A robust and versatile parallel FFT-based mechanical solver for general non-periodic and periodic boundary conditions

Abstract

General boundary conditions are implemented within a fast Fourier transform framework for linear and non-linear
mechanical problems using small or finite transformation formulations. In the context of parallel computing
(distributed memory), we present a framework that enables the combination of periodic and non-periodic
(Dirichlet or Neumann) boundary conditions. Taking advantage of the link between non-periodic boundary
conditions and the symmetries of the relevant components of the fluctuation displacement and stress fields,
discrete trigonometric transforms are employed to adapt the classical Moulinec–Suquet fast Fourier transform
approach. The present study employs an original displacement-based fixed-point algorithm in combination with
a convergence acceleration method in order to solve boundary value problems. Finite difference approaches are
used to build the discrete Green operators associated with a pre-conditioner (reference material), whose choice
depends on the loading type and the small or finite transformation frameworks. The newly developed double
tetrahedron scheme is employed to investigate non-periodic problems. Outcomes are compared to those of the
classical hexahedral scheme. The robustness and computational efficiency of the presented parallel solver is
demonstrated through numerical experiments of non-trivial loading scenarios (tension, bending, normal-mixed
loading, torsion-bending), complex and densely discretized microstructures and diverse behavior laws (elasticity,
isotropic plasticity, crystal plasticity), within small and finite transformation frameworks.