FFT‐based homogenization for microstructures discretized by linear hexahedral elements

Abstract

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

An FFT‐based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

“…We consider two classical discretization schemes used in FFT‐based computational homogenization: The discretization by trigonometric polynomials of Moulinec and Suquet, where, in addition, the “energy” is approximated by the trapezoidal rule, cf. Vondřjec et al The rotated staggered grid of Willot et al, which may be interpreted as a discretization by Q1‐(trilinear hexahedral)‐finite elements with reduced integration …”

“…Due to formal similarities to the homogenization of thermal conductivity problems, we investigate two popular discretization schemes used in FFT‐based methods: the under‐integrated Fourier‐Galerkin discretization, pioneered by Moulinec and Suquet and the rotated staggered grid discretization, which may also be interpreted as an under‐integrated finiteelement discretization …”

Fast finite element calculation of effective conductivity of random continuum microstructures: The recursive Poincaré–Steklov operator method

“…There are mathematically elegant ways to access the effective properties of simple microstructures [4][5][6][7][8][9][10][11]; however, for more arbitrary composite microstructures a straightforward approach to determining γ eff is to:…”

Contact Mechanics in Computational Homogenization

“…The superscript denotes the restriction of a function to the corresponding domain or surface and σ, u and f are the Cauchy stress tensor, displacement and external force vector respectively; cf [1] for details about (1) - (2). The contact enforcement is achieved by the Karush-Kuhn-Tucker conditions along the contact surface Γ C ⊂ ∂Ω in (3): gap function g n (u (i) , u (j) ) and contact pressure σ n (u (i) ) are linked by complementary slackness.…”

“…With regard to that, we will demonstrate the treatment of contact between each of those by means of an implicit boundary description, used for both surface reconstruction and contact detection. Furthermore, a new technique for solving contact problems is introduced which can be easily integrated as material law routine into so called fast solvers [1], performing on Cartesian grids and exploiting the fast Fourier transform (FFT). In the end, we will present numerical results for a standard problem from contact mechanics, obtained by this new approach.…”