DBFFT: A displacement based FFT approach for homogenization of the mechanical behavior
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All the FFT-based methods available for homogenization of the mechanical response share a common point: the unknown field is the strain/deformation gradient, second order tensors which compatibility is imposed using Green's function or projection operators. This implies the allocation of redundant information and, when the method is based in a solving a linear system, the rank-deficiency of the resulting systems. In this work we propose a new, fast, robust and memory-efficient FFT approach in which the displacement field on the Fourier space is the unknown: the displacement based FFT (DBFFT) algorithm. In the linear case, the method results in a linear equation defined in terms of linear operators in the Fourier space and that does not require a reference medium. The system has an associated full rank Hermitian matrix and can be solved using iterative Krylov solvers and allows the use of preconditioners. The method is extended to non-linear problems and strain, stress and mixed control. A preconditioner is proposed to improve the efficiency of the system resolution. Finally, some numerical examples are solved to check the accuracy and efficiency of the method. The computational cost reduction respect the Galerkin-FFT was around 30%.
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