resulting

designs and one-node connected hinges, are unexpected in industrial

manufacturing.

Figure 2.3 Optimal configurations of compliant gripper with one output and two outputs using the density-based method

Figure 2.3 Optimal configurations of compliant gripper with one output and two

27

outputs using the density-based method (Sigmund, 1997)

Larsen et al. (1997) also proposed a density-based way to design the topology and

shape of compliant mechanisms by specifying the elastic properties of materials.

The researchers used a least squared formulation, which allows designers to create

mechanisms with a minimum error between the measured and required

mechanical advantage. Despite satisfactory kinematic requirements, resulting

topologies exhibited thin flexure-like components.

To solve the topological design of compliant mechanisms, Lau et al. (2001)

suggested the optimization method on the basis of the combination of the moving

asymptotes technique and the SIMP model without the filtering scheme. Luo et al.

(2005) formulated the optimization problem for compliant mechanism designs

using the rational approximation of material properties density, which is opposed to

the SIMP model. A set of optimal results of compliant mechanisms are exhibited

to illustrate that the optimization algorithm using either the density or sensitivity

filter scheme is efficient at eliminating checkerboards, but one-node connected

hinges can only be prevented for a small output export. Finally, the researchers

employed a hybrid-filtering scheme, and concluded that no one-node connected

hinges appear in final designs. However, in fact, one-node connected hinges can

only be prevented to some degree, because there are still obvious hinge regions in

resulting solutions.

In addition, Lin et al. (2010) proposed a method for implementing multi-objective

optimization of compliant mechanisms on the basis of the combination of the SIMP

model and the physical programming. The proposed framework aims to bring

flexibility, robustness and adaptability to multi-objective optimization. A set of

resulting designs were presented to demonstrate the effect of multi-objective

28

optimization on topologies in compliant mechanism design.

To capture the natural behavior of nonlinear compliant mechanisms, the SIMP

model and related density-based methods have been advocated and extended to

deal with synthesis of large-displacement compliant mechanisms. Bruns and

Tortorelli (1998) began to address geometrical nonlinearity for topology synthesis

for structure design. In 2001, they used the method of moving asymptotes to solve

the topology optimization problem of nonlinear compliant mechanisms (Bruns and

Tortorelli, 2001). Furthermore, Pedersen et al. (2001) incorporated geometrical

nonlinearity in topology optimization of compliant mechanism design based on

the variable density method. These authors first solved the nonlinear

force-displacement curve and path generation problems. Then they proposed a

topology design formulation of large-displacement compliant mechanisms based

on a multi-criteria objective function, describing the derivation of sensitivity of

output displacement in relation to the design variables. In addition, Yoon and Kim

(2005) described topology optimization of geometrically nonlinear structures,

including compliant mechanisms, using the density-based method. In order to

avoid the numerical instability caused by the SIMP formulation, these researchers

reformulated the optimization problem as seeking the optimal inter-element

connectivity distribution by element connectivity parameterization. In 2008, Du et

al. (2008) employed a topology optimization of geometrical nonlinear complaint

mechanisms using the element-free Galerkin method, in which design domain

was discretized by nodes rather than elements. Then the SIMP scheme is

addressed to represent the nonlinear dependence between material properties and

regularized discrete densities. The mathematical model of a compliant mechanism

was expressed as maximizing its output displacement.

To sum up, previous engineering researchers have taken advantage of the SIMP

29

method to improve topology optimization of compliant mechanisms. The

techniques have been extended to solve more complicated problems of designing

compliant mechanisms and are capable of producing relatively complex designs.

However, vague resulting topologies also frequently appear in previous work

when considering the SIMP method.

2.2.4 Optimal Design of Compliant Mechanisms Using Level-set Method

Most recently, the level-set method has been presented to perform shape and

topology optimization of elastic compliant mechanism. For example, Wang et al.

(2005) proposed the level-set method for the optimal design of monolithic

compliant mechanisms with multiple materials. A multiphase model adopted in

their research aims to specify material regions and sharp interfaces of single-input

and single-output multi-material mechanisms.

In addition, Luo et al. (2007) employed the parameterization level set technique for

the shape and topology optimization using a compactly supported radial basis

function. In contrast to conventional level-set method, authors discretized the

Hamilton-Jacobi equation into a set of algebraic equations so that the initial

topology optimization could be considered as a parameterization problem. The

design problem of compliant mechanisms was then formulated as maximizing

mechanical efficiency. Finally, authors proposed the topology optimization of a

compliant inverter. Related level-set surfaces are given in Figure 2.4. The design

domain initialized with a number of holes. The mechanical efficiency increased

during the first 70 iterations. The final design was achieved at Iteration 322 as

shown in Figure 2.4c.

30

Figure 2.4 Evolution of the level-set surface

Figure 2.4 Evolution of the level-set surface (a) initial surface; (b) intermediate

surface; (c) final surface (Luo et al., 2007)

Subsequently, Jouve and Mechkour (2008) added two significant extensions into

the basic level-set method for the optimal design of compliant mechanisms, namely

new cost functions in variational form and multi-load cases. The proposed method

is capable of handling multiple load cases in the optimal design of compliant

mechanisms. Next, Wang (2009) adopted the level-set method to investigate the

structural analysis of linear elasticity for topology optimization of compliant

mechanisms. The author took advantage of conventional formulations to reveal

the relationship between input and output displacements. Then author presented

the global stiffness matrix for a mechanism structure by adding external springs

into the formulations, and explained this stiffness matrix should not be singular to

ensure that the elastic system of the mechanism is better conditioned. Besides, Zhu

and Zhang (2012) presented new objective functions by taking two types of mean

compliances into consideration in the optimal design of compliant mechanisms.

Researchers took advantage of extra energy to make the level-set function close to

a signed distance function, so that the evolutionary process can be accelerated by

eliminating the re-initialization procedure in the new level-set method.

Some other techniques were also introduced into the level-set approach for

designing compliant mechanisms. For examples, a moving boundary

representation was used to describe the structural boundary of mechanisms (Luo

31

et al., 2008). The element connectivity parameterization method was integrated in

a level-set framework of designing compliant mechanisms (Dijk et al., 2010). The

phase field method was taken into consideration in optimizing shape and topology

of compliant mechanisms based on the level-set method (Takezawa et al., 2010;

Yamada et al., 2010). The method of moving asymptotes was applied to update

the level-set function for designing compliant mechanisms, considering a mutual

mean compliance constraint and a stress constraint (Otomori et al., 2011).

In addition, several researchers attempted to utilize the level-set method to

eliminate de facto hinges in compliant mechanisms because topology

optimization of hinge-free compliant mechanisms has received more attention and

has undergone considerable developments in recent years. For example, Luo et al.

(2008) took advantage of a semi-implicit scheme on the basis of an additive

operator-splitting algorithm to solve the Hamilton-Jacobi partial differential

equation in the level-set method. Then a quadratic energy functional derived

mainly from image active contour technique was introduced into the optimization

algorithm. It can generate hinge-free compliant mechanisms by controlling

structural shape features. To reduce stress concentration and fatigue breakage, Zhu

et al. (2013) employed the weighted sum method to integrate flexibility and two

kinds of mean compliances into the formulation for the design of hinge-free

compliant mechanisms. The proposed weighting factors are determined by a

self-adjusting scheme and updated during each evolutionary step. The shape

derivative and gradient method is then used to produce final topologies of

compliant mechanisms. Figure 2.5 comes up with an optimal configuration from

the mentioned study. It can be observed that the formation of hinges can be

successfully precluded due to the proposed algorithm. To sum up, researchers have

successfully exploited various techniques for designing hinge-free compliant

mechanisms based on the level-set method. However, optimization methods of

32

previous works have not proven the viability and feasibility of designing 3D

hinge-free compliant mechanisms.

Figure 2.5 An optimal configuration of hinge-free compliant using the level-set method

Figure 2.5 An optimal configuration of hinge-free compliant using the level-set

method (Zhu et al., 2013)

Furthermore, the level-set method has also been used to capture the topology of

large displacements of compliant mechanisms. For example, Luo and Tong (2008)

extended the parameterization level set technique to the structural shape and

topology optimization of compliant mechanisms with geometrically nonlinear

behavior. The authors took advantage of one weak material to fill the void areas

based on the ersatz material scheme during the evolutionary procedure. To reduce

the effects of the scheme on geometrically nonlinear designs and overcome the

convergence difficulty resulting from the indefiniteness of the tangent stiffness

matrix, authors tended to relax this criterion by using the strategy derived from

previous research (Pedersen et al., 2001; Sigmund, 2001). In other words, the

iterative process ends until both the change of nodal displacements and the

residual are smaller than a certain value. A benchmark example of compliant

inverter mechanism is shown to illustrate the features of the proposed level-set

method. Figure 2.6 depicts the final designs of compliant inverter with

33

geometrically nonlinear behavior when using different stiffness values of artificial

springs. Resulting solutions are very similar when comparing final designs with

linear and geometrically nonlinear analysis. Meanwhile, authors investigated the

influence on final results from different mesh size in initial design, presenting

identical solutions.

Figure 2.6 Optimal configurations of nonlinear compliant inverter with different spring stiffness using the level-set method

Figure 2.6 Optimal configurations of nonlinear compliant inverter with different

spring stiffness using the level-set method (Luo and Tong, 2008).

Within their work, authors have proposed a stable optimization algorithm and

demonstrated the capability of the current method for designing

large-displacement compliant mechanisms. However, it should be pointed out that

the current optimization technique cannot ensure elimination of the de facto hinge

regions. In addition, more complex topologies had not been produced when using

different constraint values. The mentioned optimization method cannot be viewed

as a high computational efficient tool as the optimization procedure often

experiences quite a long evolutionary history.

All in all, based on the level-set method, engineering researchers succeeded in

optimizing various compliant mechanism designs including hinge-free and

geometrically nonlinear compliant mechanisms. However, a high-efficient

34

optimization method is advised for both 2D and 3D hinge-free compliant

mechanism designs. Reasonable constraints are also indispensable to balance

effectively flexibility and stiffness for complex topologies of compliant

mechanisms with linear or geometrically nonlinear behavior.

2.2.5 Optimal Design of Compliant Mechanisms Using Other Methods

With the improvement in computational capability, discrete methods have been

applied in the design of compliant mechanisms. Saxena (2002) took advantage of

the genetic algorithm to lay out the optimal material distribution in a solid

continuum and obtained compliant mechanism with multiple materials. Parsons

and Canfield (2002) introduced a multi-objective scheme in the genetic algorithm

for a frame-element ground structure to search for the optimal compliant

mechanism topology. Saxena (2005) employed the barrier assignment of design

variables for multiple materials and used the non-dominated sorting in genetic

algorithm for the optimal design of compliant mechanisms.

In addition, the ground structure method has been employed for optimal truss

layout design (Prager, 1970; Prager, 1977), and has been extended to compliant

mechanism synthesis. Sigmund (1994) parameterized the design domain based on

a truss ground structure from Bendsøe‟s work, in which the optimal microstructures

have mechanism-type motions subject to loads differing from the design load. A set

of grid nodes are firstly used to discretize the design domain. Then the optimization

technique connects each grid node with every other node by truss or frame to create

a fully grounded structure. In fact, ground structure is typically used as a general

term including both fully and partially grounded structures. Both truss and frame

ground structures have also been extended to the topology synthesis of compliant

mechanisms with using the element‟s cross sections as the design variables.

35

However, behaviors of compliant mechanisms can be more accurately captured by

frame elements, as the bending of beam-like segments can produce better the

mobility than the lengthening and shortening of trusses.

Material removal strategies have been adopted in the structural optimization

(Rodriguez- Velazquez and Seireg, 1985; Xie and Steven, 1993). For example, a

modified evolutionary structural optimization (Additive ESO) procedure with an

additive strategy was proposed by Ansola et al. (2007) , and it was applied to 3D

compliant mechanisms (Ansola et al., 2010) and thermal effects (Ansola et al.,

2010). In addition, Veguería et al. (2008) proposed a new objective function

combined with Additive ESO for the optimal design of compliant mechanisms with

spring models.

Overall, the selection of an optimization method is vitally important for the

topology optimization of complaint mechanisms. Thus the current structural

optimization techniques will be described in the next sections. In particular,

efficient design tools, ESO and BESO, will be reviewed in this dissertation as they

are closely related to this research.

2.3 Topology Optimization Methods

The use of topology optimization methods is critical to compliant mechanism

design. Over the last three decades, researchers have made great efforts to develop

structural optimization techniques. Analytical and numerical techniques were

exploited to solve a multitude of structural design problems. The original structural

optimization technique can be traced back to Galileo who obtained the optimum

shapes of variable depth beams by designing solids of equal resistance. Barnett

(1966) and Hemp (1973) described the highlights of the history of this subject from

36

Galileo to the 1960s.

Generally speaking, the present structural optimization techniques concentrated

mainly on the idea of obtaining the stiffest or strongest structural optimums given a

fixed volume or weight limitation. Related methodologies were mainly approached

using parametric geometry-based and material distribution problems. For the

former, parametric geometry-based design techniques are used for the design of the

shape and size or boundary conditions for a structure. The techniques are often

required when taking into account local constraints like stress constraints. For

material distribution problems, they aim to seek the optimal location of material

within a design domain. Related approaches are primarily employed for solving

topology or layout synthesis problems and offer the largest potential for increases

in structural performance.

Originally, Topping (1983), Levy and Lev (1987) proposed a comprehensive

reviews of previous techniques. The previous theory was then developed to

perform topology optimization of discrete structural systems and examined the

optimal layout structure. In 1904, Michell presented a developed analytical

technique for layout design of frame structure (Michell, 1904). So far, various

structural topology optimization methods, e.g. homogenization method (Bendsøe

and Kikuchi, 1988), Solid Isotropic Material with Penalization (SIMP) (Zhou and

Rozvany, 1991; Rozvany et al., 1995), level-set method (Sethian and Wiegmann,

2000; Wang et al., 2003; Allaire et al., 2004), genetic algorithm method (Chapman

et al., 1994; Chapman and Jakiela, 1996), Evolutionary Structural Optimization

(ESO) (Xie and Steven, 1993; Xie and Steven, 1997) and Bidirectional

Evolutionary Structural Optimization (BESO) (Huang and Xie, 2007; Huang and

Xie, 2009; Huang and Xie, 2010) have been developed and improved mainly for

finding the lightest, stiffest and strongest structures.

37

For the homogenization method, Cheng and Olhoff (1981) first discussed the

mathematical formulation for the optimal design of plates based on the thin plate

theory. They concluded that a global optimal solution does not generally exist

within both the class of smooth functions and the class of smooth functions with a

finite number of discontinuities. Consequently, a series of works on optimal design

problems were solved by using a microstructure. Kohn and Strang (1986) presented

the concept of relaxation for the ill-posed variational problem of formulating for

the optimal design. Rozvany et al. (1987) then investigated the implication related

to the relaxation concept with the design of perforated elastic plates. In 1988,

Bendsøe and Kikuchi proposed the homogenization method for topology

optimization and created a paradigm shift in this field (Bendsøe and Kikuchi,

1988).

Unlike the homogenization approach utilizing the sizes and orientation of the micro

void, the SIMP approach uses the material density to describe each element. From

Figure 2.7, it can be seen that the SIMP approach employs intermediate densities

for each element. The material density is penalized in order to discourage the

formation of intermediate densities in structural optimization. Bendsøe (1989) first

considered the SIMP approach, in which an artificial density was introduced for

predicting the topology of a mechanical element. Zhou and Rozvany (1991)

developed the SIMP approach independently and illustrated it with examples.

Bendsøe and Sigmund (1999) analyzed and compared the SIMP and similar

material models in the effective properties of composite materials. Their analysis

proved that the power-law approach is considered to be physically permissible as

long as simple conditions on the power are satisfied.

38

Figure 2.7 Solid continua parameterizations

Figure 2.7 Solid continua parameterizations

The level-set method as a computational technique has been developed and applied

to structural shape and topology optimization. Sethian and Wiegmann (2000) first

introduced the level-set method into the structural optimization on a fixed Eulerian

grid. The method can track the motion of structural boundaries and handle the

presence of potential topological changes. Figure 2.8 illustrates its important ideas.

The left figure displays the topology shape of a bounded region. From the right

figure, the surface is the graph of a level-set function used to determine the left

shape. So far, two main categories based on the level-set method have been

developed in structural optimization. One category takes advantage of explicit

schemes to calculate the Hamilton-Jacobi partial differential equation when

advancing the implicitly represented design boundary. In 2002, Allaire et al.

integrated the classical shape gradient into the level-set algorithm to handle

topological changes (Allaire et al., 2002). The method can be applied to linear or

nonlinear models in 2D and 3D (Allaire et al., 2004). The other category, called the

parameterization level set method, does not need to solve the Hamilton-Jacobi

partial differential equation using explicit schemes (Belytschko et al., 2003; Haber,

2004; Wang and Wang, 2006).

39

Figure 2.8 An illustration of level-set method

Figure 2.8 An illustration of level-set method

In addition, the presence or absence status for each element is a discrete variable

after the design domain is discretized into finite elements. Discrete optimization

approaches adopt discrete optimization algorithms to solve discrete optimization

problems. There are two used methods, namely the genetic algorithm (Holland,

1975; Goldberg, 1989) and simulated annealing (Metropolis et al., 1953). Chapman

et al. (1994) employed the former to optimize the topology of a cantilever beam

with minimum deflection under an end load. Shim and Manoochehri (1997)

adopted the latter to generate optimal configuration in structural design. These two

methods have been utilized to seek the optimal topology in a truss ground structure

(Dhingra and Bennage, 1995; Hajela and Lee, 1995; Ohsaki, 1995; Rajan, 1995;

Topping et al., 1996).

During the last few decades, the ESO method has emerged as one of the most

popular techniques in the field (Xie and Steven, 1993). The ESO method was first

introduced by Xie and Steven in the early 1990s (Xie and Steven, 1992). Initially,

the method was to seek solutions to fully stressed design problems. Thereafter, the

ESO technique had been consistently improved and extended into a wide range of

topology optimization designs, such as stiffness constraints (Chu et al., 1996) and

frequency problems (Xie and Steven, 1996; Zhao et al., 1997). There are two main

element-removing criteria applied in the ESO approach, namely the stress criterion

and sensitivity number.

40

Stress Criterion: this method is based on a simple concept that inefficient material

is gradually removed from the design domain so that the resulting topology evolves

towards an optimum. Initially, the method was to seek to solutions of fully stressed

design problems (Xie and Steven, 1993). The stress was chosen as the element

removing criteria. Elements with small von Mises stress were seen as inefficient

material to be removed. Two types of loops, inner loop and outer loop, are

incorporated in the procedure. In the loop, the procedure aims to measure the stress

level of each element after a finite element analysis. The level can be determined by

using a ratio, for example, the ratio of the von Mises stress of the element 𝜎𝑒𝑣𝑀 to

maximum von Mises stress from the whole structure 𝜎𝑚𝑎𝑥𝑣𝑀 . The indicator, the

rejection ratio (RR), is selected as the threshold stress. If the stress level of one

element is lower than the threshold stress, such as 𝜎𝑒𝑣𝑀 𝜎𝑚𝑎𝑥

𝑣𝑀 <⁄ 𝑅𝑅𝑖 during the ith

iteration step, the element will be removed for the structure. The strategy of

removing material with inefficient use is based on the idea that the stress in every

part in the structure should be close to the same and safe level. The procedure in the

outer loop is used to consider the coming rejection ratio, which can be formulated

as 𝑅𝑅𝑖+1 = 𝑅𝑅𝑖 + 𝐸𝑅 with the evolutionary ratio ER. The loop action aims to

seek the stress of quasi-uniform distributions. In 1996, this method was extended to

the optimization of structures with material and geometric nonlinearities (Querin et

al., 1996). One of these variations for the stress-based ESO technique is Nibbling

ESO (Xie and Steven, 1997). It considers only whether the elements from the

structural boundary should be removed or not instead of ones from inside the

structure domain. The ESO method was also adopted to solve thermal stress

optimization problems under thermal loadings (Li et al., 1997), and optimal contact

shape designs for elastic bodies under the multiple load cases (Li et al., 2005). This

research exhibits the resulting ESO topologies related to the stress design in

Figures 2.9.

41

Figure 2.9 Design domain and its ESO topology with RR=25% for a Michel-type structure optimization with von Mises stress criterion

Figure 2.9 Design domain and its ESO topology with RR=25% for a Michel-type

structure optimization with von Mises stress criterion (Xie and Steven, 1993)

Sensitivity Number: the significant improvement of the classical ESO method is

that the sensitivity number is adopted and regarded as the element-removing

criterion. Elements will be removed on the basis of the value of the element

sensitivity number so that the algorithm can well capture the change in the

objective function or constraint for each evolutionary procedure. Elements with

lower sensitivity number from finite element analysis will be removed to drive

structures towards optimums. Various design problems can be solved by defining

specific objective function or constraint. For example, the ESO approach was

proposed to solve the stiffness design problem, having the capability of designing

structures with multiple displacement constraints, multiple load cases and moving

loadings (Chu et al., 1996). Frequency optimization designs were presented based

on the ESO model (Xie and Steven, 1996).

Overall, the ESO method starts with the full design. Resulting topologies can be

achieved through the removal of inefficient material such as lower stress or strain

energy levels of elements. 3D design problems can also be solved using the ESO

technique, though this dissertation only exhibits 2D resulting designs. However,

several drawbacks are presented by researchers, as follows: Firstly, the removed

H

2H

42

elements are unable to be re-introduced into design domains so that reasonable

solutions cannot be obtained in some cases. This is because the material that has

been permanently deleted in previous iterations could be necessary to be part of the

optimal design of the current iteration (Huang and Xie, 2010). Secondly, the

topology may change dramatically when compared with the optimal design from

the last iteration (Rozvany, 2001). Important numerical problems related to

topology optimization were being neglected in the traditional ESO method such as

checker-board and mesh-dependency (Huang and Xie, 2010). In addition, the ESO

technique is not suitable for compliant mechanism optimization (Veguería et al.,

2008).

