reliability-based design optimization using response

Structural & Multidisciplinary Optimization Group

EGM 6934 Experimental Engineering Optimum Design

Reliability-based Design Optimization Using Response Surface Approximations

Xueyong QuRaphael T. Haftka

Nov 13, 2002

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O tliOutline• Background and MotivationBackground and Motivation

– Deterministic design Vs. Reliability-based design

M C l Si l i (MCS)• Monte Carlo Simulation (MCS)

• Reliability-based Design Optimization (RBDO) UsingReliability based Design Optimization (RBDO) Using Response Surface Approximations

C l di R k• Concluding Remarks

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B k dBackground

• Deterministically optimized laminates are sensitive toDeterministically optimized laminates are sensitive to uncertainties– Deterministic design push design to the boundary of the constraints.

ib li d l h di i f idi i l l di– Fibers aligned along the direction of unidirectional loading – Poor performance to loading transverse to fiber direction

F

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D t i i ti d i R li bilit b d d iDeterministic design vs. Reliability-based design

• Deterministic design (Safety factor or worst case)Deterministic design (Safety factor or worst case) Typically use experience-based safety factor to knock down allowables

1.4 for space applications D i t ll t d i i i li ti Design concepts well accepted in engineering application Unknown actual safety level Usually not optimal in terms of weight and safety

• Reliability-based design (Probabilistic design) Computationally expensive

• Computational cost of a single reliability analysis is on par with aComputational cost of a single reliability analysis is on par with a deterministic optimization

• Total cost of RBDO is similar to multi-level optimization Complex to apply

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Complex to apply


D t i i ti D i f C it L i tDeterministic Design of Composite Laminates

• Design of angle-ply laminateDesign of angle-ply laminate– Maximum strain failure criterion

4 tth minimize NAxial 214 tth minimize

such thattc

NAxial

NHoop

y

1 2

1212

222

111

u

tc

x

2

1

1212

005.00.005

tt

Load induced by internal pressure:

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2NHoop = 4,800 lb./in., NAxial = 2,400 lb./in.


Summary of Deterministic DesignSummary of Deterministic Design

• Optimal ply angles are 27 from hoop direction• Optimal ply-angles are 27 from hoop direction

• Laminate thickness is 0.1 inchLaminate thickness is 0.1 inch

• Probability of failure (510-4) is high with safety factor 1 4• Probability of failure (510 ) is high with safety factor 1.4.

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R li bilit b d L i t D iReliability-based Laminate Design

4hi i i i i bl 214 tth minimize

that such

• 4 Design Variables– 1, 2, t1, t2

1005.0 tPP t

• 12 Normal Random Variables

– Tzero (CV = 0.03)– 1, 2 (CV = 0.035)

2005.0 t

Pt = 10-4

1, 2 (CV )– E1, E2, G12, 12 (CV = 0.035)– 1

c, 1t (CV = 0.06)

c t u (CV 0 09)

• First ply failure principle

Pt 10 – 2c, 2

t, 12u (CV = 0.09)

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Probability of Failure and Limit StateProbability of Failure and Limit Statex2 G(x)<0G(x) 0

Unsafe Region

G(x)>0

G(x)=0Limit State

x1

( )Safe Region

• Probability of failure xxx

X dfPG

0)(

)(

• Integral is hard to evaluate: unknown integration domain and

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• Integral is hard to evaluate: unknown integration domain and high dimension.


Probability Calculation• Monte Carlo simulation (MCS)

– Large number of samples for small probability, computationally expensive

– Random errors (noise) due to limited sample size

• Moment-based methods– Computationally efficient for single failure mode problem– Difficult for problems with multiple failure modes

• Response surface approximation (RSA)Response surface approximation (RSA)– Reduce computational cost of MCS– Filter out noise

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M t C l Si l tiMonte Carlo Simulation

P b bilit l l ti insidePP• Advantages

simple robust and accurate with large enough samples

Probability calculation:total

inside

PP

– simple, robust, and accurate with large enough samples.

• Disadvantages– Computational cost is high for small probability.

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– Random errors (noise) hinder design optimization.


Estimating Probability Using MCSg y gSample size Analytical

solution = /400MCS solution Relative errors

1e2 0 007854 0 00 100%1e2 0.007854 0.00 100%

1e3 0.007854 0.004 49%

1 4 0 007854 0 0085 8%1e4 0.007854 0.0085 8%

1e5 0.007854 0.00847 7.8%

1e6 0.007854 0.007851 0.04%

• Results of MCS changes from simulation to simulation• Do not use the same seed for random number generators,

unlessDebug program


– Debug program– Sensitivity study


Response Surface Approximation (RSA)Response Surface Approximation (RSA)

A i l i hi b d

y

• Approximate relationship between y and x– Assume basis function z(x), least square estimate

to obtain b– Low order polynomials as basis

x1

x2bxzx T)()(y

• Statistical design of experiments (DOE)– select design points, to obtain desired accuracy at

i i

y=f(x1,x2,…,xn)+e

minimum cost.– Face center central composite design (FCCCD)– Latin Hypercube sampling (LHS)



Central Composite Design (CCD)Central Composite Design (CCD)• Classical DOE for quadratic RS• FCCCD

– Move axial points to surface of the hypercube



Response Surface Optionsp p Design response surface approximation (DRS)

Response design ariables: W=W(d)– Response v.s. design variables: W=W(d)– Used in optimization: Venter and Haftka (96, 97), Roux et al. (96), etc.

