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Research Papers

Turning Black-Box Functions Into White Functions

[+] Author and Article Information
Songqing Shan

Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canadashans@cc.umanitoba.ca

G. Gary Wang1

School of Engineering Science, Simon Fraser University, Surrey, BC, V3T 0A3, Canadagary_wang@sfu.ca

1

Corresponding author.

J. Mech. Des 133(3), 031003 (Feb 22, 2011) (10 pages) doi:10.1115/1.4002978 History: Received May 03, 2010; Revised September 03, 2010; Published February 22, 2011; Online February 22, 2011

A recently developed metamodel, radial basis function-based high-dimensional model representation (RBF-HDMR), shows promise as a metamodel for high-dimensional expensive black-box functions. This work extends the modeling capability of RBF-HDMR from the current second-order form to any higher order. More importantly, the modeling process “uncovers” black-box functions so that not only is a more accurate metamodel obtained, but also key information about the function can be gained and thus the black-box function can be turned “white.” The key information that can be gained includes: (1) functional form, (2) (non)linearity with respect to each variable, and (3) variable correlations. The black-box “uncovering” process is based on identifying the existence of certain variable correlations through two derived theorems. The adaptive process of exploration and modeling reveals the black-box functions until all significant variable correlations are found. The black-box functional form is then represented by a structure matrix that can manifest all orders of correlated behavior of the variables. The resultant metamodel and its revealed inner structure lend themselves well to applications such as sensitivity analysis, decomposition, visualization, and optimization. The proposed approach is tested with theoretical and practical examples. The test results demonstrate the effectiveness and efficiency of the proposed approach.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

A simplified flow of RBF-HDMR metamodeling

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Figure 2

An example of component correlation matrix indicating a function having all significant bivariate terms

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Figure 3

Process for high-order component identification

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Figure 4

The structure matrix of the example

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Figure 5

Deterioration of f(x) when decreasing coefficients α4 and α5

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Figure 6

Structure matrices and correlation matrices of problem 12

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