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Reformulation and influence of discretization in Fourier-series-based FFT methods for heterogeneous materials
[Reformulation et influence de la discrétisation dans les méthodes FFT basées sur les séries de Fourier pour les matériaux hétérogènes]
Abdoul Magid Amadou Sanoko1, 2  ; Simon Essongue1, 2  ; Joseph Paux3  ; Léo Morin1, 2 
1Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400 Talence, France
2Arts et Metiers Institute of Technology, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400 Talence, France
3Université Sorbonne Paris Nord, Laboratoire des Sciences des Procédés et des Matériaux, LSPM, CNRS, UPR 3407, F-93430, Villetaneuse, France
Comptes Rendus. Mécanique, Volume 354 (2026), pp. 633-662

The aim of this work is to revisit the formulation of FFT-based methods for heterogeneous materials in the periodic setting. These numerical methods are based on the iterative resolution of an auxiliary problem involving a reference homogeneous material and a polarization tensor. Assuming a description of the fields by Fourier series, we show the equivalence between three discretization approaches based either on the strong and the weak formulation of the problem. A special emphasis is put on the representation of the local fields described by Fourier series including the possibility of using non-uniform grid. Numerical experiments are performed on a model problem of conductivity with a checkerboard microstructure for which an analytical solution allows to assess the effect of Fourier modes together with the grid discretization. The occurrence of oscillations is finally addressed by studying (generalized) discrete Green operators (still in the context of Fourier series) and interface spreading approaches based on smoothing techniques.

L’objectif de ce travail est de revisiter la formulation des méthodes basées sur la FFT pour les matériaux hétérogènes dans un cadre périodique. Ces méthodes numériques reposent sur la résolution itérative d’un problème auxiliaire impliquant un matériau homogène de référence et un tenseur de polarisation. En supposant une description des champs par des séries de Fourier, nous montrons l’équivalence entre trois approches de discrétisation fondées soit sur la formulation forte, soit sur la formulation faible du problème. Une attention particulière est portée à la représentation des champs locaux décrits par des séries de Fourier, incluant la possibilité d’utiliser un maillage non uniforme. Des expériences numériques sont réalisées sur un problème modèle de conductivité avec une microstructure en damier, pour lequel une solution analytique permet d’évaluer l’effet des modes de Fourier conjointement à la discrétisation du maillage. L’apparition d’oscillations est enfin étudiée en s’appuyant sur des opérateurs de Green discrets (généralisés) toujours dans le cadre des séries de Fourier, ainsi que sur des approches d’étalement d’interfaces basées sur des techniques de lissage.

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Révisé le :
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DOI : 10.5802/crmeca.369
Keywords: FFT-based solvers, Strong and weak formulations, Fourier series, Green operators
Mots-clés : Solveurs basés sur la FFT, formulations forte et faible, séries de Fourier, opérateurs de Green

Abdoul Magid Amadou Sanoko  1 , 2   ; Simon Essongue  1 , 2   ; Joseph Paux  3   ; Léo Morin  1 , 2

1 Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400 Talence, France
2 Arts et Metiers Institute of Technology, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400 Talence, France
3 Université Sorbonne Paris Nord, Laboratoire des Sciences des Procédés et des Matériaux, LSPM, CNRS, UPR 3407, F-93430, Villetaneuse, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Abdoul Magid Amadou Sanoko; Simon Essongue; Joseph Paux; Léo Morin. Reformulation and influence of discretization in Fourier-series-based FFT methods for heterogeneous materials. Comptes Rendus. Mécanique, Volume 354 (2026), pp. 633-662. doi: 10.5802/crmeca.369
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     journal = {Comptes Rendus. M\'ecanique},
     pages = {633--662},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
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