At present, a new development in ESO has been proposed by researchers. The

additive evolutionary strategy was introduced into the method. The improved

version of ESO was named bi-directional ESO (BESO) (Querin et al., 1998). The

method aims to overcome the deficiencies that exist in the ESO method (Huang and

Xie, 2007). In this dissertation, the research employed the BESO method and its

modified method for designing compliant mechanisms. Therefore, the BESO

method will be described and discussed in detail in the next section.

2.4 Bi-directional Topology Optimization Method

2.4.1 The Early Bi-directional Evolutionary Structural Optimization Method

As later version of the ESO method, the early BESO method, not only removes

inefficient elements from structures but also recovers the removed elements in

structures. This BESO procedure starts with a minimum ground structure and

evolves towards a fully stressed structure via a stress-based optimality criterion.

Compared to the ESO method, the early version of the BESO method is more

43

robust and efficient in structural optimization (Yang et al., 2002).

In 1999, this approach was extended to stiffness and displacement constraints

(Yang et al., 1999). In their work, a linear extrapolation of the displacement field

was used to estimate the sensitivity of the void elements. The parameters, the

rejection ratio (RR) and inclusion ratio (IR), were considered as the element

removal/addition criterion. That is to say, after finite element analysis, the solid

elements with the lower von Mises stress levels are removed from the structure, and

the void elements with the higher von Mises stress levels can be changed into solid

elements. This initial BESO method has been employed to solve the frequency

optimization problem (Yang et al., 1998). The same principle has also been

adopted in the topology optimization of fully stressed designs (Querin et al., 2000).

In addition, Yang et al. (2002) and Li et al. (2001) introduced the perimeter control

and a smoothing algorithm into evolutionary structural optimization respectively.

Design area and its BESO topology for a Michel-type structure optimization are

depicted in Figure 2.10. Final material volume can be smaller than the volume of

the initial guess design. The optimization algorithm can also reach mesh

independent solutions and improve the numerical stability. In 2007, further work

on the BESO method was presented by Zhu et al. (2007). The element replaceable

approach was employed in BESO. Solid elements are replaced by orthotropic

cellular microstructure elements when they are removed from structures. It means

that the BESO procedure can directly calculate the sensitivity number of void

elements. This version of BESO should be regarded as a microstructure approach.

44

Figure 2.10 Design area and its ESO topology with the performance index PI=0.01211 and perimeter limit P*=193L for a Michel-type structure optimization

Figure 2.10 Design area and its BESO topology with the performance index

PI=0.01211 and perimeter limit P*=193L for a Michel-type structure optimization

(Yang et al., 2002).

However, the above-mentioned version of BESO was not appropriate for obtaining

optimal solutions, especially the early BESO method (Rozvany, 2001; Zhou and

Rozvany, 2001; Rozvany, 2009). This dissertation summarizes some deficiencies

as follows:

1. Computational Efficiency: the early version of the BESO method is regarded

as a low efficient algorithm. The evolutionary procedure tackles element

removal and addition separately. Consequently, resulting designs are

produced through a number of iterations.

2. Accuracy: the algorithm cannot guarantee the accuracy of topology

optimization. It is inaccurate to measure the sensitivity number of void

elements so that the method cannot accurately estimate the change in the

objective function. As a result, researchers have to select better final designs

from a lot of resulting topologies.

3. Convergence: the element removal/addition strategy relies mainly on the

empirical idea. There is a lack of the reliable math explanation that leads the

L

26L 3L 3L

Initial Design

Design Area

45

evolutionary procedure towards an optimum. Researchers have to determine

parameters RR and IR carefully. It often results in a chaotic convergence

history.

4. Checker-board Pattern: it should be noted that the checker-board pattern is

regarded as an undesirable feature in the optimal design as it leads to final

topologies that are not appreciated in the practical design (Sigmund and

Petersson, 1998; Bendsøe and Sigmund, 2003). The phenomenon results from

the mesh-dependence. In the early BESO with 0/1 optimization technique,

dense holes appears in topologies when a finer finite element mesh is

employed.

2.4.2 Introduction to the Current Bi-directional Evolutionary Structural

Optimization Method

As a new development in BESO, an advanced version of the BESO method has

been proposed (Huang and Xie, 2007). This current BESO method still takes

advantage of discrete design variables to solve optimization problems. After

removing individual elements, the optimization method will utilize the

approximate variation of objective function to estimate the sensitivity numbers for

elements. The optimal topology can be obtained according to the relative ranking

of sensitivity numbers.

For the current BESO method, some new techniques have been introduced into its

optimization procedure. For example, material can be removed and added

simultaneously in the design domain. Void elements are replaced by soft elements

by utilizing the material interpolation scheme (Huang and Xie, 2009).

46

In addition, the filter scheme was established as an effective mechanism for

extrapolating sensitivity numbers in the current BESO method (Huang and Xie,

2007). The so-called checkerboard pattern is the area where the density jumps

frequently from 0 (or xmin) to 1 between neighboring elements in a resulting

topology. Figure 2.11 is showing the typical checkerboard regions in a resulting

topology. It results from unreasonable numerical modeling (Díaz and Sigmund,

1995; Jog and Haber, 1996). The feature is also unexpected in industrial

manufacturing. Next, the mesh-dependency means that result solution varies

according to different finite element meshes. In fact, the finer the structure is

meshed, the better finite element modeling should be obtained (Sigmund and

Petersson, 1998).

Figure 2.11 Typical checkerboard regions in an optimum solution

Figure 2.11 Typical checkerboard regions in an optimum solution (Kim et al.,

2002)

In the current BESO method, the optimization technique utilizes a filter scheme to

prevent unstable phenomena such as checkerboard and mesh-dependency in

structural topology optimization (Huang and Xie, 2007). Element density can be

modified and becomes a function of its neighboring design variables. There are

several advantages for the current filter. Firstly, the technique can effectively

overcome numerical problems in topology optimization such as checkerboard and

mesh-dependency. Its effectiveness has been demonstrated in Figure 2.12 when

47

taking stiffness optimization for instance. Figure 2.12a shows the design domain,

in which a cantilever beam is loaded by a concentrated force. Figures 2.12b and

2.12c depict corresponding solutions using different mesh sizes. It can be

observed that the resulting topology does not rely on the finite element mesh, and

there is no any checkerboard region in resulting solutions. Secondly, the scheme

uses less computational time. Thirdly, it is easy to apply this filter scheme to

optimization algorithms (Huang and Xie, 2010).

Figure 2.12 Design domain and its BESO topologies with 32×20 and 160×100 meshes

Figure 2.12 Design domain and its BESO topologies with 32×20 and 160×100

meshes (Huang and Xie, 2007)

Furthermore, in void-solid or soft-solid design, the evolutionary procedure

calculates the sensitivity number of elements at each iteration. However, elements

may frequently switch between void (or soft) and solid status, which can result in

unstable evolution process. In order to stabilize the evolution process, the current

BESO method will further modify elemental sensitivity by averaging it with its

historical information. Figure 2.13 is showing the stable evolutionary history of

5H

8H

48

stiffness optimization for the above-mentioned cantilever while considering the

evolutionary stabilization. Both topology shape and structural compliance are

very stable after reaching the prescribed volume.

Figure 2.13 The evolutionary history of stiffness optimization for a cantilever

Figure 2.13 The evolutionary history of stiffness optimization for a cantilever

(Huang and Xie, 2007)

2.4.3 Hard-kill Based and Soft-kill Based Bi-directional Evolutionary Structural

Optimization Method

In topology optimization, the element should be seen as the design variable 𝑒

during the evolution procedure. There are two main types of approaches in the

current BESO method, namely the hard-kill and soft-kill BESO. For the hard-kill

based BESO, binary values 0 and 1 were designated to be the absence and presence

of an element respectively. That is, the hard-kill based BESO method can

completely remove ineffective elements in the design domain as opposed to the

density-based method. The method can reduce the computational time

considerably. This is because the removed elements will not be taken into

49

consideration in the finite element analysis, especially a great deal of removed

elements during the late period of the evolution process. However, it is possible

that really removing elements can result in inaccurate sensitivity calculation for

void elements.

Therefore, the soft elements with very low density were used to replace the void

elements in a Sequential Element Rejection and Admission method (Rozvany,

2009). As a result, the thickness problem appears in topology optimization. In 2009,

Huang and Xie proposed a soft-kill BESO method utilizing the material

interpolation scheme with penalization (Huang and Xie, 2009). In this method,

design variable 𝑒 is seen as the relative density of an element. Normally, a small

value of 𝑚𝑖𝑛 e.g. 0.001 is used to denote the void elements in the material

interpolation scheme. That means that the design variable 𝑒 change from 1 to

𝑚𝑖𝑛 for elements with lower sensitivity and from 𝑚𝑖𝑛 to 1 for elements with a

higher one during the evolution process. Elements with low sensitivity numbers

are not really removed from the design domain. This is the explicit characteristic

that distinguishes the soft-kill based BESO from the hard-kill based BESO.

In a word, the hard-kill BESO method can be seen as a special case of the soft-kill

BESO method (Huang and Xie, 2010). As low-density material can be really

removed in design domain, this optimization method is beneficial to overcome the

convergence difficulty caused by extreme deformation of low-density elements in

the large-displacement finite element analysis. Therefore, this dissertation will

present an optimization approach for designing geometrically nonlinear compliant

mechanisms with the aid of the hard-kill BESO method.

2.4.4 Applications of the Current Bi-directional Evolutionary Structural

Optimization Method

50

This advanced BESO method has been applied to a wide range of topology

optimization designs. In 2007, Huang and Xie took advantage of this new

approach to solve several stiffness designs (Huang and Xie, 2007). The authors

provided some benchmark examples to demonstrate the effectiveness of the new

BESO method including 2D beams and a 3D bridge, showing the capability of

this method to obtain checkerboard-free and mesh-independent solutions.

After that, this BESO method has been applied to more complex stiffness

optimization designs. For instance, Huang and Xie (2008a) proposed topology

optimization of geometrically and materially nonlinear structures under

displacement loading, in which the optimization problem was considered as

maximizing the structural stiffness subject to prescribed deformation. In addition,

Huang and Xie (2009) exhibited stiffness optimization of continuum structures

using multiple materials. Figure 2.14 sketches the resulting topology of a MBB

beam composed with void and two solid materials, which are respectively set to

be 60%, 25% and 15% material volumes of the design domain.

Figure 2.14 The BESO topology for a MBB beam using multiple materials

Figure 2.14 The BESO topology for a MBB beam using multiple materials

(Huang and Xie, 2009)

Next, the BESO method was extended to the automotive industry. Design

optimization of energy-absorbing structures was proposed to avoid serious

consequences caused by crashworthiness. Within the work, absorbed energy per

unit volume and absorbed energy ratio were considered as the main design

51

parameters. The adjoint method was then employed to derive the sensitivity

numbers (Huang et al., 2007).

In 2008, this advanced BESO technique was developed to solve an optimization

problem for periodic structures (Huang and Xie, 2008b). That work adopts a

certain quantity of unit cells in the design domain, and determines these cells on

basis of the element removal/addition strategy. In 2010, Huang et al. presented a

new BESO method for the topology optimization of vibrating continuum

structures (Huang et al., 2010). The optimization algorithm combined with a

modified SIMP model was used to avoid the artificial localized modes, and the

rigorous optimality criteria were addressed to satisfy a prescribed weight

constraint.

As for topology optimization of material microstructure, the current BESO

technique has also been exploited and developed for the topological design of

cellular and composite materials, considering a wide range of design objectives.

For example, Huang et al. (2011) described the optimal design of microstructures

of cellular materials for maximum bulk or shear modulus. In 2012, Huang et al.

achieved the topology optimization of periodic composites for external magnetic

permeability and electrical permittivity (Huang et al., 2012). The design

objectives aim to search for maximizing magnetic permeability, maximizing

electrical permittivity, and maximizing or minimizing a combination of the above

properties.

So far, it has been demonstrated that the current BESO method is capable of

generating reliable and practical topologies with high computational efficiency for

various optimization problems. It is worthwhile to extend the BESO method to

optimizing compliant mechanisms. Therefore, this dissertation will adopt and

52

modify the advanced BESO method for the optimal design of compliant

mechanisms.

2.4.5 Summary of the Bi-directional Evolutionary Structural Optimization

Method

This chapter has reviewed the BESO method. In the current BESO method, a filter

scheme is introduced to prevent checkerboard and mesh-dependency. Averaging

the sensitivity number is used to improve the convergence histories of topology

optimization.

This advanced BESO method makes up effectively for the deficiencies of the ESO

and the early BESO methods. Firstly, the current BESO method can achieve

structural optimums based on any possible initial design domain (Huang et al.,

2007). Secondly, this optimization method saves significant computational time.

Thirdly, its evaluation procedure is more robust when compared to the ESO and

initial BESO methods affected significantly by parameters. Fourthly, the current

BESO method is capable of generating reliable and practical topologies for more

optimization design problems. Fifthly, the new BESO procedure can obtain clear

resulting topologies so that final designs are easily measured and manufactured in

the industry (Huang et al., 2007).

Overall, this new BESO reduces computational time significantly. Resulting

topologies have also demonstrated that the current BESO method is capable of

generating reliable and practical topologies with high computational efficiency for

various optimization design problems (Huang and Xie, 2007; Huang and Xie,

2008b). Thus it is worthwhile to take the BESO method into consideration in the

optimal design of compliant mechanisms.

53

2.5 General Remarks

This chapter has reviewed previous research of designing compliant mechanisms

based on both the kinematics-based approach and the structural optimization-based

approach. As continuum-based models reflect the real artifact more closely, there

is a large emphasis on the description and discussion of topology optimization of

compliant mechanisms on the basis of various optimization-based methods. As

previous designs have shown, several deficiencies have been summarized in this

chapter such as vague resulting topologies of compliant mechanisms, structural

connectivity problems, unreasonable constraints cited in optimization algorithms

and the control of hinge regions in topologies.

In addition, this chapter also reviewed some topology optimization methods and

introduced fundamental theories of the BESO method. It has been proved that the

current BESO method is capable of generating reliable and practical topologies

with high computational efficiency for various optimization design problems. Thus,

it is worthwhile to extend the BESO method to the optimal design of compliant

mechanisms. However, the discrete material distribution pattern often results in an

ill-posed problem so that the current BESO method cannot be directly applied to

the optimal design of compliant mechanisms.

Therefore, this dissertation aims to propose a modified BESO method for topology

optimization of compliant mechanisms. To achieve the goal, several efforts will be

made in this work:

• Firstly, this research will first introduce the BESO method into the systematic

design field of compliant mechanisms. The traditional BESO procedure will

54

be revolutionized by eliminating the binary material distribution theory and

introducing intermediate densities. This modified BESO method will be

beneficial to accurately capture the change of design variables for topology

optimization of compliant mechanisms.

• Secondly, this dissertation will establish new computational algorithms for

optimal design of compliant mechanisms. Compliance constraints are used to

effectively control the energy stored in compliant mechanisms with linear or

geometrically nonlinear behaviors after loading. These new optimization

algorithms will benefit in producing a series of optimal solutions with the

balance of the flexibility and stiffness necessities. In addition, a new objective

optimization will be modeled by considering desirable displacement and

simultaneously precluding the formation of hinges in order to design

hinge-free compliant mechanisms.

• Thirdly, nonlinear analysis procedures of compliant mechanisms will be

incorporated into the topology synthesis of compliant mechanisms as linear

analysis is inaccurate in modeling large deformation.

• Fourthly, this dissertation will exhibit a number of 2D and 3D numerical

examples to verify the proposed optimization techniques and demonstrate the

effectiveness of presented methods for compliant mechanism designs, such as

hinge-free and nonlinear compliant mechanisms.

• Fifthly, developed techniques will be shown to be suitable for industrial

applications. The technique involved in the control of hinge regions will help

mechanism designs reduce stress concentration and fatigue breakage. Clear

resulting mechanism topologies generated here will be more acceptable for

55

industrial manufacturability. In addition, the optimal design of nonlinear

compliant mechanisms will be performed for real applications as the

simulation is more accurate.

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and a level-set method", Journal of Computational Physics 194(1), 363-393.

Allaire, G., Jouve, F. and Toader, A.-M., 2004, "Structural optimization using sensitivity analysis

and a level-set method ", Journal of Computational Physics 194(1), 363-393.

Ananthasuresh, G. K., 1994. A new design paradigm for micro-electro-mechanical systems and

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67

Chapter 3

Modified Bi-directional Evolutionary

Structural Optimization

68

Chapter Overview

To design ideal compliant mechanisms, flexibility should be considered in the

created compliant mechanisms. As a result, the optimal design of compliant

mechanisms can be quite sensitive to the classical removal/addition criterion of the

BESO method so that topology optimization may result in an ill-posed problem.

The traditional BESO method cannot be directly applied to the compliant

mechanism design. Therefore, this research presents the modified BESO method

using different volume constraints for designing compliant mechanisms. This

chapter first introduces its basic ideas and explains relative techniques. Then

numerical examples are provided to demonstrate capability and effectiveness of

solving optimization design problems.

3.1 Problem Statement

To seek optimums with a given volume of material, the BESO method formulates a

general optimization problem as follow:

Minimize 𝑕( ) = 𝐇( 1, 2, 3, … , 𝑛) (3.1)

Subject to: 𝑉∗ − ∑ 𝑉𝑒𝑁𝑒=1 𝑒 = 0

where h(x) denotes the objective function of the optimization problem. H is defined

as an n-dimension function. In addition, 𝑉𝑒 and 𝑉∗ represent respectively the

volumes for an individual element and the prescribed total structure in the volume

constraint. Design variable 𝑒 can be equal to 1 for solid element in the

traditional BESO method. Otherwise, 𝑒 will be set as 0 in solid-void design or

𝑚𝑖𝑛 in solid-soft design.

69

For the optimal design of compliant mechanisms, optimization techniques

naturally lead to the introduction of de facto hinges into final topologies. To ensure

structural continuity of compliant mechanisms, this research employs

intermediate densities to modify the traditional BESO method for optimizing

compliant mechanisms. Specifically, the optimal design can be viewed as the

special continuum containing m types of materials. Their corresponding density

variables of eth element can be defined as 0 ≤ 𝑒,𝑚 < 𝑒,𝑚−1 < ⋯ < 𝑒,𝑗 < ⋯ <

𝑒,2 < 𝑒,1 ≤ 1 when introducing intermediate densities in the interval [ 𝑒,𝑚, 1].

𝑒,𝑚 is defined as the density of soft elements 𝑚𝑖𝑛 . 𝑒,1 and 𝑒,𝑚−1 donate

upper and lower densities of solid elements respectively. Therefore, Design

variable 𝑒 is equal to 𝑒,1, 𝑒,2, … , 𝑒,𝑚−1 𝑜𝑟 𝑒,𝑚 in the modified BESO

method.

3.2 Sensitivity Calculation

The sensitivity from elements denotes the gradient of the objective function. It aims

to measure the changing trend owing to the small change in the design variable

when the whole design domain is meshed with finite elements. Generally, the

calculation can be written as Equation 3.2 by differentiating its objective function

𝛼𝑖 =𝜕ℎ(𝑥)

𝜕𝑥𝑒=𝜕𝐇(𝑥1,𝑥2,𝑥3,…,𝑥𝑛)

𝜕𝑥𝑒 (3.2)

When the eth element is completely removed from a design domain meshed

uniformly, the sensitivity number can estimate the change of the corresponding

design objective and can be expressed as Equation 3.3.

𝛼𝑒 = ∆𝑕𝑒 (3.3)

70

For the SIMP model, intermediate designs are penalized by using a constant

penalty in a matrix of constants of elasticity in Hooke‟ law, as shown in Equation

3.4 (Bendsøe and Sigmund, 2003).

=𝑥𝑒𝑝

1− 2[

1 0 1 0

0 01−

2

] (3.4)

The effective Young‟s modulus can be formulated as a function of 𝑒 for different

values of p, namely 𝑒𝑝𝐸 . For each element, the density of material varies

continuously between 0 and 1. The elastic properties for intermediate densities can

be determined according to the density using a power-law interpolation which leads

to nearly 0/1 optimum solution during the evolutionary process (Rietz, 2001).

In the traditional BESO method, Young‟s modulus can be interpolated as the

function with the density variable as 𝐸( 𝑒) = 𝑒𝑝𝐸1 according to the above

solid-void design principle, where 𝐸1 denotes Young‟s modulus for solid material

and the power p is referred to as the penalty exponent. After meshing the whole

design domain, the stiffness matrix K can be expressed by the elemental stiffness

matrix and design variables as = ∑ 𝑒𝑝 𝑒

1𝑒 , in which 𝑒

1 represents the element

stiffness matrix for solid elements.

As described in the previous literature (Huang and Xie, 2010), it is supposed that

the soft-kill method is employed to solve the stiffness design problem. The

sensitivity of the mean compliance can be simply expressed at two types of

elemental levels, solid and soft elements, by introducing this material interpolation

scheme. (Huang and Xie, 2010b)

71

𝜕𝐶

𝜕𝑥𝑒= {

−𝑝𝐮𝑒𝑇 𝑒

1𝐮𝑒 2⁄ 𝑤𝑕𝑒𝑛 𝑒 = 1

−𝑝 𝑚𝑖𝑛𝑝−1𝐮𝑒

𝑇 𝑒1𝐮𝑒 2⁄ 𝑤𝑕𝑒𝑛 𝑒 = 𝑚𝑖𝑛

(3.5)

In the modified BESO method, the interpolation function, which addresses

intermediate densities, can be expressed as 𝐸( 𝑒,𝑗) = ( 𝑒,𝑗 𝑒,0⁄ )𝑝𝐸0, where 𝑒,0

and 𝐸0 are respectively the base density and base Young‟s modulus in the initial

design domain. The global stiffness matrix can be written as

= ∑ ( 𝑒,𝑗 𝑒,0⁄ )𝑝 𝑒0

𝑒 using an initial element stiffness matrix. It can be noted

that the design problem is including a unique constraint 𝑉∗ = ∑ 𝑉𝑒 𝑒 ( 𝑒 =𝑛𝑒=1

𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚).

