Sensitivity Analysis of Structural Response Uncertainty Propagation Using Evidence Theory

1. Sensitivity Analysis of Structural Response Uncertainty Propagation Using Evidence Theory Ha-Rok Bae, Ramana V. Grandhi andRobert A. Canfield hbae@cs.wright.edurgrandhi@cs.wright.edurobert.canfield@afit.af.mil Department of Mechanical and Materials Engineering Air Force Institute of Technology Wright State University, Dayton, OH 45435 WPAFB, OH 45433

2. Uncertainty Quantification (UQ) in an engineering system = Prediction of Probability (Traditional definition)

3. Pf =  fX(X)dX Limit state function g(U) MPP  U Probability theory, how it works ? • Failure Probability Estimation • : Failure region • fx(X) : Joint PDF Pf Pf = () Sample number Monte Carlo Simulation (MCS) ISM, FORM, SORM, …

4. Two basic Assumptions in Probability theory • The law of large data • Randomness of uncertainty

5. 100000 samples 1000 samples Two basic Assumptions in Probability theory • The law of large data Histograms: of Input parameter g(x) Limit State PDF

6. Two basic Assumptions in Probability theory • Randomness of Uncertainty • Numerical data should be free from subjective opinion • Uncertainty should not be caused by deficient knowledge • Lack of parameter data • Boundary conditions • Exact failure mode • 100% confidence in a system model of physical behavior

7. Information • Insufficiency • Incompleteness • Interval information (w/o PDF) No longer valid Uncertainties in a complex system Challenging problems are … • Basic assumptions • in Probability Theory • The law of large data • Randomness

8. Evidence theory – Why ? Parameter Material properties Loads Geometries Physical system Modeling Initial conditions Model form Scenario abstraction Failure modes No Sufficient data ? Yes Aleatory (Random) Uncertainty Epistemic (Subjective) Uncertainty Probability Theory Possibility Theory Evidence Theory

9. Evidence theory Dempster-Shafer Theory (DST)

10. The set of mutually exclusive elementary propositions from given possible sets 0 1 2 3 x1 x2 x3 x • Frame of Discernment (, X) Given possible sets - Interval information : Frame of discernment : Elementary proposition : All possible propositions of our interest.

11. The portion of total belief that is assigned exactly to a proposition through basic belief assignment function - m Axioms I. m(A)0 for any A2X II. m()=0 III. • Basic Belief Assignment (BBA) Available Evidence Possible Events {x1} {} {x1} {x3} {x2, x3} {x3} Evidence for {x3} {x1, x3} {x2} {x1, x2} {x1, x2} Evidence for {x1, x2} X X

12. BBA structure x1 x2 x3 {x3} {x1} m({x1})=0.5 m({x3})=0.3 m({x1, x2})=0.1 {x1, x2} m(X)=0.1 BBA  Probability X Probability – Uniform Distribution • Additivity does not necessarily hold • m({x1})+m({x2})m({x1, x2}) • Monotonicity does not necessarily hold • m({x1})  m({x1, x2}) even though {x1}{x1, x2} • It is not required that m(X)=1, but m(X)1 x1 x2 x3 0.5 0.3 0.05 +0.05 +0.033 +0.033 +0.033 =0.583 =0.333 =0.083 • Basic Belief Assignment (BBA) • Incomplete information from evidence 0.10.30.5 0.1

13. m1 m1 BPA structures mc Combined BPA structures • Dempster’s rule of combining Dempster’s rule disregards every contradiction Algebraic properties : commutative and associative

14. 0 1 Bel(A) Bel(A) Pl(A) • Belief & Plausibility functions Likelihood for event A lies in the interval [Bel(A), Pl(A)] Uncertainty Uncertainty Uncertainty Uncertainty Uncertainty Uncertainty

15. Pl & Bel Pl Pl_dec Bel Parameter, x Plausibility decision function – Pl_dec • Discontinuous Continuous measure • Bound result [Bel, Pl]  Single measure for decision making situation By the generalized insufficient reason principle [Savage, 1972]

16. Sensitivity Analysis Sensitivity of plausibility w.r.t a proposition - useful for determining future data acquisition Sensitivity of plausibility w.r.t a deterministic parameter - useful for improving the system design

17. Sensitivity Analysis Sensitivity of plausibility for a given BBA of a proposition

18. f(X) Function Limit State x Sensitivity Analysis Sensitivity of plausibility for a deterministic parameter where,

