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
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
Excitation at engine mounts
Vibratory door displacement
Monte Carlo Simulation (10,000 replications)
Reliability analysis
Probability of failure
Search for optimum
( Monte Carlo Simulations)
RBDO
Optimum

5. Outline Objectives and Scope Probabilistic Re-analysis Example
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
Alternative designs
Failure set
Failure region x1

9. Estimation of 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
PDF when mean values of design variables = µX
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. RBDO with Probabilistic Re-analysis
Feasible Region
Increased Performance x2 x1
Optimum
Failure subset
MAXIMIZING the objective function
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. Exhaust valve diameter
Sampling PDF
Average Values
Bore
82.13
Intake valve diameter
35.84
Exhaust valve diameter
30.33
Compression ratio
9.34
RPM
5.31

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