Here the stiffness design is still cited to demonstrate the sensitivity calculation in

the modified BESO method. The sensitivity of the mean compliance can be

expressed at all elemental levels including solid and soft elements as

𝜕𝐶

𝜕𝑥𝑒= −𝑝 𝑒

𝑝−1𝐮𝑒𝑇 𝑒

0𝐮𝑒 (2 0𝑝)⁄ 𝑤𝑕𝑒𝑛 𝑒 = 𝑒,1, 𝑒,2, … , 𝑒,𝑚−1 𝑜𝑟 𝑒,𝑚 (3.6)

3.3 Filter Scheme and Stability Process

The filter scheme aims to prevent unstable phenomena such as checkerboard and

mesh-dependency in the traditional BESO method. The nodal sensitivity number

αn has no real physical meaning during the optimization process. It can be

obtained by averaging the elemental sensitivity numbers in Equation 3.7

𝛼𝑑𝑛 = ∑ 𝜔𝑒𝛼𝑒

𝑀𝑒=1 (3.7)

M is the total number of nodes in the circular domain. The weight factor of the eth

72

element ωe can be defined as (1 − 𝑟𝑒𝑑 ∑ 𝑟𝑒𝑑𝑀𝑒=1⁄ ) (𝑀 − 1)⁄ , in which red denotes

the distance between the center of the element e and node d. Figure 3.1 shows the

distribution from this element sensitivity number to nodal sensitivity numbers in a

design domain with a uniform mesh.

Figure 3.1 The filter scheme in the BESO method

Figure 3.1 The filter scheme in the BESO method

After the element sensitivity number is averaged to its neighboring nodes, the

nodal sensitivity will be converted back into the sensitivity of this element by

using the following filter scheme as

�̂�𝑒 =∑ 𝜔(𝑟𝑒𝑑)𝛼𝑑

𝑛𝑀𝑑=1

∑ 𝜔(𝑟𝑒𝑑)𝑀𝑑=1

(3.8)

where (𝑟𝑒𝑑) is a weight factor and equal to 𝑟𝑚𝑖𝑛 − 𝑟𝑒𝑑 when 𝑟𝑒𝑑 is smaller

than 𝑟𝑚𝑖𝑛. Otherwise, the weight factor equals zero. In addition, the filter radius

𝑟𝑚𝑖𝑛 is a scale parameter in BESO. It aims to identify the nodes that impact on

the sensitivity of the element located at the center of the circle. In other words,

𝛼1𝑛 = 𝛼1 4⁄

rmin red

𝛼2

𝛼1

𝛼2𝑛 = (𝛼1 + 𝛼2) 4⁄

�̂�𝑒

73

nodes inside the circular domain contribute to the calculation of the sensitivity

number for the central element in BESO as shown in Figure 3.1.

A filter scheme should also be employed in the modified BESO method. Otherwise,

topologies may appear discontinuous scattered points. The elemental sensitivity

number 𝛼𝑒 should be tackled through a simple filter scheme formulated in

Equation 3.9

�̂�𝑒 =∑ 𝜔(𝑟𝑒𝑑)𝛼𝑒𝑀𝑑=1

∑ 𝜔(𝑟𝑒𝑑)𝑀𝑑=1

(3.9)

where the filter radius 𝑟𝑒𝑑 denotes the distance between centers of element e and

element d. Figure 3.2 illustrates the procedure of smoothing element sensitivities.

Figure 3.2 The filter scheme in the modified BESO method

Figure 3.2 The filter scheme in the modified BESO method

rmin

red

𝛼2 𝛼1

�̂�𝑒

74

Same to the traditional BESO technique, the modified BESO method will further

modify the elemental sensitivity by averaging the historical information to

stabilize the evolution process. That can be written as Equation 3.10

�̃�𝑒 = (�̂�𝑒,𝑘 + �̂�𝑒,𝑘−1) 2⁄ (3.10)

where k represents the current iteration number. Then let �̂�𝑒,𝑘 = �̃�𝑒, thus the

modified sensitivity number considers the sensitivity information in the previous

iterations.

3.4 Volume Constraint and Convergence Criterion

The BESO method is an iterative approach. The volume of the structure changes

before it reaches the required volume constraint or prescribed volume 𝑉∗. The

target volume for the next iteration 𝑉𝑘+1 should be determined in advance when

the evolutionary procedure considers adding or removing elements in the current

iteration. The target volume can be formulated as

𝑉𝑘+1 = {

𝑉𝑘 × (1 + 𝐸𝑅) 𝑤𝑕𝑒𝑛 𝑛𝑜𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 𝑉∗ 𝑏𝑢𝑡 𝑉𝑘+1 > 𝑉𝑘

𝑉𝑘 × (1 − 𝐸𝑅) 𝑤𝑕𝑒𝑛 𝑛𝑜𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 𝑉∗ 𝑏𝑢𝑡 𝑉𝑘+1 < 𝑉𝑘

𝑉∗ 𝑤𝑕𝑒𝑛 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 𝑉∗ (3.11)

where evolutionary ratio ER is used to identify the proportion of the volume

reduction of solid elements to the total volume of solid elements and 𝑉𝑘 means the

current material volume of the kth iteration. Meanwhile, Equation 3.11 implies that

the volume constraint can be larger or smaller than the volume of the initial guess

design. The prescribed volume can be achieved by increasing or decreasing

material step by step before the volume of the structure keeps constant at the level

of the objective volume during the evolution procedure.

75

To address material removal or addition, all elements including void (or soft) and

solid elements will be arranged in a descending order according to the values of

their sensitivity numbers. For solid elements, the elemental density is switched

from 1 to 0 (or 𝑚𝑖𝑛) if the criterion, 𝛼𝑒 ≤ 𝛼𝑡ℎ, is satisfied. As for void (or soft)

elements, the elemental density is changed from 0 (or 𝑚𝑖𝑛) to 1 if the criterion,

𝛼𝑒 > 𝛼𝑡ℎ, is satisfied. The reference variable 𝛼𝑡ℎ represents the threshold of the

sensitivity number which can be easily determined by the target material volume

and ranking of the sensitivity number. It is supposed that the optimization design is

a maximization problem. There are 1000 elements in the model. Sensitivity

numbers can be sorted as 𝛼1000 ≤ 𝛼 ≤ ⋯ ≤ 𝛼3 ≤ 𝛼2 ≤ 𝛼1 . 𝛼𝑡ℎ is equal to

𝛼 00 if 800 elements should remain in the model in order to achieve the target

volume Vk+1. (Huang and Xie, 2007).

In the modified BESO method, the evolutionary process can adopt the constant or

variable material volume pattern. For the former, the material volume can be kept

constant as 𝑉𝑘+1 = 𝑉∗ during the evolutionary process. Volume constraint can

be expressed as 𝑉∗ = ∑ 𝑉𝑒 𝑒,0𝑛𝑒=1 = ∑ 𝑉𝑒 𝑒 ( 𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚)

𝑛𝑒=1 . In

addition, the variable material volume pattern employed in the traditional BESO

method can also be applied to the modified BESO method. According to Equation

3.11, the target volume 𝑉𝑘+1 changes and 𝑉𝑘+1 𝑉∗ before it reaches the

required volume constraint or prescribed volume.

To achieve the volume constraint in the modified BESO method, the elements are

firstly sorted according to the values of their sensitivity number before

determining the appropriate threshold 𝛼𝑡ℎ . For example, it is assumed that

𝛼 1 ≤ 𝛼 0 ≤ ⋯ ≤ 𝛼3 ≤ 𝛼2 ≤ 𝛼1 in a design domain consisting of 9×9 elements.

The initial bound values of 𝛼𝑡ℎ are set as 𝛼𝑡ℎ,𝑙𝑜𝑤𝑒𝑟 = 𝛼 1 and 𝛼𝑡ℎ,𝑢𝑝𝑝𝑒𝑟 = 𝛼1.

76

The density variation criterion starts from an initial guess with 𝛼𝑡ℎ = 𝛼( 1+1) 2⁄ =

𝛼41. The procedure will be repeated in terms of the bisection method to determine

the threshold 𝛼𝑡ℎ, by which density variable of eth element satisfies ∑ 𝑉𝑒 𝑒𝑛𝑒=1 =

𝑉𝑘+1 after it is switched from 𝑒,𝑗 to 𝑒,𝑗−1 if 𝛼𝑒 > 𝛼𝑡ℎ or from 𝑒,𝑗 to 𝑒,𝑗+1

if 𝛼𝑒 < 𝛼𝑡ℎ.

To complete the whole optimization process, both traditional and modified BESO

methods employ Equation 3.12 to satisfy the convergence criterion after the

objective volume 𝑉∗ is achievedV*.

𝑒𝑟𝑟𝑜𝑟 =|∑ (𝐶𝑘−𝑖+1−𝐶𝑘−𝑁−𝑖+1)𝑁𝑖=1 |

∑ 𝐶𝑘−𝑖+1𝑁𝑖=1

≤ 𝜏 (3.12)

where k is the current iteration number, τ is an allowable convergence error and N is

integral number which are set to be 0.01% and 5 throughout the research.

3.5 Evolutionary Procedure of Modified Bi-directional Evolutionary

Structural Optimization Methods

The evolutionary procedure of the modified BESO methods is outlined in Table

3.1.

77

Table 3.1 The evolutionary procedure of the modified BESO method

Step 1 Discretize the whole design domain using a finite element mesh with

given boundary and loading conditions.

Step 2 Define parameters relative to modified BESO methods such as

element densities, penalty exponent p and the filter radius 𝑟𝑚𝑖𝑛

Step 3 Carry out a finite element analysis (FEA) which is conducted by

ABAQUS in this research.

Step 4

Calculate elemental sensitivities using Equation 3.2, then update

sensitivity numbers using the filter scheme Equation 3.9 and average

with their historical information using Equation 3.10.

Step 5

Target volume is determined according to 𝑉𝑘+1 = 𝑉∗ for constant

material volume or Equation 3.11 for variable material volume. Reset

and rank all design variables, and set the threshold of the sensitivity

number, 𝛼𝑡ℎ . The corresponding density variable of eth element

should be switched from 𝑒,𝑗 to 𝑒,𝑗−1 if 𝛼𝑒 > 𝛼𝑡ℎ, or from 𝑒,𝑗 to

𝑒,𝑗+1 if 𝛼𝑒 < 𝛼𝑡ℎ

Step 6 Repeat Steps 3-5 until the performance of the objective function is

convergent.

3.6 Numerical Implementation of Modified Bi-directional Evolutionary

Structural Optimization Methods

This section presents the stiffness optimization design to demonstrate the

effectiveness of the modified BESO method using constant material volume. The

design domain, a cantilever beam under a concentrated loading, has been depicted

in Figure 2.12. Its length and height are set as 160mm and 100mm respectively. It

has been discretized with 16000 4-node quadrilateral elements. A downward force

78

is applied at the center of left edge with the magnitude of 100N. The material has a

Young‟s modulus of 100GPa and a Poisson‟s ratio of 0.3. The filter radius is

𝑟𝑚𝑖𝑛 = 6 . The lower bound of the material 𝑒,𝑚 (or 𝑚𝑖𝑛) is set to be

0.001 as void elements in this simulation. 0.5 is selected as the density 𝑒,0.

Penalty exponent is p=3.

Figure 3.3 shows the evolutionary history of the mean compliance for the

cantilever beam. The modified BESO method starts with the full design, in which

all elements have same base density and their densities are gradually polarized

until both the topology and objective function are convergent. Unlike the

evolutionary process in the traditional BESO method, the objective is gradually

decreasing until the convergence criterion is reached. The modified BESO

method is quite efficient since the topology obtains its basic shape at about

Iteration 20 and the evolutionary procedure finalizes its process at Iteration 58.

Figure 3.4 shows the optimal topologies using different mesh sizes. Resulting

topologies demonstrates that the optimal topology does not rely on mesh size

when using the modified BESO method.

Figure 3.3 The evolutionary history of the mean compliance using the modified BESO method

Figure 3.3 The evolutionary history of the mean compliance using the modified

79

BESO method

Figure 3.4 Resulting topologies with 80×50 and 160×100 meshes

Figure 3.4 Resulting topologies with 80×50 and 160×100 meshes

The research adopts the modified BESO method with variable material volume to

solve the stiffness optimization design problem. As shown in Figure 3.5, the beam

is loaded at the center of the bottom edge by P=100N. The design domain, a

symmetric beam, is discretized by 120×40 four node plane stress elements. 50% of

the material volume is used to construct the final structure. The evolution ratio is

set as 2%. The material has a Young‟s modulus of 100GPa and a Poisson‟s ratio of

0.3. The filter radius is 𝑟𝑚𝑖𝑛 = 6 . The lower bound of the material 𝑒,𝑚 (or

𝑚𝑖𝑛) is set to be 0.001 as soft elements in this simulation. Penalty exponent is

p=3.

Figure 3.5 Dimensions of the design domain and boundary conditions for a beam

Figure 3.5 Dimensions of the design domain and boundary conditions for a beam

Figure 3.6 displays the evolution histories of the mean compliance and the volume

fraction. Initially, the material occupies the full design domain. The objective is

80

convergent at a stable value via the change of element densities. Comparing

different volume constraints, the BESO technique with the constant volume is

more efficient. The other technique is more stable and accurate for structural design

by gradually changing volume and gradually updating element densities.

Figure 3.6 Evolution histories of mean compliance and volume fraction using the modified BESO method

Figure 3.6 Evolution histories of mean compliance and volume fraction using the

modified BESO method

3.7 Conclusions

The new BESO procedure has been proposed in this chapter. The current BESO

method introduces intermediate element densities which will be used for dealing

with de facto hinges in the design of compliant mechanisms. In order to prevent

the checkerboard phenomenon and mesh-dependency, elemental sensitivity

number is tackled through averaging directly sensitivity numbers of neighboring

elements. The new BESO algorithm gradually updates design variables until a

convergent solution is obtained. Numerical experiments have been implemented to

demonstrate that the modified BESO method is a quite efficient optimization

algorithm for stiffness optimization problem.

81

References


Springer, Berlin.

Huang, X. and Xie, Y. M., 2007, "Numerical stability and parameters study of an improved

bi-directional evolutionary structural optimization method", Structural Engineering and

Mechanics 27(1), 49-61.

Huang, X. and Xie, Y. M., 2010b, "A further review of ESO type methods for topology


Rietz, A., 2001, "Sufficiency of a finite exponent in SIMP (power law) methods", Structural and


82

Chapter 4

Effects of Spring Stiffness on Topology

Design of Linear Compliant

Mechanisms

83

Chapter Overview

The dissertation has reviewed previous work related to the optimal design of

compliant mechanisms. The dissertation has also demonstrated that the traditional

BESO method and modified BESO methods are capable of generating reliable and

practical topologies with high computational efficiency for stiffness optimization

problems.

In this chapter, the research presents topology optimization approaches for the

optimal design of linear compliant mechanisms. The optimization approach uses

several objective functions and employs the modified BESO method with constant

material volume. The sensitivity analysis is performed by applying the adjoint

method. The evolutionary procedure gradually switches elements‟ densities until

optimal topologies are achieved. This chapter then analyzes the stored energy

after deformation, output displacement and spring effect when using different

objective function.

4.1 Optimization Problem and Structural Analysis

Using structural topology optimization in the optimal design of monolithic

compliant mechanisms is an innovative method (Ananthasuresh, 2003). Within

the topology optimization framework, a variety of objective functions have been

defined to find a compliant mechanism with the desired performance (Sigmund,

1997; Saxena and Ananthasuresh, 2001; Ansola et al., 2007; Luo et al., 2007; Lin

et al., 2010). For designing compliant mechanisms, it is quite necessary to

consider and quantify the performance characteristics of compliant mechanisms,

which have been reviewed in Section 2.2.1, such as the output displacement,

mechanical advantage (MA), geometric advantage (GA) and mechanical

84

efficiency (ME). Here, this dissertation aims to employ different design criteria

using the modified BESO method with the constant material volume for the

optimal design of compliant mechanisms.

To formulate the topology optimization problem of compliant mechanisms, this

research considers a general design domain Ω under given loadings and boundary

conditions as shown in Figure 4.1a. It is assumed that the applied force at the input

port i is Fin and the reaction force at the output port j is Fout acted on a workpiece

which is modeled by a spring with a constant stiffness kout. The resulting input

displacement is △in at the input port i and the output displacement △out at the output

port j. This research mathematically formulates an optimization problem as

Functions A and B for the design of a compliant mechanism as follows:

Function A:

Maximize 𝑕( ) = 𝐺𝐴 (4.1)


𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚

where GA represents the ratio of the output displacement △out to input

displacement △in.

Function B:

Maximize 𝑕( ) = 𝑀𝐸 = 𝑆𝑖𝑔𝑛(𝐺𝐴) × (𝑀𝐴 × 𝐺𝐴) (4.2)


𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚

where ME is equal to the ratio of the output work Fout×△out to input work

Fin×△in when loading at input port, and 𝑆𝑖𝑔𝑛(𝐺𝐴) is used to indicate the

85

expected direction of the output displacement. Mechanical efficiency (ME), as a

combination of the geometric advantage (GA) and the mechanical advantage (MA),

exhibits well the expected mechanical performance in that it can quantify the

relationship between the applied work at the input port and the desired work at

output port (Sigmund, 1997; Wang, 2009).

(a)

(b) (c)

Figure 4.1 Model A for the optimal design of compliant mechanisms

Figure 4.1 Model A for the optimal design of compliant mechanisms (a)

mechanism design; (b) input load case; (c) output load case

The linear combination method is employed to simulate the mechanism design

with the spring attached at output port. According to the principle of the

superposition, the original compliant mechanism (refer to Figure 4.1a) can be

divided into two structures as shown in Figures 4.1b and 4.1c. In other words, the

Input Port i

Fin , △in

Output Port j

kout

Fout

Ω △out

F1, △11 △21

Ω

△12

F2, △22

Ω

86

compliant mechanism is equivalent to the linear combination of two structures

when the following displacement relationships are satisfied:

∆𝑖𝑛= ∆11 + 𝑐 × ∆12 (4.3)

∆𝑜𝑢𝑡= ∆21 + 𝑐 × ∆22

where △ij denotes the displacement at the port i due to the applied force Fj. c is the

combination coefficient to be determined. Meanwhile, the forces should satisfy the

following relationships

𝐹𝑖𝑛 = 𝐹1 (4.4)

𝐹𝑜𝑢𝑡 = −𝑐 × 𝐹2 = 𝑘𝑜𝑢𝑡 × ∆𝑜𝑢𝑡

Thus, the combination coefficient c can be identified as

𝑐 = −∆21𝑘𝑜

2+∆22𝑘𝑜 (4.5)

Substituting Equation 4.5 into Equations 4.3 and 4.4, the input and output

displacements can be can be expressed as

∆𝑖𝑛= ∆11 −𝑘𝑜 ∆21∆12

2+∆22𝑘𝑜 (4.6)

∆𝑜𝑢𝑡= 2∆21

2+∆22𝑘𝑜

Thus Function A using the geometric advantage can be stated as Equation 4.7

𝐺𝐴 = ∆𝑜𝑢𝑡 ∆𝑖𝑛⁄ = (𝐹2∆21) (𝐹2∆11 + ∆11∆22𝑘𝑜𝑢𝑡 − ∆21∆12𝑘𝑜𝑢𝑡)⁄ (4.7)

87

Function B with using the mechanical efficiency can be expressed as Equation 4.8

𝑀𝐸 = 𝑀𝐴 × 𝐺𝐴 = (𝐹𝑜𝑢𝑡𝐹𝑖𝑛

) × (∆𝑜𝑢𝑡∆𝑖𝑛

) =𝑘𝑜𝑢𝑡∆𝑜𝑢𝑡

2

𝐹1∆𝑖𝑛

= 𝑘 ( 2∆21

2+∆22𝑘𝑜 )2

( 1 2∆11+ 1∆11∆22𝑘𝑜 − 1∆21∆12𝑘𝑜

2+∆22𝑘𝑜 )⁄ (4.8)

Equation 4.8 should be pre-multiplied by 𝑆𝑖𝑔𝑛(𝐺𝐴) when taking into account the

desired direction of the output displacement.

According to the dummy load approach, displacements △ij can be expressed as

∆𝑖𝑗= 𝐮𝑗 𝐮𝑖 𝐹𝑖⁄ (4.9)

where K is the global stiffness matrix of the structure. 𝐮1 and 𝐮2 represent the

displacement fields of two structures shown in Figures 4.1b and 4.1c respectively.

In addition, other formulations have been applied to the optimal design of

compliant mechanisms (Frecker et al., 1997). For the systematic design of

compliant mechanisms, the optimization problem can also be mathematically

formulated as Functions C and D.

Function C:

Maximize 𝑕( ) = ∆𝑜𝑢𝑡 (4.10)


𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚

Function D:

Maximize 𝑕( ) = −𝑀𝐴 (4.11)

88


𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚

where MA represents the ratio of the output force Fout to input force Fin.

The topology may suffer a material disconnection between input and output ports

when simply using Function C (Wang, 2009). For Model A in which an external

spring is attached to the output port, the determinant of the mechanism stiffness

matrix, det (Km), cannot be equal to zero so that the matrix is no longer singular. To

avoid the presence of structural discontinuity, this research considers either

employing a corresponding input constraint or attaching an external spring at input

port for maximizing the magnitude ∆𝑜𝑢𝑡. Appendix A presents its elastic analysis.

Model B is proposed for the optimal design of compliant mechanisms by attaching

a spring at input port with a constant stiffness kin when using Function C, shown in

Figure 4.2. In addition, Function D can be written as = 𝐹𝑜𝑢𝑡 𝐹𝑖𝑛⁄ =

𝑘𝑜𝑢𝑡∆𝑜𝑢𝑡 𝐹𝑖𝑛⁄ , the sensitivity rank of which is the same as that of Function C due to

constant variables kout and Fin. Therefore, this research takes advantage of Model B

to optimize the compliant mechanism design using Function C or D.

89

(a)

(b) (c)

Figure 4.2 Model B for the optimal design of compliant mechanisms

Figure 4.2 Model B for the optimal design of compliant mechanisms (a)

mechanism design; (b) input load case; (c) output load case

According to the dummy load approach, Function C using single output

displacement can be written as Equation 4.12 based on Model B

∆𝑜𝑢𝑡= ∆21 (4.12)

4.2 Sensitivity Number

Solving the optimization problem based on these mentioned objective models

requires sensitivity computation. As mentioned in Section 3.2, the stiffness matrix

K can be expressed by the elemental stiffness matrix and design variables as

Input Port i

Fin , △in

Output Port j kin

Fout

Ω △out

kout

F1, △11 △21

Ω

△12 F2, △22

Ω

90

= ∑ 𝑒𝑝 𝑒

0𝑒 when the binary design variable 𝑒 can switch between 𝑚𝑖𝑛 and 1.

