An Efficient Unified Approach for Reliability and Robustness in Engineering Design

1. An Efficient Unified Approach for Reliability and Robustness in Engineering Design Zissimos P. Mourelatos Jinghong Liang Mechanical Engineering Department Oakland University Rochester, MI

2. Outline • Design under uncertainty • Uncertainty types and theories • Definition of optimality, reliability and robustness • RBDO formulations • Problem definition • PMA approach • Single – Loop RBDO • Robust design problem formulation • Reliable / Robust design problem formulation

3. Outline (Cont.) • Examples • A mathematical example • A cantilever beam example • Summary and conclusions

4. Model (Transfer Function) Output Input Uncertainty (Quantified) Uncertainty (Calculated) Uncertainty (Quantified) Design Under Uncertainty

5. 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)

6. Evidence Theory Possibility Theory Probability Theory Uncertainty Theories

7. Reliable & Robust Optimum Feasible Region Reliable Optimum Optimality, Reliability & Robustness x1 g1(x1,x2)=0 g2(x1,x2)=0 Increased Performance f(x1,x2) contours x2

8. 99.99% reliable design 99.99% of design realizations meet performance targets Reliable Design A design is reliable if performance targets are met in the presence of uncertainty.

9. Design Parameter Robust Design A design is robust if performance is not sensitive to inherent variation/uncertainty.

10. Optimized, Reliable and Robust Design Optimized and insensitive performance under uncertainty. Design Goal

11. Reliable/Robust Design • Reliable (Probabilistic) design vs Robust design • Probabilistic Design : Maintains design feasibility by changing means (e.g. RBDO) • Robust Design : Reduces variability by changing std dev. or other variability measure

12. RBDO Formulations • Double loop (classical formulation) • Decoupled (or sequential) approach; e.g. SORA • Single loop • Performs two optimization loops simultaneously! • Enforces KKT conditions of reliability loop as a constraint of the outer design loop

13. s.t. , Single Objective where : : vector of deterministic design variables : vector of random design variables : vector of random design parameters RBDO Problem

14. s.t. , where : s.t. Performance Measure Approach (PMA) Formulation

15. s.t. , Single-Loop RBDO Formulation where :

16. , s.t. Multi Objective where : : vector of deterministic design variables : vector of random design variables : vector of random design parameters Robust Design Problem

17. , s.t. Multi Objective where : : vector of deterministic design variables : vector of random design variables : vector of random design parameters Reliable / Robust Design Problem

18. Reliable / Robust Design Problem: Issues • Percentile Difference (Variability Measure) Calculation • Trade – offs in Multi – Objective Optimization • Percentile calculation using AMV method • Preference aggregation method to handle trade - offs

19. 1 0.95 0 Percentile Calculation using AMV Percentile Definition

20. Form limit state • CDF definition gives • Limit state is linearized around mean point • Calculate MPP in U-space • Convert MPP to original space (5%) (95%) Percentile Calculation using AMV

21. AMV method states that • Thus, R-percentile is 1 0 Percentile Calculation using AMV

22. Annihilation : Idempotency : Monotonicity : if Commutativity : Continuity : Preference Aggregation Method: Aggregation Properties

23. satisfies annihilation for only. Fully compensating : For For : Non - Compensating Preference Aggregation Method

24. s.t. s.t. R = 99.87% A Mathematical Example Reliable/Robust Problem

25. RBDO Problem s.t. Reliable Problem s.t. A Mathematical Example

26. For h2 the “cut-off” value is “cut-off” , ; A Mathematical Example Final Optimization Problem Single-Loop RBDO

27. A Mathematical Example

28. A Mathematical Example a: Robust design; b: Reliable design

29. s.t. A Mathematical Example Weighted Sum Approach . R=99.87%

30. A Mathematical Example

31. w,t : Normal R.V.’s • y, E,Y,Z : Normal Random Parameters s.t. , • L : fixed where: • R = 99.87% A Cantilever Beam Example Reliable/Robust Formulation

32. s.t. , where: A Cantilever Beam Example RBDO Problem

33. s.t. , where: A Cantilever Beam Example Robust Problem

34. A Cantilever Beam Example

35. Summary and Conclusions • A unified approach was presented for reliability and robustness • A multi – objective optimization formulation is used • Preference aggregation method handles trade – offs • Variationis reduced by minimizing a percentile difference • AMV method is used to calculate percentiles • An efficient single – loop probabilistic optimization algorithm identifies the reliable / robust design • Examples demonstrated the feasibility, efficiency and accuracy of the proposed method

36. Q & A

37. OBJECTIVE FUNCTION (COST or WEIGHT) Reliability Evaluation of mth Limit State Reliability Evaluation of 1st Limit State DETERMINISTIC OPTIMIZATION LOOP RELIABILITY ASSESSMENT LOOP Coupled Loop Methods

38. 2nd Iteration 1st Iteration RL1-1st LS RL2-1st LS OL2 OL1 RL1-mth LS RL1-mth LS DETERMINISTIC OPTIMIZATION LOOP RELIABILITY ASSESSMENT LOOP Decoupled Method

39. Min Cost(d) s.t. Min Cost(d) s.t. : Design Vector X : Random Vector : Indicates Failure Reliability – Based Design Optimization (RBDO) or where :

40. Minimize G(U) min Cost(d) min Cost(d) s.t. s.t. s.t. where where calculated from calculated from Minimize s.t. G(U)=0 Reliability – Based Design Optimization (RBDO) RBDO based on RIA RBDO based on PMA

41. Propagation of Uncertainty Since functions are generally nonlinear, use first-order approximation (Taylor series expansion around the means of the random variables)

42. Validity of Linearization Y(X) mY mX X

43. A Mathematical Example a: Robust design; b: Reliable design

44. Aggregation Procedure • Common characterization of all • performance measures • manufacturing tolerances • design alternatives, based on “expected” cost. • Aggregation of (commonly characterized) • performance • Manufacturing • design measures.

45. Theorem from Functional Theory (Aczel; 1996) if : is strictly monotonic, continuous with inverse

46. Optimized and Reliable Design Reliability – Based Design Optimization Design Goal Optimized performance considering uncertainty.

47. Feasible Region Reliable Optimum Optimality, Reliability & Robustness x1 g1(x1,x2)=0 g2(x1,x2)=0 Increased Performance f(x1,x2) contours x2

48. Increased Performance Feasible Region Deterministic Optimum High Probability of Failure Deterministic Optimization

49. Increased Performance Feasible Region Reliability-Based Optimum Low Probability of Failure Reliability-Based Design Optimization