Uncertainty Quantification Using Evidence Theory With A Cost-Effective Algorithm

1. Uncertainty Quantification Using Evidence Theory With A Cost-Effective Algorithm Ha-Rok Bae, Ramana V. Grandhi and Robert A. Canfield Department of Mechanical and Materials Engineering Air Force Institute of Technology Wright State University, Dayton, OH 45435 WPAFB, OH 45433 Second M.I.T. Conferenceon Computational Fluid and Solid Mechanics Cambridge, MA, June 17-20, 2003

2. Presentation Outline Motivations & objectives • The benefits of uncertainty quantification analysis • Multiple types of uncertainty and propagation Proposed methodology & Numerical example • Engineering structural problem description in evidence theory • Improvement of computational performance and accuracy • Intermediate Complexity Wing Example Summary Remarks

3. Input variables(X) Responses (Y) x1 y1 x2 y2   xn ym Benefits Two Basic Assumptions • Better understanding of real system behaviors • Enhanced confidence in system analysis results • Improvement in a decision making situation • Robust system for possible uncertainties • The law of large data • Randomness of uncertain variables Uncertainty Quantification (UQ) Analysis System Analysis

4. 4500 50 45 4000 40 3500 35 3000 30 2500 25 2000 20 1500 15 1000 10 500 5 0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Two basic assumptions in probabilistic analysis • The law of large data Histograms: 100000 samples 1000 samples of Input parameter • Randomness of Uncertainty • Numerical data should be free from imprecise subjective opinion • Uncertainty should not be caused by deficient knowledge

5. Scenario abstraction • Unrealized & unexpected failure modes • Undefined system behaviors Model form • Faults in a conceptual and mathematical modeling • Imperfect representation of boundary conditions System parameter • Natural variability (dimensions, modulus, …) • Imprecise statistical data from poor observations Uncertainty Sources Challenging problems are … Various uncertainty sources due to the restrictions of budget, resources, knowledge and time

6. Probabilistic Analysis Non-probabilistic Analysis - MCS, ISM - FORM, SORM - Spectral stochastic FEM - Interval analysis - Possibility Theory - Evidence Theory UQ Techniques Parameter Model form Scenario abstraction No Sufficient data ? Yes Aleatory Uncertainty Epistemic Uncertainty

7. Research Objectives & Challenges • Reliable and accurate UQ analysis technique for an engineering structural system in an imprecise informative situation (Epistemic uncertainty). • Appropriate representation of intrinsic uncertainties of an engineering system • Robust methodology for a complex and large scale system • Modern structural design problem requires multi-disciplinary Analyses (high computational cost) • Iterative nature of UQ analysis

8. Research Approaches • Investigate evidence theory for an UQ analysis in an engineering structural system. • Develop a robust UQ methodology using evidence theory. • Employ a surrogate model for - reducing the computational cost of repetitive engineering simulations - capturing the non-linearity of limit state functions

9. Parameter Model form Scenario abstraction No Sufficient data ? Yes Aleatory Uncertainty Epistemic Uncertainty Probabilistic Analysis Non-probabilistic Analysis Spectral stochastic FEM Evidence Theory Evidence Theory (Dempster-Shafer Theory)

10. 0 1 2 3 x1 x2 x3 x I. m(A)0 for any A2X II. m()=0 III. Evidence Theory • Frame of Discernment (X) The set of mutually exclusive elementary propositions from given possible sets : Frame of discernment • Basic Belief Assignment (BBA) The portion of total belief that is assigned exactly to a proposition through basic belief assignment function - m Axioms

11. x1 x2 x3 {x3} {x1} m({x1})=0.5 m({x3})=0.3 m({x1, x2})=0.1 {x1, x2} m(X)=0.1 X1 X2 X3 X Probability – Uniform Distribution x1 x2 x3 0.5 0.3 0.05 +0.05 +0.033 +0.033 +0.033 =0.583 =0.333 =0.083 Fundamental Differences • Incomplete information from evidence BBA structure 0.10.30.5 0.1 • Additivitydoes not necessarily hold • m({x1})+m({x2})m({x1, x2}) • Monotonicitydoes not necessarily hold • m({x1})  m({x1, x2}) even though {x1}{x1, x2} • It is not requiredthat m(X)=1, but m(X)1 BBA  Probability

12. Various BBA Structures • Bayesian Belief Structure • Consonant Belief Structure m(x1) m(x2) m(x1) m(x2) … m(xn) … m(xn) • General Belief Structure BBA structure gives much more flexibility to express our partial belief. m(x1) m(xn) m(x2) … Partially and totally overlap, discontinuous, disconnected, …