The global stiffness matrix can be written as = ∑ ( 𝑒,𝑗 𝑒,0⁄ )𝑝 𝑒0

𝑒 when

intermediate densities are introduced in the modified BESO method.

The sensitivity of displacement variables shown in Equation 4.9 can be obtained

by differentiating expression given in Equation 4.13

∆𝑖

𝑥𝑒= (

𝜕𝐮

𝜕𝑥𝑒 𝐮𝑖 + 𝐮𝑗

𝜕

𝜕𝑥𝑒𝐮𝑖 + 𝐮𝑗

𝜕𝐮𝑖

𝜕𝑥𝑒) 𝐹𝑖⁄ (4.13)

The following equations can be found since external forces are

design-independent in equilibrium equations 𝐮𝑖 = 𝑖 and 𝐮𝑗 = 𝑗.

𝜕

𝜕𝑥𝑒𝐮𝑖 +

𝜕𝐮𝑖

𝜕𝑥𝑒= 0 (4.14)

𝜕

𝜕𝑥𝑒𝐮𝑗 +

𝜕𝐮

𝜕𝑥𝑒= 0

Substituting Equation 4.13 into Equation 4.14, the sensitivity of displacements

ij can be found as

∆𝑖

𝑥𝑒= −(𝐮𝑗

𝜕

𝜕𝑥𝑒𝐮𝑖) 𝐹𝑖⁄ (4.15)

The sensitivity of the displacement can be expressed at elemental levels for solid

and soft elements as discrete design variables are employed in the modified BESO

method.

∆𝑖

𝑥𝑒= −𝑝 𝑒

𝑝−1𝐮𝑗,𝑒 𝑒

0𝐮𝑖,𝑒 ( 0𝑝𝐹𝑖)⁄ 𝑤𝑕𝑒𝑛 𝑒 = 𝑒,1, 𝑒,2, … , 𝑒,𝑚−1 𝑜𝑟 𝑒,𝑚

(4.16)

91

With the help of Equation 4.12, the sensitivity of Function C can be obtained

according to Model B

𝛼𝑒 = ∆𝑜

𝑥𝑒= ∆21

𝑥𝑒 (4.17)

For the sensitivity numbers of Functions A and B, the linear combination method is

employed to simulate the mechanism design by coupling two load cases depicted

in Model A. According to Equation 4.5, the derivatives of the combination

coefficient c can be defined as

𝑐

𝑥𝑒= −

𝑘𝑜 ( 2+∆22𝑘𝑜 ) ∆21 𝑒

−𝑘𝑜 2 ∆21

∆22 𝑒

( 2+∆22𝑘𝑜 )2 (4.18)

and the derivatives of the input and output displacements according to Equation 4.3

are

∆𝑜

𝑥𝑒=𝜕∆21

𝜕𝑥𝑒+

𝜕𝑐

𝜕𝑥𝑒∆22 + 𝑐

𝜕∆22

𝜕𝑥𝑒 (4.19)

∆𝑖𝑛

𝑥𝑒=𝜕∆11

𝜕𝑥𝑒+

𝜕𝑐

𝜕𝑥𝑒∆12 + 𝑐

𝜕∆12

𝜕𝑥𝑒

Thus, the sensitivity of the geometric advantage of Function A can be obtained.

𝛼𝑒 =

𝑥𝑒= (∆𝑖𝑛 ×

𝜕∆𝑜

𝜕𝑥𝑒− ∆𝑜𝑢𝑡 ×

𝜕∆𝑖𝑛

𝜕𝑥𝑒) ∆𝑖𝑛

2⁄ (4.20)

With the help of Equation 4.8, the sensitivity of the mechanical efficiency of

Function B can be obtained

92

𝛼𝑒 = 𝑆𝑖𝑔𝑛(𝐺𝐴) × 𝑀

𝑥𝑒= 𝑆𝑖𝑔𝑛(𝐺𝐴) × 𝑘𝑜𝑢𝑡 (2∆𝑖𝑛∆𝑜𝑢𝑡

∆𝑜

𝑥𝑒− ∆𝑜𝑢𝑡

2 ∆𝑖𝑛

𝑥𝑒) (𝐹1∆𝑖𝑛

2 )⁄

(4.21)

4.3 Numerical Implementation

Here, the post-processing techniques of the modified BESO method are employed

in the optimal design of linear compliant mechanisms. To avoid checkerboard and

mesh-dependency, a filter scheme is used by Equation 3.9. The elemental

sensitivity number is tackled by averaging sensitivity numbers from neighboring

elements. Secondly, Equation 3.10 is adopted for the stabilization of the

evolutionary process. Finally, the modified BESO will complete the whole

optimization process of design compliant mechanisms once the performance of

the objective function satisfies the convergence criterion (refer to Equation 3.12).

4.4 Iterative Procedure

The procedure of optimizing compliant mechanisms with using the modified

BESO method is listed in the following steps, summarizing the previous sections

in this chapter. In addition, Figure 4.3 describes visually and briefly its iterative

procedure.

Step 1 Present the design domain with boundary conditions for optimizing

compliant mechanism. Discretize the whole design domain using a finite element

mesh. Set material properties such as Young‟s modulus and Poisson‟s ratio.

Define BESO parameters like penalty exponent p, the filter radius 𝑟𝑚𝑖𝑛 , the

allowable convergence error τ and intermediate densities

Step 2 Carry out a finite element analysis which is conducted by ABAQUS. Output

93

required data, such as displacements.

Step 3 The sensitivity number is calculated according to Equation 4.20 for

Function A, Equation 4.21 for Function B and Equation 4.17 for Function C.

Step 4 Update sensitivity numbers by the filter scheme Equation 3.9 and average

with their historical information by Equation 3.10.

Step 5 Reset and rank all design variables, and set the threshold of the sensitivity

number, 𝛼𝑡ℎ , which is determined according to the target material volume

constraint 𝑉∗ = ∑ 𝑉𝑒 𝑒,0 =𝑛𝑒=1 ∑ 𝑉𝑒 𝑒 ( 𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚)

𝑛𝑒=1 . The

corresponding density variable of eth element should be switched from 𝑒,𝑗 to

𝑒,𝑗−1 if 𝛼𝑒 > 𝛼𝑡ℎ, or from 𝑒,𝑗 to 𝑒,𝑗+1 if 𝛼𝑒 < 𝛼𝑡ℎ.

Step 6 Repeat Steps 2-5 until the volume fraction reaches target volume and the

performance of objective function satisfies its convergence criteria according to

Equation 3.12.

94

Figure 4.3 Flow chart of the modified BESO procedure for the optimal design of linear compliant mechanisms

Figure 4.3 Flow chart of the modified BESO procedure for the optimal design of

linear compliant mechanisms

4.5 Numerical Examples and Discussion

No

Yes

Yes

End the optimal design of


Start the optimal design of


Define design domain, boundary conditions, finite

element mesh and set BESO parameters

Carry out finite element analysis

Calculate sensitivity numbers of optimizing


Filter sensitivity number and average historical

information

Calculate and satisfy the volume constraint

Construct a new design

Is the volume constraint

satisfied?

Is the procedure

convergent?

No

95

4.5.1 2D Numerical Examples

In a benchmark example, the proposed topology optimization approach is applied

to the optimal design of an inverter mechanism which outputs the displacement in

an opposite direction to an actuating force. A 100μm×100μm design domain

sketched in Figure 4.4 is meshed with 100×100 4-node quadrilateral elements. Its

left-upper and left-lower corners are simply fixed. An input force Fin=0.1N is

horizontally applied at the center of the left edge. An artificial spring with

stiffness kout=0.01N/μm is attached to the output port. The output port at the center

of the right edge is expected to produce a horizontal displacement △out to the left.

The spring stiffness kin is set as 0.01N/μm when using Model B, and kin=0 when

using Model A. The volume constraint is limited to 20% of the design domain

during the whole evolution process. The material properties are Young‟s modulus

E=100GPa and Poisson‟s ratio ν =0.3. The BESO parameters are the filter radius

𝑟𝑚𝑖𝑛 = 7 , penalty exponent p=3, allowable convergence error τ=0.001%

and 𝑚𝑖𝑛 = 0 001.

Figure 4.4 Design domain and boundary conditions of the inverter mechanism

Figure 4.4 Design domain and boundary conditions of the inverter mechanism

The optimized designs using different objective functions for the inverter

mechanism are shown in Figure 4.5. The final output displacements of resulting

△out Fin , △in

96

solutions are respectively 9.568μm, 19.517μm and 0.767μm for topologies with

20% material volume. Although different design models are employed for the

optimal design of compliant mechanisms, the resulting solution using Function A

is very similar to that based on Model B. As shown in Figure 4.5c, the topology of

input port is slightly different with counterpart in Figure 4.5a. Under the same

design domain and boundary conditions, the evolution process can produce

relatively larger output displacement due to the presence of de facto hinge regions,

comparing Function B with Function A. There will be quite a small output

displacement owing to an artificial spring attached at input port for topology

obtained based on Model B. In addition, Figures d-f present the final designs when

30% of material volume is available. Topologies are relatively stable since they do

not greatly change with different material volumes. All resulting topologies are

comparable with those obtained using the density optimization approach

(Sigmund, 1997) but without much “grey area”. The examples demonstrated that

the proposed formulations are effective for solving different design problems of

compliant mechanisms.

97

(a) (b)

(c) (d)

(e) (f)

Figure 4.5 Optimum topologies for the inverter mechanism

Figure 4.5 Optimum topologies for the inverter mechanism (a) optimum topology

based on Function A, kout=0.01N/μm and 20% material volume; (b) optimum

topology based on Function B, kout=0.01N/μm and 20% material volume; (c)

optimum topology based on Function C, kin=0.01N/μm, kout=0.01N/μm and 20%

material volume; (d) optimum topology based on Function A, kout=10N/μm and 30%

material volume; (e) optimum topology based on Function B, kout=10N/μm and 30%

material volume; (f) optimum topology based on Function C, kin=10N/μm,

kout=10N/μm and 30% material volume

Figure 4.6 shows the evolution histories of different objective functions and the

98

output displacements for the inverter mechanism of 0.2 volume rate. The

evolution process starts with the full design, satisfies its volume constraint value

of 20% during the whole evolutionary procedure, and experiences 452 iterations

based on using Function A, 349 iterations based on using Function B and 325

iterations based on using Function C under the very small convergence factor

0.001%.

(a)

(b)

99

(c)

Figure 4.6 Evolution histories of the inverter mechanism using different design problems

Figure 4.6 Evolution histories of the inverter mechanism using different design

problems: (a) evolution history based on Function A; (b) evolution history based on

Function B; (c) evolution history based on Function C

Figure 4.7 proposes the trend lines for plotting the relationship between the spring

stiffness of output port and values of objective functions of final design,

demonstrating the effects on objective functions from the output spring stiffness.

The optimization process also presents the resulting solutions when using different

spring stiffness for Functions A and B with the filter radius 𝑟𝑚𝑖𝑛 = 3 and 20%

material volume constraint.

Obviously, it can be seen that the geometrical advantage and mechanical

efficiency are decreasing with the spring stiffness becomes greater. In addition,

the resulting solution also tends to be a simpler topology. The difference is that

the former tends to be an exponential decrease, but the mechanical efficiency

shows a linear decrease. The smaller the output spring stiffness, the more de facto

100

hinge regions the topology produces when considering the mechanical efficiency

as an objective function. However, less material is distributed around the input

port for the resulting topologies produced by using Function A. The geometrical

advantage and mechanical efficiency can reach 5.8007 and 0.9848 respectively

due to the number and position of hinge regions when spring stiffness is set as

0.00001 N/μm. The geometrical advantage and the corresponding final topologies

are more sensitive to small spring stiffness of output port. Overall, in comparison to

previous research, more complex topologies can be obtained. The effects from

attached springs on topologies can be analyzed. That is, the small spring stiffness

has a strong influence on the result of the optimal design.

(a)

101

(b)

Figure 4.7 Output spring‟s effects on objective functions

Figure 4.7 Output spring‟s effects on objective functions: (a) spring effects on

Function A; (b) spring effects on Function B.

Table 4.1 lists the resulting solutions when the optimization process employs

Function C using different stiffness values of input and output springs. The filter

radius is set as 𝑟𝑚𝑖𝑛 = 5 and 20% of material volume is available.

It can be observed that spring stiffness has a strong effect on the structure

connectivity of compliant mechanisms. Small spring stiffness results in point

flexures (or de facto hinges). Even about the topology with kin=0.00001N/μm and

kout=0.00001N/μm, mechanism parts are still connected by intermediate density

material. The main parts of these compliant mechanisms undergo rigid-body

movements. The energy and force are mainly transmitted through the bending of

the hinges or regions with the intermediate density material rather than elastic

deformation when loading. Furthermore, the presence of stress concentration can

lead to the fragility of compliant mechanisms. Resulting topologies placed in the

102

upper left have the significant reduction of hinge regions due to greater spring

stiffness values. Therefore, the stiffness values of the artificial springs attached to

the input and output ports have a significant effect on the preclusion of de facto

hinges.

Table 4.1 Resulting topologies of the inverter mechanism using various spring

stiffness values

kout=10N/μm kout=0.1N/μm kout=0.001N/μm kout=0.00001N/μm

k in=

10N

/μm

k in=

0.1

N/μ

m

k in=

0.0

01N

/μm

k in=

0.0

0001N

/μm

The second example demonstrates the design of a gripper mechanism. The design

103

domain is shown in Figure 4.8 as a 100mm×100mm square with a 25mm×26mm

gap which allows the workpiece to be gripped. The whole design domain is

discretized with 9350 4-node quadrilateral elements. A linear spring with kout=0.1

N/μm is defined to simulate the stiffness of the workpiece. The spring stiffness kin

is set as 0.01N/μm when using Model B. This design domain is simply supported

at the top and the bottom corners of the left edge. The input load Fin=0.1N is

applied at the center of the left edge. The objective is to synthesize the design so

that the gripper produces an efficient gripping force Fout or output displacement

△out on the workpiece under the action of input force. The volume constraint is

limited to 25% of the design domain during the whole evolutionary process. The

material properties are Young‟s modulus E=100GPa and Poisson‟s ratio ν =0.3.

The BESO parameters are the filter radius 𝑟𝑚𝑖𝑛 = 5 . penalty exponent p=3

and 𝑚𝑖𝑛 = 0 001.

Figure 4.8 Design domain and boundary conditions of the gripper mechanism

Figure 4.8 Design domain and boundary conditions of the gripper mechanism

The final designs of the gripper mechanism are displayed in Figure 4.9. The output

displacements of topologies can reach 0.8885μm, 2.6039μm and 0.00918μm.

Obviously, the evolution process, the objective optimization of which is

100

Fin, △in △out

△out

26

25

104

formulated as maximizing mechanical efficiency, can produce the resulting

solution with a relatively large output displacement due to the presence of de

facto hinges. On the other hand, attaching a spring at the input port results in a

very small output displacement.

(a) (b) (c)

Figure 4.9 Optimum topologies for the gripper mechanism

Figure 4.9 Optimum topologies for the gripper mechanism (a) optimum topology

based on Function A, kout=0.1N/μm; (b) optimum topology based on Function B,

kout=0.1N/μm; (c) optimum topology based on Function C, kin=0.1N/μm and

kout=0.1N/μm

Figure 4.10 shows the evolution histories of different objective functions for the

gripper mechanism. The evolution process starts with the full design, satisfies its

volume constraint value 25% during the whole evolutionary procedure, and

experiences 247 iterations based on using Function A, 343 iterations based on

using Function B and 234 iterations based on using Function C.

105

(a) (b) (c)

Figure 4.10 Evolution histories of the gripper mechanism using different design problems

Figure 4.10 Evolution histories of the gripper mechanism using different design

problems (a) evolution history based on Function A; (b) evolution history based on

Function B; (c) evolution history based on Function C.

Table 4.2 lists the various resulting solutions of the gripper mechanism using

Functions A and B with the filter radius 𝑟𝑚𝑖𝑛 = 5 and 30% material volume

constraint. The table also indicates the ratio of the stored energy to the total strain

energy Ws/Win, namely (𝐹𝑖𝑛∆𝑖𝑛 − 𝐹𝑜𝑢𝑡∆𝑜𝑢𝑡) 𝐹𝑖𝑛∆𝑖𝑛⁄ . The stored energy accounts

for a larger proportion of total strain energy when a stiffer spring is attached to the

output port of this gripper mechanism. However, for topologies based on Function

A, the increase of the proportion results from the reduction of hinge regions and

the promotion of the elasticity. For example, hinge is absent in the topology with

kout=0.05N/μm. Hinge regions exist in proposed topologies optimized by using

Function B, contributing more to rigid-body movement. Therefore, a significant

reduction in the output displacement causes the greater ratio of Ws to Win.

106

Table 4.2 Resulting topologies of the gripper mechanism using various spring

stiffness values

Function: GA

Function: ME

kout: 0.00005 kout: 0.00005

Ws/Win 46% Ws/Win 18.3%

Function: GA

Function: ME

kout: 0.0001 kout: 0.0001

Ws/Win 61.3% Ws/Win 19.3%

Function: GA

Function: ME

kout: 0.0005 kout: 0.0005


Function: GA

Function: ME

kout: 0.001 kout: 0.001


Function: GA

Function: ME

kout: 0.005 kout: 0.005

Ws/Win 88.9% Ws/Win 73%

Function: GA

Function: ME

kout: 0.01 kout: 0.01

Ws/Win 93.2% Ws/Win 84.2%

Function: GA

Function: ME

kout: 0.05 kout: 0.05

Ws/Win 98.3% Ws/Win 96.4

Figure 4.11 shows the resulting topologies optimized by using Function C. The

107

filter radius is set as 𝑟𝑚𝑖𝑛 = 3 , and 40% of material volume is available.

Topologies are stable because shape does not rely on different material volumes.

In addition, material tends to concentrate on the port, which is attached with the

stiffer artificial spring. It can be observed from Figure 4.11a that balancing and

promoting the stiffness constants are beneficial to eliminate de facto hinges.

Figure 4.11d indicates that soft springs lead to poor structural connection.

(a) (b)

(c) (d)

Figure 4.11 Resulting topologies for the gripper mechanism by using Function C

Figure 4.11 Resulting topologies for the gripper mechanism by using Function C (a)

topology with kin=5N/μm and kout=5N/μm; (b) topology with kin=0.0005N/μm and

kout=5N/μm; (c) topology with kin=5N/μm and kout=0.0005N/μm; (d) topology with

kin=0.0005N/μm and kout=0.0005N/μm.

The proposed optimization approach is applied to another force inverter

mechanism with two output ports. Figure 4.12 shows the design domain and

boundary conditions. The design domain is defined as a rectangle with

108

100μm×60μm size and meshed with 100×60 4-node quadrilateral elements. It is

simply supported at the top and the bottom corners of the left edge. An input force

Fin=0.2N is applied at the center of the left edge in the horizontal direction. Two

output ports placed at the top and the bottom corners of the left edge are expected

to produce horizontal displacements △out, the direction of which is in the opposite

direction. Artificial springs with stiffness kout=0.5N/μm are attached to the output

ports. The volume constraint is limited to 25% of the design domain during the

whole evolutionary process. The material properties are Young‟s modulus

E=100GPa and Poisson‟s ratio ν =0.3. The BESO parameters are the filter radius

𝑟𝑚𝑖𝑛 = 3 , penalty exponent p=3 and 𝑚𝑖𝑛 = 0 001.

Figure 4.12 Design domain and boundary conditions of the compliant force inverter with two output ports

Figure 4.12 Design domain and boundary conditions of the compliant force inverter

with two output ports

The resulting topologies for the force inverter mechanism are shown in Figure 4.13.

The output displacement changes from the initial displacement to the final

displacement 0.316μm, 0.9682μm, 0.0004μm and 16.667μm. The current

optimization method changes effectively the direction of output displacements and

succeeds in obtaining final solutions.

△out

△out Fin , △in

109

(a) (b)

(c) (d)

Figure 4.13 Optimum topologies for the force inverter with two output ports

Figure 4.13 Optimum topologies for the force inverter with two output ports (a)

optimum topology based on Function A, kout=0.5N/μm; (b) optimum topology

based on Function B, kout=0.5N/μm; (c) optimum topology based on Function C,

kin=1N/μm and kout=0.5N/μm; (d) optimum topology based on Function C,

kin=0.01N/μm and kout=0.0005N/μm.


The proposed algorithm is extended to the 3D compliant mechanism design. The

design domain is depicted in Figure 4.14 and is defined as a 45μm×30μm×42μm

cuboid with a 9μm×30μm×18μm gap. It is discretized with 8-node brick elements

with the uniform sizes 1μm×1μm×1μm. Two forces Fin=1N are applied to the input

ports located at centers of left edges. Centers of upper and lower edges of left face

are fixed. A linear spring with kout=0.5 N/μm is defined to simulate the stiffness of

the workpiece. kin=1 N/μm and kout=1 N/μm when using Model B. The material

properties and BESO parameters are set as E = 290GPa, Poisson‟s ratio ν =0.3,

110

filter radius rmin = 5μm, penalty exponent p=3 and 𝑚𝑖𝑛 = 0 001. The volume

constraint is limited to 14% of the design domain during the whole evolutionary

process.

Figure 4.14 Design domain and boundary conditions of 3D gripper mechanism

Figure 4.14 Design domain and boundary conditions of 3D gripper mechanism

Figure 4.15 shows the resulting topologies and their deformed configurations. The

output displacement changes from the initial displacement to the final

displacement 2.479μm, 8.541μm and 0.0066μm. Similarly, the final design based

on Function B has significant hinge regions and large output displacements.

Therefore, the current algorithm appears valid for 3D mechanism design.

