optimization and stochastic analysis - towards robust designoptimization and stochastic analysis -...
TRANSCRIPT
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Optimization andstochastic analysis -towards robust design
Christian Bucher
Center of Mechanics and Structural DynamicsVienna University of Technology
& DYNARDO GmbH, Vienna
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Overview
IntroductionRobustness conceptReliability analysisReliability-based design optimizationResponse surface methodApproximation strategiesExample - ten bar truss
Deterministic optimizationRobust optimization
Concluding remarks
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Uncertainties in optimizationDesign variables (e.g. manufacturing tolerances)Objective function (e.g. tolerances, external factors)Constraints (e.g. tolerances, external factors)
Design Variable 1
Contour lines of objective functionDesign Variable 2
Infeasible Domain
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Traditional design approachIntroduce ”safety factors” into the constraintsLeads to results satisfying safety requirement, but notnecessarily optimal designs
Design Variable 1
Design Variable 2
Infeasible Domain Safety
margin
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Robustness
Optimal design should not be “sensitive” to smallvariations of uncertain parametersFor the case of stochastic uncertainty, robustness can bemeasured in statistical termsChoice of robustness measures
Variance-based: Global behavior, relatively frequenteventsProbability-based: Specific behavior (oftensafety-related), rare events
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Robustness measuresUse probabilistic quantities in objective and/orconstraintsAdd multiple of standard deviation of objective to themean values to form new objective function
R = f̄+ kσf
Case k = 0 corresponds to game theory (large portfoliomanagement, insurance, bank loans, fair gambling, ...)Use probability of constraint violation as new constraint
P[fk(x∗)] > 0] ≤ ε
In safety related constraints, the probability level εmaybe very small
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Optimization strategies
Conventional strategyOptimizecheck robustness/reliabilityif not sufficient, optimize again with modified constraintsEasily implemented, may lead to non-optimal results
Full RDO strategyincorporate robustness/reliability measure intooptimization problemSolve directly for design satisfying robustness/reliabilityconstraintLeads directly to desired solution, may be very expensive
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Mathematical exampleTwo local minima, located at x = -1.96 (f = 15.22) and x =0.48 (f = 15.90)
-2.50 -1.25 0.00 1.25 2.500
10
20
30
40
50
60
x
f(x)
f(x) = 20 + x4 − 10.25 + (x− 0.5)2
− 10.05− (x + 2)2
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Variability of designDesign variable is random with a mean value of x∗ and astandard deviation of 0.5Assume Gaussian distributionGenerate random samples of xCompute statistics of objective function f(x) (mean,standard deviation)Deterministic global minimum is much worse on averageand has larger scatterRobust minimum appears to be located near the secondlocal minimum
x∗ f̄ σf-1.96 32.56 23.160.48 17.83 2.30
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StatisticsFluctuations due to sampling uncertaintySmoothing is helpful for optimization (here: polynomialof 5th order)
-2.50 -1.25 0.00 1.25 2.500
10
20
30
40
50
60
x
f(x)
f̄
σf
Simulation
Smoothed
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Variance-based robustness
Combine mean value and standard deviation
-2.50 -1.25 0.00 1.25 2.500
25
50
75
100
125
150
x
f(x)
f̄
f̄ + σf
f̄ + 2σf R = f̄+ kσf
f xR Rmin0 0.34 17.971 -0.14 18.892 -0.21 19.433 -0.23 19.94
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Reliability analysisMechanical system
Mpl
F
Me
L
Failure condition
F = {(F, L,Mpl) : FL ≥ Mpl} = {(F, L,Mpl) : 1−FLMpl
≤ 0
Failure probabilityP(F) = P[{X : g(X) ≤ 0}
P(F) =∫ ∞−∞
. . .
∫ ∞−∞
Ig(x)fX1...Xndx1 . . . dxn
Ig(x1 . . . xn) = 1 if g(x1 . . . xn) ≤ 0 and Ig(.) = 0 else12/34 © Christian Bucher 2010-2012
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First-order reliability method (FORM)Rosenblatt-Transformation, e.g. for Nataf model
Yi = Φ−1[FXi(Xi)]; i = 1n
U = L−1Y; CYY = LLT
Inverse transformation
Xi = F−1Xi
[Φ
( n∑k=1
LikUk
)]
Computation of “design point”
u∗ : uTu → Min.; subject to: g[x(u)] = 0
Linearize at the design point (in standard Gaussian space)13/34 © Christian Bucher 2010-2012
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FORM procedureFind point u∗ with minimal distance β from origin instandard Gaussian space
u1
u2
β
s1g(u) = 0
ḡ(u) = 0u∗
s2
ḡ : −n∑
i=1
uisi+ 1 = 0;
n∑i=1
1s2i
=1β2
P(F) = Φ(−β)14/34 © Christian Bucher 2010-2012
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Optimization loopOuter optimization loop controls the structural designProbability of constraint violation computed by FORMInner optimization driven by random variablesBoth loops need gradients
Compute probability ofconstraint violationPk = P[fk(xj) > 0]
Compute objective f0(xj)
Start optimization loop
Create one design xj
Check convergence
FORM optimization loop
FE analysis
Repe
atforg
radien
ts
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Response surface methodReduce computational effort by replacing expensive FEanalysesEstablish meta-models in terms of simple mathematicalfunctionsFit model parameters to FE solution using regressionanalysis
x
yη
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RegressionAdjust a model to experiments
Y = f(X,p)
Set of parameters
p = [p1, p2, . . . , pn]T
Experimental values for input X and output Y
(X(k),Y(k)), k = 1 . . .m
Search for best model by minimizing the residual
S(p) =m∑
k=1
[Y(k) − f(X(k),p)
]2; p∗ = argmin S(p)
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Linear regressionLinear dependence on parameters (not on variables!)
