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
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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