Reliability and Robustness in Engineering Design Zissimos P. Mourelatos, Associate Prof.

1. Reliability and Robustness in Engineering Design Zissimos P. Mourelatos, Associate Prof. Jinghong Liang, Graduate Student Mechanical Engineering Department Oakland University Rochester, MI 48309, USA mourelat@oakland.edu

2. Outline • Definition of reliability-based design and robust design • Reliable / Robust design • Problem statement • Variability measure • Multi-objective optimization • Preference aggregation method • Indifferent designs • Examples • Summary and conclusions

3. Reliability Reliable Design Problem Statement Maximize Mean Performance subject to : Probabilistic satisfaction of performance targets

4. Robust Design Problem Statement Minimize Performance Variation subject to : Deterministic satisfaction of performance targets

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

6. Robustness Reliability Reliable & Robust Design under Uncertainty: Problem Statement Maximize Mean Performance Minimize Performance Variation subject to : Probabilistic satisfaction of performance targets

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

8. Reliable / Robust Design Problem: Issues • Variability Measure Calculation • Variance • Percentile Difference • Trade – offs in Multi – Objective Optimization • Preference Aggregation Method

9. PDFf f ΔRf Percentile Difference Approach Advanced Mean Value (AMV) method is used

10. f Pareto set min f utopia pt min g g Multi – Objective Optimization: Min – Min Problem min f min g subject to constraints

11. Expensive Multi – Objective Optimization: Issues • Must calculate whole Pareto set • Series of RBDO problems • Visualize Pareto set • Choose “best” point on Pareto set (How??)

12. Preference Aggregation Method • Capable of calculating whole Pareto set • Use of Indifferent Designs to only get the “best” point on Pareto set

13. hr hw 1 1 0 0 reliability weight Preference Functions Example: Trade – off between weight and reliability Aggregate h(hw,hr) is maximized

14. Annihilation : Idempotency : Monotonicity : if Commutativity : Continuity : Preference Aggregation Axioms

15. satisfies annihilation for only. Fully compensating : For For : Non - Compensating Aggregation is defined by Preference Aggregation Method

16. Preference Aggregation Properties • For any Pareto optimal point, there is always a set (s,w) to select it. • For any fixed s, there are Pareto sets for which some Pareto points can never be selected for any choice of w.

17. h h2=1 h1=1 1 href h2=a2 h1=a1 0 Indifferent Designs • Two designs are indifferent if they have the same overall preference

18. The calculated (s,w) pair will select the “best” design on the Pareto set Indifferent Designs resulting in and

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

20. RBDO Problem s.t. Robust Problem s.t. A Mathematical Example

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

22. Chosen Design Performance Optimum Robust Optimum

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

24. A Mathematical Example Performance

25. 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

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

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

28. Chosen Design Performance Optimum Robust Optimum

29. Summary and Conclusions • A methodology was presented for trading-off performance and robustness • A multi – objective optimization formulation was used • Preference aggregation method handles trade – offs • Variationis reduced by minimizing a percentile difference • AMV method is used to calculate percentiles • A single – loop probabilistic optimization algorithm identifies the reliable / robust design • Examples demonstrated the feasibility of the proposed method

30. Q & A

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

32. Feasible Region Reliable Optimum Deterministic Design Optimization and Reliability-Based Design Optimization (RBDO) x1 g1(x1,x2)=0 g2(x1,x2)=0 Increased Performance f(x1,x2) contours x2

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

34. Indifferent Designs • Two designs are indifferent if they have the same overall preference • Designer provides specific preferences a1=h1(xi) and a2=h2(xi) so that :