Separable Monte Carlo Separable Monte Carlo is a
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 Bharani’s slides. 1
Probability of Failure is defined by “Limit State Function” Limit state function is defined as Response depends on a set of random variables X 1 Capacity depends on a set of random variables X 2 R C Potential failure region For small probabilities of failure & computationally expensive response calculations, MCS can be expensive! 2
Crude Monte Carlo Method Assuming Response ( ) involves Expensive computation (FEA) • isotropic material • diameter d, thickness t • Pressure P= 100 k. Pa Limit state function z Failure y Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y x 3
Crude Monte Carlo Method Assuming Response ( ) involves Expensive computation (FEA) • isotropic material • diameter d, thickness t • Pressure P= 100 k. Pa Limit state function z Failure y Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y x I – Indicator function takes value 0 (not failed) or 1( failed) 4
Crude Monte Carlo Method Assuming Response ( ) involves Expensive computation (FEA) • isotropic material • diameter d, thickness t • Pressure P= 100 k. Pa Limit state function z Failure y Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y x 0. 23 estimate 0. 21 0. 19 0. 17 0. 15 Pf 0. 13 0. 11 0. 09 0. 07 -2000 I – Indicator function takes value 0 (not failed) or 1( failed) 3000 Number of samples 5 8000
Separable Monte Carlo Method Simple Limit state function 0. 23 CMC 0. 21 Response - Stress = f (P, d, t) Capacity - Yield Strength, Y Pf estimate G (X 1, X 2) = R (X 1) – C (X 2) 0. 19 0. 17 SMC 0. 15 0. 13 0. 11 0. 09 Example: 0. 07 0 2000 4000 6000 Number of samples 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 Nx N 6 8000
Separable Monte Carlo Method • If response and capacity are independent, we can look at all of the possible combinations of random samples • An extension of the conditional expectation method Empirical CDF Example: 7
Separable Monte Carlo Method • If response and capacity are independent, we can look at all of the possible combinations of random samples • An extension of the conditional expectation method Empirical CDF Example: 8
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
Reliability for Bending in a Composite Plate • Maximum deflection • Square plate under transverse loading: from Classical Lamination Theory (CLT) where, Limit State: • RVs: Load, dimensions, material properties, and allowable deflection 10
Using the Flexibility of Separable MC • Plate bending random variables: [90°, 45°, -45°]s t = 125 mm Limit State: • Large uncertainty in expensive response • Reformulate the problem! 11
Reformulating the Limit State • Reduce uncertainty linked with expensive calculation • Assume we can only afford 1, 000 D* simulations CVR CVC 17% 3% _______________ 7. 5% 16. 5% 12
Comparison of Accuracy • pf = 0. 004 • Empirical variance calculated from 104 repetitions 13
Varying the Sample Size N = 1000 (fixed) 104 reps pf = 0. 004 14
Accuracy of probability of failure For CMC, accuracy of pf 0. 23 Pf estimate 0. 21 For SMC, Bootstrapping – resampling with replacement Re-sampling with replacement, N …. …. . . ‘b’ bootstrap samples………. . pf estimate from bootstrap sample, SMC 0. 17 0. 15 0. 13 0. 11 0. 07 -2000 k= b k=2 0. 19 0. 09 Initial Sample size N k=1 CMC Re-sampling with replacement, N pf estimate from bootstrap sample, ‘b’ estimates of bootstrapped standard deviation/ CV = error in pf estimate 15 3000 Number of samples 8000
SMC – non separable limit state z Composite pressure vessel problem y Uncertainties considered Material Properties – 5%, P – Pressure Loads – 15%, S – Strengths – 10% x Actual Pf = 0. 012 Tsai- Wu Criterion - non separable limit state { } = { 1, 2, 12}T S = {S 1 T S 1 C S 2 T S 2 C S 12 } S – Strength in different directions u – Stress per unit load 16
SMC Regrouped- Improved accuracy Using statistical independence of random variables Regrouped limit state N M N Stress per unit load M Shift uncertainty away from the expensive component furthers helps in accuracy gains. Error in pf estimate - bootstrapping CV of pf estimate (N=500) CMC SMC Original G SMC Regrouped G 40% 16% 4% 17
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
- Slides: 18