13. Dempster’s Rule of Combining BBA structures m1 m1 mc Combined BBA structure Algebraic properties : commutative and associative Dempster’s rule disregards every contradiction

14. 0 1 Bel(A) Bel(A) Pl(A) Belief & Plausibility Functions Likelihood for event A lies in theinterval [Bel(A), Pl(A)] Uncertainty

15. Structural Problem Description in Evidence Theory • Epistemic parameter uncertainty with interval information • Definingasystem failure setwith limit state value, Yfail Uf ={Y : Yfail - Y< 0 , Y=f(ck) and ck=[x1, x2, x2,…,xn] C} where, x is an uncertain parameter vector • Resulting measurements,Belief & Plausibility functions

16. Assessing the measurements [ Bel, Pl] Given Information Constructing & Combining BBA Structure Defining System Failure Set (Uf) & Function Evaluation Space (FES) • Vertex method • Sampling method (MCS) Assessing Bel & Pl For Entire FES System responses (FEM Analyzer) Evaluating Bel and Pl functions [ Bel , Pl ]

17. Pl Pl_dec Bel & Pl 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]

18. I = f - yfail> 0 1 otherwise 0 , where Plausibility decision function – Pl_dec f(x) yfail f is fail f is safe x1 x2 c=[x1 , x2]

19. f(x) yfail ck=[x1 , x2] x1 x2 Improvement of Computational Efficiency Constructing Surrogate modelusing approximation concept • to reduce the computational cost • to capture the non-linearity of a system response • to be useful for a sensitivity analysis and an iterative analysis

20. Improvement of Computational Efficiency MPA method with TANA2local approximation • Approximation methods • Single Point Approximation • Two Point Approximation • Response Surface Method • Multiple Point Approximation : local approximation : weight function P : MPA design points Two-point Adaptive Non-linear Approximation (TANA2) method [Grandhi,1995]

21. X1 Safe region Failure region Failure boundary point Sub-optimization Initial point X2 Function evaluation points The defined function evaluation space Constructed approximation Improvement of Computational Efficiency • Sub-optimization problem within the defined function space (Unconstrainted, only bounds of DVs) • Crude optimum point of failure boundary • Deploying function evaluation points for MPA based on valid bound of a local approximation.

22. A Cost-Effective Algorithm Given Information Constructing & Combining BBA Structure Defining System Failure Set (Uf) & Function Evaluation Space (FES) Constructing Surrogate Model (MPA) FEM Analyzer Assessing Bel & Pl For Failure FES Evaluating Bel and Pl functions Surrogate Model [ Bel , Pl_dec, Pl ]

23. Tip displacement Wing Root Intermediate Complexity Wing (ICW) • Epistemic parameter uncertainty • Epistemic information Sources  Elastic modulus • Static loads  Two different experts  Interval information • System failure set Uf ={( d,  ): } where, Tip displacement: d=Disp(x,b) Fundamental frequency: =Freq(x,b)

24. Source 1 Source 2 PID: P21 P23 P24 P25 PID: P11 P13 P15 BBA: 0.04 0.7 0.14 0.02 0.5 0.025 P22 P14 P12 BBA: 0.025 0.1 0.25 0.2 0.2 0.2 0.7 0.7 0.8 0.8 1.0 1.0 1.1 1.1 1.2 1.2 1.5 1.5 0.9 0.9 Numerical Example (ICW) Elastic Modulus Uncertainty Cumulative Normal distributions  = [0.8 1.0],2=0.12 Averaging Discretization Method Force Uncertainty

25. Bel Pl_dec Pl Function Evaluations Sampling Method 0.0000 - 0.0526 100000 Vertex Method 0.0000 - 0.0101 512 Belief and plausibility Proposed Method 0.0000 0.0014 0.0526 76 Bel Pl Pl Probability Bel DispLimit Numerical Example (ICW) Belif & Plausibility Complementary Cumulative Plausibility and Belief for the varying values of tip displacement limit state, DispLimit

26. Summary • Evidence theory is investigated for an uncertainty quantification analysis in an engineering structural system. • Evidence theory provides suitable representation (BBA structure) to handle an imprecise informative situation. • A supplementary measurement, Plausibility decision function, is introduced. • Surrogate model based on MPA method reduces the computational cost without sacrificing the accuracy of UQ analysis.

27. Thank you !