[Une méthode d'éléments en base réduite]
Les méthodes de base réduite sont particulièrement attractives par la diminution du nombre de degrés de liberté qu'elles entraı̂nent pour l'approximation d'un système d'équations aux dérivées partielles. L'idée principale sur laquelle repose cette approche est la définition de fonctions de base ad'hoc contenant une grande part d'information sur le système considéré. Dans cette Note nous proposons et analysons une méthode de base réduite pour la simulation d'écoulements dans un système hiérarchique comme ce peut être le cas dans un réseau de distribution de fluides ou le système sanguin. Nous proposons de décomposer la géométrie du domaine en un assemblage de formes génériques (des conduites et des bifurcations par exemple) et d'associer à ces sous parties des fonctions de bases réduites obtenues comme des instantanés géométriques représentatifs. Le système global est alors construit en recollant ces fonctions locales par des multiplicateurs de Lagrange.
Reduced basis methods are particularly attractive to use in order to diminish the number of degrees-of-freedom associated with the approximation to a set of partial differential equations. The main idea is to construct ad hoc basis functions with a large information content. In this Note, we propose to develop and analyze reduced basis methods for simulating hierarchical flow systems, which is of relevance for studying flows in a network of pipes, an example being a set of arteries or veins. We propose to decompose the geometry into generic parts (e.g., pipes and bifurcations), and to construct a reduced basis for these generic parts by considering representative geometric snapshots. The global system is constructed by gluing the individual basis solutions together via Lagrange multipliers.
Révisé le :
Publié le :
Yvon Maday 1 ; Einar M. Rønquist 2
@article{CRMATH_2002__335_2_195_0,
author = {Yvon Maday and Einar M. R{\o}nquist},
title = {A reduced-basis element method},
journal = {Comptes Rendus. Math\'ematique},
pages = {195--200},
publisher = {Elsevier},
volume = {335},
number = {2},
year = {2002},
doi = {10.1016/S1631-073X(02)02427-5},
language = {en},
}
Yvon Maday; Einar M. Rønquist. A reduced-basis element method. Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 195-200. doi : 10.1016/S1631-073X(02)02427-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02427-5/
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