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. Lecture based on Bharanis slides. 1 2 Probability of Failure RC Potential failure region Response depends on a set of random variables X 1 Capacity depends on a set of random variables X 2 Failure is defined by Limit State Function For small probabilities of failure & computationally expensive response calculations, MCS can be expensive! Limit state function is defined as 3 Crude Monte Carlo Method x y z isotropic material diameter d, thickness t Pressure P= 100 kPa Limit state function Failure Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y Assuming Response ( ) involves Expensive computation (FEA) 4 Crude Monte Carlo Method x y z isotropic material diameter d, thickness t Pressure P= 100 kPa Limit 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) 5 Crude Monte Carlo Method x y z isotropic material diameter d, thickness t Pressure P= 100 kPa Limit 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, Y CMC SMC Example: G (X 1, X 2 ) = R (X 1 ) C (X 2 ) 6 Nx N Advantages of SMC Looks at all possible combinations of limit state R.V.s Permits different sample sizes for response and capacity Improves the accuracy of the probability of failure estimated 7 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 8 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 Problems SMC You 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 calculate Have more samples of the wider distribution Both 9 10 Reliability for Bending in a Composite Plate Maximum deflection Square plate under transverse loading: RVs: Load, dimensions, material properties, and allowable deflection where, from Classical Lamination Theory (CLT) Limit State: 11 Using the Flexibility of Separable MC Plate bending random variables: [90, 45, -45]s t = 125 m Large uncertainty in expensive response Reformulate the problem! Limit State: 12 Reformulating the Limit State Reduce uncertainty linked with expensive calculation Assume we can only afford 1,000 D * simulations CV R CV C _____________________________ 17%3% 7.5%16.5% 13 Comparison of Accuracy p f = Empirical variance calculated from 10 4 repetitions 14 N = 1000 (fixed) 10 4 reps p f = Varying the Sample Size Accuracy of probability of failure 15 For SMC, Bootstrapping resampling with replacement = error in p f estimate Initial Sample size N Re-sampling with replacement, N bootstrapped standard deviation/ CV .... b bootstrap samples.. p f estimate from bootstrap sample, b estimates of k=1 k=2 k= b CMC SMC For CMC, accuracy of p f SMC non separable limit state Tsai- Wu Criterion - non separable limit state Actual P f = { } = { 1, 2, 12 } T S = {S 1T S 1C S 2T S 2C S 12 } 16 x y z Uncertainties considered Material Properties 5%, P Pressure Loads 15%, S Strengths 10% S Strength in different directions u Stress per unit load Composite pressure vessel problem SMC Regrouped- Improved accuracy 17 Regrouped limit state NMNM Shift uncertainty away from the expensive component furthers helps in accuracy gains. CMCSMC Original G SMC Regrouped G 40%16%4% CV of p f estimate (N=500) Error in pf estimate - bootstrapping Using statistical independence of random variables Stress per unit load Additional problems SMC The following samples were taken of the stress and strength of a structural component Stress: 9, 10, 11, 12 Strength: 10.5, 11.5, 12.5, 13.5 Give the estimate of the probability of failure using crude Monte Carlo and SMC What is the accuracy of the Monte Carlo estimate? How would you estimate the accuracy of SMC from the data? 18