International Journal for Numerical Methods in Engineering, vol. 62, No. 1, pp 122-157, 2005
A. Sutradhar1, G. H. Paulino1, and L. J. Gray2
1Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, IL 61801, U.S.A.
2Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.
Abstract
A symmetric Galerkin formulation and implementation for heat conduction in a three-dimensional functionally graded material is presented. The Green's function of the graded problem, in which the thermal conductivity varies exponentially in one co-ordinate, is used to develop a boundary-only formulation without any domain discretization. The main task is the evaluation of hypersingular and singular integrals, which is carried out using a direct limit to the boundary approach. However, due to complexity of the Green's function for graded materials, the usual direct limit procedures have to be modified, incorporating Taylor expansions to obtain expressions that can be integrated analytically. Several test examples are provided to verify the numerical implementation. The results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations.
Keywords: Symmetric Galerkin, functionally graded materials, diffusion, hypersingular integrals.