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16.810 (16.682)16.810 (16.682)Engineering Design and Rapid PrototypingEngineering Design and Rapid Prototyping

Design Optimization- Structural Design OptimizationInstructor(s)

Prof. Olivier de Weck Dr. Il Yong Kim

January 23, 2004


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Course Concept

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Course Flow Diagram

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CAD/CAM/CAE Intro

Overview

ManufacturingTraining

Structural TestTraining

Design Optimization

Hand sketching

CAD design

FEM analysis

Produce Part 1

Test

Produce Part 2

Optimization

Problem statement

Final Review

Test

Learning/Review Deliverables

Design Sketch v1

Part v1

Experiment data v1

Design/Analysis

output v2

Part v2

Experiment data v2

Drawing v1

Design Intro

today

Wednesday

FEM/Solid MechanicsAnalysis output v1

3


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What Is Design Optimization?

Selecting the best design within the available means

1. What is our criterion for best design? Objective function

2. What are the available means? Constraints

(design requirements)

3. How do we describe different designs? Design Variables

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Optimization Statement

MinimizeSubject to

f

g

h(x)

( ) 0x( ) 0x

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Constraints

- Design requirementsInequality constraints

Equality constraints

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Objective Function- A criterion for best design (or goodness of a design)

Objective function

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Design VariablesParameters that are chosen to describe the design of a system

Design variables are controlled by the designers

The position of upper holes along the design freedom line

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Design VariablesFor computational design optimization,

Objective function and constraints must be expressedas a function of design variables (or design vector X)

Objective function: f(x) Constraints:g(x), h(x)Cost = f(design)Displacement = f(design)

What is f for each case?Natural frequency = f(design)Mass = f(design)

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Optimization Statement( )

( ) 0

( ) 0

f

h

x

x

x

Minimize

Subject to g

f(x) : Objective function to be minimized

g(x) : Inequality constraints

h(x) : Equality constraints

x : Design variables

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Optimization Procedure

Improve Design Computer Simulation

START

Converge ?

Y

N

END

( )

Subj ( ) 0

( ) 0

f

g

h

x

x

x

Change x

Determine an initial design (x0)

termination criterion?

Minimize

ect to

Evaluate f(x), g(x), h(x)

Does your design meet a

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Structural OptimizationSelecting the beststructuraldesign

- Size Optimization - Shape Optimization- Topology Optimization

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Structural Optimization

( )

j ( ) 0

( ) 0

f

g

h

x

x

x

BCs are given Loads are given

minimize

sub ect to

1. To make the structure strong Min. f (x)

e.g. Minimize displacement at the tip

g(x) 02. Total mass MC

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Size OptimizationBeams

( )j ( ) 0

( ) 0

fg

h

xx

x

minimizesub ect to

Design variables (x) f(x) : compliance

x : thickness of each beam g(x) : mass

Number of design variables (ndv)

ndv = 5

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Size Optimization

- Shapeare given

Topology- Optimize cross sections

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Shape OptimizationB-spline

( )j ( ) 0

( ) 0

fg

h

xx

x

Hermite, Bezier, B-spline, NURBS, etc.

minimizesub ect to

Design variables (x) f(x) : compliancex : control points of the B-spline

g(x) : mass(position of each control point)

Number of design variables (ndv)

ndv = 8

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Shape OptimizationFillet problem Hook problem Arm problem

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Shape OptimizationMultiobjective & Multidisciplinary Shape OptimizationObjective function

1. Drag coefficient, 2. Amplitude of backscattered wave

Analysis

1. Computational Fluid Dynamics Analysis

2. Computational Electromagnetic WaveField Analysis

Obtain Pareto Front

Raino A.E. Makinen et al., Multidisciplinary shape optimization in aerodynamics and electromagnetics using genetic

algorithms, International Journal for Numerical Methods in Fluids, Vol. 30, pp. 149-159, 1999

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Topology OptimizationCells

( )j ( ) 0

( ) 0

fg

h

xx

x

minimizesub ect to

Design variables (x) f(x) : compliance

x : density of each cell g(x) : massNumber of design variables (ndv)

ndv = 27

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Topology OptimizationShort Cantilever problem

Initial

Optimized

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Topology Optimization

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Topology OptimizationBridge problem

Obj = 4.16105Distributed

loading

Obj = 3.29105Minimize

i idzF ,

)toSubject ( dx M ,o0 (x) 1 Obj = 2.88105

Mass constraints: 35%

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Topology OptimizationDongJak Bridge in Seoul, Korea

H

L

H

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Structural OptimizationWhat determines the type of structural optimization?

