An algorithm for stress and mixed control in Galerkin‐based FFT homogenization

Lucarini, Sergio; Segurado, J.

doi:10.1002/nme.6069

Cited by 32 publications

(38 citation statements)

References 13 publications

(44 reference statements)

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“…for E ∈ Sym(d) and u ∈ H 1 # (Y ; R d ). Kabel et al [187] and Lucarini and Segurado [188] showed that the problem (4.2) is equivalent to a Lippmann-Schwinger equation, which takes the form…”

Section: Mixed Boundary Conditionsmentioning

confidence: 99%

A review of nonlinear FFT-based computational homogenization methods

Schneider

2021

Acta Mech

126

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Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform.

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Section: Mixed Boundary Conditionsmentioning

confidence: 99%

A review of nonlinear FFT-based computational homogenization methods

Schneider

2021

Acta Mech

126

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“…Journal Pre-proof average planar stretch is prescribed -see for example [37] for a comprehensive discussion of mixed boundary conditions). Above, • denotes spatial average.…”

Section: Implementation In the Periodic Settingmentioning

confidence: 99%

Accelerated computational micromechanics and its application to polydomain liquid crystal elastomers

Zhou¹,

Bhattacharya²

2021

Journal of the Mechanics and Physics of Solids

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This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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“…Classically, FFT methods are conceived to have the macroscopic strain as input and resolve the stress and mixed cases by iterative approaches. Very recently, the authors have proposed a method to include directly the prescribed terms of the macroscopic stress in the formulation [17] for the Galerkin-FFT approach. In this paper, we propose a similar approach for the DBFFT.…”

Section: Dbfft: Stress Controlmentioning

confidence: 99%

DBFFT: A displacement based FFT approach for non-linear homogenization of the mechanical behavior

Lucarini

Segurado

2019

International Journal of Engineering Science

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All the FFT-based methods available for homogenization of the mechanical response share a common point: the unknown field is the strain/deformation gradient, second order tensors which compatibility is imposed using Green's function or projection operators. This implies the allocation of redundant information and, when the method is based in a solving a linear system, the rank-deficiency of the resulting systems. In this work we propose a new, fast, robust and memory-efficient FFT approach in which the displacement field on the Fourier space is the unknown: the displacement based FFT (DBFFT) algorithm. In the linear case, the method results in a linear equation defined in terms of linear operators in the Fourier space and that does not require a reference medium. The system has an associated full rank Hermitian matrix and can be solved using iterative Krylov solvers and allows the use of preconditioners. The method is extended to non-linear problems and strain, stress and mixed control. A preconditioner is proposed to improve the efficiency of the system resolution. Finally, some numerical examples are solved to check the accuracy and efficiency of the method. The computational cost reduction respect the Galerkin-FFT was around 30%.

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An algorithm for stress and mixed control in Galerkin‐based FFT homogenization

Cited by 32 publications

References 13 publications

A review of nonlinear FFT-based computational homogenization methods

A review of nonlinear FFT-based computational homogenization methods

Accelerated computational micromechanics and its application to polydomain liquid crystal elastomers

DBFFT: A displacement based FFT approach for non-linear homogenization of the mechanical behavior

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