f(X,p) =n∑
i=1pigi(X)
Necessary condition for a minimum
∂S∂pj
= 0; j = 1 . . . n
Solutionm∑
k=1
{gj(Xk)[Yk −
n∑i=1
pigi(Xk)]}
= 0; j = 1 . . . n
Qp = q18/34 © Christian Bucher 2010-2012
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Example
Adapt polynomial function to 6 data points
Training set
0 1 2 3 4 5 6 7 8 9 10 110123456789
10
Variable X
Varia
bleY
(Xk, Yk)
Test set
0 1 2 3 4 5 6 7 8 9 10 110123456789
10
Variable X
Varia
bleY
(Xk, Yk)
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Regression result
Adjust model to training data, cross-check with test data
Training set
0 1 2 3 4 5 6 7 8 9 10 110123456789
10
Variable X
Varia
bleY Quadratic
Cubic
Test set
0 1 2 3 4 5 6 7 8 9 10 110123456789
10
Variable X
Varia
bleY Quadratic
Cubic
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What to approximate?For reliability analysis, the limitstate function g(X) is neededImmediate approximation of gmayintroduce unwanted nonlineardependencies on input variablesExample: shear stresses in aconsole with square cross section
τxy =3FH2B2 ; τxz =
3FV2B2
Failure criterion (v. Mises)
g(FH, FV) = βF −√
3(τ 2xy + τ 2xz) ≤ 0
L B
B
FH
FV
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Comparison
Stresses τxy and τxz are linear functions of FH and FvLimit state function g(FH, FV) is highly nonlinear andcontains a singularity
τxy τxz g
FH FH FH FVFVFV
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Example - ten bar trussConfiguration: L = 360, F1 = F2 = 100000
L L
L
F1 F2
1 2
3 4
5 67
8
9
10
1 2 3
4 5 6
Objective: Minimize structural massConstraints:
All member stresses < 25000All nodal displacements < 2
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Deterministic optimizationGradient-based solver (Conmin)
(opt_det.mov)
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opt_det.movMedia File (video/quicktime)
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Optimal design
Objective function: m = 5048Cross sectional areas
Member Area Member Area1 5.53 6 0.102 0.10 7 2.953 4.91 8 4.554 3.83 9 4.555 0.10 10 0.12
Active constraints: no member stress, displacements innodes 3 and 6Effort: ≈400 FE analyses
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Robustness evaluationTake into account randomness of loads (F1 and F2independent, log-normally distributed, coefficient ofvariation = 0.3)Compute variability of constraintsEstimate probability of constraint violation(s) using adistribution hypothesis (Gaussian)High probability of violating active constraints, but also 3inactive onesEffort: 1000 FE analyses
Member Pσ Member Pσ1 0.00 6 0.002 0.00 7 0.043 0.00 8 0.004 0.00 9 0.005 0.35 10 0.00
Node Pu Node Pv2 0.00 2 0.003 0.00 3 0.515 0.00 5 0.096 0.00 6 0.51
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Robust optimization
Model randomness of loads (F1 and F2 independent,log-normally distributed)Accept constraint violation(s) with a probabilitycorresponding to a reliability index β = 3Use FORM to obtain β for all designs during theoptimization processBest designs depend on the coefficient of variation of F1and F2Effort: ≈40.000 FE analyses (factor 100 vs. deterministiccase)
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Robust optima
COV 0.10 0.2 0.30mopt 6540 8512 10942
0.00 0.10 0.20 0.305000
6000
7000
8000
9000
10000
11000
Coefficient of variation of load
Objective
Computational effort: 120.000 FE runs for 3 COVs28/34 © Christian Bucher 2010-2012
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Approximation of stresses
Stress in member 5Compare stresscomputed from FEanalysis to responsesurface results(1000 randomsamples)
0 10000 20000 300000
10000
20000
30000
True stress 5
Fitte
dstress
5
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Approximation of displacements
Verticaldisplacement ofnode 6Comparedisplacementcomputed from FEanalysis to responsesurface results(1000 randomsamples) 0 1 2 3
0
1
2
3
True disp 17
Fitte
ddisp
17
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Robust optima from RSMCOV 0.10 0.2 0.30mopt 6711 8517 11571
0.00 0.10 0.20 0.305000
6000
7000
8000
9000
10000
11000
Coefficient of variation of load
Objective
Direct
RSM
Computational effort reduced by a factor of 40 for oneCOV (120 for 3 COVs)
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Compare deterministic and robustdesigns
All elements are strengthenedRelative member cross section ratios remain similar
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Benefits of robust optimization
Avoids highly “specialized” designsReduces imperfection sensitivityNaturally includes statistical uncertainties into theprocessAllows the inclusion of quality control measures(manufacturing, maintenance) into the design processBUT: computationally very expensive unless based onapproximations such as response surface models
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Further readingChristian Bucher: Computational Analysis of Randomnessin Structural Mechanics, Taylor & Francis, 2009.
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