[Un solveur FFT précis et robuste pour la diffusion transitoire dans les matériaux hétérogènes]
Cette note propose un solveur FFT efficace, précis et robuste pour la diffusion transitoire dans des matériaux hétérogènes avec des conditions aux limites (CL) dites « réalistes » . Cette approche exploite deux avancées récentes : le traitement des conditions aux limites non périodiques et l’utilisation d’un nouveau schéma de différences finies limitant les oscillations parasites. Il s’agit d’une étape essentielle vers le couplage entre la mécanique et d’autres phénomènes physiques (comme la diffusion d’espèces) à l’aide de solveurs FFT. Les Transformées Trigonométriques Discrètes sont utilisées pour implémenter des conditions aux limites non périodiques, tandis qu’un schéma de différences finies (DF) récemment proposé par Finel est comparé avantageusement au schéma DF classique hexaédrique.
Le terme « précis » se réfère à deux aspects : (1) la précision locale, grâce à un schéma de DF de petite taille permettant de capturer les fluctuations autour des hétérogénéités du matériau, et (2) la précision en termes de fidélité, avec l’absence d’oscillations spatiales parasites (au moins dans les cas étudiés avec des inclusions bien séparées). La « robustesse » fait référence à la stabilité du solveur, notamment grâce une méthode d’intégration temporelle implicite. Bien que la description se concentre sur la diffusion thermique, la méthode s’applique à tout processus de diffusion similaire (régie par une équation parabolique du même type).
En complément, le schéma DF proposé par Finel est présenté de manière plus générale (pour mélanger différents schémas de différences finies) et simplifiée (sans introduction de sous-grilles FCC), élargissant également son usage à tout type de parité de grille (et pas seulement aux grilles paires).
The purpose of the present letter is to propose an efficient, accurate and robust FFT-based solver for transient diffusion in heterogeneous materials with “realistic” BC, taking advantage of two recent advances in terms of boundary conditions and finite difference schemes to overcome their actual limitations (periodic BC and spurious oscillations). It is an essential step towards couplings between mechanics and other physics (such as the diffusion of species) through FFT-based solvers. Discrete Trigonometric Transform are used to implement non-periodic boundary conditions, and a finite difference (FD) scheme recently proposed by Finel is advantageously compared to the common hexahedral FD scheme. “Accurate” refers to two properties: accurate in term of locality with a small size of Finite Difference pencil to capture fluctuations around material heterogeneities, and accurate in term of precision with the absence of spurious spatial oscillations (at least in the reported cases with well-separated inclusions). The “robustness” is here associated to the stability of the solver, especially associated to the implicit time integration method. The description of the method focuses on thermal diffusion but applies to any other similar diffusion process (with the same type of parabolic equation). As a by-product, the FD scheme proposed by Finel is introduced in a more general (for mixing finite different schemes) and simple way (no introduction of FCC subgrids), extending its usage to any type of grid parity (not only even grids).
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Mots-clés : FFT, Matériaux hétérogènes, Diffusion transitoire, Différences finies, Tétraèdres
Lionel Gélébart 1
CC-BY 4.0
@article{CRMECA_2025__353_G1_113_0,
author = {Lionel G\'el\'ebart},
title = {An accurate and robust {FFT-based} solver for transient diffusion in heterogeneous materials},
journal = {Comptes Rendus. M\'ecanique},
pages = {113--125},
publisher = {Acad\'emie des sciences, Paris},
volume = {353},
year = {2025},
doi = {10.5802/crmeca.281},
language = {en},
}
Lionel Gélébart. An accurate and robust FFT-based solver for transient diffusion in heterogeneous materials. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 113-125. doi : 10.5802/crmeca.281. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.281/
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