An FFT-based Galerkin method for homogenization of periodic media

Vondřejc, Jaroslav; Zeman, Jan; Marek, Ivo

doi:10.1016/j.camwa.2014.05.014

Cited by 140 publications

(280 citation statements)

References 39 publications

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“…(A.1), from which it follows that G is a self-adjoint operator; see, e.g., [20,Lemma 2]. Notice that no approximation is made in (5), because all quantities are Ω-periodic and the sum is infinite.…”

Section: Compatibilitymentioning

confidence: 99%

A finite element perspective on nonlinear FFT-based micromechanical simulations

Zeman

Geus

Vondřejc

et al. 2017

Int. J. Numer. Meth. Engng

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107

137

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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 fixed-point solutions of the Lippmann-Schwinger type integral equation. Their computational efficiency results from handling the kernel of this equation by the Fast Fourier Transform (FFT). However, the kernel is derived from an auxiliary homogeneous linear problem, which renders the extension of FFT-based schemes to non-linear problems conceptually difficult. This paper aims to establish a link between FE-and FFT-based methods, in order to develop a solver applicable to general history-and time-dependent material models. For this purpose, we follow the standard steps of the FE method, starting from the weak form, proceeding to the Galerkin discretization and the numerical quadrature, up to the solution of non-linear equilibrium equations by an iterative Newton-Krylov solver. No auxiliary linear problem is thus needed. By analyzing a two-phase laminate with non-linear elastic, elasto-plastic, and visco-plastic phases, and by elasto-plastic simulations of a dual-phase steel microstructure, we demonstrate that the solver exhibits robust convergence. These results are achieved by re-using the non-linear FE technology, with the potential of further extensions beyond small-strain inelasticity considered in this paper.

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Section: Compatibilitymentioning

confidence: 99%

A finite element perspective on nonlinear FFT-based micromechanical simulations

Zeman

Geus

Vondřejc

et al. 2017

Int. J. Numer. Meth. Engng

Self Cite

107

137

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“…defined on a trial space V = ∇H The Fourier-Galerkin method, described for homogenisation in [1][2][3][4][5], belongs to FFT-based methods introduced in [6] and investigated and developed to many different schemes such as [7][8][9]. is based on Galerkin approximation with trigonometric polynomials of uniform order N = (n, .…”

Section: Application To Numerical Homogenisation Within Fourier-galermentioning

confidence: 99%

“…where the linear operator G : R d×N → R d×N enforces the compatibility condition (curl-free and zero-mean condition of gradient functions) and its action is again provided by FFT algorithm, for details see [1][2][3][4][5].…”

Section: Application To Numerical Homogenisation Within Fourier-galermentioning

confidence: 99%

Double Grid Integration with Projection (DoGIP): Application to numerical homogenization by Fourier‐Galerkin method

Vondřejc

2016

Proc Appl Math and Mech

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In this contribution, the DoGIP approach is introduced as a method for decomposition of a linear system into a (block) diagonal matrix based on a double grid integration and an interpolation/projection operator that is never assembled but optimised for fast matrix-vector multiplication. The method reduces memory requirements, especially when higher order basis functions are used for discretisations. The method is explained for Fourier-Galerkin method within a numerical homogenisation.

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“…Next, the method does not use the Green operator, but a periodized Green operator that is not explicitly known and must be evaluated numerically. In another recent contribution [33,34], a link has been established between the Lippmann-Schwinger equation and the Galerkin discretization of the weak form of the unit cell problem.…”

Section: Introductionmentioning

confidence: 99%

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

Monchiet¹

2015

Computer Methods in Applied Mechanics and Engineering

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Highlights• FFT based iterative schemes are reformulated in a unified variational approach.• The approach uses classic and Hashin-Shtrikman variational principles. • The method delivers rigorous bounds for the effective elastic properties of composites. • A hierarchy between the different FFT solutions is established. AbstractIn this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin-Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin-Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin-Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound.

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An FFT-based Galerkin method for homogenization of periodic media

Cited by 140 publications

References 39 publications

A finite element perspective on nonlinear FFT-based micromechanical simulations

A finite element perspective on nonlinear FFT-based micromechanical simulations

Double Grid Integration with Projection (DoGIP): Application to numerical homogenization by Fourier‐Galerkin method

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

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