19. Engineering Structural Application

20. Deterministic Function : f(X) Y X Problem Description in evidence theory Epistemic parameter uncertainty Physical system : Responses : Y = [ y1, y2, …, yn] Input data : X = [ x1, x2, …, xn]

21. Combining Information Construct BBA structure & Function Evaluation Space [Bel, Pl] & Sensitivity Results Numerical Methodology Information from sources FEM Analyzer Evaluation of Limit State Function & Bounds [Bel, Pl] Construct Approximation & Pl_dec evaluation

22. Numerical Example (ICW) Epistemic parameter uncertainty Epistemic information Sources  Elastic modulus • Static load • Root and tip part’s skin thicknesses • Two different experts • Interval information Tip displacement Limit State dispfail={disptip : | disptip | < 5} Upper wing skin Tip Region (t1) Spars and Ribs Lower wing skin RootRegion (t2) Wing Root

23. Numerical Example • Expert 1 Elastic modulus factor Static load factor 0.8 0.7 0.2 0.9 1.2 1.0 1.5 0.5 0.6 0.8 0.9 1.0 1.1 1.3 1.5 1.6 6.0 1.1 P13 PID: P15 P16 P11 PID: E11 E13 E15 0.3 0.005 BBA: 0.02 BBA: 0.025 0.5 0.025 0.2 P12 P14 E14 E12 0.4 0.075 0.2 0.25 • Expert 2 Static load factor Elastic modulus factor 0.8 0.7 0.2 0.9 1.2 1.0 0.5 0.6 0.8 0.9 1.0 1.1 1.3 1.5 1.6 6.0 1.5 1.1 PID: E24 E25 E21 E23 P24 PID: P26 P21 P23 BBA: 0.1 0.001 0.005 0.7 0.02 BBA: 0.01 0.4 0.4 P22 P25 E22 0.07 0.1 0.1

24. Expert1 Expert1 Interval Interval [0.8, 1.0] [0.7,0.9] [0.95,1.05] [0.9,1.1] [1.0, 1.2] [1.1,1.3] BBA BBA 0.05 0.08 0.9 0.82 0.05 0.1 Expert2 Interval [0.8,0.95] [0.95,1.05] [1.05 1.2] Expert2 Interval [0.7,0.9] [0.9,1.0] [1.0,1.1] [1.1,1.3] BBA 0.1 0.85 0.05 BBA 0.03 0.2 0.7 0.07 Numerical Example Skin thicknesses Tip part’s skin thickness factor Root part’s skin thickness factor

25. Combined BBA for Static Load factor ‘P’ 0.5 0.6 0.8 0.9 1.0 1.1 1.3 1.5 1.6 6.0 Pc4 PID: Pc1 Pc3 Pc4 Pc4 Pc4 Pc4 BBA: 0.0005 0.0621 0.0034 0.0136 0.0002 0.4744 0.4427 Pc2 0.0032 Numerical Example • Combined BBA for Elastic modulus factor ‘E’ 0.8 0.7 0.2 0.9 1.2 1.0 1.5 1.1 Ec3 Ec3 Ec3 PID: Ec1 Ec3 0.0057 0.0007 0.1393 BBA: 0.0014 0.8173 Ec2 0.0355

26. 0.0018 0.0007 0.0001 Pl Bel Bel [Bel, Pl]=[0.0001,0.0018] Pl Pl v v=5.0 Probability Numerical Example Belief and Plausibility Complementary Cumulative Plausibility and Belief Function for the occurrence of limit tip displacement value y > v Plausibility of y < v

27. Expert1 Expert2 Expert2 Expert1 Expert1 Expert2 Expert1 Expert2 Numerical Example Sensitivities of plausibility for given expert’s opinion  Elastic modulus factor.  Tip part thickness factor.  Static load factor.  Root part thickness factor.

28. Numerical Example Sensitivities of plausibility for deterministic parameters Tip part (t1) Root part (t2)

29. Summary Evidence theory provides useful tool for partial and incomplete information situation. Evidence theory provides a Bound [Bel, Pl] for a uncertainty quantification problem, which has consistency with given incomplete information. Analytical sensitivities are defined and they can be useful in iterative sequence to improve a design with respect to the degree of plausibility

30. Thank you ! Acknowledgment This work has been sponsored by the Air Force Office of Scientific Research (AFOSR) through the Grant F49620-00-1-0377.