Optimal FFT-accelerated Finite Element Solver for Homogenization

Ladecký, Martin; J., Leute, Richard; Falsafi, Ali; Pultarová, Ivana; Pastewka, Lars; Junge, Till; Zeman, Jan

doi:10.48550/arxiv.2203.02962

Cited by 4 publications

(10 citation statements)

References 36 publications

(91 reference statements)

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“…Our implementation of the near-equilibrium method is novel but the others are standard, and for all three methods we employ staggered solutions of the phase field and mechanics subproblems. We use an FFT-accelerated strain-based micromechanics solver for the mechanics sub-problem [55,79,80], and implement a bound-constrained conjugate gradients algorithm [59] to solve for the phase field while directly enforcing irreversibility constraints.…”

Section: Discussionmentioning

confidence: 99%

“…This convolution is symmetric and sparse in Fourier space as the Fourier-space operator Ĝ is block diagonal (see e.g., Refs. [81,55,79,82] for its precise form). Now, taking the discrete Fourier transform of Eq.…”

Section: Elasticity Sub-problemmentioning

confidence: 99%

“…This numerical method is highly efficient due to the sparsity of the linear operations in real space (calculation of the stresses) and Fourier space (application of the projection operator) and the efficiency of the only dense operation, the FFT [55]. It also benefits from almost optimal conditioning [83,80].…”

Section: Elasticity Sub-problemmentioning

confidence: 99%

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Crack-path selection in phase-field models for brittle fracture

Andrews¹,

Pastewka²

2022

Preprint

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This work presents a critical overview of the effects of different aspects of model formulation on crack path selection in quasi-static phase field fracture. We consider different evolution methods, mechanics formulations, fracture dissipation energy formulations, and forms of the irreversibility condition. The different model variants are implemented with common numerical methods based on staggered solution of the phase-field and mechanics sub-problems via FFT-based solvers. These methods mix standard approaches with novel elements, such as the use of bound-constrained conjugate gradients for the phase field sub-problem and a heuristic method for near-equilibrium evolution. We examine differences in crack paths between model variants in simple model systems and microstructures with randomly heterogeneous Young's modulus. Our results indicate that near-equilibrium evolution methods are preferable for quasi-static fracture of heterogeneous microstructures compared to minimization and time-dependent methods. In examining mechanics formulations, we find distinct effects of crack driving force and the model for contact implicit in phase field fracture. Our results favor the use of a strain-spectral decomposition for the crack driving force but not the contact model. Irreversibility condition and fracture dissipation energy formulation were also found to affect crack path selection, but systematic effects were difficult to deduce due to the overall sensitivity of crack selection within the heterogeneous microstructures. Our findings support the use of the AT1 model over the AT2 model and irreversibility of the phase field within a crack set rather than the entire domain. Sensitivity to these differences in formulation was reduced but not eliminated by reducing the crack width parameter relative to the size scale of the random microstructures.

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Section: Discussionmentioning

confidence: 99%

Section: Elasticity Sub-problemmentioning

confidence: 99%

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Crack-path selection in phase-field models for brittle fracture

Andrews¹,

Pastewka²

2022

Preprint

Self Cite

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“…41,49 Alternative discretization schemes were studied to replace this discretization with the hope to improve the quality of both the local solution fields and the accuracy of the computed effective properties. These alternatives include fully integrated trigonometric polynomials, 50,51 voxel-wise constant fields, 34,35 finite-difference, [52][53][54] finite-volume [55][56][57] and finite-element [58][59][60] discretizations. More recently, higher-order discretizations 61,62 were also studied.…”

Section: State Of the Artmentioning

confidence: 99%

“…As a consequence, for linear elastic materials, the computational effort that comes with multiple integration points is avoided, at the expense of a much higher memory footprint. Recently, Ladecký et al 60 proposed to evaluate the action of the FE stiffness matrix on the fly.…”

Section: State Of the Artmentioning

confidence: 99%

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

Schneider

2022

Numerical Meth Engineering

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The power of fast Fourier transform (FFT)-based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory-efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill-conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite-volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite-element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier-type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT-based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass-control parameter can be adjusted to "soften" the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement-based implementation. The article exposes an efficientThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials

Schneider

Wicht

2022

Numerical Meth Engineering

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We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial-scale microstructures, computational homogenization methods often utilize a rather crude approximation of the microstructure, favoring regular grids over accurate boundary representations. As the accuracy of such an approach has been under continuous verification for decades, it appears astonishing that this strategy is successful in homogenization, but is seldom used on component scale. A part of the puzzle has been solved recently, as it was shown that the effective elastic properties converge with twice the rate of the local strain and stress fields. Thus, although the local mechanical fields may be inaccurate, the averaging process leads to a cancellation of errors and improves the accuracy of the effective properties significantly. Unfortunately, the original argument is based on energetic considerations. The straightforward extension to the inelastic setting provides superconvergence of (pseudoelastic) potentials, but does not cover the primary quantity of interest: the effective stress tensor. The purpose of the work at hand is twofold. On the one hand, we provide extensive numerical experiments on the convergence rate of local and effective quantities for computational homogenization methods based on the fast Fourier transform. These indicate the superconvergence effect to be valid for effective stresses, as well. Moreover, we provide theoretical justification for such a superconvergence based on an argument that avoids energetic reasoning.

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Optimal FFT-accelerated Finite Element Solver for Homogenization

Cited by 4 publications

References 36 publications

Crack-path selection in phase-field models for brittle fracture

Crack-path selection in phase-field models for brittle fracture

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

Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials

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