Fin, △in △out

18

45

9

30

42

111

(a) (b) (c)

(d) (e) (f)

Figure 4.15 Optimized topologies and CAD models of 3D gripper mechanisms with using different design criteria

Figure 4.15 Optimized topologies and CAD models of 3D gripper mechanisms

with using different design criteria: (a) FE optimized topology based on Function

A; (b) FE optimized topology based on Function B; (c) FE optimized topology

based on Function C; (d) deformed configuration of CAD model for (a); (e)

deformed configuration of CAD model for (b); (f) deformed configuration of CAD

model for (c).

4.6 Conclusions

This research proposes the modified BESO method for the systematic design field

of compliant mechanisms. The sensitivity analysis is conducted by applying the

adjoint method to optimize different design problems such as geometrical

advantage, mechanical efficiency and output displacement. The research produces

112

various resulting designs with the change in the springs‟ constants. Examples

proposed by the new BESO procedure demonstrate the springs‟ effects in the

optimal design of compliant mechanisms. Some conclusions can be drawn in this

section.

Firstly, numerical examples have demonstrated the capability and validity of the

new BESO algorithm. Secondly, there is little undesirable „grey area‟ in resulting

designs. Thirdly, resulting topologies can produce relatively larger output

displacement due to the presence of de facto hinge regions when using the

optimization problem of maximizing mechanical efficiency. Fourthly, when

attaching a stiffer artificial spring to input port, the geometrical advantage and

mechanical efficiency decrease, and resulting topology become simpler. In

addition, the former tends to be an exponential decrease, but the mechanical

efficiency shows a linear decrease. Fifthly, for topologies based on using the

optimization problem of maximizing output displacement, balancing and

promoting stiffness constants are beneficial to eliminate de facto hinges. Soft

springs can lead to poor structural connection. Overall, spring stiffness has strong

effects on the optimal design of compliant mechanisms.

References

Ananthasuresh, G. K., 2003. Optimal Synthesis Methods for MEMS, Kluwer Academic Publisher,

Boston.



Design 44(1-2), 53-62.



113




40(1-6), 241-255.

Luo, Z., Tong, L., Wang, M. Y. and Wang, s., 2007, "Shape and topology optimization of compliant

mechanisms using a parameterization level set method", Journal of Computational Physics

227(1), 680-705.

Saxena, A. and Ananthasuresh, G. K., 2001, "Topology optimization of compliant mechanisms with

strength considerations", Mechanics of Structures and Machines 29(2), 199-221.




Frontiers of Mechanical Engineering in China 4(3), 229-241.

114

Chapter 5

Desired Structural Stiffness in

Topology Design of Linear Compliant

Mechanisms

115

Chapter Overview

The dissertation has applied the modified BESO method to the optimal design of

compliant mechanisms and demonstrated the capability of solving several

optimization problems. This chapter presents a new algorithm for the effective

control of the desired structural stiffness in the topology design of linear

compliant mechanisms. The design problem is reformulated as maximizing the

flexibility of the compliant mechanisms subject to the mean compliance constraint,

considering the mechanism flexibility and structural stiffness simultaneously.

This research employs the modified BESO method, in which element densities are

gradually switched before a convergent solution and material volume is gradually

reduced until the prescribed volume. Several 2D and 3D examples are then

presented to demonstrate the effectiveness of the proposed BESO method. Finally,

this research summarizes the theory of flexibility and hinge-related properties of

optimized compliant mechanisms.


Generally, an efficient compliant mechanism should be flexible enough to

produce the expected kinematic motion (flexibility) but should also be stiff

enough to resist external forces (stiffness). As shown in Figure 4.1a, the

performance characteristics of flexibility and stiffness for a compliant mechanism

can be quantified using relationships among the applied forces, the resulting

displacements at the input port of the mechanism, and the resulting displacements

and reaction forces at the output port of the mechanism.

The topology optimization problem has been formulated in a number of

alternative ways through the use of assorted objective and constraint functions due

116

to the inherent multi-objective performance demand. In the light of the resulting

topologies outlined in the previous chapter, designing compliant mechanisms using

topology optimization methods typically results in de facto hinge regions in the

design models due to the problem formulation. The existence of de facto hinge

regions makes compliant mechanisms function as rigid-link mechanisms so as to

maximize their capability of transferring kinematic motion. Due to the difficulties

in manufacturing reliable hinges especially for micro-scale mechanical systems,

designing monolithic and hinge-free compliant mechanisms has attracted extensive

attention and undergone considerable development in recent years.

Among previous techniques, reformulating the problem as a multi-criteria

optimization might be an effective way for entirely circumventing de facto hinge

regions which generally lie along the force path from the mechanism input port to

the output port. For example, simultaneously maximizing the flexibility and

minimizing the stiffness of the input-restrained structure can achieve hinge-free

compliant mechanisms (Frecker et al., 1997; Rahmatalla and Swan, 2005), and

the resulting compliant mechanisms are also stiff and can resist the additional load

exerted by the workpiece once it has been secured. Recently, Zhu et al. (2013)

incorporated this approach to optimize hinge-free compliant mechanisms with

multiple outputs. Nevertheless, it should be noted that the stiffness of a compliant

mechanism is only equivalent to that of the input-restrained structure when the

stiffness of the workpiece tends to infinity.

With a given stiffness of the workpiece, the formation of de facto hinge regions

must be correlated with the structural stiffness of compliant mechanisms.

However, there is no related literature to reveal the inherent relationship between

the formation of de facto hinge regions and the structural stiffness. This

dissertation proposes a new BESO method for optimally designing the flexibility

117

of compliant mechanisms by altering the desired structural stiffness, which

includes the influence of external loads exerted by the workpiece.

As shown in Figure 4.1a, consider the design domain of a compliant mechanism

where Fin is the applied force at the input port and △out is the expected output

displacement at the output port. A spring with a constant stiffness, kout, is used to

simulate the interaction between the workpiece and the compliant mechanism. △in

is the resulting displacement at the input port and Fout = kout×△out is the output

force. From Section 4.1, we can note that the displacement field of the mechanism

can be obtained through coupling the displacements caused by the input unit

dummy load case (refer to Figure 4.1b) and the output unit dummy load case

(refer to Figure 4.1c) via a linear elastic fashion. △11 and △21 denote the

displacements at the input port and the output port of the input unit dummy load

case, and △12 and △22 represent the displacements at the input port and the output

port of the output unit dummy load case. The input displacement △in and output

displacement △out can be found in Equation 4.3 through the superposition of the

input unit dummy load case and the output unit dummy load case. They can be

explicitly expressed by Equation 4.6.

The performance of compliant mechanisms can be measured by the characteristics

of their flexibility and structural stiffness. The flexibility of a compliant mechanism

can be quantified by the resulting displacement at the output port, △out, which

reflects the force or motion transmission ability of compliant mechanisms. While

the structural stiffness or mean compliance is one of key factors that must be taken

into account in the design of structures, the mean compliance of the compliant

mechanism, C, can be defined as the work done by external forces as

𝐶 =𝐹𝑖𝑛∆𝑖𝑛2

−𝐹𝑜𝑢𝑡∆𝑜𝑢𝑡

2=𝐹𝑖𝑛∆𝑖𝑛2

−𝑘 ∆𝑜𝑢𝑡

2

2

118

= 1 2∆11+ 1∆11∆22𝑘 − 1∆21∆12𝑘

2( 2+∆22𝑘 )−𝑘

2(

2∆21

2+∆22𝑘 )2

(5.1)

It can be seen that the mean compliance considers the influence of the stiffness of the

workpiece, which exerts external forces to the compliant mechanism. It should be

noted that the definition of the above mean compliance is different from the standard

mean compliance which is simply 𝐹𝑖𝑛∆𝑖𝑛 (Lin et al., 2010) or the input-restrained

compliance (Rahmatalla and Swan, 2005).

Instead of using the combination of the flexibility and stiffness as a single objective

function (Frecker et al., 1997; Rahmatalla and Swan, 2005; Ansola et al., 2007) or

multiple objective functions (Lau et al., 2001; Lin et al., 2010), the research

reformulates the design of a compliant mechanism by maximizing the output

displacement at the output port, and constraining its mean compliance and the

amount of structural material that can be used. Mathematically, the optimization

problem of a compliant mechanism is expressed by

Maximize 𝑕( ) = ∆𝑜𝑢𝑡 (5.2)

Subject to: 𝐶 ≤ 𝐶∗

𝑉∗ − ∑ 𝑉𝑒𝑁𝑒=1 𝑒 = 0

𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚

where C* and V

* are the prescribed mean compliance and volume of the compliant

mechanism respectively. The design domain is discretized with finite elements, and

the volumetric density of each element, xe, is used as the design variable.


In order to satisfy the additional constraint, the prescribed mean compliance, the

119

original objective function is modified by adding the constraint and introducing a

Lagrange multiplier λ (Huang and Xie, 2010a). The modified objective function is

expressed by

Maximize 𝑕( ) = ∆𝑜𝑢𝑡 + × (𝐶∗ − 𝐶) (5.3)

Equation 5.3 points out that the modified objective function is equivalent to the

original one if the mean compliance is equal to its prescribed value. Otherwise λ = 0

if 𝐶 < 𝐶∗, which means the compliance constraint is already satisfied, and λ equals

to a certain value if 𝐶 > 𝐶∗ , which means the constraint algorithm should

minimize the mean compliance (or maximize −𝐶) to satisfy the constraint in the

later iterations. Therefore, the Lagrange multiplier is employed to design compliant

mechanisms with the compromise between maximizing △out and minimizing C so

as to satisfy the compliance constraint.

Consequently, the sensitivity number of the modified objective function can be

established as

𝛼𝑒 =𝑑ℎ

𝑑𝑥𝑒= ∆𝑜

𝑥𝑒−

𝑑𝐶


𝑥𝑒− (

𝑖𝑛

2

∆𝑖𝑛

𝑥𝑒− 𝑘 ∆𝑜𝑢𝑡

∆𝑜

𝑥𝑒) (5.4)

The derivatives of the input and output displacements can be obtained by using

Equations 4.18 and 4.19. The sensitivity of displacements ∆𝑖𝑗 can be found in

Equation 4.16.

5.3 Determination of Lagrange Multiplier

For the numerical implementation, the Lagrange multiplier λ can be defined as

= (1 − 𝜔) 𝜔⁄ , the objective function can be rewritten as

120

Maximize 𝑕( ) = 𝜔∆𝑜𝑢𝑡 + (1 − 𝜔)(𝐶∗ − 𝐶) (5.5)

where ω is a constant ranging from 0 to 1 which will be determined according to the

compliance constraint. The relative ranking of the sensitivity number is sorted

according to Equation 5.4.

Specifically, the volume constraint is firstly satisfied in the algorithm. The

procedure should determine appropriately the volume threshold of sensitivity

numbers, 𝛼𝑡ℎ . Then the elemental density is switched from 𝑒,𝑗 to 𝑒,𝑗−1 if

𝛼𝑒 > 𝛼𝑡ℎ, or from 𝑒,𝑗 to 𝑒,𝑗+1 if 𝛼𝑒 < 𝛼𝑡ℎ in the proposed BESO algorithm.

Secondly, to satisfy the compliance constraint in the algorithm, the variation of

the compliance ∆𝐶 is measured according to the variation of the design variables.

The mean compliance in the next iteration Ck+1

can be approximately estimated by

Equation 5.6, in which Ck represents the value of the mean compliance in the

current iteration and ∆ 𝑒 = 𝑒,𝑗−1 − 𝑒,𝑗 = 𝑒,𝑗 − 𝑒,𝑗+1, namely the difference of

two adjacent densities for each element. Here, ∆xe is set as 0.02, which means a

number of intermediate element densities.

𝐶𝑘+1 𝐶𝑘 + ∑𝜕𝐶

𝜕𝑥𝑒× ∆ 𝑒𝑒 (5.6)

Finally, the Lagrange multiplier can be determined by comparing the approximate

value and the maximum acceptable value. Therefore, 𝜔 will be replaced by

�̂� = (𝜔 + 𝜔𝑙𝑜𝑤𝑒𝑟) 2⁄ if 𝐶𝑘+1 > 𝐶∗. At the same time, 𝜔 will be selected as the

upper bound 𝜔𝑢𝑝𝑝𝑒𝑟 . Otherwise the procedure will update 𝜔 with �̂� =

(𝜔 + 𝜔𝑢𝑝𝑝𝑒𝑟) 2⁄ if 𝐶𝑘+1 < 𝐶∗ . Also, 𝜔 will be chosen as the lower bound

𝜔𝑙𝑜𝑤𝑒𝑟. With �̂� is continuously updated, the above procedure is repeated until

121

the difference of two bound values of ω is less than 10-5

(Huang and Xie, 2010a).


Here, the post-processing techniques of the modified BESO method are employed

in the optimal design of linear compliant mechanisms. To avoid checkerboard and

mesh-dependency, a filter scheme is used by Equation 3.9. The elemental


elements. Secondly, Equation 3.10 is adopted for the stabilization of the

evolutionary process. Finally, the new BESO algorithm will complete the whole

optimization process of designing compliant mechanisms once the performance of

the objective function satisfies the convergence criterion in Equation 3.12 and

material volume reaches the objective volume.


The following steps list the procedure of optimizing compliant mechanisms based

on the modified BESO method with variable material volume. In addition, Figure

5.1 is used to visually and briefly describe its iterative procedure.


compliant mechanism and loads for both the input and output dummy load cases.

Set material properties and mechanism parameters such as Poisson‟s ratio and

linear spring stiffness kout.

Step 2 Discretize the whole design domain using a finite element mesh. Define

relative BESO parameters like ER, penalty exponent p and the filter radius 𝑟𝑚𝑖𝑛.

Set key algorithm parameters such as the compliance constraint and element

122

densities.

Step 3 Carry out a finite element analysis and output the displacement fields from

both dummy load cases.

Step 4 Calculate sensitivity numbers with the help of Equation 5.4,

Step 5 Determine appropriately the volume threshold of sensitivity numbers for

achieving the volume constraint.

Step 6 Determine the Lagrange multiplier based on the procedure detailed in

Section 5.3



Step 8 Reset and rank all design variables, and set the threshold of the sensitivity

number, 𝛼𝑡ℎ , which is determined according to the target material volume

constraint Equation 3.11 and 𝑉∗ = ∑ 𝑉𝑒 𝑒 ( 𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚)𝑛𝑒=1 . The

corresponding density variable of eth element should be switched from 𝑒,𝑗 to

𝑒,𝑗−1 if 𝛼𝑒 > 𝛼𝑡ℎ, or from 𝑒,𝑗 to 𝑒,𝑗+1 if 𝛼𝑒 < 𝛼𝑡ℎ

Step 9 Repeat Steps 3-8 until both the volume fraction reaches objective volume

and the performance of objective function satisfies its convergence criteria

according to Equation 3.12.

123

Figure 5.1 Flow chart of the proposed BESO procedure with the compliance constraint for the optimal design of compliant mechanisms

Figure 5.1 Flow chart of the proposed BESO procedure with the compliance

constraint for the optimal design of compliant mechanisms

No

No

Yes

Yes






element mesh and set parameters about BESO





information

Calculate and satisfy the volume constraint



satisfied?

Is the procedure

convergent?

Update Lagrange multiplier of the mean

compliance constraint

124



In this example, the research applies the proposed method to the optimal design of

the compliant inverter, which has been shown in Figure 4.4. The design domain is

fixed at the upper and the lower corners of the left edge. Size of the design domain

is defined as 120μm×120μm and discretized with 120×120 4-node quadrilateral

elements. An input force Fin=1N to the right is applied at the center point of the

left edge. An artificial spring with stiffness kout=0.01μN/μm is attached at the

output port to simulate the resistance from the workpiece, but there is no attached

spring at input port. The material properties are assumed to be Young‟s modulus

E = 1 MPa and Poisson‟s ratio ν =0.3.

BESO starts from the initial full design with the mean compliance C0 and the output

displacement ∆𝑜𝑢𝑡0 . In the following discussion, both the mean compliance C and

the output displacement ∆𝑜𝑢𝑡 are normalized by C0 and ∆𝑜𝑢𝑡

0 . The volume fraction

of the final design is limited to be 30% of the full design domain and the

compliance constraint is set to be 𝐶 𝐶0⁄ ≤ 5. The used BESO parameters are

evolution rate ER=1% and filter radius 𝑟𝑚𝑖𝑛 = 3 . Figure 5.2 shows the

evolution history of the topology. It can be seen that there are some grey areas in

the intermediate topologies which are similar to that from other density-based

optimization methods (Bendsøe and Sigmund, 2003), due to the existence of a

number of intermediate density values. With the help of these intermediate

density values, stable hinge regions are formed in the final design as shown in

Figure 5.2f. However, unlike other density-based optimization methods with

continuous design variables, BESO utilizes discrete design variables so that the

possible local optimum can be easily avoided and the solution is quickly

125

convergent to an almost black and white design. Figure 5.3 shows the evolution

histories of the output displacement, mean compliance and volume fraction. It can

be seen that the magnitude of the output displacement gradually increases to its

negative maximum value until both the volume and compliance constraints are

stably satisfied. The convergent solution is achieved with 209 iterations under the

convergence factor of 0.1% for both the output displacement and the mean

compliance.

(a) (b) (c)

(d) (e) (f)

Figure 5.2 Evolution history of the topology for the compliant inverter

Figure 5.2 Evolution history of the topology for the compliant inverter (a) initial

topology; (b) topology at Iteration 50; (c) topology at Iteration 100; (d) topology at

Iteration 150; (e) topology at Iteration 200; (f) resulting topology.

126

(a) (b)

Figure 5.3 Evolution histories of the objective function and constraints

Figure 5.3 Evolution histories of the objective function and constraints (a)

evolution history of output displacement; (b) evolution histories of mean

compliance and volume fraction.

To demonstrate the effect of the compliance constraint on the optimized design, the

research sets the compliance constraint 𝐶 𝐶0⁄ ≤ 1.2, 2, 2.5, 3, 4, 5, 7.5, 10 and 20

respectively. The optimized topologies for nine different cases are shown in Figure

5.4. With a very small value of the compliance constraint such as 𝐶 𝐶0⁄ ≤ 1 2, the

resulting topology is the same to that of the traditional stiffness optimization

(Huang and Xie, 2010) resulting in the mean compliance 𝐶 𝐶0⁄ ≤ 1 34. Obviously,

it is impossible to satisfy such a low compliance constraint which is less than 1.34,

and BESO therefore would minimize the mean compliance only by automatically

applying ω = 0 in Equation 5.5. With a larger compliance constraint, BESO would

maximize the output displacement and minimize the mean compliance

simultaneously through gradually adjusting the value of ω, so as to satisfy the

specified compliance constraint. It can be observed that the optimized topologies

are free of hinges for the cases with 𝐶 𝐶0⁄ ≤ 2, 2.5, 3 and 4. With the increase of

the compliance constraint, the optimized topologies show numerous hinge regions

which lie along the force path from the mechanism input port to the output port.

Such hinge regions would favorably increase the flexibility of compliant

127

mechanisms to efficiently transfer the energy/motion from the input port to the

output port. However these hinge regions would greatly decreases the stiffness of

the compliant mechanism, apart from the existing difficulties in manufacturing.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 5.4 Optimized topologies and output displacements of the compliant inverters for various compliance constraints


inverters for various compliance constraints (a) 𝐶 𝐶0⁄ ≤ 1 2 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = 0;

(b) 𝐶 𝐶0⁄ ≤ 2 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −2 015 ; (c) 𝐶 𝐶0⁄ ≤ 2 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ =

−3 747 ; (d) 𝐶 𝐶0⁄ ≤ 3 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −6 108 ; (e) 𝐶 𝐶0⁄ ≤ 4 and

∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −10 766; (f) 𝐶 𝐶0⁄ ≤ 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ = −14 806; (g) 𝐶 𝐶0⁄ ≤

7 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −21 515; (h) 𝐶 𝐶0⁄ ≤ 10 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ = −26 253; (i)

128

𝐶 𝐶0⁄ ≤ 20 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −37 169.

As mentioned above, the design of compliant mechanism is inherently a

multi-objective optimization problem. The resulting output displacement

∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ is plotted against the values of mean compliances in Figure 5.5 which

shows the Pareto front with a convex curve. Generally, the magnitude of the output

displacement increases with the increase of the compliance constraint. The design

of compliant mechanisms is therefore a matter of making a trade-off decision from

a set of compromising solutions. The solid line shown in Figure 5.5 gives the

possible range within which the optimized topologies are free of hinges. In other

words, the problematic hinges can be effectively eliminated by adopting a low

compliance constraint. Compared with the approach for eliminating hinges by

introducing the input-restrained compliance (Yoon et al., 2004; Rahmatalla and

Swan, 2005), the current approach compromises the output displacement with the

improved load-carrying capacity of the compliant mechanism rather than that of the

input-restrained structure.

Figure 5.5 Relationship between the output displacement and mean compliance of optimized compliant mechanisms.

Figure 5.5 Relationship between the output displacement and mean compliance of

optimized compliant mechanisms.

129

The design domain and boundary conditions of the second example are shown in

Figure 5.6 as a 120μm×120μm square with a 30μm×24μm gap which allows the

workpiece to be gripped. The design domain is meshed with 4-node quadrilateral

elements with the uniform sizes 1μm×1μm. An input force Fin=1N to the right is

applied at the center point of the left edge. The material properties are assumed to

be Young‟s modulus E = 1 MPa and Poisson‟s ratio ν =0.3. An artificial spring

with stiffness kout=0.01μN/μm is attached at the output port to simulate the

resistance from the workpiece. The vertical displacement, ∆out at the output port

will be maximized to grip the workpiece firmly. The BESO parameters used are

evolution rate ER=1%, filter radius rmin = 3μm. The objective volume fraction is

set to be 30% of the full design domain.

