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Research Papers: Design Automation

Generalized Radial Basis Function-Based High-Dimensional Model Representation Handling Existing Random Data

[+] Author and Article Information
Haitao Liu

School of Energy and Power Engineering,
Dalian University of Technology,
Dalian 116024, China
e-mail: lht@mail.dlut.edu.cn

Xiaofang Wang

School of Energy and Power Engineering,
Dalian University of Technology,
Dalian 116024, China
e-mail: dlwxf@dlut.edu.cn

Shengli Xu

School of Energy and Power Engineering,
Dalian University of Technology,
Dalian 116024, China
e-mail: xusl@dlut.edu.cn

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 3, 2016; final manuscript received September 2, 2016; published online November 11, 2016. Assoc. Editor: Gary Wang.

J. Mech. Des 139(1), 011404 (Nov 11, 2016) (13 pages) Paper No: MD-16-1098; doi: 10.1115/1.4034835 History: Received February 03, 2016; Revised September 02, 2016

The radial basis function-based high-dimensional model representation (RBF–HDMR) is very promising as a metamodel for high dimensional costly simulation-based functions. But in the modeling procedure, it requires well-structured regular points sampled on cut lines and planes. In practice, we usually have some existing random points that do not lie on cut lines or planes. For this case, RBF–HDMR cannot utilize the information of these random points because of its inner regular sampling process. To utilize the existing random points, this article presents two strategies to build a generalized RBF–HDMR (GRBF–HDMR) model. The GRBF–HDMR model using the error model (EM) strategy, called GRBF–HDMREM, constructs an error RBF model based on the prediction errors at all the sampled points to improve the RBF–HDMR predictions. While the GRBF–HDMR model using the error allocation (EA) strategy, called GRBF–HDMREA, employs the virtual regular points projected from the random points and the estimated virtual responses to update the component RBF predictions, which thereafter improves the overall RBF–HDMR predictions. Numerical experiments on eight functions and an engineering example reveal that the error allocation strategy is more effective in utilizing the random data to improve the RBF–HDMR predictions, since it creates the virtual points that follow the sampling rule in RBF–HDMR and estimates the virtual responses accurately for most cases.

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Topics: Modeling , Errors
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Figures

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Fig. 1

Flowchart of the second-order RBF–HDMR metamodeling process

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Fig. 3

Contour plot of the 2D function

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Fig. 4

Contour plot of the RBF–HDMR model for the 2D function

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Fig. 5

Ten existing random points that do not lie on the two cut lines

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Fig. 6

Contour plot of the GRBF–HDMREM model for the 2D function

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Fig. 7

Contour plot of the GRBF–HDMREA model for the 2D function

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Fig. 8

Stepped cantilever beam with d steps

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Fig. 9

Boxplots of the (a) RAAEN and (b) RMAEN values of GRBF–HDMREM and GRBF–HDMREA for the test functions

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Fig. 10

The average modeling results of the first-order and second-order GRBF–HDMREM and GRBF–HDMREA for F1, F3, F5, and F7 with different random sample sizes

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Fig. 11

The modeling results of GRBF–HDMREA for F1, F3, F5, and F7 with β, respectively, being 0.5, 1, 2, and 4

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