FFT-based homogenisation accelerated by low-rank tensor approximations

Vondřejc, Jaroslav; Liu, Dishi; Ladecký, Martin; Matthies, Hermann G.

doi:10.1016/j.cma.2020.112890

Cited by 19 publications

(11 citation statements)

References 57 publications

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“…Numerical tests are made on a 2-dimensional test case [5] consists of a homogenisation of stochastic material property which is expressed by a Karhunen-Loève expansion. We compared the computational time cost of the solution of (2) in cases of using full and low rank tensors.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Reduced‐order FFT‐based homogenisation by tensor approximation

Liu

Vondřejc

Ladecký

et al. 2019

Proc Appl Math and Mech

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We enhanced the efficiency of Fast Fourier transform (FFT) based Galerkin methods on numerical homogenisation problems by exploiting low-rank tensor approximations in canonical, Tucker, and tensor train formats. This leads to a significant reduction in computational complexity and memory requirement. The advantages of the approach are demonstrated in a numerical example of a model homogenisation problem with stochastic heterogeneous material coefficients. Homogenisation by Fourier-Galerkin methodsWe consider microscopic homogenisation of a scalar linear elliptic problem on a d-dimensional cell Y = (− 1 2 , 1 2 ) d with bounded, symmetric, and uniformly elliptic material coefficients A : Y → R d×d . The variational formulation of finding effective material properties A H ∈ R d×d for any macroscopic gradient fieldis discretised with the Fourier-Galerkin method [1,2] that incorporates trigonometric polynomials as basis functions.The space H 1 0 (Y) consists of the Y-periodic scalar function v : Y → R with a square integrable gradient and zero mean. The Fourier coefficientsû of the minimisers can be identified by solving the following linear system of equations [2] in discretised Fourier domainThe tensorP is an inverse Laplacian preconditioner [3], N ∈ N d is the discretization size, F N stands for Fourier transform, ∇ N and ∇ * N for tensors of differentiation and divergence operators in Fourier domain.Ã ∈ R d×d×N ×N is a block diagonal tensor obtained by a weighted projection of A onto the dicretised space, and E a constant vector.We speed up the numerical solution by exploiting tensor decomposition techniques [4]. In the low-rank tensor format the multidimensional Fast Fourier transform (FFT) can be implemented at the cost of a series of 1-dimensional FFTs. For tensors with moderate representation rank this alleviates the "curse of dimension". The system is solved by a Minimal Residual iteration method. Low-rank tensor approximationLow rank tensor approximation techniques is applied to handle the huge number of degrees of freedom in the solution of (2). We use three formats, namely canonical, Tucker, and Tensor Train (TT) format of low-rank tensors. As an example, a canonical approximation a tensor v ∈ K N1×···×N d (K is R or C) is a sum of r rank-1 tensors in the form:where c ∈ R r stores the coefficients with respect to basis vectors b (j) ∈ K r×Nj in spatial direction j; operator ⊗ denotes tensor product.The linearity of Fourier transform facilitates to express d-dimensional FFT of a tensor as the sum of tensor products of 1-dimensional FFTs, i.e.,

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Section: Numerical Resultsmentioning

confidence: 99%

“…The latter is therefore significantly faster despite it is calculated on a larger grid. More details can be found in [5]. (3N, .…”

Section: Numerical Resultsmentioning

confidence: 99%

Reduced‐order FFT‐based homogenisation by tensor approximation

Liu

Vondřejc

Ladecký

et al. 2019

Proc Appl Math and Mech

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“…In addition, the Galerkin nature of the FE method connected with the minimization of the related energy functional allows us to use a well-built theory on the FE method for error estimation, convergence analysis, and other useful tools. In the future, the extension of the equivalence of DB and SB schemes to a general reference material and the fusion of low-rank tensor approximation technique of Vondřejc et al [41] with our FE scheme are of primal interest.…”

Section: Discussionmentioning

confidence: 99%

Optimal FFT-accelerated Finite Element Solver for Homogenization

Ladecký¹,

J.²,

Falsafi³

et al. 2022

Preprint

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We propose a specialized preconditioned FE homogenization scheme that is significantly more efficient than generic FE implementations.• Our formulation offers computational efficiency equivalent to spectral homogenization solvers.• Our formulation is devoid of ringing phenomena intrinsic to spectral methods.• All results are reproducible in an open-source project µSpectre available at https: //muspectre.gitlab.io/muspectre.

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“…Possible solutions include the use of sparse sampling techniques [337], tensor methods [338] or AI-based surrogate models [339].…”

Section: Concluding Remarks and Future Directionsmentioning

confidence: 99%

A review of nonlinear FFT-based computational homogenization methods

Schneider

2021

Acta Mech

148

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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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FFT-based homogenisation accelerated by low-rank tensor approximations

Cited by 19 publications

References 57 publications

Reduced‐order FFT‐based homogenisation by tensor approximation

Reduced‐order FFT‐based homogenisation by tensor approximation

Optimal FFT-accelerated Finite Element Solver for Homogenization

A review of nonlinear FFT-based computational homogenization methods

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