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Separable Monte Carlo. Separable 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 . - PowerPoint PPT Presentation
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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