Assessment of Imprecise Reliability Using Efficient Probabilistic Reanalysis

1. Assessment of Imprecise Reliability Using Efficient Probabilistic Reanalysis Farizal Efstratios Nikolaidis SAE 2007 World Congress

2. Outline • Introduction • Objective • Approach • Example Calculation of Upper and Lower Reliabilities of System with Dynamic Vibration Absorber • Conclusion

3. Introduction Challenges in Reliability Assessment of Engineering Systems: • Scarce data, poor understanding of physics • Difficult to construct probabilistic models • No consensus about representation of uncertainty in probabilistic models • Calculations for reliability analysis are expensive

4. Introduction (continued) • Modeling uncertainty in probabilistic models Probability • Second-Order Probability: Parametric family of probability distributions. Uncertain distribution parameters, ,arerandom variables with PDF fΘ(θ) • Reliability - random variable CDF R()

5. Introduction (continued) Interval Approach to Model Uncertainty Given ranges of uncertain parameters find minimum and maximum reliability • Finding maximum or minimum reliability: Nonlinear Programming, Monte Carlo Simulation, Global Optimization • Expensive – requires hundreds or thousands reliability analyses

6. Objective • Develop efficient Monte-Carlo simulation approach to find upper and lower bounds of Probability of Failure (or of Reliability) given range of uncertain distribution parameters

7. ApproachGeneral formulation of global optimization problem Max (Min) PF() Such that:

8. Solution of optimization problem • Monte-Carlo simulation • Select a sampling PDF for the parameters θand generate sample values of these parameters. Estimate the reliability for each value of the parameters in the sample. Then find the minimum and maximum values of the values of the reliabilities. • Challenge: This process is too expensive

9. Using Efficient Reliability Reanalysis (ERR) to Reduce Cost • Importance Sampling True PDF Sampling PDF

10. Efficient Reliability Reanalysis • If we estimate the reliability for one value the uncertain parameters θ using Monte-Carlo simulation, then we can find the reliability for another value θ’ very efficiently. • First, calculate the reliability, R(θ), for a set of parameter values, θ. Then calculate the reliability, R(θ’), for another set of values θ’ as follows:

11. Efficient Reliability Reanalysis (continued) • Idea: When calculating R(’), use the same values of the failure indicator function as those used when calculating R (). • We only have to replace the PDF of the random variables, fX(x,θ), in eq. (1) with fX(x,θ’). • The computational cost of calculating R(’) is minimal because we do not have to compute the failure indicator function for each realization of the random variables.

12. Using Extreme Distributions to Estimate Upper and Lower Reliabilities PDF of smallest reliability in sample PDF Parent PDF (Reliabilities in a sample follow this PDF) Reliability • If we generate a sample of N values of the uncertain parameters θ, and estimate the reliability for each value of the sample, then the maximum and the minimum values of the reliability follow extreme type III probability distribution.

13. Algorithm for Estimation of Lower and Upper Probability Using Efficient Reliability Reanalysis Path A Information about Uncertain Distribution Parameters Estimate of Global Min and Max Failure Probabilities Reliability Analysis Repeated Reliability Reanalyses Path B Fit Extreme Distributions To Failure Probability Values Estimate of Global Min And Max Failure Probability From Extreme Distributions

14. Path B: Estimation of Lower and Upper Probabilities

15. Normalized system amplitude y m, n2 Dynamic absorber Original system M, n1 F=cos(et) Example: Calculation of Upper and Lower Failure Probabilities of System with Dynamic Vibration Absorber

16. Objectives of Example • Evaluate the accuracy and efficiency of the proposed approach • Determine the effect of the sampling distributions on the approach • Assess the benefit of fitting an extreme probability distribution to the failure probabilities obtained from simulation

17. Displacement vs. normalized frequencies Displacement β2 β1

18. Why this example • Calculation of failure probability is difficult • Failure probability sensitive to mean values of normalized frequencies • Failure probability does not change monotonically with mean values of normalized frequencies. Therefore, maximum and minimum values cannot be found by checking the upper and lower bounds of the normalized frequencies.

19. Problem Formulation Max (Min) R() Such that : 0.9 ≤ i ≤ 1.1, i = 1, 2 0.05 ≤ i ≤ 0.2, i = 1, 2 0 ≤ R() ≤ 1 i: mean values of normalized frequencies i:standard deviations of normalized frequencies

20. PFmax vs. number of replications per simulation (n), groups of failure probabilities (N), and failure probabilities per group (m) 2000 replications True value of PFmax 5000 replications 10000 replications

21. Comparison of PFmin and PFmax for n = 10,000True PFmax=0.332

22. Effect of Sampling Distribution on PFmax Two sampling distributions One sampling distribution Monte Carlo

23. CPU TimeCPU time for simulation with n= 10000

24. Fitted extreme CDF of maximum failure probability vs. dataN=120, m=1000, n=10000 Fitted, ERR Fitted MC

25. Conclusion • The proposed approach is accurate and yields comparable results with a standard Monte Carlo simulation approach. • At the same time the proposed approach is more efficient; it requires about one fiftieth of the CPU time of a standard Monte Carlo simulation approach. • Sampling from two probability distributions improves accuracy. • Extreme type III distribution did not fit minimum and maximum values of failure probability