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FFT‐based homogenization for microstructures discretized by linear hexahedral elements
Author(s) -
Schneider Matti,
Merkert Dennis,
Kabel Matthias
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5336
Subject(s) - homogenization (climate) , discretization , conjugate gradient method , fast fourier transform , hexahedron , finite element method , linear elasticity , solver , mathematics , nonlinear system , computer science , mathematical analysis , algorithm , mathematical optimization , physics , structural engineering , engineering , biodiversity , ecology , quantum mechanics , biology
Summary The FFT‐based homogenization method of Moulinec–Suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems. In this work, we generalize the method to problems discretized by trilinear hexahedral elements on Cartesian grids and physically nonlinear elasticity problems. We present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation. For benchmark problems in linear elasticity, the solver exhibits mesh‐ and contrast‐independent convergence behavior and enables the computational homogenization of complex structures, for instance, arising from computed tomography computed tomography (CT) imaging techniques. There exist 3D microstructures involving pores and defects, for which the original FFT‐based homogenization scheme does not converge. In contrast, for the proposed scheme, convergence is ensured. Also, the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes. We demonstrate the power of the approach by computing the elasto‐plastic response of a long‐fiber reinforced thermoplastic material with 172 × 10 6 (displacement) degrees of freedom. Copyright © 2016 John Wiley & Sons, Ltd.