Overview

1. Non-Probabilistic Design Optimization with Insufficient Data using Possibility and Evidence Theories Zissimos P. Mourelatos Jun Zhou Mechanical Engineering Department Oakland University Rochester, MI 48309, USA mourelat@oakland.edu

2. Overview • Introduction • Design under uncertainty • Uncertainty theories • Possibility – Based Design Optimization (PBDO) • Uncertainty quantification and propagation • Design algorithms • Evidence – Based Design Optimization (EBDO) • Examples • Summary and conclusions

3. Propagation Output Input Analysis / Simulation Design Quantification Uncertainty (Quantified) Uncertainty (Calculated) 2. Propagation 3. Design Design Under Uncertainty

4. Uncertainty Types • Aleatory Uncertainty (Irreducible, Stochastic) • Probabilistic distributions • Bayesian updating • Epistemic Uncertainty (Reducible, Subjective, Ignorance, Lack of Information) • Fuzzy Sets; Possibility methods (non-conflicting information) • Evidence theory (conflicting information)

5. Evidence Theory Probability Theory Possibility Theory Uncertainty Theories

6. Power Set (All sets) Element Universe (X) A B C B A Evidence Theory Non-Probabilistic Design Optimization: Set Notation

7. Possibility-Based Design Optimization (PBDO)

8. Evidence Theory No Conflicting Evidence (Possibility Theory) Possibility-Based Design Optimization (PBDO)

9. - cut provides confidence level At each confidence level, or -cut, a set is defined as convex normal set Quantification of a Fuzzy Variable: Membership Function

10. The “extension principle” calculates the membership function (possibility distribution) of the fuzzy response from the membership functions of the fuzzy input variables. If where then Practical Approximations of Extension Principle • Vertex Method • Discretization Method • Hybrid (Global-Local) Optimization Method Propagation of Epistemic Uncertainty Extension Principle

11. 1.0 1.0 1.0 1.0 1.0 1.0 a a a a a a 0.0 0.0 0.0 0.0 0.0 0.0 Global Global s.t. s.t. Optimization Method where : and

12. What is possible may not be probable • What is impossible is also improbable (Possibility Theory) If feasibility is expressed with positive null form then, constraint g is ALWAYS satisfied if or for Possibility-Based Design Optimization (PBDO)

13. , we have Considering that Possibility-Based Design Optimization (PBDO)

14. ; OR s.t. ; Double Loop Possibility-Based Design Optimization (PBDO) s.t.

15. s.t. Triple Loop , , with ; s.t. and s.t. PBDO with both Random and Possibilistic Variables

16. Evidence-Based Design Optimization (EBDO)

17. 5 9.5 8 6.2 11 Y “Expert” A If m(A)>0 for then A is a focal element 0.3 0.2 0.1 0.4 5 8.7 7 11 “Expert” B Y 0.4 0.3 0.3 Combining Rule (Dempster – Shafer) 6.2 8.7 7 8 5 9.5 11 0.x1 0.x4 0.x2 0.x6 0.x5 0.x3 Y Evidence-Based Design Optimization (EBDO) Basic Probability Assignment (BPA): m(A)

18. Evidence-Based Design Optimization (EBDO) define: where For Assuming independence, where

19. Evidence-Based Design Optimization (EBDO) BPA structure for a two-input problem

20. Evidence-Based Design Optimization (EBDO) Uncertainty Propagation If we define, then where and

21. Contributes to Plausibility Contributes to Belief Evidence-Based Design Optimization (EBDO) Position of a focal element w.r.t. limit state

22. Evidence-Based Design Optimization (EBDO) Design Principle If non-negative null form is used for feasibility, feasible infeasible failure is satisfied if Therefore, OR is satisfied if

23. Evidence-Based Design Optimization (EBDO) Formulation ,

24. Calculation of Evidence-Based Design Optimization (EBDO)

25. Geometric Interpretation of PBDO and EBDO

26. Possibility-Based Design Optimization (PBDO) Implementation Feasible Region Objective Reduces x1 initial design point g1(x1,x2)=0 frame of discernment g2(x1,x2)=0 PBDO optimum deterministic optimum x2

27. Evidence-Based Design Optimization (EBDO) Implementation x1 initial design point g1(x1,x2)=0 Feasible Region frame of discernment g2(x1,x2)=0 Objective Reduces EBDO optimum deterministic optimum x2

28. Evidence-Based Design Optimization (EBDO) Implementation x1 hyper-ellipse initial design point g1(x1,x2)=0 Feasible Region frame of discernment g2(x1,x2)=0 B Objective Reduces MPP for g1=0 deterministic optimum x2

29. Evidence-Based Design Optimization (EBDO) Implementation x1 hyper-ellipse initial design point g1(x1,x2)=0 Feasible Region frame of discernment g2(x1,x2)=0 B Objective Reduces EBDO optimum MPP for g1=0 deterministic optimum x2

30. where : Cantilever Beam Example: RBDO Formulation s.t.

31. RBDO PBDO s.t. s.t. Cantilever Beam Example: PBDO Formulation

32. EBDO s.t. BPA Structure Cantilever Beam Example: EBDO Formulation

33. Cantilever Beam Example: EBDO Formulation BPA structure for y, Y, Z, E

34. Cantilever Beam Example: Comparison of Results

35. s.t. Thin-walled Pressure Vessel Example yielding

36. Thin-walled Pressure Vessel Example BPA structure for R, L, t, P and Y

37. Thin-walled Pressure Vessel Example

38. Deterministic PBDO EBDO RBDO Less Information More Conservative Design Summary and Conclusions • Possibility and evidence theories were used to quantify and propagate uncertainty. • PBDO and EBDO algorithms were presented for design with incomplete information. • EBDO design is more conservative than the RBDO design but less conservative than PBDO design.

39. Q & A