Numerical homogenization method for heterogeneous materials based on Hill tensor for cuboids and Fourier Transform

In this paper, the collocation and Galerkin techniques are used to solve the Lippmann–Schwinger equation for periodic heterogeneous materials with pixelized/voxelized microstructures. In the context of conduction behavior, the integral equation is formulated using the periodic Green’s function generated by a neutralized Dirac comb and the associated periodic Green’s tensor in real space. After discretization, the Hill tensor for cuboids and the average Hill tensor representing the interaction between two distant cuboids naturally appear in the matrix of coefficients. Notably, these distance-dependent tensors can be determined accurately through analytical expressions, and the action of the coefficient matrix can be evaluated using the Fourier transform. Different integral equation formulations and iterative solution schemes can be employed. The main improvement compared to results of previous methods in the same field is that the resulting solution fields do not exhibit the spurious artifacts typically found in the original Fourier series–based approaches. The method is also successfully applied to both linear and nonlinear behaviors; in the latter case, the combination of the polarization formulation with the Anderson acceleration algorithm has demonstrated an excellent performance. • Collocation and Galerkin formulation of Lippmann Schwinger volume integral equation. • Accurate analytical expression of the Hill and average Hill tensors for cuboids. • Smooth solution compared with Fourier series-based methods. • Efficient iteration scheme for both linear and non linear behavior.