Figure 5.6 Design domain and boundary conditions of the compliant gripper

Figure 5.6 Design domain and boundary conditions of the compliant gripper

BESO starts from the full design and the resulting optimized topologies are shown

in Figure 5.7 for various compliance constraints. When the compliance constraint

𝐶 𝐶0⁄ ≤ 1 2, the optimized design is the same as that of the traditional stiffness

optimization. For 𝐶 𝐶0⁄ ≤ 2, the optimized topology show that most of materials

are distributed near the input port and no hinges have formed due to the low value

120

Fin, △in △out

△out 24

30

130

of the compliance constraint. As the compliance constraint increases, more and

more materials are shifted from a position close to the input port to a position close

to the output port. The middle connections become narrower and narrower, and

clear hinge regions are formed when the compliance constraint 𝐶 𝐶0⁄ ≤ 4 by the

inspection. With the further increase of the compliance constraint, the hinge regions

become even clearer as shown in Figure 5.7h. It is also observed that the hinges are

located at the output port side for 𝐶 𝐶0⁄ ≤ 4, 5 and 7.5, exactly at the connections

for 𝐶 𝐶0⁄ ≤ 10 and at the input port side for 𝐶 𝐶0⁄ ≤ 20. For 𝐶 𝐶0⁄ ≤ 20, the

hinge regions are completely composed by grey areas which also demonstrate the

importance of intermediate density values.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 5.7 Optimized topologies and output displacements of the compliant grippers for various compliance constraints

131


grippers for various compliance constraints (a) 𝐶 𝐶0⁄ ≤ 1 2 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = 0;

(b) 𝐶 𝐶0⁄ ≤ 2 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −16 092 ; (c) 𝐶 𝐶0⁄ ≤ 2 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ =

−30 047 ; (d) 𝐶 𝐶0⁄ ≤ 3 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −43 762 ; (e) 𝐶 𝐶0⁄ ≤ 4 and

∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −70 928; (f) 𝐶 𝐶0⁄ ≤ 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ = −90 058 (g) 𝐶 𝐶0⁄ ≤

7 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −134 340 (h) 𝐶 𝐶0⁄ ≤ 10and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ = −158 970 ;

(i) 𝐶 𝐶0⁄ ≤ 20 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −252 328

The above examples with kout=0.01μN/μm show that hinge-free optimized designs

can be obtained only when the compliance constraint is less than 3. It is necessary

to examine the effect of the stiffness of the workpiece on the optimized topology.

Figure 5.8 shows the optimized designs by imposing the constraint 𝐶 𝐶0⁄ ≤ 5 but

for kout=0.03μN/μm, 0.1μN/μm, 1μN/μm and 10μN/μm respectively. It can be seen

that these optimized topologies have no significant difference from the optimized

topology for kout=0.1μN/μm except that optimized topologies are hinge-free. The

connections between the input port and the output port become stronger and

stronger with the increase of the stiffness of the spring. The set of design

computation results presented in Figure 5.9 is intended to identify the region of

free-hinge mechanism designs by considering the effects of kout and imposed

compliance constraint. It can be seen that it is only possible to obtain hinge-free

compliant mechanisms when the compliance constraint is less or equal than about

10. Even when the compliance constraint is less than 10, hinge regions still occur at

the output port side of the connections, as shown by the inset for a spring with low

stiffness. Such hinges can be effectively eliminated by increasing the stiffness of

the spring, which consequently increases the stiffness at the output part side.

However, when the compliance constraint is larger than 10, hinge regions normally

occur at the input port side of the connections as shown by the inset. Increasing the

stiffness of the spring has little effect on the stiffness of the input port side and

132

therefore fails to preclude the formation of these hinges.

(a) (b)

(c) (d)

Figure 5.8 Optimized topologies of the compliant grippers for various stiffness of the spring

Figure 5.8 Optimized topologies of the compliant grippers for various stiffness of

the spring (a) kout=0.03μN/μm; (b) kout=0.1μN/μm; (c) kout=1μN/μm; (d)

kout=10μN/μm;

Figure 5.9 Optimized designs with hinges or without hinges under various compliance constraints and stiffness of the spring

133

Figure 5.9 Optimized designs with hinges or without hinges under various

compliance constraints and stiffness of the spring (corresponding output

displacement ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ is shown with the value in the brackets).

The above results indicate that it is difficult to design a hinge-free compliant

mechanism for a soft workpiece such as kout=0.01μN/μm. In order to obtain a

hinge-free compliant mechanism for a soft workpiece, we may select a soft material

for the compliant mechanism. Numerical examples in Figure 5.10 demonstrate that

hinge-free optimized mechanisms for kout=0.01μN/μm and 𝐶 𝐶0⁄ ≤ 5 can can

still be achieved by using a soft material such as E = 0.1MPa or 0.01MPa. It also

indicates that the selection of materials is a very important factor for designing

compliant mechanisms, which is completely different from the stiffness

optimization in which the optimized topology is independent from the stiffness of

the material (Bendsøe and Sigmund, 2003).

(a) (b)

Figure 5.10 Optimized topologies of the compliant gripper with soft materials

Figure 5.10 Optimized topologies of the compliant gripper with soft materials (a)

E = 0.1; (b) E = 0.01.


The current BESO algorithm can be easily extended for 3D cases. Figure 5.11

134

shows the design domain and boundary conditions of a 3D gripper mechanism. The

force Fin=1kN is applied to the input port located at the center of the left face. A

spring with kout=1kN/mm is attached to the output port as shown in Figure 5.11.

Upper and lower edges of left face are fixed. The design domain is discretized with

8-node brick elements with a uniform size of 1mm×1mm×1mm. Here, the material

considered is nylon with Young‟s modulus E = 3GPa and Poisson‟s ratio ν =0.4.

The BESO parameters are evolution rate ER =1% and filter radius rmin = 3mm. The

objective volume fraction is set to be 10% of the full design domain.

Figure 5.11 Design domain and boundary conditions of a 3D compliant gripper

Figure 5.11 Design domain and boundary conditions of a 3D compliant gripper

The optimized designs are shown in Figure 5.12 when the compliance constraint

𝐶 𝐶0⁄ ≤ 2, 3 and 5 is adopted respectively. The resulting optimized topologies for

these three cases are different due to the different compliance constraint. The

corresponding output displacements ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ are -236, -412 and -688

respectively. Similar to 2D cases, the magnitude of the output displacement

increases with the increase of the compliance constraint. The deformed

configurations of the smooth CAD models for these optimized topologies are also

Fin, △in △out

20

100

20

40

100

135

given in Figure 5.12

(a) (b) (c)

(d) (e) (f)

Figure 5.12 Optimized topologies and CAD models of 3D compliant grippers with various compliance constraints

Figure 5.12 Optimized topologies and CAD models of 3D compliant grippers with

various compliance constraints: (a) 𝐶 𝐶0⁄ ≤ 2 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ ≤ −236 ; (b)

𝐶 𝐶0⁄ ≤ 3 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ ≤ −412; (c) 𝐶 𝐶0⁄ ≤ 5 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ ≤ −688; (d)

deformed configuration of CAD model for (a); (e) deformed configuration of CAD

model for (b); (f) deformed configuration of CAD model for (c).

5.7 Conclusions

The research proposed a new BESO method for topology optimization of

136

compliant mechanisms by maximizing the flexibility subject to the mean

compliance and volume constraints. The compliance constraint is imposed on the

optimization algorithm by introducing a Lagrange multiplier. Based on the derived

sensitivity numbers, the discrete design variables are gradually updated until a

convergent solution is achieved.

Using the proposed BESO method, a set of optimized designs with and without

hinges has been obtained by altering the desired structural stiffness of compliant

mechanisms. The flexibility of the optimized compliant mechanism generally

increases with the compliance constraint, so that the resulting design has the

improved capacity of transferring the energy/motion from the input port to the

output port. However, such an optimized compliant mechanism may have low

stiffness to resist external forces and also contain hinge regions which are difficult

to manufacture especially for a micro system. To achieve a hinge-free compliant

mechanism, it is recommended that the flexibility of the compliant mechanism be

optimized by specifying a low compliance constraint or increasing the stiffness of

the workpiece. For an extreme soft workpiece, selecting a soft material for

compliant mechanisms can also preclude the formation of hinge regions during the

optimization process.

References



Design 44(1-2), 53-62.

Bendsøe, M. P. and Sigmund, O., 2003. Topology Optimization: Theory, Method and Application,

Springer, Berlin.


137





Huang, X. and Xie, Y. M., 2010a, "Evolutionary topology optimization of continuum structures with

an additional displacement constraint", Structural and Multidisciplinary Optimization

40(16), 409-416.






40(1-6), 241-255.



Engineering 62(12), 1579-1605.

Yoon, G. H., Kim, Y. Y., Bendsøe, M. P. and Sigmund, O., 2004, "Hinge-free topology optimization

with embedded translation-invariant differentiable wavelet shrinkage", Structural and


Zhu, B., Zhang, X. and Wang, N., 2013, "Topology optimization of hinge-free compliant

mechanisms with multiple outputs using level set method", Structural and


138

Chapter 6

Evolutionary Topology Optimization

of Hinge-free Compliant Mechanisms

139

Chapter Overview

This chapter aims to develop a BESO method for the design of hinge-free

compliant mechanisms. A new objective function will be proposed to maximize

the desirable displacement and simultaneously preclude the formation of hinges.

Sensitivity numbers are derived according to the variation of the objective

function with respect to the design variables. Based on the resulting sensitivity

numbers, the BESO procedure is established by gradually removing and adding

elements until an optimal topology is achieved. Several 2D and 3D examples are

given to demonstrate the effectiveness of the proposed BESO method for the

design of various hinge-free compliant mechanisms.


The designs of compliant mechanisms using topology optimization techniques

naturally lead to the introduction of hinges into the models, making them function

essentially as rigid-body mechanisms (Rahmatalla and Swan, 2005). Such hinge

zones cause high stress concentration and are difficult to fabricate for micro-scale

systems.

Recently, several different techniques have been established and developed for

eliminating de facto hinges in the design of compliant mechanisms such as

embedding wavelet-base functions (Poulsen, 2002; Yoon et al., 2004), imposing a

minimum length constraint (Poulsen, 2003) and filter schemes (Sigmund, 2007).

Such morphology-based approaches could greatly reduce the occurrence of

one-node connected hinges, but were not entirely effective due to the nature of the

optimization problem. Reformulating the problem as a multi-criteria optimization

might be an effective way for entirely circumventing de facto hinge regions which

140

generally lie along the force path from the mechanism input port to the output port.

Therefore, this research develops a new BESO algorithm for optimally designing

hinge-free compliant mechanisms. The strain energy of the structure is introduced

into the formulation of the optimization problem. Based on the finite element

analysis and sensitivity analysis, a BESO procedure is established to evolve the

compliant mechanism to an optimum. Several examples are presented to

demonstrate the effectiveness of the proposed method for designing various

hinge-free compliant mechanisms. Different from other density-based topology

optimization methods, the traditional BESO can provide clear topological design of

compliant mechanisms without any „grey‟ areas.

Consider a general design domain Ω under given loading and boundary conditions,

which have been depicted in Figure 6.1a. It is assumed that the applied force at the

input port i is Fin and the reaction force at the output port j is Fout. The latter force is

acting on a workpiece which is modeled by a spring with a constant stiffness, kout.

The resulting input displacement is △in at the input port i and the output

displacement △out at the output port j. As mentioned, an efficient compliant

mechanism should be flexible enough to produce expected kinematic motion under

the action of applied loads to satisfy the flexibility requirement (Joo and Kota,

2004). Therefore, the design objective could be maximizing a displacement ratio,

which is called geometric advantage out inGA . Meanwhile, the compliant

mechanism becomes a structure as shown in Figure 6.1b when the output port is

fixed. Such a structure should possess a certain stiffness to limit the input

displacement at the input port. In other words, a compliant mechanism should be

flexible enough to produce the expected kinematic motion and stiff enough to resist

applied forces (Nishiwaki et al., 1998; Joo et al., 2000). The stiffness of the

structure can be inversely measured by the mean compliance or total strain energy

in the structure. Furthermore, maximizing the structural stiffness or minimizing the

141

total strain energy can also preclude the formation of hinges in the design of

compliant mechanism (Rahmatalla and Swan, 2005). Considering those factors, we

can mathematically formulate an optimization problem for the design of a

hinge-free compliant mechanism as follows

Maximize 𝑕 = 𝐺𝐴 𝑆𝐸⁄ (6.1)


𝑒 = 𝑚𝑖𝑛 𝑜𝑟 1

where 𝑆𝐸 = 𝐮 𝐮 2⁄ is the total elastic strain energy in the continuum. The

structure has been shown in Figure 6.1b. A small value of 𝑚𝑖𝑛, e.g. 0.001, is used

to denote the soft elements.

(a) (b)

(c) (d)

Figure 6.1 A general design model for hinge-free compliant mechanism

Input Port i

Fin , △in

Output Port j

kout

Fout

Ω △out

Ω

F1, △11 △21

Ω

△12 F2, △22

Ω

142

Figure 6.1 A general design model for hinge-free compliant mechanism (a)

mechanism design; (b) structure design; (c) input load case; (d) output load case

The analysis of the compliant mechanism can be the superposition of two load

cases as shown in Figures 6.1c and d. The compliant mechanism is equivalent to the

linear combination of two structures when the displacement relationships are

satisfied in Equation 4.3. The forces should satisfy the relationship of Equation 4.4,

and the input and output displacements can be found in Equation 4.6. According to

the dummy load approach, displacements ∆𝑖𝑗 can be expressed as Equation 4.9,

in which 𝐮1 and 𝐮2 represent the displacement fields of two structures shown in

Figures 6.1c and 6.1d respectively. Similarly, the displacement field of the structure

in Figure 6.1b, u can be expressed by 𝐮1 and 𝐮2 as

𝐮 = 𝐮1 −∆11

∆12× 𝐮2 (6.2)


The sensitivity of displacements ∆𝑖𝑗 can be found in Equation 4.15. The

sensitivity of the displacement can be expressed at elemental levels for solid and

soft elements, as two discrete design variables, 𝑒 = 1 (solid elements) and 𝑚𝑖𝑛

(soft elements), are employed in the soft-kill BESO (Huang and Xie, 2010).

∆𝑖

𝑥𝑒= {

−𝑝𝐮𝑗,𝑒 𝑒

0𝐮𝑖,𝑒 𝐹𝑖 𝑤𝑕𝑒𝑛 𝑒 = 1⁄

−𝑝 𝑚𝑖𝑛𝑝−1𝐮𝑗,𝑒

𝑒0𝐮𝑖,𝑒 𝐹𝑖 𝑤𝑕𝑒𝑛 𝑒 = 𝑚𝑖𝑛⁄

(6.3)

With the help of Equations 4.18 and 4.19, the sensitivity of the geometric

advantage can be obtained.

143

𝑥𝑒= (∆𝑖𝑛 ×

𝜕∆𝑜

𝜕𝑥𝑒− ∆𝑜𝑢𝑡 ×

𝜕∆𝑖𝑛

𝜕𝑥𝑒) ∆𝑖𝑛

2⁄ (6.4)

According to the adjoint method (Buhl et al., 2000), the sensitivity of the strain

energy for the structure of Figure 6.1b can be easily obtained by

𝑥𝑒= −

1

2𝐮

𝜕

𝜕𝑥𝑒𝐮 = −

1

2𝑝 𝑒

𝑝−1𝐮𝑒 𝑒

0𝐮𝑒 (6.5)

With the help of Equation 6.2, the sensitivity of the total strain energy can be

explicitly expressed for solid and void elements by

𝑥𝑒= {

−1

2𝑝 (𝐮1,𝑒

−∆11

∆12𝐮2,𝑒 ) 𝑒

0 (𝐮1,𝑒 −∆11

∆12𝐮2,𝑒) 𝑤𝑕𝑒𝑛 𝑒 = 1

−1

2𝑝 𝑚𝑖𝑛

𝑝−1 (𝐮1,𝑒 −

∆11

∆12𝐮2,𝑒 ) 𝑒

0 (𝐮1,𝑒 −∆11

∆12𝐮2,𝑒)𝑤𝑕𝑒𝑛 𝑒 = 𝑚𝑖𝑛

(6.6)

As the result, the elemental sensitivity number which measures the sensitivity of

the objective function is established accordingly

𝑒 = (𝑆𝐸 ×𝜕

𝜕𝑥𝑒− 𝐺𝐴 ×

𝜕

𝜕𝑥𝑒) 𝑆𝐸2⁄ (6.7)


The post-processing techniques of the BESO method are employed in the optimal

design of compliant mechanisms. Firstly, a filter scheme is used to avoid

checkerboard and mesh-dependency. The nodal sensitivity number is obtained

based on Equation 3.7. The element smooth technique is to convert the nodal

sensitivity back to the elemental sensitivity according to Equation 3.8. Secondly,

Equation 3.10 is adopted for the stabilization of the evolutionary process. Finally,

BESO will terminate the optimization process once the variation of the objective

144

function satisfies the convergence criterion (refer to Equation 3.12).


The research summarizes the steps of the proposed algorithm of designing

hinge-free compliant mechanism with using the traditional BESO approach in

detail. In addition, the iterative procedure is briefly and visually described in

Figure 6.2.

Step 1 Present the design domain with boundary conditions of compliant

mechanism. Set material properties and define mechanism parameters such as

Young‟s modulus, Poisson‟s ratio, value of input force and linear spring stiffness

kout.


relative BESO parameters such as objective volume 𝑉∗, evolutionary ratio ER,

penalty exponent p and the filter radius 𝑟𝑚𝑖𝑛.

Step 3 Carry out a finite element analysis which is conducted by ABAQUS.

Produce required data such as nodal displacements.

Step 4 Calculate sensitivity numbers according to Equation 6.7, update sensitivity

numbers by the filter scheme Equation 3.8 and average with their historical

information by Equation 3.10.

Step 5 Determine the target volume for the next design iteration. Equation 3.11 is

used to calculate the target volume for the next iteration 𝑉𝑖+1. 𝑉𝑖+1will be set as

𝑉∗ if the calculated volume for the next design satisfies the objective volume.

145

Step 6 Rank all elements based on their sensitivity numbers in descending order

and set the threshold of the sensitivity number 𝛼𝑡ℎ so that the total volume of

elements with 𝛼𝑖 > 𝛼𝑡ℎ equals the target volume of the next iteration Vi+1.

Step 7 Reset the design variables of the design domain. That is, for solid elements,

the elemental density is switched from 1 to 𝑚𝑖𝑛 if 𝛼𝑖 ≤ 𝛼𝑡ℎ, and the element

density for soft elements will be changed from 𝑚𝑖𝑛 to 1 if 𝛼𝑖 > 𝛼𝑡ℎ.

Step 8 Repeat Steps 3-7 until both the volume fraction reaches target volume and

the performance of objective function satisfies its convergence criteria according to

Equation 3.12.

146

Figure 6.2 Flow chart of the BESO procedure for the optimal design of compliant mechanisms

Figure 6.2 Flow chart of the BESO procedure for the optimal design of compliant

mechanisms



No

Yes

Yes






element mesh and set BESO parameters





information

Determine the target volume for the next design



satisfied?

Is the procedure

convergent?

No

147

In this example, the proposed topology optimization approach is applied to the

optimal design of an inverter mechanism, which outputs the displacement in an

opposite direction from that of the input force. The design domain is

200mm×200mm as shown in Figure 6.3, which is discretized with 100×100

4-node quadrilateral elements. It is simply supported at the top and the bottom

corners of the left edge. An input force 𝐹𝑖𝑛 = 1 is applied at the center of the

left edge in horizontal direction. The output port at the center of the right edge is

expected to produce a horizontal displacement △out to the right. The material

properties are Young‟s modulus 𝐸 = 200 and Poisson‟s ratio = 0 3. The

BESO parameters for this simulation are evolutionary rate 𝐸𝑅 = 2 , the filter

radius 𝑟𝑚𝑖𝑛 = 6 , penalty exponent 𝑝 = 3 and 𝑚𝑖𝑛 = 0 001.

Figure 6.3 Design domain and boundary conditions of designing hinge-free inverter mechanism

Figure 6.3 Design domain and boundary conditions of designing hinge-free

inverter mechanism

The optimized designs for the inverter mechanism are shown in Figure 6.4. It

indicates that the optimized topology and the deformed shape change with the

volume constraint and spring constant. The deformed shapes are clearly shown

that the output port (the center of the right edge) moves in the desired direction

while transmitting the input force to the output port. The final displacements are

△out Fin , △in

148

respectively 1.14×10-2

mm, 5.07×10-2

mm, 1.09×10-2

mm and 4.78×10-2

mm. The

resulting hinge-free topologies are very similar to those obtained using the

density-based optimization approach (Sigmund, 1997) but without any “grey

area”. This example demonstrated that the proposed formulation is effective for

solving the problem of the design of compliant mechanisms.

(a)

(b)

(c)

(d)

Figure 6.4 Optimum topologies for the hinge-free inverter mechanism

149

Figure 6.4 Optimum topologies for the hinge-free inverter mechanism (a) optimum

topology and deformed configuration, 𝑉∗ = 40 and 𝑘 = 0 ; (b) optimum

topology and deformed configuration, 𝑉∗ = 20 and 𝑘 = 0 ; (c) optimum

topology and deformed configuration, 𝑉∗ = 40 and 𝑘 = 0 2 / ; (d) optimum

topology and deformed configuration, 𝑉∗ = 20 and 𝑘 = 0 2 / .

Figure 6.5 shows the evolution histories of volume fraction and objective function

for the inverter mechanism of 0.2 volume rate with 𝑘 = 0 2 / . BESO starts

with the full design and gradually decreases the volume fraction until it satisfies

its constraint value 20%. After 86 iterations, both the topology and objective

function are convergent.

Figure 6.5 Evolution histories of the hinge-free inverter mechanism

Figure 6.5 Evolution histories of the hinge-free inverter mechanism

The second example demonstrates the design of a gripper mechanism. The design

domain is shown in Figure 6.6a as a 200mm×400mm rectangle with a

50mm×100mm gap which allows the workpiece to be gripped. The whole design

150

domain is discretized with 18750 4-node quadrilateral elements. A linear spring

with 𝑘 = 0 2 / is defined to simulate the stiffness of the workpiece. This

design domain is simply supported at the top and the bottom corners of the left

edge. The input load 𝐹𝑖𝑛 = 1 is applied at the centre of the left edge. The

objective is to synthesize the design so that the gripper produces efficiently

gripping force Fout or output displacement ∆𝑜𝑢𝑡 on the workpiece under the

action of input force. The material properties are Young‟s modulus 200E GPa

and Poisson‟s ratio 3.0v .

