FFT-based simulations of heterogeneous conducting materials with combined non-uniform Neumann, Periodic and Dirichlet boundary conditions
Résumé
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 non-uniform 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 and their relation with the Discrete Fourier
Transform on 4 times extended signals. 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.
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