Improved guaranteed computable bounds on homogenized properties of periodic media by Fourier-Galerkin method with exact integration

  title={Improved guaranteed computable bounds on homogenized properties of periodic media by Fourier-Galerkin method with exact integration},
  author={Jaroslav Vondvrejc},
Moulinec and Suquet introduced FFT-based homogenization in 1994, and twenty years later, their approach is still effective for evaluating the homogenized properties arising from the periodic cell problem. This paper builds on the author’s (2013) variational reformulation approximated by trigonometric polynomials establishing two numerical schemes: Galerkin approximation (Ga) and a version with numerical integration (GaNi). The latter approach, fully equivalent to the original Moulinec-Suquet… 

Voxel‐based finite elements with hourglass control in fast Fourier transform‐based computational homogenization

  • M. Schneider
  • Computer Science
    International Journal for Numerical Methods in Engineering
  • 2022
The work at hand introduces FFT‐based solution techniques for underintegrated trilinear finite elements with hourglass control that stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field.

An FFT‐based Galerkin method for the effective permeability of porous material

A numerical scheme which is a combination of FFT and Galerkin methods is proposed for calculating the effective permeability of periodic porous material. First, a periodic stress field whose

A finite element perspective on nonlinear FFT‐based micromechanical simulations

Fourier solvers have become efficient tools to establish structure–property relations in heterogeneous materials. Introduced as an alternative to the finite element (FE) method, they are based on

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure.

Efficient numerical method for reliable upper and lower bounds on homogenized parameters

. A numerical procedure providing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators is presented. The upper bounds are obtained in a standard manner

A two‐scale FE‐FFT approach to nonlinear magneto‐elasticity

Fourier‐based approaches are a well‐established class of methods for the theoretical and computational characterization of microheterogeneous materials. Driven by the advent of computational

FFT‐based homogenization for microstructures discretized by linear hexahedral elements

This work generalizes the FFT‐based homogenization method of Moulinec–Suquet to problems discretized by trilinear hexahedral elements on Cartesian grids and physically nonlinear elasticity problems.

A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstructure Evolution and Macroscopic Material Behavior

The overall, macroscopic constitutive behavior of most materials of technological importance such as fiber-reinforced composites or polycrystals is very much influenced by the underlying



Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method

An FFT-based Galerkin method for homogenization of periodic media

FFT-based method for homogenization of periodic media: Theory and applications

This dissertation is devoted to an FFT-based homogenization scheme, a numerical method for the evaluation of the effective (homogenized) matrix of periodic linear heterogeneous materials. A problem

A Fast Fourier-Galerkin Method for Solving Singular Boundary Integral Equations

A convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis is proposed and a fast Fourier-Galerkin method is developed for solving a class of singular boundary integral equations.

Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients

Fast numerical computation of precise bounds of effective elastic moduli

A fast numerical solver to compute precise bounds of effective properties of multi-phase elastic composites is presented in contrast to analytical estimates like Hashin-Shtrikman bounds. Integral

A Modified Fourier–Galerkin Method for the Poisson and Helmholtz Equations

A modified Fourier–Galerkin method for the numerical solution of the Poisson and Helmholtz equations in a d-dimensional box is presented, based on an extension of the Fourier spaces by adding appropriate functions.