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

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

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

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

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

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

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

9 PDF f f ΔRfΔRf Percentile Difference Approach Advanced Mean Value (AMV) method is used

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

11 Multi – Objective Optimization: Issues Must calculate whole Pareto set  Series of RBDO problems  Visualize Pareto set Choose “best” point on Pareto set Expensive (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 Preference Functions 1 0 weight hwhw 1 0 reliability hrhr Example: Trade – off between weight and reliability Aggregate h(h w,h r ) is maximized

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

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

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 Indifferent Designs h h 1 =1 h ref 1 0 h 2 =a 2 h 1 =a 1 h 2 =1 Two designs are indifferent if they have the same overall preference

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

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

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

21 A Mathematical Example “cut-off” For h 2 the “cut-off” value is Final Optimization Problem Single-Loop RBDO

22 Performance Optimum Robust Optimum Chosen Design

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

24 A Mathematical Example Performance

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

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

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

28 Robust Optimum Performance Optimum Chosen Design

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  Variation is 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 Design Under Uncertainty Analysis / Simulation Input Output Uncertainty (Quantified) Uncertainty (Calculated) 1. Quantification Propagation 2. Propagation Design 3. Design

32 Feasible Region Increased Performance x2x2 x1x1 f(x 1,x 2 ) contours g 1 (x 1,x 2 )=0 g 2 (x 1,x 2 )=0 Deterministic Design Optimization and Reliability-Based Design Optimization (RBDO) Reliable Optimum

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

34 Indifferent Designs Two designs are indifferent if they have the same overall preference Designer provides specific preferences a 1 =h 1 (x i ) and a 2 =h 2 (x i ) so that :