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

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

3. 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. 4 Introduction (continued) Modeling uncertainty in probabilistic models Probability Second-Order Probability: Parametric family of probability distributions. Uncertain distribution parameters, , are random variables with PDF f Θ (θ) Reliability - random variable R(  ) CDF

5. 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. 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. 7 Approach General formulation of global optimization problem Max (Min)  PF(  ) Such that:

8. 8 Solution of optimization problem Monte-Carlo simulation –Select a sampling PDF for the parameters θ and 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. 9 Using Efficient Reliability Reanalysis (ERR) to Reduce Cost Importance Sampling Sampling PDF True PDF

10. 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. 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, f X (x,θ), in eq. (1) with f X (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. 12 Using Extreme Distributions to Estimate Upper and Lower Reliabilities Reliability PDF Parent PDF (Reliabilities in a sample follow this PDF) PDF of smallest reliability in sample 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. 13 Algorithm for Estimation of Lower and Upper Probability Using Efficient Reliability Reanalysis Information about Uncertain Distribution Parameters Reliability Analysis Repeated Reliability Reanalyses Estimate of Global Min and Max Failure Probabilities Fit Extreme Distributions To Failure Probability Values Estimate of Global Min And Max Failure Probability From Extreme Distributions Path A Path B

14. 14 Path B: Estimation of Lower and Upper Probabilities

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

16. 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. 17 Displacement vs. normalized frequencies β1β1 Displacement β2β2

18. 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. 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. 20 PF max vs. number of replications per simulation (n), groups of failure probabilities (N), and failure probabilities per group (m) 2000 replications 5000 replications 10000 replications True value of PF max

21. 21 Comparison of PF min and PF max for n = 10,000 True PF max =0.332 Nm Proposed Method with ERR MC PF min (  PFmin ) PF max (  PFmax ) PF min (  PFmin ) PF max (  PFmax ) 36 250.03069 (0.0021) 0.27554 (0.0144) 0.032 (0.0017) 0.2763 (0.0045) 10000.02333 (0.0016) 0.30982 (0.0190) 0.0251 (0.0016) 0.3106 (0.0046) 120 250.03069 (0.0021) 0.3008 (0.0170) 0.032 (0.0018) 0.3004 (0.0046) 10000.02333 (0.0016) 0.32721 (0.0200) 0.0239 (0.0016) 0.3249 (0.0047)

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

23. 23 CPU Time CPU time for simulation with n= 10000 Nm CPU Time (sec) Proposed Method with ERR MC 36252.70151 10001006061 120258.61503 100034220198

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

25. 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