(a)

(b) (c)

Figure 6.6 Topology optimization for the hinge-free gripper mechanism

Figure 6.6 Topology optimization for the hinge-free gripper mechanism (a) design

200

△out

△out Fin, △in

100

50

150

151

domain and boundary conditions; (b) optimum topology and deformed

configuration subjected to a 40% volume constraint; (c) optimum topology and

deformed configuration subjected to a 20% volume constraint

Figures 6.6b and 6.6c illustrate the resulting topologies and their deformed

configurations, the final volumes of which are restricted to be 40% and 20% of

the design domain respectively. It could be observed that parts around output

ports have formed topology shapes which are stiff enough to grip the workpiece.

For the gripper mechanism with a 0.2 volume rate, the resulting output

displacement reaches 1.177×10-2

mm inward, which is much larger than the initial

value of 2.401×10-5

mm. The resulting topology subjected to 40% volume

constraint can achieve an output displacement of 2.477×10-3

mm.


Here, the proposed optimization method is extended to the application of 3D

hinge-free compliant mechanisms, which is unusual in previous topology

optimization research. To illustrate the capability and effectiveness of

three-dimensional applications, a gripper mechanism is presented by this research

and depicted in Figure 6.7a. This domain with dimensions

150mm×150mm×100mm and with a hollow is equally divided into 32000

eight-node cubic elements. A linear spring with 𝑘 = 0 2 / is used to

simulate the stiffness of the workpiece. This design domain is supported at the top

and the bottom centers of the left face. The input load 𝐹𝑖𝑛 = 1 is applied at the

center of the left face.

152

(a)

(b) (c)

Figure 6.7 Topology optimization for the 3D hinge-free gripper mechanism

Figure 6.7 Topology optimization for the 3D hinge-free gripper mechanism (a)

Design domain and boundary conditions; (b) FE optimized topology; (c) Deformed

configuration of the CAD model.

The objective is to obtain the gripper efficiently produce a gripping force Fout or

output displacement ∆𝑜𝑢𝑡. The material properties are a Young‟s modulus of

150

100

100

50

50

△out

△out

Fin, △in

153

200GPa and a Poisson‟s ratio of 0.3. The final optimal design‟s 20% effective

volume is obtained after 89 iterations and shown in Figure 6.7b, corresponding to

the deformation in Figure 6.7c. Inward displacements are obviously achieved

when the input force is applied at the resulting topology. It can be concluded that

it is similar to the final optimal design proposed by the previous 2D case.

Although there is no hole formed at right side when comparing it to the 2D

example, a number of elements have been removed in the corresponding part. The

study considers that it is feasible to apply the BESO method to 3D hinge-free


The design problem for a 3D elevation mechanism is sketched in Figure 6.8a. The

design domain is a 200mm×70mm×100mm rectangular cuboid which is

discretized with 89600 eight-node brick elements. It is supported at the four lower

corners and subjected to unit horizontal loads acting at the center of both side

surfaces. An upward displacement expected is produced at the center of the top

surface. The material properties are assumed to be Young‟s modulus, 200GPa and

Poisson‟s ratio, 0.3. The BESO parameters for this simulation are evolutionary

rate 𝐸𝑅 = 2 , the filter radius 𝑟𝑚𝑖𝑛 = 8 , penalty exponent 𝑝 = 3 and

𝑚𝑖𝑛 = 0 001.

154

(a)

(b) (c)

Figure 6.8 Topology optimization for the 3D hinge-free elevation mechanism

Figure 6.8 Topology optimization for the 3D hinge-free elevation mechanism (a)


configuration of the CAD model.

The design objective of the elevation mechanism is to maximize the desired

output displacement but also possess a certain structural stiffness. Figure 6.8b

gives the optimized topology using the proposed topology optimization approach,

and the deformation of the CAD model is presented in Figure 6.8c. Upward

displacements are obviously achieved when the input forces are applied to the

resulting topology. This example also demonstrates that it is feasible to apply the

proposed topology optimization approach to 3D hinge-free compliant mechanisms

with high computational efficiency.

70

Fin, △in

200

Fin, △in

100

△out

155

Finally, this research analyses the design problem for a 3D contraction

mechanism. The design domain is shown in Figure 6.9a and equally divided into

76032 eight-node cubic elements. The mechanism is supported at the four lower

corners and is subjected to a vertical input load Fin=1N acting at the center of the

bottom face. Four corners situated at the top face of the 90mm×90mm×45mm

groove are expected to produce a contraction inwards when a unit input force is

applying at the structure. Iterative process reaches the allowable volume, namely

10% for solid elements, after 118 iterations. The final optimal topology and the

deformation of the CAD model for the contraction mechanism are presented in

Figures 6.9b and 6.9c. It can be seen that the desired displacements are obtained

through the elastic deformation of the mechanism which is composed of clear

members without any hinges.

(a)

(b) (c)

Figure 6.9 Topology optimization for the 3D hinge-free contraction mechanism

180

180

135

Fin, △in

△out

156

Figure 6.9 Topology optimization for the 3D hinge-free contraction mechanism (a)


configuration of the CAD model

6.6 Conclusions

This research proposed a BESO procedure for the design of 2D and 3D hinge-free

compliant mechanisms. The optimization problem is formulated as maximizing

the geometry advantage and structural stiffness of the compliant mechanisms

simultaneously. The sensitivity analysis is conducted by applying the adjoint

method to both the kinematical function and the structural function and then the

BESO procedure is adopted based on the resulting sensitivity numbers. Through

gradually removing and adding material, the optimized topology of compliant

mechanisms is achieved for the defined design. The formation of hinges has been

successfully precluded due to the introduction of the strain energy into the

objective function. Numerical results show that desirable output displacements are

obtained through the elastic deformation of the mechanism designs and

demonstrate the effectiveness of the proposed BESO algorithm for the design of

various 2D and 3D hinge-free compliant mechanisms. Compared with other

density-based optimization methods, the current BESO algorithm provides clear

solid-void topologies of compliant mechanisms without any “grey” areas.

Overall, based on BESO methods, new formulations proposed in this dissertation

greatly promote topology optimization of linear compliant mechanisms. However,

it should be pointed out that the mentioned methods are limited in the framework of

linearized elasticity analysis. A nonlinear analysis is necessary in real applications

due to practical accuracy. Certainly, some concepts presented in the linear analysis

can be extended to the design of compliant mechanisms with large-displacement

157

behavior. Therefore, the next chapters will present the related nonlinear finite

element analysis and topology synthesis of large-displacement compliant

mechanisms.

References

Buhl, T., Pedersen, C. B. W. and Sigmund, O., 2000, "Stiffness design of geometrically nonlinear

structures using topology optimization", Structural and Multidisciplinary Optimization

19(2), 93-104.



Joo, J. and Kota, S., 2004, "Topological synthesis of compliant mechanisms using nonlinear beam

elements", Mechanics Based Design of Structures and Machines 39(1), 17-38.

Joo, J., Kota, S. and Kikuchi, N., 2000, "Topological synthesis of compliant mechanisms using

linear beam elements", Mechanics of Structures and Machines 28(4), 245-280.




Poulsen, T. A., 2002, "Topology optimization in wavelet space", International Journal for

Numerical Methods in Engineering 53(3), 567-582.

Poulsen, T. A., 2003, "A new scheme for imposing a minimum length scale in topology

optimization", International Journal for Numerical Methods in Engineering 57(6),

741-760.



Engineering 62(12), 1579-1605.



158

Sigmund, O., 2007, "Morphology-based black and white filters for topology optimization",

Structural and Multidisciplinary Optimization 33(4-5), 401-424.

Yoon, G. H., Kim, Y. Y., Bendsøe, M. P. and Sigmund, O., 2004, "Hinge-free topology optimization

with embedded translation-invariant differentiable wavelet shrinkage", Structural and


159

Chapter 7

Topology Design of Nonlinear

Compliant Mechanisms

160

Chapter Overview

This dissertation has proposed several approaches for the systematic design field of

linear compliant mechanisms, demonstrating the capability and validity of the

improved algorithms. However, the optimal design of nonlinear compliant

mechanisms is necessary in real applications as the simulation is more accurate.

In this chapter, the research focuses on optimizing compliant mechanisms with

large-deformation behavior by extending related concepts from linear compliant

mechanism designs of Chapter 5.

7.1 Differences between Structural Linearity and Nonlinearity

Sometimes, the solutions to engineering problems are based on linear

approximations. Linear approximations can be considered and used to represent the

nonlinear structural analysis in many cases. For example, the structure produces a

small deformation, which can be neglected in equilibrium. The strain is

proportional to the stress. Forces are conservative and independent on

displacements in structural analysis. Structural supports remain unchanged when

loading. However, there are marked differences between linear and nonlinear

analysis as described in Table 7.1.

161

Table 7.1 Comparison of linear and nonlinear structure analysis (Becker, 2001)

Performances Linear Structure Nonlinear Structure

The relationship

between load and

displacement

The load-displacement

relationships are linear in the

structural analysis

Displacements are nonlinearly

dependent on the applied loads

The relationship

between stress and

strain

The structural analysis

assumes a linear relationship

between stress and strain

The relationship between stress and

strain is formulated as a nonlinear

function for material nonlinearity

Magnitude of

displacement

Changes in geometry owing

to displacement are identified

as small deformation.

The structure produces large

displacements.

Material properties Linear elastic material

properties Nonlinear material properties

Reversibility

The structural behavior is

reversible after removing the

external loads

The structural behavior is not

reversible in that the structural state is

different from the initial state after the

removal of the external loads

Boundary conditions Boundary conditions remain

same during the analysis

Boundary conditions vary for

boundary nonlinearity

Loading sequence

Loading sequence does not

play an important role and

affect the final state

The load history affects structural

behavior

Iterations and

increments

There is no iteration in a load

step when the force is applied

at the structure

The load is divided into small

increments. Iterations are presented to

solve the equilibrium problem at each

load increment.

Computation time

The structural analysis spends

less computation time than

nonlinear problem

More time is required as load

incrementation and iterations need

more solution steps

In addition, the set of equations can be described as Equation 7.1 for linear

structural behaviors in the Finite Element Analysis.

𝐮 = (7.1)

K, u and F represent the stiffness matrix of the structure, nodal displacements

vector and the external nodal force vector respectively.

162

In fact, real engineering problems often involve nonlinear behavior, like thin shell

and slender beam. As for the optimal design of compliant mechanisms, the

deformation may be often so large that configuration changes cannot be neglected.

As a result, structural behaviors should be considered as nonlinearity. To

incorporate such phenomena into finite element analysis, a set of nonlinear

algebraic equations can be expressed as

(𝐮) = (7.2)

It denotes the equilibrium equation of vectors of externally applied nodal loads and

internally generated nodal forces. The stiffness equation is switched into (𝐮)𝐮 =

, in which the stiffness matrix (𝐮) is a function of the nodal displacements

vector and named as secant stiffness matrix (or Ks). Tangent stiffness matrix

represents the slope of the tangent to the force-displacement curve at any point and

can be expressed as Equation 7.3. Figure 7.1 displays the secant and the tangent

stiffness matrices by using the slope of straight lines.

=𝜕

𝜕𝑢=𝜕 𝐮

𝜕𝑢 (7.3)

Figure 7.1 Tangent and secant stiffness u0

Linear stiffness KL

Secant

stiffness Ks

Tangent stiffness KT

163

Figure 7.1 Tangent and secant stiffness

7.2 Types of Structural Nonlinearity

Nonlinear behaviors can be grouped into three main behaviors, namely boundary

nonlinearity, material nonlinearity and geometrical nonlinearity. One of them,

boundary nonlinearity involves in the change of the status in the structural analysis.

Deformation is not independent on external boundary conditions when loading. In

other words, component contacts produce stresses and friction, resulting in

disproportionate changes in deformation.

In addition, material nonlinearity exhibits nonlinear stress-strain relationship.

Possible material models include nonlinear elastic, elastoplastic, viscoelastic and

viscoplastic (Bhashyam, 2002). It is different with linear elasticity as the material is

no longer constant. For nonlinear elastic behavior regarding with materials, a

general equation can be formulated as 𝜎 = 𝐸 ( ) , in which the terms of the secant

modulus is used to define the relationship between stress and strains.

Geometrical nonlinearity is based on the nonlinearity in kinematic quantities like

the relationship of strain and displacement in solids. Large displacements, large

strains and large rotations are main reasons causing nonlinearities. For small

displacement analysis, the change in element stiffness contributing to the overall

structural stiffness can be ignored because of the very small deformation of the

element and change in its spatial orientation. On the other hand, the change in the

element geometrical shape, e.g. length and area, can lead to the change in the local

element stiffness in geometrically nonlinear structures (Wriggers, 2009). For

topology optimization of compliant mechanisms, previous research reviewed in

Chapter 2 has also addressed related geometrically nonlinear behaviors in

164

structural analysis.

7.3 Geometrical Nonlinearity in Designing Compliant Mechanisms

Designing compliant mechanisms has been facing many challenges. One of them

is to address geometrically nonlinear analysis (Zhaokun and Xianmin, 2006).

However, previous optimization designs of compliant mechanism are derived

mainly from linear finite element model based on the assumption that the structures

can only undergo small deformation. In fact, the optimal design of compliant

mechanisms using the linear analysis is not very accurate, and the optimal design of

compliant mechanisms with geometrically nonlinear behaviors is essential to

capture the large-displacement behavior in real applications. This is because the

nonlinearities are associated with the deformation of the constituting members in

the optimal design of compliant mechanisms. Compliant mechanisms generate

force and motion transmission through elastic deformation, in particular via large

deformation. Certainly, various objective functions of designing linear compliant

mechanisms are not suitable for nonlinear compliant mechanisms, and the

nonlinear finite element analysis may experience convergence difficulty (Luo and

Tong, 2008). Consequently, there is relatively little research focusing on topology

design of large-displacement compliant mechanisms using geometrically nonlinear

finite element models (Jinqing and Xianmin, 2011).

In this dissertation, the research will investigate the topology optimization of

compliant mechanisms undergoing a large deflection based on the modified BESO

method. A total Lagrange finite element formulation will be employed for the

geometrically nonlinear structural response (Buhl et al., 2000; Gea and Luo, 2001).


165

As depicted in Figure 4.1a, we consider a general design domain Ω under given

loadings, boundary conditions and spring workpiece. Fin means the applied force,

△out is the expected output displacement and the output force can be expressed as

Fout = kout×△out. Similar to linear compliant mechanism designs, topology

optimization should quantify and measure the performance characteristics of

mechanism flexibility and structural stiffness. To reflect the motion transmission

ability, the flexibility of a compliant mechanism can be quantified by the resulting

displacement at the output port, △out.

To achieve the goal of the topology optimization problem of maximizing the

stiffness of a structure undergoing large nonlinear deformation, previous works

often consider three main objective functions, namely minimization of

end-compliance, minimization of a weighted sum of end-compliances and

minimization of the complementary elastic work (Buhl et al., 2000). Here,

end-compliance C (=PTU) is taken into account in this design of nonlinear

compliant mechanism because of the low cost calculation or no requirement of an

iterative procedure. Consequently, we reformulate the design of a compliant

mechanism by maximizing the output displacement at the output port, and

constraining its end-compliance and the amount of structural material that can be

used. Mathematically, the optimization problem of a compliant mechanism is

expressed by

Maximize 𝑕 = ∆𝑜𝑢𝑡 (7.4)

Subject to: 𝐶 ≤ 𝐶∗

𝑉∗ − ∑ 𝑉𝑒𝑁𝑒=1 𝑒 = 0

𝑒 = 𝑒,1, 𝑒,2, … , 𝑜𝑟 𝑒,𝑚

166

where C* is the prescribed end-compliance and V

* is used to restrict the allowable

material usage. The design domain is discretized with finite elements and the

volumetric density of each element, xe, is used as the design variable.

7.5 Geometrically Nonlinear Finite Element Analysis

To solve the topology optimization problem of designing large-displacement

compliant mechanisms, the equilibrium should be obtained using an iterative

procedure as opposed to the linear problem (Bendsøe and Sigmund, 2003). The key

assumption is that the mechanisms undergo large displacements and rotations but

remain small strains (Pedersen et al., 2001). Thus the strain-displacement relation,

the strain tensor ηij, can be expressed by using the nonlinear Green-Lagrange strain

measure.

𝑖𝑗 = (𝑢𝑖,𝑗 + 𝑢𝑗,𝑖 + 𝑢𝑘,𝑖𝑢𝑘,𝑗) 2⁄ (7.5)

Subscript j represents differentiation with respect to coordinate j in the point-wise

displacement 𝑢𝑖,𝑗. Equation 7.6 can be obtained by using the displacement matrix

B, which transforms a change in displacement dU into a change in strain and

depends on the current deformed state.

= ( ) (7.6)

The second Piola-Kirchhoff stress tensor is calculated using the linear Hooke‟s law

as 𝑠𝑖𝑗 = 𝐸𝑗𝑖𝑘𝑙𝑒 𝑘𝑙 with the components of the constant elasticity tensor 𝐸𝑗𝑖𝑘𝑙

𝑒 for

Element e. By introducing material densities, Hooke‟s law is written as

𝑠𝑖𝑗 = ( 𝑒)𝑝𝐸𝑗𝑖𝑘𝑙

0 𝑘𝑙 (7.7)

167

in which p is the penalization factor, 𝐸𝑗𝑖𝑘𝑙0 represents the constitutive tensor for

solid isotropic material and xe is the relative material density for Element e.

If R denotes the sum of the internal and external force vectors, finite element

equilibrium equation can be identified as

( ) = − ∫ 𝑇( ) 𝑑𝑉

(7.8)

in which P means the external force vector and s is the internal stress vector for the

second Piola-Kirchhoff. The integration in this equation is performed over the

original undeformed volume in terms of total Lagrange formulation. At equilibrium,

the residual vector is equal to zero R(U)=0, and its solution with respect to the

displacement determines the deformed state of the system. The values of the

displacement vector can be obtained iteratively by the Newton-Raphson method

with the tangent stiffness matrix calculated as

= −𝜕

𝜕 (7.9)

The Newton-Raphson iterative method employs iteration to satisfy the equilibrium

condition between the applied loads and the internally generated nodal forces for

each load step. The tangential stiffness matrix is presented for each iteration within

a particular load step in the Newton-Raphson iterative method, shown in Figure 7.2.

This means that the tangent stiffness matrix should be calculated at each iteration.

In the Newton-Raphson method, the basic problem is formulated as Equation 7.10.

168

∆𝐮1 ∆𝐮2

∆𝐮3

𝐅1

𝐅1𝑡+∆𝑡

𝐅2𝑡+∆𝑡

𝐅3𝑡+∆𝑡

𝐅𝑡+∆𝑡

𝐊0𝑡+∆𝑡

𝐊1𝑡+∆𝑡

𝐊1𝑡+∆𝑡

1st iteration 2nd iteration

𝑖−1𝑡+∆𝑡∆𝐮𝑖 =

𝑡+∆𝑡 − 𝑖−1𝑡+∆𝑡 (7.10)

𝑡+∆𝑡 is the vector of externally applied nodal loads. 𝑖−1𝑡+∆𝑡 indicates the vector of

generated nodal forces at the ith iteration. 𝑖−1𝑡+∆𝑡 points out the tangent stiffness

matrix at the beginning of the ith iteration. ∆𝐮𝑖 represents the vector of

incremental nodal displacement during the ith iteration. It is used to obtain the next

displacement approximation by 𝐮𝑖𝑡+∆𝑡 = 𝐮𝑖−1

𝑡+∆𝑡 + ∆𝐮𝑖 (Khosravi, 2007). This

iterative procedure will be repeated until the displacement increment ∆𝐮𝑖 or the

out-of-balance load vector ∆ 𝑖−1 is small enough.

Figure 7.2 Newton-Raphson iterative method

Figure 7.2 Newton-Raphson iterative method (Becker, 2001)


In this dissertation, we derive the design sensitivities for compliant mechanisms

undergoing displacement through the adjoint method. The displacement ui of a

𝐏𝑡+∆𝑡

𝐏𝑡

𝐮0𝑡+∆𝑡 = 𝐮𝑡

𝐮1𝑡+∆𝑡 𝐮2

𝑡+∆𝑡 𝐮3𝑡+∆𝑡

𝐮𝑡+∆𝑡

Load

Displacement

169

specified degree of freedom i can be expressed as

𝑢𝑖 = (7.11)

in which L represents a unit load vector. Its value is equal to one at Position i,

namely Li=1, and values are zero for other positions. U means the displacement

vector generated by the load vector F applied at the input port.

The sensitivity of the displacement ui with respect to a change in design variable

can be expressed as Equation 7.12 assuming design-independent loads.

𝑑𝑢𝑖

𝑑𝑥𝑒=

𝑑

𝑑𝑥𝑒 (7.12)

The sensitivity analysis employs the adjoint method to determine 𝑑 𝑑 𝑒⁄ .

By introducing a vector of Lagrange multiplier λ, the previous displacement

function is augmented as a new function without changing anything by the term

, which is equal to zero and related to the residual term R defined in Equation

7.8.

𝑢𝑖 = + (7.13)

And the design sensitivity of the augmented function can be written as

𝑑𝑢𝑖

𝑑𝑥𝑒=

𝑑

𝑑𝑥𝑒+ (

𝜕

𝜕

𝑑

𝑑𝑥𝑒+

𝜕

𝜕𝑥𝑒) (7.14)

where 𝜕

𝜕 is equal to a negative tangent stiffness matrix KT according to the

170

previous discussions.

The Lagrange multiplier vector λ can be chosen freely as R=0. To eliminate the

unknown term 𝑑

𝑑𝑥𝑒 from Equation 7.14, λ is chosen such that

( − )𝑑

𝑑𝑥𝑒= 0 (7.15)

That corresponds to solving the system of linear = . We end up with the

sensitivity by using the symmetry of the tangent stiffness = and inserting

the solution of this system λ.

𝑑𝑢𝑖

𝑑𝑥𝑒=

𝜕

𝜕𝑥𝑒 (7.16)

The sensitivity of the residual with respect to design changes is found by

differentiation of Equation 7.8. λ is the displacement vector obtained from the

linear equation by applying the unit load vector L at Position i.

