An accelerated FFT algorithm for thermoelastic and non‐linear composites

Vinogradov, Vladimir; Milton, Graeme W.

doi:10.1002/nme.2375

Cited by 75 publications

(87 citation statements)

References 11 publications

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“…We now come to the definition of the space of trial and test functions, V h , which is to remain unchanged throughout this section. The origin of the success of numerical schemes based on the discretization of the Lippmann-Schwinger equation [9,10,13,15,16,25,26] lies in the fact that the matrix of the underlying linear system is the sum of two matrices with noteworthy properties. The first matrix is block-diagonal, and its product with a vector is performed efficiently in the real space.…”

Section: General Settingmentioning

confidence: 99%

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Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

Brisard

Dormieux

2012

Computer Methods in Applied Mechanics and Engineering

119

206

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To cite this version:Sébastien Brisard, Luc Dormieux. Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2012, 217-220, pp.197-212 AbstractWe report on the mathematical analysis of two different, FFT-based, numerical schemes for the homogenization of composite media within the framework of linear elasticity: the basic scheme of Suquet (1994, 1998), and the energy-based scheme of Brisard and Dormieux (2010). Casting these two schemes as Galerkin approximations of the same variational problem allows us to assert their well-posedness and convergence. More importantly, we extend in this work their domains of application, by relieving some stringent conditions on the reference material which were previously thought necessary. The origins of the flaws of each scheme are identified, and a third scheme is proposed, which seems to combine the strengths of the basic and energy-based schemes, while leaving out their weaknesses. Finally, a rule is proposed for handling heterogeneous pixels/voxels, a situation frequently met when images of real materials are used as input to these schemes.

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Section: General Settingmentioning

confidence: 99%

“…Proof of the second statement is not needed, as the first statement is necessary and sufficient when the bilinear form a is symmetric. (25), as well as function F, defined by (26).…”

Section: Appendix A2 Proof Of Theoremmentioning

confidence: 99%

Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

Brisard

Dormieux

2012

Computer Methods in Applied Mechanics and Engineering

119

206

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“…(31) with i = 0, see [8]. However, as follows from our developments in Section 2 and also from the discussion in [16,Section 3], this choice rather depends on the iterative algorithm used to solve the following linearized system for δε * (i+1) ,…”

Section: Collocation Fft Schemesmentioning

confidence: 99%

A finite element perspective on nonlinear FFT-based micromechanical simulations

Zeman

Geus

Vondřejc

et al. 2017

Int. J. Numer. Meth. Engng

104

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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“…Since then, several authors proposed variants or improvements of the initial method (among others: Müller [3], Lebensohn [4], Vinogradov & Milton [5], Zeman et al [6], Brisard & Dormieux [7], ...). In particular, Eyre & Milton [8,9], Michel et al [10] and Monchiet & Bonnet [11] proposed accelerated algorithms to overcome the low convergence rate of the basic scheme for highly-contrasted materials.…”

Section: Introductionmentioning

confidence: 99%

Comparison of three accelerated FFT‐based schemes for computing the mechanical response of composite materials

Moulinec

Silva

2014

Numerical Meth Engineering

130

160

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To cite this version:Hervé Moulinec, Fabrice Silva. Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials. International Journal for Numerical Methods in Engineering, Wiley, 2014Wiley, , 97 (13), pp.960-985. 10.1002 Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials This study shows that the scheme which minimizes this upper bound is the scheme of Eyre-Milton. The paper discusses the choice of the convergence test used in the schemes.

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An accelerated FFT algorithm for thermoelastic and non‐linear composites

Cited by 75 publications

References 11 publications

Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

A finite element perspective on nonlinear FFT-based micromechanical simulations

Comparison of three accelerated FFT‐based schemes for computing the mechanical response of composite materials

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