Bharani Ravishankar, Benjamin Smarslok Advisors Dr. Raphael T. Haftka, Dr. Bhavani V. Sankar

1. SEPARABLE SAMPLING OF THE LIMIT STATE FOR ACCURATE MONTE CARLO SIMULATION Bharani Ravishankar, Benjamin Smarslok Advisors Dr. Raphael T. Haftka, Dr. Bhavani V. Sankar

2. Motivation - Probability of Failure Problems • Monte Carlo simulation-based techniques can require expensive calculations to obtain random samples • To improve the accuracy of pf estimate for complex limit states without performing additional expensiveresponse computation? Capacity R C

3. Outline & Objectives • Review Monte Carlo simulation techniques - Crude Monte Carlo method - Separable Monte Carlo method • Simple limit state example - Explain the advantage of regrouping random variables • Complex (non-separable) limit state example - Tsai Wu Criterion -Demonstrate regrouping & separable sampling of stress and strength • Compare the accuracy of the Monte Carlo methods • Conclusions

4. Monte Carlo Simulations Common way to propagate uncertainty from input to output & calculate probability of failure • Limit state function is defined as • Crude Monte Carlo (CMC) - most commonly used Response depends on a set of random variables X1 Capacity depends on a set of random variables X2 R C Potential failure region

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

6. Separable Monte Carlo Method If response and capacity are independent, we can use all of the possible combinations of random samples Empirical CDF CMC SMC Example:

7. Regrouping the random variables Random variables Regrouping the random variables Response - Stress  = f (P, d, t) Capacity -Yield Strength, Y Stress is a linear function of load P P, d, t and Y are independent random variables  u – Stress per unit load Regrouped variables Stresses per unit load  u Pressure load P Yield Strength Y

8. Monte Carlo Simulation Summary • Crude MC traditional method for estimating pf • Looks at one-to-one evaluations of limit state • Expensive for small pf • Separable MC uses the same amount of information as CMC, but is inherently more accurate • Use when limit state components are independent • Looks at all possible combinations of limit state R.V.s • Permits different sample sizes for response and capacity For a complex limit state, the accuracy of the pf estimate could be improved by regrouping and separable sampling of the RVs

9. Complex limit state problem Determination of Stresses z • Pressure vessel -1m dia. (deterministic) • Thickness of each lamina 0.125 mm (deterministic) • Lay up- [(+25/-25)]s • Internal Pressure Load, P= 100 kPa y Material Properties E1,E2,v12,G12 x Laminate Stiffness (FEA) Loads P Strains Stress Stress in each ply

10. Limit State - Tsai-Wu Failure Criterion Non-separable limit state No distinct response and capacity Random Variables F = f (Strengths S) =f (Laminate Stiffness aij, Pressure P) obtained from Classical Laminate Theory (CLT) F – Strength Coefficients S – Strengths in Tension and Compression in the fiber and transverse direction Limit state G = f (F, ); G < 0 safe G ≥ 0 failed

11. Estimation of probability of failure RVs - Uncertainty Separable Monte Carlo { } = {1, 2,12}TS = {S1T S1C S2T S2C S12 } Crude Monte Carlo N N N M CV(Pressure) > CV(Strengths) > CV(Stiffness Prop.) All the properties are assumed to have a normal distribution

12. CMC and SMC Comparison N=500, repetitions = 10000 Actual Pf = 0.012 Expensive Response limited to N=500 (CLT) Cheap Capacity varied M= 500, 5000 samples

13. Tsai – Wu Limit State Function Original limit state Stresses  Stresses per unit load Load P Strengths S Finite Element Analysis From Statistical distribution Finite Element Analysis Expensive From Statistical distribution Expensive Cheap Cheap Regrouping the expensive and inexpensive variables N M Regrouped limit state Expensive Cheap Pressure Load P Strengths S Stresses per unit load u – Material Properties, P – Pressure Loads, S – Strengths

14. Regrouping the random variables

15. Comparison of the Methods Expensive RVs limited to N=500 (CLT) Cheap RVs varied M= 500-50000 samples N=500 repetitions = 10000 Actual Pf = 0.012

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

17. Summary & Conclusions • Separable Monte Carlo was extended to non-separable limit state - Tsai-Wu failure criterion. • In Tsai-Wu Limit State, uncertainty in load affects the expensive stresses. By calculating response to unit loads, we can sample the effect of random loads more cheaply. • Statistical independence of the random variables enables appropriate sampling, thereby improving the accuracy of the estimate. • Shift uncertainty away from the expensive component furthers helps in accuracy gains. • Accuracy of the methods - for the same computational cost,