Separable Monte Carlo

1. 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 BharaniRavishankar. • Lecture based on Bharani’s slides.

2. Probability of Failure Failure is defined by “Limit State Function” • Limit state function is defined as Response depends on a set of random variables X1 Capacity depends on a set of random variables X2 R C Potential failure region For small probabilities of failure & computationally expensive response calculations, MCS can be expensive!

3. Crude Monte Carlo Method AssumingResponse () involves Expensive computation (FEA) z y • isotropic material • diameter d, thickness t • Pressure P= 100 kPa x Failure Random variables Response - Stress  = f (P, d, t) Capacity -Yield Strength, Y Limit state function

4. Crude Monte Carlo Method AssumingResponse () involves Expensive computation (FEA) z y • isotropic material • diameter d, thickness t • Pressure P= 100 kPa x Failure Random variables Response - Stress  = f (P, d, t) Capacity -Yield Strength, Y Limit state function I – Indicator function takes value 0 (not failed) or 1( failed)

5. Crude Monte Carlo Method AssumingResponse () involves Expensive computation (FEA) z y • isotropic material • diameter d, thickness t • Pressure P= 100 kPa  x Failure Random variables Response - Stress  = f (P, d, t) Capacity -Yield Strength, Y Limit state function I – Indicator function takes value 0 (not failed) or 1( failed)

6. Separable Monte Carlo Method CMC G (X1, X2) = R (X1) – C (X2) SMC  Response - Stress  = f (P, d, t) Capacity -Yield Strength, Y Example: Simple Limit state function 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

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. 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:

9. 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

10. Reliability for Bending in a Composite Plate Maximum deflection Square plate under transverse loading: RVs:Load, dimensions, material properties,and allowable deflection from Classical Lamination Theory (CLT) where, Limit State:

11. Using the Flexibility of Separable MC Plate bending random variables: [90°, 45°, -45°]s t = 125 mm 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 CVRCVC _____________________________ 17%3% 7.5%16.5%

13. Comparison of Accuracy pf = 0.004 Empirical variance calculated from 104 repetitions

14. N = 1000 (fixed) 104 reps pf = 0.004 Varying the Sample Size

15. Accuracy of probability of failure For CMC, accuracy of pf CMC Initial Sample size N SMC k=1 k= b k=2 Re-sampling with replacement, N Re-sampling with replacement, N For SMC, Bootstrapping – resampling with replacement ….…... ‘b’ bootstrap samples……….. pf estimate from bootstrap sample, pfestimate from bootstrap sample, bootstrapped standard deviation/ CV = error in pf estimate ‘b’ estimates of

16. 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}TS = {S1T S1C S2T S2C S12} S – Strength in different directions  u – Stress per unit load

17. SMC Regrouped- Improved accuracy Using statistical independence of random variables N M Stress per unit load Shift uncertainty away from the expensive component furthers helps in accuracy gains. Error in pf estimate - bootstrapping N M Regrouped limit state CV of pfestimate (N=500)

18. 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?