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.

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