Monte Carlo Based Reliability Analysis

Transcription

1 Monte Carlo Based Reliability Analysis Martin Schwarz 15 May 2014 Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

2 Plan of Presentation Description of the problem Monte Carlo Simulation Sensitivity based Importance Sampling Subset Simulation Comparison Prospects Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

3 Reliability Probability of failure: Use a random R n valued random variable X for describing the parameters of an input-output model of an engineering structure. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

4 Reliability Probability of failure: Use a random R n valued random variable X for describing the parameters of an input-output model of an engineering structure. P f := P(X F ), F := {x R n : x is a parameter combination leading to failure} Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

5 Reliability Probability of failure: Use a random R n valued random variable X for describing the parameters of an input-output model of an engineering structure. P f := P(X F ), F := {x R n : x is a parameter combination leading to failure} We define F by a function Φ(x): x F Φ(x) > 1 Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

6 The Small Launcher Model FE-model of the Ariane 5 frontskirt. 35 input parameters: Loads, E-moduli, yield stresses etc. Considered as uniformly distributed with spread ±15% around nominal value. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

7 The Small Launcher Model Φ(x) := max n PEEQ(x) > 0.07 SP(x) > 180 MPa EV (x) < Martin Schwarz PEEQ(x) SP(x) , 180, EV (x) o plastification of metallic part rupture of composite part buckling Monte Carlo Based Reliability Analysis 15 May / 19

8 Monte Carlo Simulation Theorem The probability of failure can be estimated by P MC f P f = P(X F ) P MC f := 1 N N 1 F (X ). i=1 is an unbiased estimator and V(P MC f ) = P f (1 P f ) N Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

9 Monte Carlo Simulation Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

10 Monte Carlo Simulation Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

11 Monte Carlo Simulation 1 Monte Carlo Simulation with samplesize 5000 N t = 5000 Pf SS = 1.16% Bootstrap CoV κ BS = 13.03% Bayesian CoV κ BA = 12.94% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

12 Monte Carlo Simulation 1 Monte Carlo Simulation with samplesize 5000 N t = 5000 Pf SS = 1.16% Bootstrap CoV κ BS = 13.03% Bayesian CoV κ BA = 12.94% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

13 Monte Carlo Simulation 1 Monte Carlo Simulation with samplesize 5000 N t = 5000 Pf SS = 1.16% Bootstrap CoV κ BS = 13.03% Bayesian CoV κ BA = 12.94% 2 Monte Carlo Simulation with samplesize 1500 N t = 1500 Pf SS = 0.93% Bootstrap CoV κ BS = 26.7% Bayesian CoV κ BA = 25.7% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

14 Monte Carlo Simulation 1 Monte Carlo Simulation with samplesize 5000 N t = 5000 Pf SS = 1.16% Bootstrap CoV κ BS = 13.03% Bayesian CoV κ BA = 12.94% 2 Monte Carlo Simulation with samplesize 1500 N t = 1500 Pf SS = 0.93% Bootstrap CoV κ BS = 26.7% Bayesian CoV κ BA = 25.7% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

15 Importance Sampling Use the following idea: 1 F f dx = 1 F f g g dx Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

16 Importance Sampling Use the following idea: 1 F f dx = 1 F f g g dx Theorem P f can be estimated by g-iid random variables (Y 1,..., Y N ). P f P IS f := 1 N N i=1 1 F (Y i ) f (Y i) g(y i ), where P IS f is unbiased and V(Pf IS f 2 F g ) = dx P2 f N. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

17 Importance Sampling How to find a good g? Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

18 Importance Sampling How to find a good g? Idea: use correlation coefficient between parameters and output, use a function that pushes the realizations towards the critical area. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

19 Importance Sampling How to find a good g? Idea: use correlation coefficient between parameters and output, use a function that pushes the realizations towards the critical area. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

20 Importance Sampling h(x, θ) Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

21 Importance Sampling First approach: Use (rank) correlation coefficients from reference solution (Monte Carlo with N = 5000). Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

22 Importance Sampling First approach: Use (rank) correlation coefficients from reference solution (Monte Carlo with N = 5000). Promising results: N = 780 P IS f = 0.84% Bootstrap CoV κ = 19% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

23 Importance Sampling Second approach: Estimate correlation with 99 realizations and do Importance Sampling with these coefficients. N = 780 P IS f = 1.24% Bootstrap CoV κ = 25% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

24 Importance Sampling Second approach: Estimate correlation with 99 realizations and do Importance Sampling with these coefficients. N = 780 P IS f = 1.24% Bootstrap CoV κ = 25% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

25 Subset Simulation Let α 0 α 1... α m = 1. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

26 Subset Simulation Let α 0 α 1... α m = 1. Then P f = P(X F ) = Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

27 Subset Simulation Let α 0 α 1... α m = 1. Then m P f = P(X F ) = P(Φ(X ) > α 0 ) P(Φ(X ) > α k Φ(X ) > α k 1 ) k=1 Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

