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

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

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

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

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

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

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

, 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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

Chosen Design Performance Optimum Robust Optimum

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

A Mathematical Example Performance

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

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

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

Chosen Design Performance Optimum Robust Optimum

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

Q & A

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

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

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

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 :

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