Separable Monte Carlo

UNCERTAINTY ANALYSIS

Separable Monte CarloSeparable Monte Carlo is a method for increasing the accuracy of Monte Carlo sampling when the limit state function is sum or difference of independent random factors.Method was developed by former graduate students Ben Smarslok and Bharani Ravishankar.Lecture based on Bharanis slides.1

2Probability of Failure

RCPotential failure regionResponse depends on a set of random variables X1Capacity depends on a set of random variables X2Failure is defined by Limit State FunctionFor small probabilities of failure & computationally expensive response calculations, MCS can be expensive!Limit state function is defined as

3Crude Monte Carlo Method xyz

isotropic material diameter d, thickness t Pressure P= 100 kPaLimit state function

Failure

Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, YAssuming Response ( ) involves Expensive computation (FEA)

4Crude Monte Carlo Method xyz

isotropic material diameter d, thickness t Pressure P= 100 kPaLimit state function

Failure

Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y

I Indicator function takes value 0 (not failed) or 1( failed)Assuming Response ( ) involves Expensive computation (FEA)

5Crude Monte Carlo Method

xyz

isotropic material diameter d, thickness t Pressure P= 100 kPaLimit state function

Failure

Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y

I Indicator function takes value 0 (not failed) or 1( failed)Assuming Response ( ) involves Expensive computation (FEA)

Separable Monte Carlo Method

Simple Limit state function

Response - Stress = f (P, d, t) Capacity - Yield Strength, YCMCSMCExample:G (X1, X2) = R (X1) C (X2)6

Nx NAdvantages of SMC Looks at all possible combinations of limit state R.V.s

Permits different sample sizes for response and capacityImproves the accuracy of the probability of failure estimated

For separable MC with the simple limit state as in Eq. (1), Ref. 10 derived analytical estimates of the standard deviation via expectation calculus67

Separable Monte Carlo Method If response and capacity are independent, we can look at all of the possible combinations of random samples

Example:Empirical CDF

An extension of the conditional expectation methodFocus on uniform distributions8

Separable Monte Carlo Method If response and capacity are independent, we can look at all of the possible combinations of random samples

Example:Empirical CDF

An extension of the conditional expectation method

Focus on uniform distributionsProblems SMCYou have the following samples of the response: 8,9,10, 8,10, 11, and you are given that the capacity is distributed like N(11,1). Estimate the probability of failure without sampling the capacity.Unlike the standard Monte Carlo sampling, we can now have different number of samples for response and capacity. How do we decide which should have more samples?Have more samples of the cheaper to calculateHave more samples of the wider distributionBoth910Reliability for Bending in a Composite PlateMaximum deflection

Square plate under transverse loading:

RVs: Load, dimensions, material properties, and allowable deflection

where,

from Classical Lamination Theory (CLT)

Limit State:

11Using the Flexibility of Separable MCPlate bending random variables:[90, 45, -45]s t = 125 mm

Large uncertainty in expensive responseReformulate the problem!

Limit State:

12Reformulating the Limit StateReduce uncertainty linked with expensive calculationAssume we can only afford 1,000 D* simulations

CVRCVC_____________________________

17%3%

7.5%16.5%

13Comparison of Accuracypf = 0.004Empirical variance calculated from 104 repetitions

14N = 1000 (fixed) 104 reps pf = 0.004

Varying the Sample Size

Accuracy of probability of failure15For SMC, Bootstrapping resampling with replacement= error in pf estimate

Initial Sample size N Re-sampling with replacement, NRe-sampling with replacement, Nbootstrapped standard deviation/ CV

.... b bootstrap samples..pf estimate from bootstrap sample, pf estimate from bootstrap sample, b estimates of

k=1k=2k= b

CMCSMC

For CMC, accuracy of pf

SMC non separable limit stateTsai- Wu Criterion - non separable limit stateActual Pf = 0.012

{ } = {1, 2,12}T S = {S1T S1C S2T S2C S12 }16xyz

Uncertainties consideredMaterial Properties 5%, P Pressure Loads 15%, S Strengths 10%S Strength in different directions

u Stress per unit loadComposite pressure vessel problem

SMC Regrouped- Improved accuracy17

Regrouped limit state

N M

N MShift uncertainty away from the expensive component furthers helps in accuracy gains.CMCSMC Original G SMCRegrouped G40% 16%4% CV of pf estimate (N=500)

Error in pf estimate - bootstrappingUsing statistical independence of random variablesStress per unit load

Additional problems SMC The following samples were taken of the stress and strength of a structural componentStress: 9, 10, 11, 12Strength: 10.5, 11.5, 12.5, 13.5Give the estimate of the probability of failure using crude Monte Carlo and SMCWhat is the accuracy of the Monte Carlo estimate?How would you estimate the accuracy of SMC from the data?18