FFT-based methods for the mechanics of composites: A general variational framework

Brisard, Sébastien; Dormieux, Luc

doi:10.1016/j.commatsci.2010.06.009

Cited by 219 publications

(289 citation statements)

References 23 publications

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“…They are referred to with a 'tilde' notation as DS, AS, AL, VAR, DS D , and AL D . We emphasize that the results presented here for the variational approaches VAR and VAR make use of the Green operators G 0 and G 0 rather than of the more complex discretization proposed in [14]. Also, in the latter approaches, definite-positiveness of matrix M (see Sec.…”

Section: Classical and Modified Green Operatorsmentioning

confidence: 99%

“…The alternative 'variational' algorithm [14] relies on two distinct ideas. First, Equation (3) is written as:…”

Section: Fft Methodsmentioning

confidence: 99%

“…First, FFT methods will ultimately be less efficient when dealing with highly porous media such as foams, where voids need to be discretized. Second, interfaces are crudely rendered when using voxel grids, although smoothness can be somewhat recovered by introducing intermediate properties between phases [13,14]. This matter is the most important one for ideal microstructure models where interfaces are completely known; less so when dealing with experimental images where such information is usually absent.…”

Section: Introductionmentioning

confidence: 99%

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Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

Willot

Abdallah

Pellegrini

2014

Int. J. Numer. Meth. Engng

103

146

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Nota Bene: The present document constitutes a 'postprint' version of the published paper, in which a few errors in the proofs (on the present pages 2, 5, 15 and 17) have been corrected and an incomplete reference on p. 22 has been completed. These minor corrections are marked out in red. Results are unchanged. AbstractA modified Green operator is proposed as an improvement of Fourier-based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite. Furthermore, it is shown that the method produces much more accurate local fields inside highly conducting and quasi-insulating phases, as well as in the vicinity of phase boundaries. These good properties stem from the discretization of Green's function, which is consistent with the pixel grid while retaining the local nature of the operator that acts on the polarization field. Finally, a fast implementation of the 'direct scheme' of Moulinec et al. (1994) that allows for parsimonious memory use is proposed.

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Section: Classical and Modified Green Operatorsmentioning

confidence: 99%

“…The alternative 'variational' algorithm [14] relies on two distinct ideas. First, Equation (3) is written as:…”

Section: Fft Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

Willot

Abdallah

Pellegrini

2014

Int. J. Numer. Meth. Engng

103

146

View full text Add to dashboard Cite

show abstract

“…Recently, Monchiet and Bonnet [340] proposed a polarization-based FFT iterative scheme to determine the overall properties of multiphase composites with arbitrary phase contrast, see also Refs. [341][342][343][344]. The use of a discrete element formulation to resolve the RVE problem has been addressed in Refs.…”

Section: Analysis At the Rve Levelmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

183

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…First-order finite difference-based approximations composed onto Fourier transforms have been shown to result in significantly mitigating oscillatory artifacts (Müller, 1996;Brown et al, 2002;Berbenni et al, 2014;Brisard and Dormieux, 2010;Lebensohn and Needleman, 2016;Schneider et al, 2016), while maintaining consistency with the original governing equations with h-refinement. Willot et al (2014) showed that rotated first-order schemes show a marked reduction in oscillatory artifacts.…”

Section: Introductionmentioning

confidence: 99%

Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium

Vidyasagar

Tutcuoglu

Kochmann

2018

Computer Methods in Applied Mechanics and Engineering

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Complex microstructural patterns arise as energy-minimizers in systems having non-convex energy landscapes such as those associated with phase transformations, deformation twinning, or finite-strain crystal plasticity. The prediction of such patterns at the microscale along with the resulting, effective material response at the macroscale is key to understanding a wide range of mechanical phenomena and has classically been dealt with by simplifying energy relaxation theory or by expensive finite element calculations. Here, we discuss a stabilized Fourier spectral technique for the homogenized response at the level of a representative volume element (RVE). We show that the FFT-based method admits sufficiently high resolution suitable to predict the emergence of energy-minimizing microstructures and the resulting effective response by computing the approximated quasiconvex energy hull. We test the method in the classical single-slip problem in single-and bicrystals. Especially the latter goes beyond the scope of traditional finite element and analytical relaxation treatments and hints at mechanisms of pattern formation in polycrystals. We also demonstrate that the chosen spectral finite-difference approximation, important for removing ringing artifacts in the presence of high contrasts, adds a natural regularization to the non-convex minimization. Finally, the technique is applied to polycrystalline pure magnesium, where we account for the competition between dislocation-mediated plasticity and deformation twinning. These inelastic deformation mechanisms result in complex texture evolution paths at the polycrystalline mesoscale and are simulated within RVEs of varying grain size and texture by a constitutive crystal plasticity model with an effective, volume fraction-based description of twinning.

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FFT-based methods for the mechanics of composites: A general variational framework

Cited by 219 publications

References 23 publications

Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium

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