Metamodeling using extended radial basis functions: a comparative approach

  • Anoop A. Mullur
  • Achille Messac
Original Article


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.


Metamodeling Response surfaces Optimization Simulation-based design Radial basis functions 



Computationally expensive analysis


Computationally benign metamodel of f(x)


Vector of n p exact function values


Number of design variables (dimension of the problem)


Degree of the monomial term in nonradial basis functions


Number of data points evaluated


Number of test points for evaluating accuracy


Radial distance from a given data point


Design variable vector


i-th Design configuration, or data point


j-th Element of the vector x


Smoothness parameter in nonradial basis functions


Correlation parameters for kriging


Coordinate vector of point x relative to data point x i


j-th Element of vector ξ i

List of symbols


Extended radial basis function


Hammersley sequence sampling


Latin hypercube sampling


Multidisciplinary design optimization


Normalized root mean squared error


Normalized maximum absolute error


Radial basis function


Random sampling


Response surface methodology



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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