In addition, to satisfy the additional constraint (end-compliance), a Lagrange

multiplier λ is used in this BESO algorithm. The original objective function,

Equation 7.4, is rewritten as Equation 7.17

Maximize 𝑕 = ∆𝑜𝑢𝑡 + (𝐶∗ − 𝐶) (7.17)

Equation 7.17 points out that the modified function is identical to the previous one

when the end-compliance constraint equals its prescribed value. The value of the

Lagrange multiplier λ is correspondingly increased if the end-compliance

constraint has not been satisfied, namely 𝐶 > 𝐶∗, even approaching infinity when

171

𝐶 is far greater than its prescribed value. That means we should minimize the

end-compliance to satisfy the constraint in later iterations. The Lagrange multiplier

λ will be equal to zero when the end-compliance constraint has already been

satisfied, 𝐶 < 𝐶∗. Equation 7.17 can be seen as an unconstraint objective function

in such a situation. The procedure of determining λ will be briefly presented in the

next section.

Therefore, the sensitivity number of the modified objective function can be

established in Equation 7.18 through the adjoint method, in which derivatives of

displacements are solved by using Equation 7.16.

𝛼𝑒 =𝑑ℎ


𝑥𝑒−

𝑑𝐶


𝑥𝑒− (

𝑖𝑛

2

∆𝑖𝑛

𝑥𝑒− 𝑘 ∆𝑜𝑢𝑡

∆𝑜

𝑥𝑒) (7.18)

7.7 Determination of Lagrange Multiplier

Similar to Section 5.3, in order to determine it, the Lagrange multiplier λ can be

defined as = (1 − 𝜔) 𝜔⁄ , in which 𝜔 ranges from a small value, like 0, to 1. In

other words, initial bound values of ω can be set as 𝜔𝑙𝑜𝑤𝑒𝑟 = 𝜔𝑚𝑖𝑛 and 𝜔𝑢𝑝𝑝𝑒𝑟 =

1 to implement the program. The volume constraint is firstly satisfied in the

constraint algorithm. The relative ranking of the sensitivity number is sorted based

on Equation 7.18. The volume constraint is achieved by gradually updating

elemental densities. To satisfy the end-compliance constraint in the algorithm, it is

necessary to estimate the variation of the end-compliance due to the variation of

design variables. Based on the value of the end-compliance in the current iteration

Ck, the end-compliance in the next iteration can be approximately estimated by

𝐶𝑘+1 𝐶𝑘 + ∑𝜕𝐶

𝜕𝑥𝑒× ∆ 𝑒𝑒 (∆xe = 0.02 in this chapter). Then, the value of ω can

be determined by using the bisection method in such a way that the end-compliance

constraint will be satisfied in the subsequent iterations.

172


In the algorithm of designing compliant mechanisms with geometrically nonlinear

behaviors, the research employs post-processing techniques, which are the same as

those in the linear compliant mechanism design. Equation 3.9 is considered as a

filter scheme to avoid checkerboard and mesh-dependency. The elemental


elements. Then Equation 3.10 is adopted for the stabilization of the evolutionary

process. The evolution process will complete the whole optimization design once

the performance of the objective function satisfies the convergence criterion (refer

to Equation 3.12) and material volume reaches its objective volume.


The procedure of optimizing geometrical nonlinear compliant mechanisms with

the structural flexibility constraint based on the BESO method is presented in the

following steps by summarizing the previous sections. Figure 7.3 aims to describe

briefly and visually its iterative procedure.


geometrically nonlinear compliant mechanism. Set material properties and

mechanism parameters such as Young‟s modulus E0 and Poisson‟s ratio.


relative BESO parameters objective volume 𝑉∗, evolutionary ratio ER, penalty

exponent p and the filter radius 𝑟𝑚𝑖𝑛. Set the end-compliance constraint 𝐶∗.

173

Step 3 Carry out a finite element analysis which is conducted by ABAQUS to

output required data.

Step 4 With the help of Equation 7.18, calculate sensitivity numbers of the output

displacement and the end-compliance constraint.

Step 5 Calculate and satisfy the volume constraint.

Step 6 Determine the Lagrange multiplier based on the procedure detailed in

Section 7.7.



Step 8 Determine the target volume for the next design iteration according to

Equation 3.11.

Step 9 Rank all elements based on their sensitivity numbers in descending order

and set the threshold of the sensitivity number 𝛼𝑡ℎ so that the total volume of

elements with 𝛼𝑖 > 𝛼𝑡ℎ equals to the target volume of the next iteration Vi+1.

Step 10 Reset the design variables of the design domain. The corresponding density

variable of eth element should be switched from 𝑒,𝑗 to 𝑒,𝑗−1 if 𝛼𝑒 > 𝛼𝑡ℎ, or

from 𝑒,𝑗 to 𝑒,𝑗+1 if 𝛼𝑒 < 𝛼𝑡ℎ.

Step 11 Repeat Steps 3-10 until the volume fraction reaches objective volume and

the performance of objective function satisfies its convergence criteria according to

Equation 3.12.

174

Figure 7.3 Flow chart of the BESO procedure for the optimal design of geometrical nonlinear compliant mechanisms

Figure 7.3 Flow chart of the BESO procedure for the optimal design of

geometrically nonlinear compliant mechanisms.

No

No

Yes

Yes


nonlinear compliant mechanisms




element mesh and set parameters about BESO like

V*, ER and p





information

Calculate the target volume for the next design



satisfied?

Is the procedure

convergent?

Update Lagrange multiplier of the end-compliance

constraint

175


The proposed optimization method is applied to the design of the benchmark

example with geometrical nonlinearities, namely an inverter mechanism shown in

Figure 4.4. The design domain is also defined as 100mm×100mm and discretized

with 100×100 4-node quadrilateral elements. An input force Fin=5kN to the right

is applied at the center point of the left edge. An artificial spring with stiffness

kout=1kN/mm is attached at the output port to simulate the resistance from the

workpiece, but there is not attached spring at input port. Optimal design expects

that a horizontal displacement △out to the left can be produced from the output port

at the center of the right edge. The material properties are assumed to be Young‟s

modulus E = 100GPa and Poisson‟s ratio ν =0.3. The BESO parameters are

evolutionary rate 𝐸𝑅 = 1 , the filter radius 𝑟𝑚𝑖𝑛 = 5 , penalty exponent

𝑝 = 3 and 𝑚𝑖𝑛 = 0 001.

The optimization procedure starts with the full design with the initial output

displacement ∆𝑜𝑢𝑡0 = 2.244×10

-1mm and end-compliance C

0 = 2.898×10

4Nmm.

Both the end-compliance C and the output displacement ∆𝑜𝑢𝑡 are normalized by

C0 and ∆𝑜𝑢𝑡

0 for the whole process. The volume fraction of the final design is

limited to be 25% of the full design domain and the compliance constraint is set to

be 𝐶 𝐶0⁄ ≤ 5. Figure 7.4 displays the evolution history of the topology of the

compliant inverter with geometrical nonlinearity. De facto regions are relatively

reduced due to a more accurate simulation in the nonlinear analysis, compared with

the final design of linear compliant inverter design. Figure 7.5 shows the evolution

histories of the output displacement, mean compliance and volume fraction. The

convergent solution is achieved with 222 iterations under the convergence factor

of 0.1% for both the output displacement and the mean compliance. Figure 7.6

demonstrates performance of the topology after loading. There is a nonlinear

176

increase in the output displacement with the loading time of input force. Figure

7.6b presents the final deformed configuration.

(a) (b) (c)

(d) (e) (f)

Figure 7.4 Evolution history of the topology of designing the compliant inverter with geometrical nonlinearity

Figure 7.4 Evolution history of the topology of designing the compliant inverter

with geometrical nonlinearity (a) initial design; (b) 50 iteration; (c) 100 iteration; (d)

150 iteration; (e) 200 iteration; (f) final design.

(a) (b)

Figure 7.5 Evolution histories of the objective function and constraints for the compliant inverter with geometrical nonlinearity

Figure 7.5 Evolution histories of the objective function and constraints for the

177

compliant inverter with geometrical nonlinearity (a) evolution history of output

displacement; (b) evolution histories of mean compliance and volume fraction.

(a) (b)

Figure 7.6 Performance of the topology after loading

Figure 7.6 Performance of the topology after loading (a) relationship of output

displacement and loading time; (b) deformed configuration.

However, finite element analysis may experience convergence difficulty during

the evaluation phase when the high-density material around the input port is

switched to low-density material. The optimization algorithm cannot produce

various resulting topologies with using different end-compliance constraints. To

overcome the proposed difficulty, the research adopts the theory from the

hard-kill BESO. The elements with the lowest density will be really removed in

the design domain. In addition, the lowest density for solid elements is increased

to 𝑒,2 = 0 02. The volume fraction of the final design is limited to be 30% of the

full design domain with 130×130 4-node quadrilateral elements. Other parameters

are Fin=1kN, kout=200N/mm and 𝑟𝑚𝑖𝑛 = 7 . To demonstrate the effectiveness

of the current algorithm, the research sets the end-compliance constraint 𝐶 𝐶0⁄ ≤

2, 3, 4, 7, 10 and 20. Corresponding optimized topologies are shown in Figure 7.7.

Similarly, with the increase of the end-compliance constraint, the topology tends to

178

increase the flexibility of compliant mechanisms and efficiently transfer the motion.

Overall, the BESO algorithm of designing linear compliant mechanisms can be

successfully extended to the optimal design of compliant mechanisms with

geometrically nonlinear behavior. Compared with previous research, a series of

solutions and complex topologies can be produced by balancing flexibility and

stiffness requirements.

(a) (b) (c)

(d) (e) (f)

Figure 7.7 Optimized topologies and output displacements of the nonlinear compliant inverter for various compliance constraints

Figure 7.7 Optimized topologies and output displacements of the nonlinear

compliant inverter for various compliance constraints (a) 𝐶 𝐶0⁄ ≤ 2 and

∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −4 143; (b) 𝐶 𝐶0⁄ ≤ 3 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ = −11 683; (c) 𝐶 𝐶0⁄ ≤

4 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −24 855 ; (d) 𝐶 𝐶0⁄ ≤ 7 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ = −44 467 ; (e)

𝐶 𝐶0⁄ ≤ 10 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡0⁄ = −64 642 ; (f) 𝐶 𝐶0⁄ ≤ 20 and ∆𝑜𝑢𝑡 ∆𝑜𝑢𝑡

0⁄ =

−88 799

179

7.11 Conclusions

The research extends the BESO algorithm of designing linear compliant

mechanisms to the optimal design of compliant mechanisms with geometrically

nonlinear behavior. The optimization problem is re-formulated as maximizing the

flexibility subject to end-compliance and volume constraints. The values of the

displacement vector are obtained iteratively by the Newton-Raphson method with

the tangent stiffness matrix. In addition, the research derives the design sensitivities

for compliant mechanisms undergoing large displacement through the adjoint

method.

Based on the proposed optimization algorithm, numerical examples demonstrate

the validity and effectiveness of designing nonlinear compliant mechanisms using

BESO. Furthermore, with the aid of the hard-kill BESO method, the optimization

design can successfully overcome the convergence difficulty caused by extreme

deformation of low-density elements in the nonlinear finite element analysis. Next,

the optimization method is more computationally efficient and can produce a set

of resulting topologies with the change of the end-compliance constraint. Finally,

one-node hinge does not appear in resulting designs when the nonlinear finite

element analysis is used to capture the real behavior of large-displacement


However, this research is not intended to provide further comparisons of final

designs with linearity and nonlinearity. This is because linear analysis is not precise

when large deformations are involved, and the used compliance constraints are

actually different in linear and nonlinear analysis.

180

References

Becker, A. A., 2001. Understanding non-linear finite element analysis: Through illustrative

benchmarks, NAFEMS, Glasgow.

Bendsøe, M. P. and Sigmund, O., 2003. Topology Optimization: Theory, Method and Application,

Springer, Berlin.

Bhashyam, G. R., 2002. ANSYS Mechanical - A powerful Nonlinear Simulation Tool, ANSYS Inc,

Canonsburg.

Buhl, T., Pedersen, C. B. W. and Sigmund, O., 2000, "Stiffness design of geometrically nonlinear

structures using topology optimization", Structural and Multidisciplinary Optimization

19(2), 93-104.

Gea, H. C. and Luo, J., 2001, "Topology optimization of structures with geometrical nonlinearities",

Computers and Structures 79(20-21), 1977-1985.

Jinqing, Z. and Xianmin, Z., 2011, "Topology optimization of compliant mechanisms with

geometrical nonlinearities using the ground structure approach", Chinese Journal of

Mechanical Engineering 24(2), 257-263.

Khosravi, P., 2007. Nonlinear finite element analysis and design optimization of thin-walled

structures, Mechanical and Industrial Engineering. Concordia University, Montreal,

Quebec.




Pedersen, C. B. W., Buhl, T. and Sigmund, O., 2001, "Topology synthesis of large-displacement


50(12), 2683-2705.

Wriggers, P., 2009. Nonlinear Finite Element Methods, Spinger-Verlag, Berlin.

Zhaokun, L. and Xianmin, Z., 2006. Topology optimization of compliant mechanisms with

geometrically nonlinear. Technology and Innovation Conference, Hangzhou, China.

181

Chapter 8

Conclusions

182

This dissertation has addressed the topology optimization of linear and

geometrically nonlinear compliant mechanisms using a modified bi-directional

evolutionary structural optimization (BESO) method. Conclusions and

achievements are summarized below:

Firstly, a modified BESO method presented in this research changes the original

binary material distribution by introducing intermediate densities. The

evolutionary procedure employs a constant material volume strategy. Elemental

densities are increased or decreased according to a prescribed threshold, updated

gradually by design functions. A filter scheme is applied to resolve the problem of

mesh-dependence. The optimality criteria and solution-convergence algorithms

ensure the final optima to be obtained after an iterative procedure.

The proposed BESO method has been applied to the stiffness optimization of a

cantilever beam in order to demonstrate its feasibility. The final solution proves

that the current BESO method is an efficient optimization tool. Then, this

dissertation extends the proposed BESO method to the optimal design of compliant

mechanisms. The sensitivity analysis is conducted by applying the adjoint method

to solve various design problems such as maximizing geometrical advantage,

mechanical efficiency or output displacement. The new BESO algorithm produces

various designs with the change in the springs‟ constants. Through a large number

of numerical experiments, the research proves the capability and validity of the

new BESO algorithm and considers that attached springs have strong effects on

the topology optimization of compliant mechanisms:

1. Resulting topologies can produce relatively larger output displacement due to

the presence of de facto hinge regions when using the optimization problem

of maximizing mechanical efficiency.

183

2. The optimized geometrical advantage and mechanical efficiency are

decreasing as the spring stiffness increases and the resulting topologies tend

to be simpler. The geometrical advantage tends to be an exponential decrease,

but the mechanical efficiency shows a linear decrease.

3. For topologies based on using the optimization problem of maximizing

output displacement, balancing and promoting the stiffness constants are

beneficial to eliminate de facto hinges. Soft springs can lead to poor structural

connection.

Secondly, this dissertation presents the control of the desired structural stiffness in

the topology design of linear compliant mechanisms. The design problem is

reformulated as maximizing the flexibility of the compliant mechanisms subject

to the mean compliance and volume constraints, considering the mechanism

flexibility and structural stiffness simultaneously. 2D and 3D examples are given

to demonstrate the effectiveness of the BESO algorithm for controlling the

flexibility and hinge-related properties via the desired structural stiffness.

1. A set of different optimized topologies have been produced through the

change of the mean compliance constraint.

2. Using a relatively small compliance constraint can produce hinge-free

topologies. With the increase of the compliance constraint, the optimized

topologies show numerous hinge regions which lie along the force path from

the mechanism input port to the output port.

3. Different with stiffness optimization, material property greatly affects final

184

designs of compliant mechanisms. For an extreme soft workpiece, selecting a

soft material for compliant mechanisms can also preclude the formation of

hinge regions.

Thirdly, this research develops a BESO method for the optimal design of

hinge-free compliant mechanisms. A new objective function is proposed to

maximize the desirable displacement and simultaneously preclude the formation

of hinges. Sensitivity numbers are derived according to the variation of the

objective function with respect to the design variables. Based on the resulting

sensitivity numbers, the BESO procedure is established by gradually removing

and adding elements until an optimal topology is achieved. Several numerical

examples, especially 3D cases, are given to demonstrate the effectiveness of the

proposed BESO method for the design of various hinge-free compliant

mechanisms. There is no any “grey area” in the resulting topologies of compliant

mechanisms owing to the proposed formulation and the binary variables in the

traditional BESO method. Therefore, the current BESO algorithm provides clear

solid-void topologies.

Fourthly, the research extends the idea of designing linear compliant mechanisms,

presenting new formulation and constraint in the optimal design of geometrically

nonlinear compliant mechanisms. The optimization problem is formulated as

maximizing the flexibility subject to the end-compliance and volume constraints.

The adjoint method is employed to derive the sensitivity numbers for compliant

mechanisms undergoing a large deformation. Numerical experiment demonstrates

that the developed algorithm has the capability of designing nonlinear compliant

mechanisms.

1. The convergence difficulty from extreme deformation can be successfully

185

overcome through the hard-kill BESO method in the nonlinear finite element

analysis.

2. The proposed optimization method is more computationally efficient and can

provide a set of optimized topologies as the end-compliance constraint

changes.

3. There is no one-node hinge region generated in resulting designs when the

nonlinear finite element analysis is used to capture the real behavior of

large-displacement compliant mechanisms.

Based on the current achieved outcomes and proposed BESO method, further

research on the topic of topology optimization for compliant mechanisms and

related topics can be carried out to widen the current scope and refine the numerical

algorithms. For instance, a multi-criteria function can be established to optimize

compliant mechanisms with multiple input and output ports based on weighted

global criterion method or weighted sum method. In addition, the normal

distribution function can be considered as the materials to accommodate the use

of multiple materials in the design of compliant mechanisms. The modified BESO

method is also employed to design thermally actuated compliant mechanisms.

Furthermore, piezoelectric and electrostatic principles can be used to achieve the

expected motion for multi-physics actuators in topology optimization of

compliant mechanisms. Further research can also focus on optimizing compliant

actuators of electro-thermo-elasticity via non-uniform Joule heating induced

actuation based on the modified BESO method.

186

Appendix A

As suggested in the literature (Wang, 2009), the global stiffness matrix K is positive

definite when properly defining a mechanism structure. The relationship of three

sets of elastic displacements can be found in Equation A1.

[

𝐾𝑖𝑖 𝐾𝑖𝑜 𝑲𝑖 𝐾𝑖𝑜 𝐾𝑜𝑜 𝑲𝑜 𝑲𝑖 𝑇 𝑲𝑜

𝑇 𝑲

] [

∆𝑖∆𝑜∆

]=[𝐹𝑖𝐹𝑜𝑭

] (A1)

in which ∆i, ∆o and ∆s denote input port, output port and other internal nodal

displacements, and nodal loads are partitioned into three sets as Fi, Fo and Fs. As

the internal degree of freedoms are free of external loads, namely Fs=0, Equation

A1 can be written as Equation A2 with the mechanism stiffness Km.

[𝐾𝑖 𝐾 𝐾 𝐾𝑜

]⏞

𝑲𝑚

[∆𝑖∆𝑜] = [

𝐹𝑖𝐹𝑜] (A2)

where the input-stiffness 𝐾𝑖 = 𝐾𝑖𝑖 −𝑲𝑖 𝑲 −1𝑲𝑖

𝑇 , output-stiffness 𝐾𝑜 = 𝐾𝑜𝑜 −

𝑲𝑜 𝑲 −1𝑲𝑜

𝑇 and structure stiffness 𝐾 = 𝐾𝑖𝑜 −𝑲𝑖 𝑲 −1𝑲𝑜

𝑇 . Equation A2 can be

expressed as A3 when using Model A.

[𝐾𝑖 𝐾 𝐾 𝐾𝑜 + 𝑘𝑜𝑢𝑡

] [∆𝑖∆𝑜] = [


To ensure that the elastic system of the mechanism be better conditioned, the

determinant of the mechanism stiffness matrix, det (Km), should not be equal to

zero so that the matrix is no longer singular. The maximization of the output

displacement of Function C becomes

187

Max Function C = max |∆𝑜| = max |𝐾 ∆𝑖

𝐾𝑜+𝑘𝑜 | (A4)

The input constraint is set as 0 < ∆𝑖≤ ∆𝑚𝑎𝑥 in the optimization design. The

evolution process supposes that ∆𝑖 reaches ∆𝑚𝑎𝑥 and 𝐾𝑜 vanishes when the

maximum solution is given. | et(𝑲𝑚)| = |𝐾𝑖𝑘𝑜𝑢𝑡 − (𝐾 )2| = |𝐹𝑖|𝑘𝑜𝑢𝑡/∆𝑚𝑎𝑥 0

That means that the mechanism stiffness matrix Km is no longer singular. The

research suggests that an external spring with spring stiffness 𝑘𝑖𝑛 should be added

to input port, namely Model B, if there is no input constraint in the evolution

process. Equation A2 can be written as

[𝐾𝑖 + 𝑘𝑖𝑛 𝐾 𝐾 𝐾𝑜 + 𝑘𝑜𝑢𝑡

] [∆𝑖∆𝑜] = [


When the maximum solution is given, the determinant of the mechanism stiffness

is identified as | et(𝑲𝑚)| = |(𝐾𝑖+𝑘𝑖𝑛)𝑘𝑜𝑢𝑡 − (𝐾 )2| = |𝐹𝑖|𝑘𝑜𝑢𝑡 ∆𝑖⁄ . ∆𝑖 can be

expressed as |𝑅𝑖| 𝑘𝑖𝑛⁄ (|𝑅𝑖| |𝐹𝑖| and |𝑅𝑖| > 0) using the input spring reaction

force, so | et(𝑲𝑚)| = |𝐹𝑖||𝑅𝑖|𝑘𝑜𝑢𝑡 𝑘𝑖𝑛 0⁄ . That means also that the mechanism

stiffness matrix Km is no longer singular.

References


Frontiers of Mechanical Engineering in China 4(3), 229-241

topology optimization of compliant mechanisms based...

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