Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F15%3A43926412" target="_blank" >RIV/49777513:23520/15:43926412 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21110/15:00233279 RIV/61989100:27740/15:86095971

Výsledek na webu

<a href="http://dx.doi.org/10.1016/j.cma.2015.09.003" target="_blank" >http://dx.doi.org/10.1016/j.cma.2015.09.003</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.cma.2015.09.003" target="_blank" >10.1016/j.cma.2015.09.003</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method

Popis výsledku v původním jazyce

Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast Fourier Transform (FFT) homogenization scheme by Vondřejc et al. (2014), which is based on the conforming Galerkin approximation with trigonometric polynomials. Upper-lower bounds are obtained by adjusting the primal-dual finite element framework developed independently by Dvořák (1993) and Wi?ckowski (1995) to the FFT-based Galerkin setting. We show that the discretization procedure differs for odd and non-odd number of grid points. Thanks to the Helmholtz decomposition inherited from the continuous formulation, the duality structure is fully preserved for the odd discretizations. In the latter case, a more complex primal-dual structure is observed due to presence of the trigonometric polynomials associated with the Nyquist frequencies. These theoretical findings are confirmed with numerical examples. To conclude, the main advantage of the FFT-based approach over conventional finite-element schemes is that the primal and the dual problems are treated on the same basis, and this property can be extended beyond the scalar elliptic setting.

Název v anglickém jazyce

Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method

Popis výsledku anglicky

Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast Fourier Transform (FFT) homogenization scheme by Vondřejc et al. (2014), which is based on the conforming Galerkin approximation with trigonometric polynomials. Upper-lower bounds are obtained by adjusting the primal-dual finite element framework developed independently by Dvořák (1993) and Wi?ckowski (1995) to the FFT-based Galerkin setting. We show that the discretization procedure differs for odd and non-odd number of grid points. Thanks to the Helmholtz decomposition inherited from the continuous formulation, the duality structure is fully preserved for the odd discretizations. In the latter case, a more complex primal-dual structure is observed due to presence of the trigonometric polynomials associated with the Nyquist frequencies. These theoretical findings are confirmed with numerical examples. To conclude, the main advantage of the FFT-based approach over conventional finite-element schemes is that the primal and the dual problems are treated on the same basis, and this property can be extended beyond the scalar elliptic setting.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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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

2015

Kód důvěrnosti údajů

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

Název periodika

Computer Methods in Applied Mechanics and Engineering

ISSN

0045-7825

e-ISSN

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Svazek periodika

297

Číslo periodika v rámci svazku

December 01

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

34

Strana od-do

258-291

Kód UT WoS článku

000364061800012

EID výsledku v databázi Scopus

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