development of a design method based on ...• nash2 14 exch 246 comput • nash3 24 exch 322 comput...

1

DEVELOPMENT OF A DESIGN METHOD

BASED ON GAME THEORY AND INTERPOLATION

TECHNIQUESGraduating:

Lucia PARUSSINI

Università degli Studi di TRIESTEDIPARTIMENTO DI ENERGETICA

Supervisor:Ing. V. PEDIRODACo-supervisors:Prof. C.POLONIIng. A.CLARICH

PRESENTATION OUTLINE

• Introduction• Implementation of algorithm

Response surfaces Gradient MethodGame Theory

• ApplicationRobust Design

• Conclusion

2

INTRODUCTION

METHOD

ROBUSTQUICK

ACCURATE

3

GAME THEORY

GRADIENT METHOD

multi objective optimization

mono objective optimization



RESPONSE SURFACES

number of designswhich is need to be computed to obtain

good results by optimization

with advantage for industry

REDUCTION

integration of CAD, CAE, CFD etc. techniques with numeric optimization

methods

4



Response surfaces Gradient MethodGame Theory


• Conclusion

IMPLEMENTATION OF MULTI OBJECTIVE

OPTIMIZATION ALGORITHM

5

• study of two interpolation techniques: Ordinary Kriging and DACE

• development of mono ojective algorithmbased on Gradient Method and DACEmodel

• implementation of multi objective optimizationby two methods:Nash Theory and Stackelberg Theory



Response Surfaces Gradient MethodGame Theory


• Conclusion

6

RESPONSE SURFACES

matemathical modelwhich approximates the problem

extrapolation of the functionby few known points in the domain

great time advantages

7

The interpolation method we have studied is:

KRIGING

term used in geostatistic for the best linear predictor

of space functions.

simplifing hypotheses

linear estimators

modelling phenomena bySRF Spatial Random Fields

predictorstandard deviation of the predictor

∑=

+=k

hhh xxfxY

1

)()()( εβ

8

ORDINARY KRIGING (OK)

estimates the true value of function by aweighted linear combination of sampled points

)~(~~ 022 µσσ +−= ∑

jjjR Cw

∑=j

jj ywŷ

We regard y(x) as a stochastic process Y(x).

error R(x) stochastic process

expected value equal to zerounbiasedness condition

minimum varianceLagrange method

9

The weights are obtained by:

⎪⎪⎩

⎪⎪⎨

⎧

=

=+

∑

∑

=

=

n

ii

i

n

jijj

w

CCw

1

01

1

~~ µ ni ,...1=∀

where C is correlation function

we choose !!!

In general we consider only k nearest points:

jj

nearestk

jywy ∑

−

=

=1

ˆ

)~(~~1

022 µσσ +−= ∑

−

=

nearestk

jjjR Cw

10

DACE

We know a deterministic process y(x) depending on k variables at n points. We adopt a linear regression model:

CORRELATION FUNCTION between errors:

))(())(( ixixy εµ += ))(),...(()( 1 ixixix k=for i=1,…n with

[ ] [ ]))(),((exp))(()),(( jxixdjxixCorr −=εε

∑=

−=k

h

phhh

hjxixjxixd1

)()())(),(( θ with 0>hθ [ ]2,1∈hpand

LIKELYHOOD :

θ and p parameters

2

1

2/122/

)1()'1(exp)R(det)2(

12/

σµµ

σπ−− − yRy

nn

MAX

11

BEST LINEAR UNBIASED PREDICTOR

MEAN SQUARED ERROR OF PREDICTOR

)ˆ1y(R'rˆ*)x(ŷ 1 µ−+µ= −

⎥⎦

⎤⎢⎣

⎡ −+−σ= −

−−

1R'1)rR11(rR'r1*)x(s 121

122

TEST FUNCTION

[ ][ ]ππ

ππ,,

2

1

−∈−∈

xx

ℜ→ℜ2:1TEST

12

ORDINARY KRIGINGACCURACY OF EXTRAPOLATION

DEPENDS ON:

• number of k-nearest points used

• number of training points

• chosen correlation function

DACEACCURACY OF EXTRAPOLATION

DEPENDS ON:

• number of training points

• domain of θ and p

13

COMPARISON BETWEEN OK AND DACE

ESXTRAPOLATED FUNCTION

ABSOLUTE ERROROK DACE

14

DACE :36 sampled points

EXTRAPOLATED FUNCTION

STANDARD DEVIATION

ABSOLUTE ERROR

ADAPTIVE RESPONSE SURFACES

Real function

Extrapolated function

point of maximumstandard deviation


(simplex method)

15

ADAPTIVE DACE by σ (standard deviation)

ADAPTIVE DACE by σ (extrapolated function)

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ERROR INDEX

standard deviation

extrapolated function

local accuracy of extrapolation

Tested error indeces:

fI ⋅= σ2

fI σ=3

maxminmax

max4 σ

σ⋅

−−

=ffffI

maxminmax

min5 σ

σ⋅

−−

=ff

ffI

MAXIMUM OF FUNCTION

MINIMUM OF FUNCTION

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Adaptive DACE model by I4


ABSOLUTE ERROR

STANDARD DEVIATION

ERROR INDEX

Adaptive DACE model by I5


ABSOLUTE ERROR

STANDARD DEVIATION

ERROR INDEX

18



Responce SurfacesGradient MethodGame Theory


• Conclusion

GRADIENT METHOD

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finding the absolute optimum depends

strongly on the initial point

not robust

computing first derivatives of the function

computationally expensive when number of variables grows

DISADVANTAGES:

