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Metamodeling using extended radial basis functions: a comparative approach

  • Anoop A. Mullur
  • Achille Messac
Original Article

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.

Keywords

Metamodeling Response surfaces Optimization Simulation-based design Radial basis functions 

Nomenclature

f(x)

Computationally expensive analysis

\(\tilde{f}(x)\)

Computationally benign metamodel of f(x)

F

Vector of n p exact function values

m

Number of design variables (dimension of the problem)

n

Degree of the monomial term in nonradial basis functions

np

Number of data points evaluated

nt

Number of test points for evaluating accuracy

r

Radial distance from a given data point

x

Design variable vector

xi

i-th Design configuration, or data point

xj

j-th Element of the vector x

γ

Smoothness parameter in nonradial basis functions

θ

Correlation parameters for kriging

ξi

Coordinate vector of point x relative to data point x i

ξji

j-th Element of vector ξ i

List of symbols

E-RBF

Extended radial basis function

HSS

Hammersley sequence sampling

LHS

Latin hypercube sampling

MDO

Multidisciplinary design optimization

NRMSE

Normalized root mean squared error

NMAX

Normalized maximum absolute error

RBF

Radial basis function

RND

Random sampling

RSM

Response surface methodology

Notes

Acknowledgements

Support from the National Science Foundation Award number DMI-0354733 is gratefully acknowledged.

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Copyright information

© Springer-Verlag London Limited 2005

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace Engineering, Multidisciplinary Design and Optimization LaboratoryRensselaer Polytechnic InstituteTroyUSA

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