Metamodeling using extended radial basis functions: a comparative approach

Abstract

The process of constructing computationally benign approximations of expensive computer simulation codes, or metamodeling, is a critical component of several large-scale multidisciplinary design optimization (MDO) approaches. Such applications typically involve complex models, such as finite elements, computational fluid dynamics, or chemical processes. The decision regarding the most appropriate metamodeling approach usually depends on the type of application. However, several newly proposed kernel-based metamodeling approaches can provide consistently accurate performance for a wide variety of applications. The authors recently proposed one such novel and effective metamodeling approach—the extended radial basis function (E-RBF) approach—and reported highly promising results. To further understand the advantages and limitations of this new approach, we compare its performance to that of the typical RBF approach, and another closely related method—kriging. Several test functions with varying problem dimensions and degrees of nonlinearity are used to compare the accuracies of the metamodels using these metamodeling approaches. We consider several performance criteria such as metamodel accuracy, effect of sampling technique, effect of sample size, effect of problem dimension, and computational complexity. The results suggest that the E-RBF approach is a potentially powerful metamodeling approach for MDO-based applications, as well as other classes of computationally intensive applications.

Appendices

Appendix A: List of engineering examples

1.1 Weld design problem (WELD) [21]

A simple welded beam structure is subjected to a combined bending and shear load. The design variables are weld height h, weld length l, bar thickness t, and width b. Force acting on the beam, F=6,000 lbs; shear strength, τ_d=13,600 psi; bending strength, σ_max=30,000 psi; Young’s modulus of the beam material, E=30×10⁶ psi; shear modulus, G=12×10⁶ psi. The variable bounds are: 0.125 in ≤ h ≤ 2.0 in; 2.0 in ≤ l ≤ 10.0 in; 2.0 in ≤ t ≤ 10.0 in; 0.125 in ≤ b ≤ 2.0 in. The function approximated is the shear stress in the weld, given by

$$f_{1} = \left(\tau_{1}^{2} + 2 \tau_{1} \tau_{2} \cos \theta + \tau_{2}^{2}\right)^{1/2},$$

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where

$$\begin{aligned} \cos \theta =& l/2R; \quad \tau_{1} = \frac{F}{\sqrt{2} h l}; \quad \tau_{2} = \frac{MR}{J};\\ R =& \left[\frac{l^{2}}{4} + \left(\frac{t + h}{2}\right)^{2}\right]^{1/2};\\ M =& F \left(L + \frac{l}{2}\right);\\ J =& \sqrt{2} h l \left[\frac{l^{2}}{12}+ \left(\frac{t+h}{2}\right)^{2}\right].\\ \end{aligned}$$

1.2 Two bar truss under compressive loading (TRUSS) [21]

The second response is the safety constraint function of a two-bar truss under compressive loading. The constants are: the load, 2P=66,000 lbs, the span 2B=60 in, thickness of the members T=0.1 in, material density, ρ=0.3 lbs/in³, and the Young’s modulus, E=30×10⁶ psi. Also, the bounds on the design variables are given as 0.5 in ≤ D ≤ 5 in, and 5 in ≤ H ≤ 50 in.

The response to be approximated is the buckling constraint expression given by

$$f_{2} = \left(s/\sigma_{\rm c}\right) - 1,$$

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where σ_c is the critical buckling stress, and s is the compressive stress induced in the bar, as given below.

$$\sigma_{\rm c} = \frac{\pi^{2} E(D^{2} + T^{2})}{8(B^{2}+H^{2})}; \quad s = \frac{P \sqrt{B^2 + H^2}}{\pi TDH}.$$

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Appendix B: List of mathematical functions

Table 5 contains the list of functions used in the comparative study, along with their sources.

Table 5 Mathematical functions