multiobjective optimization of structure using modified - constraint approach

MULTIOBJECTIVE OPTIMIZATION MULTIOBJECTIVE OPTIMIZATION OF STRUCTURE USING MODIFIED OF STRUCTURE USING MODIFIED

- CONSTRAINT APPROACH- CONSTRAINT APPROACH

1) Graduate Student, Department of Civil Eng., KAIST, KOREA2) Professor, Department of Civil Eng., Sung Kyun Kwan Univ., KOREA3) Professor, Department of Civil Eng., KAIST, KOREA

Ju-Tae KimJu-Tae Kim11, Sun-Kyu Park, Sun-Kyu Park22 and In-Won Lee and In-Won Lee33

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INTRODUCTION

MODIFIED CONSTRAINT APPROACH

NUMERICAL EXAMPLE

CONCLUSIONS

CONTENTSCONTENTS

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INTRODUCTIONINTRODUCTION

COMPETING OBJECTIVES

Objectives Point of ViewsConstruction Cost Economy

Deflection of Structure Serviceability

Static Safety Factor Static Safety

Natural Frequencies Dynamic Safety

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OBJECTIVE SPACE

COSTSAFETYDEFLECTIONFREQUENCY

MULTI-OPTIMALSTRUCTURE

DECISION MAKING

Multiobjective Optimization

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Pareto Solution Set

Feasible Design Region

Pareto Solutions

f1

f2

f1, min

f2, min

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Decision Making

Deterministic

Probabilistic FuzzyRule Base

Constraint ApproachGame TheoryWeighting Method

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- Constraint Approach

(1.a)

(1.b)

(1.c)

Multiobjective Optimization Problem

JjXg j ,,2,10)(tosubject

)(,),(),(Minimize 21 XfXfXfF m

NnXhn ,,2,10)(

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- Constraint Approach

Transformed into Single Objective Problem

(2.a)

(2.b)

(2.c)

(2.d)

)(Minimize Xf p

)(,,2,1)(tosubject pmiXf ii

NnXhn ,,2,10)(

JjXg j ,,2,10)(

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(3.a)

(3.b)

(3.c)

(3.d)

(3.e)

(3.f)

MODIFIED APPROACH

)(Minimize Xf p

)(,,1)()(tosubject 0 pmiXfXf ii

JjXg j ,,2,10)(

NnXhn ,,2,10)(

m

iii XcX

1

*0

m

iic

1

1

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Differences of Two Approaches

-constraint approach Modified approach

Pareto Set

Initial ValuePareto Set

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is inside the Feasible Design RegionDue to the Convexity of the Problem Considered

Limitation and Assumption

m

iii XcX

1

*0

m

iic

1

1

(3.e)

(3.f)

0X

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Advantages

Initial values of optimization are generated independently

Each Pareto Solution can be found in Parallel

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NUMERICAL EXAMPLESteel Box Girder Bridge

80

19.5

2.75 2.757.0 7.0

0.25

D

B

tuf

tbf

tw

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Material Properties of Steel

Y ie ld S tr e s s ( y ) : 4 0 0 0 k g /c m 2

A llo w a b le T e n s ile S tr e s s ( ta ) : 1 9 0 0 k g /c m 2

A llo w a b le C o m p r e s s io n S tr e s s ( ca ) : 1 9 0 0 k g /c m 2

A llo w a b le S h e a r S tr e s s ( a ) : 1 1 0 0 k g /c m 2

Y o u n g ’s M o d u lu s ( sE ) : 61012 . k g /c m 2

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Formulation of the Problem

wufbf DtttBXf 2)()(Minimize 1

)(

10186.96)(

6

2 XIXf DL

0tosubject 1 tat

I

Myg

02 cac

I

Myg

03 aIt

VQg

02.122

4

a

m

a

mg

(4.a)

(4.b)

(4.c)

(4.d)

(4.e)

(4.f)

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0485 bft

fn

Bg

0)20(80

16 ufw ttBg

01307 wtD

g

mBm 32 mDm 38.1

cmtcm bf 0.30.1 cmtcm uf 0.30.1

cmtcm w 0.30.1

Constraints

(4.g)

(4.h)

(4.i)

(4.j) (4.k)

(4.l) (4.m)

(4.n)

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Area Minimization

0 2 4 6 8 10 12

Iteration Number

1000

1500

2000

2500

3000

3500

4000

Are

a(c

m2 )

=1699.8 cm2

Design Value( )

B : 2.00 m D : 2.23m tbf : 2.46cm tuf : 2.21cm tw : 1.72cm

min,1f

*1X

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Deflection Minimization

0 2 4 6 8 10

Iteration Number

0.00

2.00

4.00

6.00

8.00

Def

lect

ion

(cm

) = 1.82cm

Design Value( )

B : 2.66 m D : 3.0m tbf : 3.0cm tuf : 3.0cm tw : 3.0cm

min,2f

*2X

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Pareto Optimization

]1,0[,)1( *2

*10 cXccXX

0 5 10 15 20

2000

2500

3000

35 0

f =3.15552

f =2.522

f =2.0364

c=0.1c=0.3c=0.5f1

2

2

2

Iterations 0 5 10 15 20 25 30 35 40

1500

1800

2100

2400

2700

f =5.15382

f =4.00322

c=0.6c=0.7

c=0.9

2

2

f =3.548122

f1

Iterations

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Pareto Solution Set

1500 2000 2500 3000 3500

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Def

lect

ion(

cm)

Area(cm2)

Pareto Set

Original Design

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CONCLUSIONS

Independent initial values for Pareto optimization

can be generated

Pareto solution set can be found in Parallel