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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
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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
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3 Reliable Design Problem Statement Maximize Mean Performance subject to : Probabilistic satisfaction of performance targets Reliability
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4 Robust Design Problem Statement Minimize Performance Variation subject to : Deterministic satisfaction of performance targets
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5 Robust Design A design is robust if performance is not sensitive to inherent variation/uncertainty. Design Parameter
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6 Reliable & Robust Design under Uncertainty: Problem Statement Maximize Mean Performance Minimize Performance Variation subject to : Probabilistic satisfaction of performance targets Reliability Robustness
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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 :
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8 Reliable / Robust Design Problem: Issues Variability Measure Calculation Variance Percentile Difference Trade – offs in Multi – Objective Optimization Preference Aggregation Method
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9 PDF f f ΔRfΔRf Percentile Difference Approach Advanced Mean Value (AMV) method is used
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10 Multi – Objective Optimization: Min – Min Problem min f min g subject to constraints min g min f g f utopia pt Pareto set
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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??)
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12 Preference Aggregation Method Capable of calculating whole Pareto set Use of Indifferent Designs to only get the “best” point on Pareto set
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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
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14 Preference Aggregation Axioms Annihilation : Idempotency : Monotonicity : if Commutativity : Continuity :
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15 satisfies annihilation for only. For : Fully compensating For: Non - Compensating Preference Aggregation Method Aggregation is defined by
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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.
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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
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18 Indifferent Designs resulting in and The calculated (s,w) pair will select the “best” design on the Pareto set
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19 A Mathematical Example s.t. Reliable/Robust Problem R = 99.87%
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20 A Mathematical Example s.t. RBDO Problem Robust Problem s.t.
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21 A Mathematical Example “cut-off” For h 2 the “cut-off” value is Final Optimization Problem Single-Loop RBDO
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22 Performance Optimum Robust Optimum Chosen Design
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23 A Mathematical Example. s.t. Weighted Sum Approach R=99.87%
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24 A Mathematical Example Performance
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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%
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26 A Cantilever Beam Example, s.t. where: RBDO Problem
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27 A Cantilever Beam Example, s.t. where: Robust Problem
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28 Robust Optimum Performance Optimum Chosen Design
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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
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30 Q & A
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31 Design Under Uncertainty Analysis / Simulation Input Output Uncertainty (Quantified) Uncertainty (Calculated) 1. Quantification Propagation 2. Propagation Design 3. Design
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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
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33, : vector of random design variables : vector of deterministic design variables : vector of random design parameters s.t. where : RBDO Problem Statement Single Objective
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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 :
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