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Research Papers

Robust Design Optimization Under Mixed Uncertainties With Stochastic Expansions

[+] Author and Article Information
Yi Zhang

Graduate Research Assistant
e-mail: zhayi@mst.edu

Serhat Hosder

Assistant Professor
e-mail: hosders@mst.edu
Department of Mechanical and Aerospace Engineering,
Missouri University of Science and Technology,
Rolla, MO 65409

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 18, 2012; final manuscript received March 21, 2013; published online June 10, 2013. Assoc. Editor: David Gorsich.

J. Mech. Des 135(8), 081005 (Jun 10, 2013) (11 pages) Paper No: MD-12-1317; doi: 10.1115/1.4024230 History: Received June 18, 2012; Revised March 21, 2013

The objective of this paper is to introduce a computationally efficient and accurate approach for robust optimization under mixed (aleatory and epistemic) uncertainties using stochastic expansions that are based on nonintrusive polynomial chaos (NIPC) method. This approach utilizes stochastic response surfaces obtained with NIPC methods to approximate the objective function and the constraints in the optimization formulation. The objective function includes a weighted sum of the stochastic measures, which are minimized simultaneously to ensure the robustness of the final design to both inherent and epistemic uncertainties. The optimization approach is demonstrated on two model problems with mixed uncertainties: (1) the robust design optimization of a slider-crank mechanism and (2) robust design optimization of a beam. The stochastic expansions are created with two different NIPC methods, Point-Collocation and Quadrature-Based NIPC. The optimization results are compared to the results of another robust optimization technique that utilizes double-loop Monte Carlo sampling (MCS) for the propagation of mixed uncertainties. The optimum designs obtained with two different optimization approaches agree well in both model problems; however, the number of function evaluations required for the stochastic expansion based approach is much less than the number required by the Monte Carlo based approach, indicating the computational efficiency of the optimization technique introduced.

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Figures

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Fig. 1

Robustness estimation of response in the presence of aleatory uncertainties only

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Fig. 2

Robustness estimation of response in the presence of epistemic uncertainties only

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Fig. 3

Robustness estimation of response in the presence of mixed uncertainties

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Fig. 4

Robustness assessment of mixed uncertainty design

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Fig. 5

Flow chart of the robust optimization process under mixed uncertainties with combined stochastic expansions

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Fig. 6

Slider-crank mechanism used in model problem 1

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Fig. 7

Convergence of NIPC results for σ¯(10 deg) as a function of expansion order for model problem 1, case 2

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Fig. 8

Convergence of NIPC results for δσ (10  deg) as a function of expansion order for model problem 1, case 2

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Fig. 9

Schematic of the beam design problem (model problem 2)

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Fig. 10

Convergence of NIPC results as a function of expansion order for model problem 2

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Fig. 11

Error convergence of NIPC results as a function of expansion order for model problem 2

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Fig. 12

The convergence history of average mean, average standard deviation, and the standard deviation difference of the beam volume for the optimization process with stochastic expansions

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