Fourier-based computational micromechanics on boundary-conforming transformed grids

Computational homogenization accelerated by Green function preconditioning with Fast Fourier transforms (FFT) is classically performed on a uniform grid, which hinders the discretization accuracy. In this work, we consider a more accurate geometry representation obtained by transforming the uniform computational grid into a boundary-conforming one. The mechanical problem is discretized using the finite element method (FEM) with isoparametric transformation of elements. Boundary adaptation can require large localized geometrical transformations of the grid, which is naturally accounted for in the FEM discretization. Rigorous bounds on the spectrum of eigenvalues of the resulting discrete system with Green preconditioner are provided. For grid transformations with projection of the nearest nodes to boundary, the modified eigenvalues correspond to eigenvectors localized at the material phases boundaries, so that the effective spectrum remains favorable for the preconditioned conjugate gradient solver. Numerical investigations confirm that the accuracy of the homogenized properties and the local fields obtained on boundary-conforming grids are greatly improved over uniform grid ones, at the expense of a moderate increase in computational cost.