Fourier-based computational micromechanics on boundary-conforming transformed grids

François Bignonnet; Martin Ladecký; Ivana Pultarová; Jan Zeman

doi:10.5802/crmeca.354

Article de recherche

Fourier-based computational micromechanics on boundary-conforming transformed grids
[Homogénéisation par FFT sur grilles transformées conformes aux interfaces]

François Bignonnet ¹ ; Martin Ladecký ² ; Ivana Pultarová ³ ; Jan Zeman ³

¹Nantes Université, École Centrale Nantes, CNRS, GeM, UMR 6183, F-44600 Saint-Nazaire, France
²Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 103, 79110 Freiburg, Germany
³Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic

Comptes Rendus. Mécanique, Volume 354 (2026), pp. 227-256

Abstract (VO)
Résumé

Computational homogenization accelerated by Green function preconditioning with Fast Fourier transforms (FFT) is classically performed on a uniform grid, which hinders the discretization accuracy. In this work, we consider a more accurate geometry representation obtained by transforming the uniform computational grid into a boundary-conforming one. The mechanical problem is discretized using the finite element method (FEM) with isoparametric transformation of elements. Boundary adaptation can require large localized geometrical transformations of the grid, which is naturally accounted for in the FEM discretization. Rigorous bounds on the spectrum of eigenvalues of the resulting discrete system with Green preconditioner are provided. For grid transformations with projection of the nearest nodes to boundary, the modified eigenvalues correspond to eigenvectors localized at the material phases boundaries, so that the effective spectrum remains favorable for the preconditioned conjugate gradient solver. Numerical investigations confirm that the accuracy of the homogenized properties and the local fields obtained on boundary-conforming grids are greatly improved over uniform grid ones, at the expense of a moderate increase in computational cost.

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

Les méthodes d’homogénéisation accélérées par un préconditionnement par l’opérateur de Green avec transformée de Fourier rapide (FFT) sont usuellement mises en œuvre sur une grille uniforme, ce qui limite la qualité de la discrétisation. Nous considérons ici une représentation plus précise de la géométrie, obtenue en transformant la grille uniforme en une grille conforme aux interfaces. Le problème mécanique est discrétisé par la méthode des éléments finis (FEM) avec une transformation isoparamétrique des éléments. L’adaptation aux interfaces peut nécessiter localement une importante transformation géométrique de la grille, ce qui est naturellement pris en compte par la discrétisation par éléments finis. Des bornes rigoureuses sont fournies sur le spectre des valeurs propres du système discret préconditionné par l’opérateur de Green. Pour des transformations de grille où seuls les nœuds les plus près des interfaces y sont projetés, les valeurs propres modifiées correspondent à des vecteurs propres localisés aux interfaces entre constituants, de sorte que le spectre effectif reste favorable pour le solveur par gradient conjugué préconditionné. Des investigations numériques confirment que la précision des propriétés homogénéisées et des champs locaux obtenus sur une grille conforme aux interfaces est grandement améliorée par rapport à celle sur une grille uniforme, pour une augmentation modérée du temps de calcul.

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Reçu le : 2025-10-24
Révisé le : 2026-02-02
Accepté le : 2026-03-02
Publié le : 2026-03-31

DOI : 10.5802/crmeca.354

Keywords: Computational homogenization, fast Fourier transform, Green function preconditioning, finite element discretization, grid adaptation
Mots-clés : Homogénéisation numérique, transformée de Fourier rapide, préconditionnement par opérateur de Green, discrétisation par éléments finis, adaptation de grille

Affiliations des auteurs :

François Bignonnet ¹ ; Martin Ladecký ² ; Ivana Pultarová ³ ; Jan Zeman ³

¹ Nantes Université, École Centrale Nantes, CNRS, GeM, UMR 6183, F-44600 Saint-Nazaire, France
² Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 103, 79110 Freiburg, Germany
³ Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fran\c{c}ois Bignonnet and Martin Ladeck\'y and Ivana Pultarov\'a and Jan Zeman},
     title = {Fourier-based computational micromechanics on boundary-conforming transformed grids},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {227--256},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {354},
     doi = {10.5802/crmeca.354},
     language = {en},
}

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François Bignonnet; Martin Ladecký; Ivana Pultarová; Jan Zeman. Fourier-based computational micromechanics on boundary-conforming transformed grids. Comptes Rendus. Mécanique, Volume 354 (2026), pp. 227-256. doi: 10.5802/crmeca.354

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