A Galerkin view of FFT-based homogenization methods

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21110%2F16%3A00306551" target="_blank" >RIV/68407700:21110/16:00306551 - isvavai.cz</a>

Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

A Galerkin view of FFT-based homogenization methods

Popis výsledku v původním jazyce

In the field of computational micromechanics of materials, the Fourier-based homogenization solvers were introduced by Moulinec and Suquet in their seminar work. Since then, they have established themselves as a competitive alternative to finite elements in terms of accuracy, efficiency, versatility, and simplicity of implementation. In its basic version, the method works as a fixed-point iterative solution to a periodic Lippman-Schwinger integral equation, whose kernel can be efficiently handled by the Fast Fourier Transform (FFT). In our recent work, motivated by a theoretical interest, we interpreted FFT-based methods in a Galerkin framework that involves the four standard steps, namely (i) introducing a weak form of the governing equations, (ii) projecting the weak form to an approximation space, (iii) applying a numerical quadrature, and (iv) solving the ensuing system of linear equations by a suitable iterative solver. Specifically, the basic Moulinec-Suquet scheme is obtained when (i) the weak form involves the gradients gradients of the field variables, (ii) the approximation space is spanned by trigonometric polynomials, (iii) the trapezoidal rule is employed for numerical integration, and step (iv) involves the Richardson iteration. The purpose of this talk is twofold: to summarize these developments and to explain how they can be used to develop more efficient FFT-based solvers, considering scalar elliptic problems for simplicity. Specifically, I will focus on a-posteriori error estimation based on duality arguments, related to the steps (i)--(iii) above, and on the comparison of iterative solvers i.e. step (iv).

Název v anglickém jazyce

A Galerkin view of FFT-based homogenization methods

Popis výsledku anglicky

In the field of computational micromechanics of materials, the Fourier-based homogenization solvers were introduced by Moulinec and Suquet in their seminar work. Since then, they have established themselves as a competitive alternative to finite elements in terms of accuracy, efficiency, versatility, and simplicity of implementation. In its basic version, the method works as a fixed-point iterative solution to a periodic Lippman-Schwinger integral equation, whose kernel can be efficiently handled by the Fast Fourier Transform (FFT). In our recent work, motivated by a theoretical interest, we interpreted FFT-based methods in a Galerkin framework that involves the four standard steps, namely (i) introducing a weak form of the governing equations, (ii) projecting the weak form to an approximation space, (iii) applying a numerical quadrature, and (iv) solving the ensuing system of linear equations by a suitable iterative solver. Specifically, the basic Moulinec-Suquet scheme is obtained when (i) the weak form involves the gradients gradients of the field variables, (ii) the approximation space is spanned by trigonometric polynomials, (iii) the trapezoidal rule is employed for numerical integration, and step (iv) involves the Richardson iteration. The purpose of this talk is twofold: to summarize these developments and to explain how they can be used to develop more efficient FFT-based solvers, considering scalar elliptic problems for simplicity. Specifically, I will focus on a-posteriori error estimation based on duality arguments, related to the steps (i)--(iii) above, and on the comparison of iterative solvers i.e. step (iv).

Druh

O - Ostatní výsledky

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

<a href="/cs/project/GA14-00420S" target="_blank" >GA14-00420S: Kvazikontuální metody pro diskrétní disipativní soustavy</a><br>

Návaznosti

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

Rok uplatnění

2016

Kód důvěrnosti údajů

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