Efficient numerical method of solution of the one-particle problems for transverse isotropic conductive medium

Abstract

An isolated heterogeneous inclusion in an infinite homogeneous transverse isotropic conductive medium subjected to an arbitrary external electric (temperature) field is considered. This problem plays an important role in application of self-consistent methods for calculation of effective conductive properties of a wide class of composites materials. Such composites consist of a homogeneous matrix phase (host medium) and a set of isolated inclusions or they are polycrystals, i.e. agglomerations of similar grains of random orientations. Calculation of the electric field in the medium with an isolated inclusion is reduced to a volume integral equation in the region occupied by the inclusion only. For numerical solution, this equation is discretized using special class of approximating functions concentrated at the nodes of a regular grid covered the inclusion domain. For such functions, the elements of the matrices of the discretized problem have form of 1D-integrals that can be tabulated. In addition, these matrices have Toeplitz' structures, and therefore, fast Fourier transform algorithms can be used for iterative solution of the discretized problems. The method is applied to calculation of the tensors that define contribution of individual inclusions in the effective conductive properties of the composites.