1. Digital Twin Workshop Harry Millwater University of Texas at San Antonio Sept. 10-11, 2012 1

2. Topics n FAA Probabilistic Damage Tolerance Analysis n System Reliability Analysis n Sensitivity Methods 2

3. Damage Tolerance Methodology Geometry … 54 3 2 1 n Sensitivities Insp 1 Insp 2 Time RS EIFS Insp 1 Insp 2 Time Crack Size Loading EVD POD SFPOF & CTPOF Material Prop.

4. POF Calculations 4 Survival t-1POF Flight t

5. EVD Results 5 Flight 1Flight 2... Flight n Frechet Gumbel Weibull

6. EVD Results 6 UsageLocationScaleShapeDistribution Single Engine Unpress. Basic Inst.14.491.750.01Frechet Single Engine Unpress. Personal12.031.150.00Gumbel Single Engine Unpress. Executive11.680.90-0.04Weibull Twin Engine Unpress. Basic Inst.13.931.420.00Gumbel Twin Engine Unpress. General12.051.160.02Frechet Pressurized13.411.260.02Frechet Agric/Special12.070.94-0.12Weibull Special Usage Survey14.561.32-0.05Weibull

7. 7 POF contours Probabilistic Modeling of Systems L. Domyancic, D. Sparkman, and H.R. Millwater, L. Smith and D. Wieland, “A Fast First-Order Method for Filtering Limit States,” 50 th AIAA AIAA Nondeterministic Approaches Conference, May 4-7, 2009, Palm Springs, CA, AIAA 2009-2260

8. Pair-wise Filtering Criteria n Pair-wise filtering – neglect limit states that don’t contribute significantly to the system POF relative to the dominant limit states 8 i = 2,n g 1 – dominant limit state, serves as a benchmark

9. Filtering Procedure Pair-wise comparison of each limit state with dominant n Each limit state is assigned a filtering error, i.e. the relative error incurred if the limit state were filtered 9 is calculated using the bivariate normal User decides how much error to allow and set tolerance (typical values: 1E-4 – 1E-6)

10. Second Order Bounds Examine the cumulative effect of the filtered limit states using 2 nd order bounds Compare bounds from a) all limit states vs. b) only unfiltered limit states 10

11. Aircraft Model n Lower wing skin analyzed, ~900 limit states n Root radius Element 292 and 277 left unfiltered (  TOL =1E-6) n Bounds on entire system vs. only critical locations widen by 1E-5 11 g1 LS #BetaCorr.Filt. Error 2920.29710 2770.650.992.50E-04 2560.9490.982.20E-09

12. Sensitivity Methods n Numerical Differentiation (*) n Visual n Linear Regression n Score Function partial derivatives n Local probabilistic partial derivatives (*) n Global Sensitivity Analysis

13. Local Prob. Sensitivity Original CDF Modified CDF

14. ZFEM Complex Variable FE Method

15. Abaqus 2D Verification AnalyticalAbaqus

16. Strain Energy Release Rate Double Cantilever Beam (DBC) Exact = 0.800 Exact = 0.8

17. Higher Order Sensitivities Thick walled cylinder