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Research Papers

Co-Evolutionary Optimization for Multi-Objective Design Under Uncertainty

[+] Author and Article Information
Rajan Filomeno Coelho

Associate Professor
Building, Architecture & Town
planning (BATir) Department,
Brussels School of Engineering/École
polytechnique de Bruxelles,
Université libre de Bruxelles,
Avenue Franklin Roosevelt, 50 (CP 194/2),
Brussels B-1050, Belgium
e-mail: rfilomen@ulb.ac.be

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 19, 2012; final manuscript received October 22, 2012; published online January 7, 2013. Assoc. Editor: David Gorsich.

J. Mech. Des 135(2), 021006 (Jan 07, 2013) (8 pages) Paper No: MD-12-1212; doi: 10.1115/1.4023184 History: Received April 19, 2012; Revised October 22, 2012

This paper focuses on multi-objective optimization under uncertainty for mechanical design, through a reliability-based formulation referring to the concept of probabilistic nondominance. To address this problem, the implementation of a co-evolutionary strategy is advocated, consisting of the concurrent evolution of two intertwined populations optimized according to coupled subproblems: the upper level optimizer handles the design variables, whereas the corresponding values of the probabilistic thresholds for the objectives (namely the reliable nondominated front) are retrieved at the lower stage. The proposed methodology is successfully applied to six analytical test cases, as well as to the sizing optimization of two truss structures, demonstrating an improved capacity to cover wider ranges of the reliable nondominated front in comparison with all-at-once strategies tackling all types of variables simultaneously.

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Figures

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Fig. 1

Example of probability of nondominance in a bicriterion problem (joint probability): the dark region illustrates the region of the objective space dominating ζ

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Fig. 2

Example of probability of nondominance in a bicriterion problem (objective space): the shaded area illustrates the region of the objective space dominating ζ, while the contours depict isoprobability values for the objective vectors

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Fig. 3

All quantile vectors ζ  ∈  {ζA,…,ζB} are dominated by the objectives (sampled with respect to the random variables θ for a fixed value of the design vector x) with the same probability αf; the quadrants (in dashed lines) represent two regions of the objective space with identical probability of dominating each point ζ, according to the isoprobability contours of {f1(xfixed,θ),f2(xfixed,θ)}

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Fig. 4

Schematic view of the co-evolutionary framework for multi-objective reliability-based optimization

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Fig. 5

Quantile nondominated front (ζ) for test case TC1: optimization with respect to {x,ζ} (left), with respect to {x,η} (middle), and co-evolutionary optimization (right)

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Fig. 6

Quantile nondominated front (ζ) for test case TC5: optimization with respect to {x,ζ} (left), with respect to {x,η} (middle), and co-evolutionary optimization (right)

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Fig. 7

Deterministic trade-off set (left) versus co-evolutionary reliability-based trade-off set for test case TC5 (right), showing the discrepancy in the design solutions obtained by accounting or not for probabilistic nondominance

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Fig. 8

Truss configuration for test case t25b

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Fig. 9

Truss configuration for test case dome

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Fig. 10

Results obtained for test case dome: nondominated Pareto front (left), deterministic nondominated set (middle), and reliability-based nondominated set (right)

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