A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity

摘要

In this paper we derive a generalized fundamental solution for the BEM solution of problems of steady state heat conduction with arbitrarily spatially varying thermal conductivity. This is accomplished with the aid of a singular nonsymmetric generalized forcing function, D, with special sampling properties. Generalized fundamental solutions, E, are derived as locally radially symmetric responses to this nonsymmetric singular forcing function, D, at a source point ξ. Both E and D are defined in terms of the thermal conductivity of the medium. Although locally radially symmetric, E varies within the domain as the source point, ξ changes position. A boundary integral equation is formulated. Examples of generalized fundamental solutions are provided for various thermal conductivities along with the corresponding forcing function, D. Here, four numerical examples are provided. Excellent results are obtained with our formulation for variations of thermal conductivity ranging from quadratic and cubic in one dimension to exponential in two dimensions. Problems are solved in regular and irregular regions. Current work is under way investigating extensions of this general approach to further applications where nonhomogeneous property variations are an important consideration.