development of a design method based on ...• nash2 14 exch 246 comput • nash3 24 exch 322 comput...
TRANSCRIPT
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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
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INTRODUCTION
METHOD
ROBUSTQUICK
ACCURATE
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GAME THEORY
GRADIENT METHOD
multi objective optimization
mono objective optimization
mono 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
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PRESENTATION OUTLINE
• Introduction• Implementation of algorithm
Response surfaces Gradient MethodGame Theory
• ApplicationRobust Design
• Conclusion
IMPLEMENTATION OF MULTI OBJECTIVE
OPTIMIZATION ALGORITHM
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• 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
PRESENTATION OUTLINE
• Introduction• Implementation of algorithm
Response Surfaces Gradient MethodGame Theory
• ApplicationRobust Design
• Conclusion
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RESPONSE SURFACES
matemathical modelwhich approximates the problem
extrapolation of the functionby few known points in the domain
great time advantages
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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
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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
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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
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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
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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
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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
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COMPARISON BETWEEN OK AND DACE
ESXTRAPOLATED FUNCTION
ABSOLUTE ERROROK DACE
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DACE :36 sampled points
EXTRAPOLATED FUNCTION
STANDARD DEVIATION
ABSOLUTE ERROR
ADAPTIVE RESPONSE SURFACES
Real function
Extrapolated function
point of maximumstandard deviation
mono objective optimization
(simplex method)
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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
EXTRAPOLATED FUNCTION
ABSOLUTE ERROR
STANDARD DEVIATION
ERROR INDEX
Adaptive DACE model by I5
EXTRAPOLATED FUNCTION
ABSOLUTE ERROR
STANDARD DEVIATION
ERROR INDEX
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PRESENTATION OUTLINE
• Introduction• Implementation of algorithm
Responce SurfacesGradient MethodGame Theory
• ApplicationRobust Design
• 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
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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
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SIXTEEN VARIABLES :
61.626 268 computi
60.770 25 comput
61.301 22 comput
MAX ASS
61.630
PRESENTATION OUTLINE
• Introduction• Implementation of algorithm:
Responce SurfacesGradient MethodGame Theory
• ApplicationRobust Design
• Conclusion
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GAME THEORY
COMPETITION INDIVIDUALS
DESIGN PROCESS
DESIGN TEAMS
multi objective optimization problems
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• 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
OBJECTIVE 2X costant
given by leader
Player1
Player2
Optimization of
XYplayer1 player2
LEADER FOLLOWER
X
Y
Optimization of
OBJECTIVE 2X costant
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
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PRESENTATION OUTLINE
• Introduction• Implementation of algorithm
Responce SurfaceGradient MethodGame Theory
• ApplicationRobust Design
• 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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PRESENTATION OUTLINE
• Introduction• Implementation of algorithm
Responce SurfaceGradient MethodGameTheory
• ApplicationRobust Design
• Conclusion
CONCLUSION
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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
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In conclusion
efficient to implementunconstraint multi objective
optimizations
wide possibilitiesof development to implement
more difficult problems