Derivation of Higher-Order Terms in FFT-Based Numerical Homogenization

Dietrich, Felix; Merkert, Dennis; Simeon, Bernd

doi:10.1007/978-3-319-96415-7_25

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…Another kind of extension concerns higher order numerical homogenization (Tran et al 2012;Dietrich et al 2019). In that case, the homogenized behavior is not only dependent of the first gradient of the displacement but also of higher order terms.…”

Section: Introductionmentioning

confidence: 99%

A simple extension of FFT-based methods to strain gradient loadings - Application to the homogenization of beams and plates with linear and non-linear behaviors

Gélebart¹

2022

Journal of Theoretical, Computational and Applied Mechanics

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Because of their simplicity, efficiency and ability for parallelism, FFT-based methods are very attractive in the context of numerical periodic homogenization, especially when compared to standard FE codes used in the same context. They allow applying to a unit-cell a uniform average strain with a periodic strain fluctuation that is an unknown quantity. Solving the problem allows to evaluate the complete stress-strain fields. The present work proposes to extend the use of the method from uniform loadings (i.e. uniform applied strain) to strain gradient loadings (i.e. strain fields with a uniform strain gradient) while keeping the algorithm as simple as possible. The identification of a subset of strain gradient loadings allows for a minimally invasive modification of the iterative algorithm so that the implementation is straightforward. In spite of a reduced subset of 9 independent loadings among the 18 available, the second part of the paper demonstrates that it is enough for considering the homogenization of beams and plates. A first application validates the approach and compares it to another FFT-based method dedicated to the homogenization of plates. The second application concerns the homogenization of beams, for the first time considered (to author's knowledge) with an FFT-based solver. The method applies to different beam cross-sections and the proposition of using composite voxels drastically improves the numerical solution when the beam cross-section is not conform with the spatial discretization, especially for torsion loading. As a result, the massively parallel AMITEX_FFTP code has been slightly modified and now offers a new functionality, allowing the users to prescribe torsions and flexions to beam or plate heterogeneous unit-cells.

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Section: Introductionmentioning

confidence: 99%

A simple extension of FFT-based methods to strain gradient loadings - Application to the homogenization of beams and plates with linear and non-linear behaviors

Gélebart¹

2022

Journal of Theoretical, Computational and Applied Mechanics

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“…The various discretisation approaches have been studied along with linear and non-linear solvers [15,16,7,17,11,18,4,5,19]. Other research directions focus, for example, on multiscale methods [20,21,22] or highly non-linear problems in solid mechanics [23,24,25,26].…”

Section: Introductionmentioning

confidence: 99%

FFT-based homogenisation accelerated by low-rank tensor approximations

Vondřejc

Liu

Ladecký

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kind of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem. It significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation. The advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients. This approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material.

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“…Precising point (iii), the FFT‐accelerated method was initially proposed by Moulinec and Suquet to solve the first cell (or corrector) problem that appears in leading‐order homogenization, in the context of linear and nonlinear elastostatics. Although well known in the mechanical engineering literature, the method was applied only recently to the additional cell problems stemming from higher order homogenization . At our knowledge, it was never used in a microstructure optimization procedure to enhance dynamic properties.…”

Section: Introductionmentioning

confidence: 99%

Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization

Cornaggia

Bellis

2020

Numerical Meth Engineering

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This works concerns the propagation of waves in periodic media, whose microstructure is optimized to obtain specific dynamical properties (typically, to maximize the dispersion in given directions). The present study, focusing on scalar waves in two dimensions, for example, antiplane shear waves, aims at setting a generic optimization framework. The proposed optimization procedure relies on a number of mathematical and numerical tools. First, the two-scale asymptotic homogenization method is deployed up to second-order to provide an effective dispersive model. Simple dispersion indicators and cost functionals are then considered on the basis of this model. Then, the minimization of these functionals is performed thanks to an algorithm that relies on the concept of topological derivative to iteratively perform phase changes in the unit cell characterizing the material. Finally, fast Fourier transform-accelerated solvers are extensively used to solve the cell problems underlying the homogenized model. To illustrate the proposed approach, the resulting procedure is applied to the design of anisotropic media with maximal dispersion in specific directions, and to the reconstruction of unknown microstructures from effective phase velocity data. K E Y W O R D Sdispersive waves, phononic crystals, second-order homogenization, topological derivatives 1 3178

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Derivation of Higher-Order Terms in FFT-Based Numerical Homogenization

Cited by 3 publications

References 18 publications

A simple extension of FFT-based methods to strain gradient loadings - Application to the homogenization of beams and plates with linear and non-linear behaviors

A simple extension of FFT-based methods to strain gradient loadings - Application to the homogenization of beams and plates with linear and non-linear behaviors

FFT-based homogenisation accelerated by low-rank tensor approximations

Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization

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