RBF interpolation of boundary values in the BEM for heat transfer problems

Abstract

This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for the numerical solution of heat transfer problems. The quality of the estimate of boundary integrals is greatly affected by the type of functions used to interpolate the temperature, its normal derivative and the geometry along the boundary from the nodal values. In this paper, instead of conventional Lagrange polynomials, interpolation functions representing these variables are based on the “universal approximator” RBFNs, resulting in much better estimates. The proposed method is verified on problems with different variations of temperature on the boundary from linear level to higher orders. Numerical results obtained show that the BEM with indirect RBFN (IRBFN) interpolation performs much better than the one with linear or quadratic elements in terms of accuracy and convergence rate. For example, for the solution of Laplace's equation in 2D, the BEM can achieve the norm of error of the boundary solution of O(10−5) by using IRBFN interpolation while quadratic BEM can achieve a norm only of O(10−2) with the same boundary points employed. The IRBFN‐BEM also appears to have achieved a higher efficiency. Furthermore, the convergence rates are of O(h1.38) and O(h4.78) for the quadratic BEM and the IRBFN‐based BEM, respectively, where h is the nodal spacing.