Optimization and stochastic analysis - towards robust designOptimization and stochastic analysis -...

Optimization andstochastic analysis -towards robust design

Christian Bucher

Center of Mechanics and Structural DynamicsVienna University of Technology

& DYNARDO GmbH, Vienna

Overview

IntroductionRobustness conceptReliability analysisReliability-based design optimizationResponse surface methodApproximation strategiesExample - ten bar truss

Deterministic optimizationRobust optimization

Concluding remarks

2/34 © Christian Bucher 2010-2012

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


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


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


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


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


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


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


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


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


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

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

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

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


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


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)


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

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)


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


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


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


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


Deterministic optimizationGradient-based solver (Conmin)

(opt_det.mov)


opt_det.movMedia File (video/quicktime)

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


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


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)


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

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


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


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)


Compare deterministic and robustdesigns

All elements are strengthenedRelative member cross section ratios remain similar


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


Further readingChristian Bucher: Computational Analysis of Randomnessin Structural Mechanics, Taylor & Francis, 2009.


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