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

2. University of Texas at San Antonio 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. University of Texas at San Antonio 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. University of Texas at San Antonio Conditional Probabilistic Sensitivities  Enhance existing Monte Carlo algorithm to compute probabilistic sensitivities (assumes a defect is present)

5. University of Texas at San Antonio Conditional Probabilistic Sensitivities  BT - Denotes Boundary Term needed if perturbing the parameter changes the failure domain, e.g., a min, a max Thus the boundary term is f(a max ). This term is an upper bound to the true BT in N dimensions

6. University of Texas at San Antonio Conditional Probabilistic Sensitivities  Example lognormal distribution Sensitivity with respect to the Coefficient of Variation (stdev/mean) Sensitivity with respect to the Median

7. University of Texas at San Antonio Sensitivity with Respect to Median,

8. University of Texas at San Antonio Sensitivity with Respect to Coefficient of Variation,

9. University of Texas at San Antonio Sensitivities of Exceedance Curve Bounds  Perturb bounds assuming same slope at end points

10. University of Texas at San Antonio Sensitivity with Respect to assumes BT is zero

11. University of Texas at San Antonio Sensitivity with Respect to Assumes BT is f(a max )

12. University of Texas at San Antonio Zone Sensitivities Partial derivative of probability of fracture of zone with respect to parameter number of zones affected by

13. University of Texas at San Antonio Disk Sensitivities Partial derivative of probability of fracture of disk with respect to parameter number of zones affected by

14. University of Texas at San Antonio 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. University of Texas at San Antonio 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. University of Texas at San Antonio 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. University of Texas at San Antonio Loading

18. University of Texas at San Antonio Model Titanium ring 24-Zones

19. University of Texas at San Antonio Random Variable Defect Dist.

20. University of Texas at San Antonio Results

21. University of Texas at San Antonio Contd…

22. University of Texas at San Antonio 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. University of Texas at San Antonio Random Variable Definitions VariableMedianCov Life Scatter10.1 Stress Multiplier0.0010.1 Defect Dist.

24. University of Texas at San Antonio Results

25. University of Texas at San Antonio Contd…

26. University of Texas at San Antonio 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. University of Texas at San Antonio Example - Sensitivities wrt amin  14 zone AC test case  Sensitivities of the conditional POF wrt amin ZoneNumericalAnalytical 11.7881E-051.7992E-05 21.7881E-051.5664E-05 31.7881E-052.1802E-05 41.1325E-041.2494E-04 54.5300E-044.5165E-04 61.2100E-031.2134E-03 72.7239E-032.6827E-03 81.1921E-051.3060E-05 95.9604E-067.9424E-06 105.9604E-068.1728E-06 111.7881E-051.4760E-05 123.5763E-053.6387E-05 131.7881E-041.8838E-04 141.8716E-031.8278E-03

28. University of Texas at San Antonio 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. University of Texas at San Antonio Exceedance Curve

30. University of Texas at San Antonio Probabilistic Model Probability of Fracture of Disk Probability of Fracture per Zone