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RESEARCH PAPERS

Understanding the Effects of Model Uncertainty in Robust Design With Computer Experiments

[+] Author and Article Information
Daniel W. Apley1

Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 60208-3119

Jun Liu

Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 60208-3119

Wei Chen

Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208-3111

1

Corresponding author.

J. Mech. Des 128(4), 945-958 (Dec 19, 2005) (14 pages) doi:10.1115/1.2204974 History: Received September 16, 2005; Revised December 19, 2005

The use of computer experiments and surrogate approximations (metamodels) introduces a source of uncertainty in simulation-based design that we term model interpolation uncertainty. Most existing approaches for treating interpolation uncertainty in computer experiments have been developed for deterministic optimization and are not applicable to design under uncertainty in which randomness is present in noise and/or design variables. Because the random noise and/or design variables are also inputs to the metamodel, the effects of metamodel interpolation uncertainty are not nearly as transparent as in deterministic optimization. In this work, a methodology is developed within a Bayesian framework for quantifying the impact of interpolation uncertainty on the robust design objective, under consideration of uncertain noise variables. By viewing the true response surface as a realization of a random process, as is common in kriging and other Bayesian analyses of computer experiments, we derive a closed-form analytical expression for a Bayesian prediction interval on the robust design objective function. This provides a simple, intuitively appealing tool for distinguishing the best design alternative and conducting more efficient computer experiments. We illustrate the proposed methodology with two robust design examples—a simple container design and an automotive engine piston design with more nonlinear response behavior and mixed continuous-discrete design variables.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 3

Prediction intervals for the objective function values for three candidate designs, d1*, d2*, and d3*, based on the four simulation outputs in Fig. 1. The dot at the center of each bar indicated the center μf(d) of the prediction interval, and the circles indicate the objective function values obtained from the fitted model ŷ.

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Figure 4

(a) Simulation output at the first four sampling sites plus four additional sites. (b) Updated prediction intervals at the three candidate designs based on all eight simulation outputs in (a).

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Figure 5

(a) Three more simulation outputs added to Fig. 4. (b) Updated prediction intervals at the three candidate designs based on all the eleven outputs in (a).

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Figure 6

Prediction interval plot based on the eleven simulation outputs in Fig. 5

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Figure 7

True response surface for the piston design example

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Figure 8

PI plots of robust design objective after: (a) 10 initial simulation runs and (b) 18 simulation runs

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Figure 9

PI plot after 27 simulation runs. The robust design solution [PO*=1.3, SP*=2.33] has an objective value within the PI of [54.24, 54.29].

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Figure 10

Three 2D cross sections of the PI plot in Fig. 8. All dark shaded regions of the design space can be ruled out (with 95% confidence) as inferior to the unshaded design region shown in panel (a).

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Figure 1

(a) True response surface (treated as unknown), the four bullets indicate simulation output at four sampling sites. (b) Fitted surrogate model ŷ based on the four sampling sites (c) Mean μ, standard deviation σ, and robust design objective function f calculated from the true response surface. (d) Mean, standard deviation, and objective function calculated from the fitted model ŷ.

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Figure 2

(a)–(c) Three realizations of the posterior Gaussian random process G, each of which represents a potential response surface consistent with the four simulated output in Fig. 1. (d)–(f) Mean, standard deviation, and objective function calculated from the potential response surfaces in (a)–(c). Prior parameters: [ϕd,ϕw]=[2,0.1] and α=35.

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