GRADIENT METHOD based on

ADAPTIVE DACE

ACCURATE QUICK ROBUST

mono objective

optimization method

20

SIMPLE GRADIENT METHOD

where α are costant and gradient issearching function

)( )()()()1( KKKK XfXX ∇−=+ α

computing partial derivatives by

central finite difference method

on DACE model

COMPUTATIONAL SAVING

EIGHT VARIABLES :

61.568 106 comput

60.499 19 comput

61.309 19 comput

MAX ABS

61.630

21

SIXTEEN VARIABLES :

61.626 268 computi

60.770 25 comput

61.301 22 comput

MAX ASS

61.630


• Introduction• Implementation of algorithm:

Responce SurfacesGradient MethodGame Theory


• Conclusion

22

GAME THEORY

COMPETITION INDIVIDUALS

DESIGN PROCESS

DESIGN TEAMS

multi objective optimization problems

23

• COOPERATIVE GAMES

PARETO

• NON-COOPERATIVE GAMES

NASH

• SEQUENTIAL GAMES

STACKELBERG

Optimization of

OBJECTIVE 1Y costant

given by player 2

Optimization of

OBJECTIVE 2X costant

given by player 1

Player1 Player2Optimization of

XYplayer1 player2

X Y

X Y

NASH GAME

non-cooperative symmetric

…

…

…

…

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Optimization step of

OBJECTIVE 1Y costant given by follower

Optimization of


given by leader

Player1

Player2

Optimization of

XYplayer1 player2

LEADER FOLLOWER

X

Y

Optimization of


given by leader

X

Y

hierarchic competitiveSTACKELBERG GAME

…

…

CONVERGENCE AND EQUILIBRIUM DEPEND ON:

• Variables and roles assignment

• Initial point

• Parameters of gradient method: maximum number of steps number of adaptive points in the area around n

point width of the area around n point

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TEST FUNCTIONS

TEST1 TEST2

MINIMIZE

OBJECTIVES

• nash1 16 exch 356 comput• nash2 14 exch 246 comput• nash3 24 exch 322 comput• nash4 6 exch 128 comput• nash5 16 exch 224 comput

NASHSIXTEEN VARIABLES

real Pareto Front (modeFrontier2.5)

600 comput

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STACKELBERG

• stackelberg1 9 exch 140 comput• stackelberg2 6 exch 85 comput• stackelberg3 9 exch 140 comput• stackelberg4 8 exch 111 comput

SIXTEEN VARIABLES

real Pareto Front (modeFrontier2.5)

REMARKS:

LIMIT: dependency on the setting of parameters

influence

convergence speed and result

BETTER THAN

COOPERATIVE GAMESwith the same number of simulations

27



Responce SurfaceGradient MethodGame Theory


• Conclusion

APPLICATION

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DESIGN OF A TRANSONIC AIRFOIL

IN STOCHASTIC OPERATIVE CONDITIONS

ROBUST DESIGN

ROBUST DESIGN

function to optimize

where x deterministic variablesy stochastic variables

• OPTIMIZATION OF EXPECTED VALUE

• MINIMIZATION OF STANDARD DEVIATION

ℜ→ℜ +mnxyF :)(

∑=

=n

iifn

xf1

1)(

1

)()( 1

2

−

−=

∑=

n

ffx

n

ii

fσ

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STOCHASTIC VARIABLES

• ANGLE OF ATTACK

• MACH NUMBER OF FREE STREAM

DETERMINISTIC VARIABLES

OBJECTIVES AND CONSTRAINTS

⎩⎨⎧

Cd

dCσmin

min

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥≤

≥

≤≤

RAE

ClRAECl

lRAEl

CdRAECd

dRAEd

tt

CC

CC

maxmax

σσ

σσwith

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TRANSONIC AIRFOIL RAE 2822

DESIGN POINT:

73.0=∞M°= 2α

DRAG COEFFICIENT AND LIFT COEFFICIENT OF RAE2822

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EXPECTED VALUE ∑=

=n

iifn

xf1

1)(

1

)()( 1

2

−

−=

∑=

n

ffx

n

ii

fσSTANDARD DEVIATION

COMPUTING OF , , AND

COMPUTATIONALLY EXPENSIVEdC Cdσ lC Clσ

RESPONSE SURFACES

DACE MODEL OF Cd AND Cl

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EXPLORATIVE DESIGN

OPTIMIZED AIRFOIL RAE2822

PRESSURE FIELDAND73.0=∞M

°= 2α

OPTIMIZED AIRFOIL RAE2822

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Responce SurfaceGradient MethodGameTheory


• Conclusion

CONCLUSION

34

GOAL :

reductionof

number of needed designsto implement amulti objective

optimization of problems withhigh number of design variables

APPLICATION

robust design of a transonic airfoil

multi objective constraint optimization

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troubles with multi objective constraint optimization

• inital difficulties to set algorithmparameters and to assign variables

• difficulty to extrapolate by DACE

method used for constraints induces discontinuities in objective functions

FURTHER STEPS

• to reduce algorithm parameters to set• to indroduce an assignment criterion of

variables and roles to players,considering the possibility to change the assignment during the optimization

• to find a different method for constraints of optimization problem

36

In conclusion

efficient to implementunconstraint multi objective

optimizations

wide possibilitiesof development to implement

more difficult problems

development of a design method based on ...• nash2 14 exch 246 comput • nash3 24 exch 322 comput...

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