28 Subset Simulation Let α 0 α 1... α m = 1. Then P f = P(X F ) = P(Φ(X ) > α 0 ) }{{} :=P 0 m P(Φ(X ) > α k Φ(X ) > α k 1 ) }{{} :=P k k=1 Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

29 Subset Simulation Let α 0 α 1... α m = 1. Then P f = P(X F ) = P(Φ(X ) > α 0 ) }{{} :=P 0 m P(Φ(X ) > α k Φ(X ) > α k 1 ) }{{} :=P k k=1 Estimate P 0 by Monte Carlo Simulation. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

30 Subset Simulation Let α 0 α 1... α m = 1. Then P f = P(X F ) = P(Φ(X ) > α 0 ) }{{} :=P 0 m P(Φ(X ) > α k Φ(X ) > α k 1 ) }{{} :=P k k=1 Estimate P 0 by Monte Carlo Simulation. Estimate P k by Markov Chain Monte Carlo. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

31 Subset Simulation Let α 0 α 1... α m = 1. Then P f = P(X F ) = P(Φ(X ) > α 0 ) }{{} :=P 0 m P(Φ(X ) > α k Φ(X ) > α k 1 ) }{{} :=P k k=1 Estimate P 0 by Monte Carlo Simulation. Estimate P k by Markov Chain Monte Carlo. We choose α k such that P k 0.2. Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

32 Subset Simulation Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

33 Subset Simulation Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

34 Subset Simulation Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

35 Subset Simulation Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

36 Subset Simulation Theorem The estimator P k is unbiased and the CoV κ k is of order O(N 1 2 ). Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

37 Subset Simulation Theorem The estimator P k is unbiased and the CoV κ k is of order O(N 1 2 ). Theorem The estimator Pf SS O(N 1 2 ). is consistent and the CoV κ is of order Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

38 Subset Simulation 1 Subset Simulation with 900 realisations per level and 35 parameters N t = 2340 P SS f = 1.30% Bootstrap CoV κ BS = 13.1% Bayesian CoV κ BA = 10.6% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

39 Subset Simulation 1 Subset Simulation with 900 realisations per level and 35 parameters N t = 2340 P SS f = 1.30% Bootstrap CoV κ BS = 13.1% Bayesian CoV κ BA = 10.6% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

40 Subset Simulation 1 Subset Simulation with 900 realisations per level and 35 parameters N t = 2340 P SS f = 1.30% Bootstrap CoV κ BS = 13.1% Bayesian CoV κ BA = 10.6% 2 Subset Simulation with 300 realisations per level and 35 parameters N t = 780 P SS f = 1.55% Bootstrap CoV κ BS = 25.0% Bayesian CoV κ BA = 17.7% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

41 Subset Simulation 1 Subset Simulation with 900 realisations per level and 35 parameters N t = 2340 P SS f = 1.30% Bootstrap CoV κ BS = 13.1% Bayesian CoV κ BA = 10.6% 2 Subset Simulation with 300 realisations per level and 35 parameters N t = 780 P SS f = 1.55% Bootstrap CoV κ BS = 25.0% Bayesian CoV κ BA = 17.7% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

42 Subset Simulation 1 Subset Simulation with 900 realisations per level and 35 parameters N t = 2340 P SS f = 1.30% Bootstrap CoV κ BS = 13.1% Bayesian CoV κ BA = 10.6% 2 Subset Simulation with 300 realisations per level and 35 parameters N t = 780 P SS f = 1.55% Bootstrap CoV κ BS = 25.0% Bayesian CoV κ BA = 17.7% 3 Subset Simulation with 300 realisations per level and 10 most sensitive parameters N t = 780 P SS f = 1.20% Bootstrap CoV κ BS = 21.3% Bayesian CoV κ BA = 18.4% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

43 Subset Simulation 3 Subset Simulation with 300 realisations per level and 10 most sensitive parameters N t = 780 P SS f = 1.20% Bootstrap CoV κ BS = 21.3% Bayesian CoV κ BA = 18.4% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

44 Comparison MC IS SS AS Sample Size Estimated P f 0.93% 1.24% 1.55% 0.83% Bootstrap symmetric 95%-confidence interval [0.47%, 1.47%] [0.70%, 1.91%] [1.04%, 2.17%] [0.25%, 2.04%] Bootstrap CoV 26.7% 25.0% 25.0% 53.19% Bayesian symmetric 95%-credibility interval [0.56%, 1.56%] [1.12%, 2.25%] [0.35%, 2.77%] Bayesian CoV 25.7% 17.7% 50.8% Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

45 Prospects Winglet: 4.7 million DOFs Composite failure by Yamada-Sun criterion (Main Joint) Metallic failure by yielding and rupture criterion (Main Joint) Currently running on HPC-system MACH 3 parallel evaluations, 17 min P f 10 6 Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19

46 Thank you for your attention Martin Schwarz Monte Carlo Based Reliability Analysis 15 May / 19