Probabilistic Re-Analysis Using Monte Carlo Simulation

1. Probabilistic Re-Analysis Using Monte Carlo Simulation Efstratios Nikolaidis, Sirine Salem, Farizal, Zissimos Mourelatos April 2008

2. Definition and Significance Probabilistic design optimization • Find design variables • To maximize average utility RBDO • Find design variables • To minimize loss function • s. t. system failure probability does not exceed allowable value • Often average utility or system failure probability must be calculated by Monte Carlo simulation. • Vibratory response of a dynamic system: failure domain consists of multiple disjoint regions

3. Definition and Significance Challenge: High computational cost • Optimization requires probabilistic analyses of many alternative designs • Each probabilistic analysis requires many deterministic analyses • Expensive to perform deterministic analysis of a practical model

4. Definition and Significance Deterministic FEA Vibratory door displacement Excitation at engine mounts Monte Carlo Simulation (10,000 replications) Reliability analysis Probability of failure Search for optimum (100- 500 Monte Carlo Simulations) RBDO Optimum

5. Outline • Objectives and Scope • Probabilistic Re-analysis • RBDO problem formulation • Method description • Sensitivity analysis • Example • Preliminary Design of Internal Combustion Engine Conclusion • Conclusion

6. 1.Objectives and Scope • Present probabilistic re-analysis approach (PRA) for RBDO • Estimate reliability of many designs by performing a single Monte-Carlo simulation • Integrate PRA in a methodology for RBDO • Demonstrate efficacy • Design variables are random; can control their average values

7. 2. Probabilistic Re-analysis RBDO problem formulation: • Find average values of random design variables • To minimize cost function • So that

8. Reducing computational cost by using Probabilistic Re-analysis • Select a sampling PDF and perform one Monte Carlo simulation • Save sample values that caused failure (failure set) • Estimate failure probability of all alternative designs by using failure set in step 2 Sampling PDF x2 Failure set Alternative designs Failure region x1

9. Estimation of failure probability PDF when mean values of design variables = µX • Failure probability • Confidence in failure probability estimate • Similar equations are available for average value of a function of design variables (for example utility) • Values xi are calculated from one Monte-Carlo simulation, same values are used to find failure probabilities of all design alternatives Sampling PDF

10. Sensitivity analysis • Analytical expression • Can be calculated very efficiently because it is easy to differentiate PDF of a random variable

11. RBDO with Probabilistic Re-analysis • Find X • To minimize • s. t. • Solution requires only n deterministic analyses

12. x1 Feasible Region Increased Performance Optimum Failure subset x2 RBDO with Probabilistic Re-analysis Iso-cost curves

13. Efficient Probabilistic Re-analysis:Capabilities • Calculates system failure probabilities of many design alternatives using results of a single Monte-Carlo simulation • Does not require calculation of the performance function of modified designs – reuses calculated values of performance function from a single simulation. Cost of RBDO  cost of a single simulation • Non intrusive, easy to program • If PDF of design variables is continuous then system failure probability varies smoothly as function of design variables • Highly effective when design variables have large variability

14. Challenges • Works only when all design variables are random • Requires sample that fills the space of design variables • Cost of single simulation increases with design variables

15. 3. Example: RBDO of Internal Combustion Engine • Preliminary design of flat head internal combustion engine from thermodynamic point of view • Find average bore, inner and outer diameters, compression ratio and RPM • To maximize specific power • S. t. system failure probability ≤pfall (0.4% to 0.67%) • Failure: any violation of nine packaging and functional requirements

16. Design variables(all variables normal)

17. Sampling PDF

18. Effect of average bore on system failure probability (100,000 replications)

20. Specific Power and Probability of Failure

21. Comparison of efficiencies of standard Monte Carlo and PRA (narrower CI means higher efficiency of the method)

22. Observations • PRA found an optimum design almost identical as RBDO using FORM (Liang 2007). • PRA converged to same optimum from different initial designs • PRA underestimated consistently system failure probability by 5% to 11%. • 95% confidence intervals have half width = 23% to 28% of system failure probability • Confidence interval from PRA is 50% wider than that of standard Monte Carlo. This means that PRA needs 225,000 replications to yield results with same accuracy as standard Monte Carlo with 100,000 replications.

23. 4. Conclusion • Presented efficient methodology for RBDO using Monte Carlo simulation • Solves RBDO problems using a single Monte Carlo simulation • Calculates sensitivity derivatives of system failure probability • Limitation: methodology, in its present form, works only when all design variables are random