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

VIBRATION CONTROL LAB. KAIST

INTRODUCTION

MODIFIED CONSTRAINT APPROACH

NUMERICAL EXAMPLE

CONCLUSIONS

CONTENTSCONTENTS


INTRODUCTIONINTRODUCTION

COMPETING OBJECTIVES

Objectives Point of ViewsConstruction Cost Economy

Deflection of Structure Serviceability

Static Safety Factor Static Safety

Natural Frequencies Dynamic Safety


OBJECTIVE SPACE

COSTSAFETYDEFLECTIONFREQUENCY

MULTI-OPTIMALSTRUCTURE

DECISION MAKING

Multiobjective Optimization


Pareto Solution Set

Feasible Design Region

Pareto Solutions

f1

f2

f1, min

f2, min


Decision Making

Deterministic

Probabilistic FuzzyRule Base

Constraint ApproachGame TheoryWeighting Method


- Constraint Approach

(1.a)

(1.b)

(1.c)

Multiobjective Optimization Problem

JjXg j ,,2,10)(tosubject

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

NnXhn ,,2,10)(


- 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)(


(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


Differences of Two Approaches

-constraint approach Modified approach

Pareto Set

Initial ValuePareto Set


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


Advantages

Initial values of optimization are generated independently

Each Pareto Solution can be found in Parallel


NUMERICAL EXAMPLESteel Box Girder Bridge

80

19.5

2.75 2.757.0 7.0

0.25

D

B

tuf

tbf

tw


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


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)


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)


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


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


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


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


CONCLUSIONS

Independent initial values for Pareto optimization

can be generated

Pareto solution set can be found in Parallel

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MULTIOBJECTIVE OPTIMIZATION OF STRUCTURE USING MODIFIED - CONSTRAINT APPROACH