Type of the design variable(How to describe the design?)

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Optimum Solution Graphical Representation

f(x)

x: design variable

f(x): displacement

Optimum solution (x*)x

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Optimization Methods

Gradient-based methods

Heuristic methods

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Gradient-based Methodsf(x)

Start

Move

Gradient=0 Stop!

You do no c ore optimization

Check gradient

Check gradient

t know this fun tion bef

No active constraints Optimum solution (x*)x

(Termination criterion: Gradient=0)

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Gradient-based Methods

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Global optimum vs. Local optimumf(x) Termination criterion: Gradient=0

Global optimum

Local optimum

Local optimum

xNo active constraints

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Heuristic Methods A Heuristic is simply a rule of thumb that hopefully will find a

good answer.

Why use a Heuristic?

Heuristics are typically used to solve complex optimization

problems that are difficult to solve to optimality.

Heuristics are good at dealing with local optima without

getting stuck in them while searching for the global optimum.

Schulz, A.S., Metaheuristics, 15.057 Systems Optimization Course Notes, MIT, 1999.

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Genetic AlgorithmPrinciple by Charles Darwin - Natural Selection

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Heuristic Methods Heuristics Often Incorporate Randomization

3 Most Common Heuristic Techniques

Genetic Algorithms

Simulated Annealing Tabu Search

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Optimization Software- iSIGHT

- DOT

- Matlab (fmincon)

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Topology Optimization Software ANSYS

Static Topology Optimization

Dynamic Topology Optimization

Electromagnetic Topology Optimization

Subproblem Approximation Method

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Design domain

First Order Method

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Topology Optimization Software MSC. Visual Nastran FEA

Elements of lowest stress are removed gradually.

Optimization results

Optimization results illustration

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MDO

Multidisciplinary Design Optimization

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Multidisciplinary Design OptimizationCentroid Jitter on Focal Plane [RSS LOS]NASA Nexus Spacecraft Concept

60

T=5 sec

14.97 m

1 pixel

Requirement: J =5 mz,2

OTA

4020

CentroidY[m]

0-20

SunshieldInstrument -40Module

0 1 2 -60

-60 -40 -20 0 20 40 60meters

Centroid X [m]

Goal: Find a balanced system design, where the flexible structure, the optics and the control systems worktogether to achieve a desired pointing performance, given various constraints

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Multidisciplinary Design OptimizationAircraft Comparison

le

Approx. 480 passengers each

Approx. 8,700 nm range each

TakeoffBWBA3XX-50R

18%

BWBA3XX-50R

19%Total

Sea-Level

19%BWBA3XX-50R

Operators

Empty

Fuel

Burn

per Seat

32%BWBA3XX-50R

Boeing Blended Wing Body Concept

Goal

Shown to Same Sca

Maximum

Weight

Static Thrust

Weight

: Find a design for a family of blended wing aircraftthat will combine aerodynamics, structures, propulsionand controls such that a competitive system emerges - as

measured by a set of operator metrics.

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Multidisciplinary Design OptimizationFerrari 360 Spider

Goal: High end vehicle shape optimization whileimproving car safety for fixed performance level

and given geometric constraints

Reference: G. Lombardi, A. Vicere, H. Paap, G. Manacorda,Optimized Aerodynamic Design for High Performance Cars, AIAA-98-4789, MAO Conference, St. Louis, 1998

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Multidisciplinary Design OptimizationDo you want to learn more about MDO?

Take this course!

