An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21110%2F23%3A00362316" target="_blank" >RIV/68407700:21110/23:00362316 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.amc.2023.127835" target="_blank" >https://doi.org/10.1016/j.amc.2023.127835</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.amc.2023.127835" target="_blank" >10.1016/j.amc.2023.127835</a>

Jazyk výsledku

angličtina

Název v původním jazyce

An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization

Popis výsledku v původním jazyce

We generalize and provide a linear algebra-based perspective on a finite element (FE) homogenization scheme, pioneered by Schneider et al.[1] and Leuschner and Fritzen [2]. The efficiency of the scheme is based on a preconditioned, well-scaled reformulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner---a discretized Green’s function of a periodic homogeneous reference problem---has a block-diagonal structure in the Fourier space which permits its efficient inversion using fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as $mathcal{O}(n log(n))$, like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to classical spectral solvers, the proposed scheme works with FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon. We show that the scheme achieves a number of iterations that are almost independent of spatial discretization. The scheme also scales mildly with phase contrast. We also discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection.

Název v anglickém jazyce

An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization

Popis výsledku anglicky

We generalize and provide a linear algebra-based perspective on a finite element (FE) homogenization scheme, pioneered by Schneider et al.[1] and Leuschner and Fritzen [2]. The efficiency of the scheme is based on a preconditioned, well-scaled reformulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner---a discretized Green’s function of a periodic homogeneous reference problem---has a block-diagonal structure in the Fourier space which permits its efficient inversion using fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as $mathcal{O}(n log(n))$, like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to classical spectral solvers, the proposed scheme works with FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon. We show that the scheme achieves a number of iterations that are almost independent of spatial discretization. The scheme also scales mildly with phase contrast. We also discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

APPLIED MATHEMATICS AND COMPUTATION

ISSN

0096-3003

e-ISSN

1873-5649

Svazek periodika

2023

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

19

Strana od-do

—

Kód UT WoS článku

000927389700001

EID výsledku v databázi Scopus

2-s2.0-85147094097