• Stochastic response surface approximation (SRS)– Response v.s. random variables: G=G(x)– Used in probability calculation: Fox (94, 96), Romero (98), etc.– Need to construct SRS at every point encountered in optimizationy p p

Analysis response surfaces (ARS, Qu et al., 2000)– Response v.s. random variables + design variables: G=G(x, d)

Ad t i ffi i f SRS– Advantage: improve efficiency of SRS– Challenges:

• DOE for both design and random variablesC S i hi h di i l ( 10 i bl )


• Construct RS in high dimensional space ( > 10 variables)


A l i R S f (ARS)Analysis Response Surfaces (ARS)Strain

• Fit strains in terms of 12 variables• Design of experiments:

L ti H b S li (LHS)– Latin Hypercube Sampling (LHS)

D.V.R.V.

Strain = g(θ1, θ2, t1, t2, E1, E2, G12, 12, 1, 2, Tzero, Tservice)

ARS

• Probabilities calculated by MCS based on fitted polynomials– Reduce computational cost of MCS


p


Latin Hypercube Sampling (LHS)Latin Hypercube Sampling (LHS)

Uniform

NormalC

• Design of experiments for RSA– Sample the random variable space according to distributions

(Courtesy of Wyss and Jorgensen)

Sample the random variable space according to distributions– Design variables are treated as uniformly distributed within design

bounds– Space-filling feature


– More flexible than orthogonal arrays: arbitrary number of design points


R li bili b d D i O i i iReliability-based Design Optimization

• Design Response Surface (DRS) ARS• Design Response Surface (DRS)– Fit to Probability in terms of 4 D.V.– Filter out noise generated by MCS

ARS

DOE & MCS– Used in RBDO

ProbabilityDRS

Optimization

i

tiConverge?

Yes

No No


iPOF = p(θ1, θ2, t1, t2) Stop


ApproximationApproximationDesign variables 1 2 t1 (inch) t2 (inch) Range 20 to 30 20 to 30 0.0125 to 0.03 0.0125 to 0.03

Quadratic ARS based on LHS 182 points

ARS Error Statistics 2 in 1

Rsquare Adj. 0.996

RMSE Predictor (millistrain) 0.060

Mean of Response (millistrain) 8 322

FCCCD 25 points LHS 252 points DRS Error Statistics

th

Mean of Response (millistrain) 8.322

quadratic 5th order

Rsquare Adj. 0.686 0.998

RMSE Predictor (probability) 5.3e-4 0.12e-4


(p y)

Mean of Response (probability) 3.2e-4 0.44e-4


Optimization

• Deterministic, Reliability-based, and Simplified designs

Pl A l Thi k (i h) Probability of Ply Angles Thickness (inch) Probability of Failure

Deterministic [(27.0)2/(27.0)3]S 0.10 5e-4

Reliability [(24.9)3/(25.2)3]S 0.12 0.55e-4Reliability [(24.9)3/(25.2)3]S 0.12 0.55e 4

Simplified [(25)6]S 0.12 0.57e-4

• The thickness is high for application



Improving Reliability-based DesignImproving Reliability-based Design• Reliability-based design

– Thickness of 0 12 inchThickness of 0.12 inch– Probability of failure of 10-4 level

Must reduce uncertainties: Quality control (QC)

– Reject small numbers of poor specimenT t di t ib ti f ll bl t l id ( 2 )– Truncate distribution of allowables at lower side (–2 )

Reduce material scatter– Reduce Coefficient of Variation (CV)– Better manufacture process (Better curing process)

Improve allowablesI M V l f ll bl


– Increase Mean Value of allowables– New materials


Change Distribution of 2 allowable• Reduce scatter (CV) by 10%• Reduce scatter (CV) by 10%

Probability of failure (0.12 inch)

CV = 0 09 CV = 0 081 CV = 0.09 CV = 0.081

[(25)6]S 0.57e-4 0.11e-4

• Increase allowable (Mean value) by 10%

Probability of failure Probability of failure (0.12 inch)

E(ε2u) = 0.0154 E(ε2

u) = 0.01694

[(25)6]S 0.57e-4 0.03e-4



Quality Control (QC) on 2 allowable• Reduce probability of failure

Probability of failure (0.12 inch) Truncate at 3 Truncate at 2 (0.12 inch)

Normal Truncate at -3 (14 out of 10,000)

Truncate at -2 (23 out of 1,000)

[(25)6]S 0.57e-4 0.001e-4 < 1e-7

• Reduce thickness

0.12 inch 0.10 inch 0.08 inch (POF=1e-4)

Normal Truncate at –2.8 (26 out of 10,000)

Truncate at -1.35 (90 out of 1,000)

[(25)6]S 0.57e-4 0.9e-4 1e-4


[( 5)6]S 0.57e 4 0.9e 4 1e 4


Tradeoff PlotTradeoff Plot

1.0E-01

1.0E+00NominalQuality control to -2 Sigma10% increase in allo able

1.0E-03

1.0E-02

babi

lity

10% increase in allowable10% reduction in variabilityAllS i 6

1 0E 06

1.0E-05

1.0E-04

Failu

re P

rob

1.0E-08

1.0E-07

1.0E-06F

1.0E 080.06 0.08 0.1 0.12 0.14 0.16

Thickness (inch)


• To be chosen by the cost of implementing these methods


Concluding RemarksConcluding Remarks

• 2 allowable is the key parameter• Probability of failure (510-4 to 0.610-4) Vs. Thickness (0.1 to

0.12 inch)

• Three methods to improve the design– Quality control, Reducing scatter, and Increase mean value of 2

allowableallowable– Changing other parameters are not as efficient as changing 2 allowable

• Analysis RS combining random and design variables is efficient– DOE: Latin Hypercube sampling (LHS)


reliability-based design optimization using response

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