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

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

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 3

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 4 RC Potential failure region Response depends on a set of random variables X 1 Capacity depends on a set of random variables X 2

5. 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 Example: I – Indicator function takes value 0 (not failed) or 1( failed) Assuming Response ( ) involves Expensive computation (FEA)

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

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

8. Monte Carlo Simulation Summary Crude MC traditional method for estimating p f –Looks at one-to-one evaluations of limit state –Expensive for small p f 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 8 For a complex limit state, the accuracy of the p f estimate could be improved by regrouping and separable sampling of the RVs

9. Complex limit state problem Pressure vessel -1m dia. (deterministic) Thickness of each lamina 0.125 mm (deterministic) Lay up- [(+25/-25)] s Internal Pressure Load, P= 100 kPa 9 x y z Material Properties E 1,E 2,v 12,G 12 Loads P Laminate Stiffness (FEA) Strains Stress Stress in each ply Determination of Stresses

10. Limit State - Tsai-Wu Failure Criterion 10 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 Non-separable limit state obtained from Classical Laminate Theory (CLT) F = f (Strengths S) =f (Laminate Stiffness a ij, Pressure P) No distinct response and capacity Random Variables

11. RVs - Uncertainty 11 ParametersMeanCV% E 1 (GPa)159.1 5 E 2 (GPa)8.3 G 12 (GPa)3.3 12 (no unit) 0.253 Pressure P (kPa)10015 S 1T (MPa)2312 10 S 1C (MPa)1809 S 2T (MPa)39.2 S 2C (MPa)97.2 S 12 (MPa)33.2 All the properties are assumed to have a normal distribution CV(Pressure) > CV(Strengths) > CV(Stiffness Prop.) Separable Monte Carlo Crude Monte Carlo { } = { 1, 2, 12 } T S = {S 1T S 1C S 2T S 2C S 12 } NN NM Estimation of probability of failure

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

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

14. 14 Regrouping the random variables Stresses Material Properties Load P Strengths S CostExpensiveCheap Uncertainty~ 5%15%10%

15. 15 Comparison of the Methods Expensive RVs limited to N=500 (CLT) Cheap RVs varied M= 500-50000 samples N=500 repetitions = 10000 M Crude Monte Carlo Separable Monte Carlo Separable Monte Carlo regrouped RVs 500 40.0%20.6%36.3% 1000 18.4% 26.0% 5000 16.2%11.7% 10000 16.0%8.2% 50000 15.6%4.0% Actual P f = 0.012

16. Accuracy of probability of failure 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

17. Summary & Conclusions 17 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, CMCSMC -original limit stateSMC- Regrouped limit state CV%40%16%4%