Efficient numerical solution of the volume integro-differential equation for time-varying temperature fields in heterogeneous media

Abstract

An efficient numerical method for calculation of time-varying temperature fields in an infinite homogeneous medium containing heterogeneous inclusions is presented. The medium is subjected to distributed or localized heat sources. The problem is reduced to the integro-differential equation for the temperature fields in the regions occupied by the inclusions only. For numerical solution, this equation is discretized using Gaussian radial functions for approximation of spatial-dependence and piece-wise constant functions for approximation of time-dependence of the temperature fields. For Gaussian approximating functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms. The iterative minimal residual method and fast Fourier transform technique are used for solution of the discretized problems at a sequence of time moments. The numerical algorithms for solution of the discretized equations are presented for 1D, 2D, and 3D heterogeneous media. Numerical solutions are compared with exact and approximate solutions in the cases of spherical layered inclusions subjected to distributed and localized heat sources.