16.888/ESD.77Multidisciplinary System

Design Optimization (MSDO)

Prof. Olivier de Weck

Prof. Karen Willcox

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Baseline DesignPerformance

Natural frequency analysis

Design requirements

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Baseline DesignPerformance and cost

1 0.070 mm

2 0.011 mmf 245Hz

m 0.224 lbs

C 5.16 $

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Baseline Design245 Hz 421 Hz

f1=0f2=0f3=0f4=0f5=0f6=0f7=421 Hzf8=1284 Hzf9=1310 Hz

f1=245 Hzf2=490 Hzf3=1656 Hz

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Design Requirement for Each Team#

Productname

mass(m)

Cost(c)

Disp(1) Disp(2)

NatFreq

(f)

Quality

F1(lbs)

F2(lbs)

F3(lbs)

Const Optim Acc

0Baseline

0.224lbs

5.16$

0.070mm

0.011mm

245Hz

3 50 50 100 c m f

1Family

economy20% -30% 10% 10% -20% 2 50 50 100 c m f

2Familydeluxe

10% -10% -10% -10% 10% 4 50 50 100 m c f

3Crossover

20% 0% -15% -15% 20% 4 50 75 75 m c f

4 City bike -20% -20% 0% 0% 0% 3 50 75 75 c m f

5 Racing -30% 50% 0% 0% 20% 5 100 100 50 m f c6 Mountain 30% 30% -20% -20% 30% 4 50 100 50 f m c7 BMX 0% 65% -15% -15% 40% 4 75 100 75 f m c

8 Acrobatic -30% 100% -10% -10% 50% 5 100 100 100 f m c

9Motorbike

50% 10% -20% -20% 0% 3 50 75 100 f c m

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Design OptimizationTopology optimization

Design domain

Shape optimization

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Design Freedom1 bar

2.50 mm

2 bars 0.80 mm

Volume is the same.

17 bars

0.63 mm

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Design Freedom1 bar

2 bars

2.50 mm

0.80 mm

17 bars

More design freedom More complex(Better performance) (More difficult to optimize)

0.63 mm

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Cost versus Performance17 bars

0

1

2

3

45

6

7

8

9

Cost

[$]

1 bar

2 bars

0 0.5 1 1.5 2 2.5 3Displacement [mm]

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Plan for the rest of the courseClass Survey

Jan 24 (Saturday) 7 am Jan 26 (Monday) 11am

Company tour

Jan 26 (Monday) : 1 pm 4 pm

Guest Lecture (Prof. Wilson, Bicycle Science)

Jan 28 (Wednesday) : 2 pm 3:30 pm

Manufacturing Bicycle Frames (Version 2)

Jan 28 (Wednesday) : 9 am 4:30 pm

Jan 29 (Thursday) : 9 am 12 pm

Testing

Jan 29 (Thursday) : 10 am 2 pm

GA GamesJan 29 (Thursday) : 1 pm 5 pm

Guest Lecture, Student Presentation (5~10 min/team)

Jan 30 (Friday) : 1 pm 4 pm

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ReferencesP. Y. Papalambros, Principles of optimal design, Cambridge University Press, 2000

O. de Weck and K. Willcox, Multidisciplinary System Design Optimization, MIT lecture note, 2003

M. O. Bendsoe and N. Kikuchi, Generating optimal topologies in structural design using ahomogenization method, comp. Meth. Appl. Mech. Engng, Vol. 71, pp. 197-224, 1988

Raino A.E. Makinen et al., Multidisciplinary shape optimization in aerodynamics andelectromagnetics using genetic algorithms, International Journal for Numerical Methods in Fluids,

Vol. 30, pp. 149-159, 1999

Il Yong Kim and Byung Man Kwak, Design space optimization using a numerical designcontinuation method, International Journal for Numerical Methods in Engineering, Vol. 53, Issue 8,pp. 1979-2002, March 20, 2002.

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Web-based topology optimization program

Developed and maintained by Dmitri Tcherniak, Ole Sigmund,

Thomas A. Poulsen and Thomas Buhl.

Features:1.2-D

2.Rectangular design domain

3.1000 design variables (1000 square elements)

4. Objective function: compliance (F)

5. Constraint: volume

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Web-based topology optimization programObjective function

-Compliance (F)

Constraint

-Volume

Design variables

- Density of each design cell

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Web-based topology optimization programNo numerical results are obtained.

Optimum layout is obtained.

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Web-based topology optimization programP 2P 3P

Absolute magnitude of load does not affect optimum solution

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Web-based topology optimization program

http://www.topopt.dtu.dk

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