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Research Papers: Design Automation

Function Prediction at One Inaccessible Point Using Converging Lines

[+] Author and Article Information
Yiming Zhang

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611-6250
e-mail: yimingzhang521@ufl.edu

Chanyoung Park

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611-6250
e-mail: cy.park@ufl.edu

Nam H. Kim

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611-6250
e-mail: nkim@ufl.edu

Raphael T. Haftka

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611-6250
e-mail: haftka@ufl.edu

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 9, 2016; final manuscript received February 6, 2017; published online March 21, 2017. Assoc. Editor: Gary Wang.

J. Mech. Des 139(5), 051402 (Mar 21, 2017) (11 pages) Paper No: MD-16-1568; doi: 10.1115/1.4036130 History: Received August 09, 2016; Revised February 06, 2017

The focus of this paper is a strategy for making a prediction at a point where a function cannot be evaluated. The key idea is to take advantage of the fact that prediction is needed at one point and not in the entire domain. This paper explores the possibility of predicting a multidimensional function using multiple one-dimensional lines converging on the inaccessible point. The multidimensional approximation is thus transformed into several one-dimensional approximations, which provide multiple estimates at the inaccessible point. The Kriging model is adopted in this paper for the one-dimensional approximation, estimating not only the function value but also the uncertainty of the estimate at the inaccessible point. Bayesian inference is then used to combine multiple predictions along lines. We evaluated the numerical performance of the proposed approach using eight-dimensional and 100-dimensional functions in order to illustrate the usefulness of the method for mitigating the curse of dimensionality in surrogate-based predictions. Finally, we applied the method of converging lines to approximate a two-dimensional drag coefficient function. The method of converging lines proved to be more accurate, robust, and reliable than a multidimensional Kriging surrogate for single-point prediction.

Copyright © 2017 by ASME
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Figures

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Fig. 2

Illustration of (a) space-filling sampling (LHS) and (b) method of converging lines using 15 samples when the target point is at the origin and the domain at distance of less than 0.5 from the origin is inaccessible

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Fig. 1

Illustration of interpolating and extrapolating regions in design of experiments: (a) interpolating inaccessible region and (b) extrapolating inaccessible region

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Fig. 5

A typical set of interpolation results from 8D Dette function

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Fig. 4

Summary of extrapolation results from multiple lines and 8D Kriging based on Dette function

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Fig. 13

Extrapolation results of drag coefficient function using samples from reduced sampling domain, as shown in Fig. 14: (a) line 1, (b) line 2, and (c) line 3

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Fig. 3

A typical set of extrapolation results from 8D Dette function. The three lines are from randomly selected vertices to the target vertex.

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Fig. 7

A typical set of extrapolation results from 100D Styblinski–Tang function

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Fig. 6

Summary of interpolation results from multiple lines and 8D Kriging based on Dette function

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Fig. 8

One hundred sets of extrapolation results from multiple lines based on 100D Styblinski–Tang function. The target point was at xi= 3 with the function value to be −2400. One hundred sets of converging lines were randomly selected among the vertices excluding the target point.

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Fig. 9

A typical set of interpolation results from 100D Styblinski–Tang function

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Fig. 10

One hundred sets of interpolation results from multiple lines based on 100D Styblinski–Tang function. The target point was at xi= 1 with the function value to be −500. One hundred sets of converging lines were randomly selected among the vertices excluding the target point.

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Fig. 11

Dependence of drag coefficient on Re and M: (a) natural coordinate and (b) log10 coordinate

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Fig. 12

Lines and samples selection for extrapolating and interpolation of drag coefficient function. The solid dots denote the location of samples. (a) Extrapolation and (b) interpolation.

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Fig. 15

Interpolation results of drag coefficient function: (a) line 1, (b) line 2, and (c) line 3

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Fig. 16

Interpolation results using 2D and 1D Kriging: (a) 2D samples and (b) comparison of 2D and 1D interpolation

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Fig. 14

Extrapolation results of drag coefficient function using samples from reduced sampling domain, as shown in Fig. 12: (a) line 1, (b) line 2, and (c) line 3

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