Solving the problems of the conductivity of heterogeneous media by the method of the discrete Radon transform

Abstract

In the framework of periodic homogenization, the Fast Fourier Transform (FFT) homogenization method allows one to formulate the linear conduction problem as an integral equation whose solution can be represented in Fourier space. In this work, we demonstrate that this solution can also be expressed in real space, with a precise geometrical interpretation. Our homogenization method builds upon a combination of the Discrete Radon Transform proposed by Gelfand, Gindikin, and Graev and a plane and anti-plane vector decomposition in some specific directions. This new combination enables us to efficiently decompose a vector field into two orthogonal subspaces (curl and divergence-free components). Based on this proposed approach, the conduction problem is reduced to an integral equation, where the local fields and subsequently the macroscopic behaviors are expressed as a real series expansion. We introduce an iterative scheme utilizing the finite Radon transform to solve the residual integral equation. In comparison to the FFT-based scheme developed by Moulinec and Suquet, our numerical experiments demonstrate that for certain specific microstructures, our method yields accurate results at significantly lower computational cost.