FFT-based simulations of heterogeneous conducting materials with combined non-uniform Neumann, periodic and Dirichlet boundary conditions

Because of their simplicity, efficiency and ability for parallelism, FFT-based methods are very attractive in the context of numerical periodic homogenization, especially when compared to standard FE codes used in the same context. The purpose of the paper is to go beyond the use of periodic Boundary Conditions (BC), but keeping the advantages of FFT-based implementations. The present paper focuses on conductivity problems, considered as a first step towards mechanical problems. The proposed implementation is highly flexible, allowing to apply nonuniform loadings, to choose between periodic, Neumann and Dirichlet BC for each face of the unit-cell (of couple of faces for periodic BC), and to choose between different types of Finite Differences schemes (two types are considered here). The implementation relies on the use of Discrete Trigonometric Transforms (i.e. sine and cosine transforms) and their relation with the Discrete Fourier Transform on 4 times extended signals. The use of DTTs implicitly considers the signal to be anti-symmetric or symmetric with respect to each domain boundary. A direct relationship exists between the choice of a DTT (among 16 available) used in a given direction and the type of symmetry assumed on each boundary. Symmetry and anti-symmetry assumptions are respectively related to Dirichlet BC and anti-symmetry. The implementation has been precisely validated for various types of loading (uniform or non-uniform, full or combined, Dirichlet, Neumann and periodic BC) from a direct comparison with the same FE simulations.