Probabilistic Sensitivity Measures Wes Osborn Harry Millwater

1. Probabilistic Sensitivity Measures Wes Osborn Harry Millwater Department of Mechanical EngineeringUniversity of Texas at San Antonio TRMD & DUST Funding

2. Objectives • Compute the sensitivities of the probability of fracture with respect to the random variable parameters, e.g., median, cov • No additional sampling • Currently implemented: • Life scatter (median, cov) • Stress scatter (median, cov) • Exceedance curve (amin, amax) • Expandable to others

3. Probabilistic Sensitivities • Three sensitivity types computed • Zone • Conditional - based on Monte Carlo samples • SS, PS, EC • Unconditional - based on conditional results • SS, PS, EC • Disk • Stress scatter - one result for all zones • Exceedance curve - one result for all zones using a particular exceedance curve (currently one) • Life scatter - different for each zone 95% confidence bounds developed for each

4. Conditional Probabilistic Sensitivities • Enhance existing Monte Carlo algorithm to compute probabilistic sensitivities (assumes a defect is present)

5. Conditional Probabilistic Sensitivities • BT - Denotes Boundary Term needed if perturbing the parameter changes the failure domain, e.g., amin, amax Thus the boundary term is f(amax). This term is an upper bound to the true BT in N dimensions

6. Conditional Probabilistic Sensitivities • Example lognormal distribution Sensitivity with respect to the Median Sensitivity with respect to the Coefficient of Variation (stdev/mean)

7. Sensitivity with Respect to Median,

8. Sensitivity with Respect to Coefficient of Variation,

9. Sensitivities of Exceedance Curve Bounds • Perturb bounds assuming same slope at end points

10. Sensitivity with Respect to assumes BT is zero

11. Sensitivity with Respect to Assumes BT is f(amax)

12. Zone Sensitivities number of zones affected by Partial derivative of probability of fracture of zone with respect to parameter

13. Disk Sensitivities number of zones affected by Partial derivative of probability of fracture of disk with respect to parameter

14. Procedure • For every failure sample: • Evaluate conditional sensitivities • Divide by number of samples • Add boundary term to amax sensitivity • Estimate confidence bounds • Results per zone and for disk

15. DARWIN Implementation • New code contained in sensitivities_module.f90 zone_risk accumulate_pmc_sensitivities accrue expected value results compute_sensitivities_per_pmc compute_sensitivities_per_zone write_sensitivities_per_zone zone_loop sensitivities_for_disk write_disk_sensitivities

16. Application Problem #1 • The model for this example consists of the titanium ring outlined by advisory circular AC-33.14-1 subjected to centrifugal loading • Limit State:

17. Loading

18. Model Titanium ring 24-Zones

19. Random Variable

20. Results

21. Contd…

22. Application Problem #2 • Consists of same model, loading conditions, and limit state • In addition to the defect distribution, random variables Life Scatter and Stress Multiplier have been added

23. Random Variable Definitions

24. Results

25. Contd…

26. Conclusion • A methodology for computing probabilistic sensitivities has been developed • The methodology has been shown in an application problem using DARWIN • Good agreement was found between sampling and numerical results

27. Example - Sensitivities wrt amin • 14 zone AC test case • Sensitivities of the conditional POF wrt amin

28. Probabilistic Sensitivities • Sensitivities for these distributions developed • Normal (mean, stdev) • Exponential (lambda, mean) • Weibull (location, shape, scale) • Uniform (bounds, mean, stdev) • Extreme Value – Type I (location, scale, mean, stdev) • Lognormal Distribution (COV, median, mean, stdev) • Gamma Distribution (shape, scale, mean, stdev) Sensitivities computed without additional sampling

29. Exceedance Curve

30. Probabilistic Model Probability of Fracture per Zone Probability of Fracture of Disk