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Outline

Title
Abstract
Introduction
Analysis of Simplified Coupled Models
Analysis of the Coupled Problem
Nonlinearities and Newton Method
Newton Method
Newton Method for Geometrical Multiscale
Chapter 3. Nonlinearities and Newton Method
Spectral Analysis
Exact Versus Inexact Newton Method
Scalability and Results
Conclusion
Conclusions
Perspectives and Future Work
References

Fluid-Structure Interaction Problems in Hemodynamics

Profile image of Paolo CrosettoPaolo Crosetto

Abstract

Les objectifs de ce travail sont la description, l'étude et la simulation numérique du problème d'interaction fluide-structure (FSI) appliquéà la dynamique du sang (hémodynamique) dans les artères. L'étude numérique du système cardiovasculaire d'un point de vue hémodynamique est un sujet de recherche très actif, permettant, une fois validé, de prédire le développement de pathologies (par example l'athérosclérose), de mieux comprendre l'influence de facteurs (comme le wall shear stress, WSS) qu'y sont associés et de l'appliquerà la pratique clinique. Ce travail est divisé en trois parties, chacune formée de deux chapitres. Dans la première partie leséquations différentielles qui constituent le problème couplé sont introduites : leś equations de Navier-Stokes dans un domaine déformable pour le fluide (sang), visqueux et incompressible, l'équation de l'élasticité pour la structure (paroi artérielle). En particulier on décrit en détail la représentation deséquations du fluide dans un repère arbitraire Lagrangien-Eulerien (ALE), un choix fréquent en FSI qui sera adopté durant cette thèse. Ensuite on décrit les conditions de couplage : continuité des vitesses et des contraintesà l'interface fluide-structure. On introduitégalement dans la première partie les discrétisations spatiale, enéléments finis, et temporelle du système FSI. Ces discrétisations permettent de représenter le système d'équations dans un espace de dimension finie, ce qui mèneà un problème discret dont la solution est unique. Le choix de la discrétisation temporelle influence deux aspects : la discrétisation en temps des deux problèmes (fluide et structure) et celle des conditions de couplage (continuité des vitesses et des contraintes). Pour les deux aspects le choix peut affecter la stabilité du système discret. Une description concernant en particulier la stabilité du système se trouveà la fin de la première partie de ce travail. Le système d'équations discrétisé n'est pas linéaire, le fait que le domaine du fluide dépende du déplacement de la structure, ainsi que la formulation ALE pour le fluide, introduisent une forte nonlinéarité dont le traitement est un des deux arguments principaux de la deuxième partie de cette thèse. Il est souvent proposé dans la littérature de résoudre la nonlinéarité du système FSI en appliquant la méthode du point fixe. Elle a pour avantages d'être robuste et d'avoir une implémentation assez simple. Par contre, dans sa forme classique, cette méthode présente le désavantage de ne pasêtre efficace dans tous les cas, notamment l'hémodynamique. Un algorithme plus performant, qui est utilisé dans ce cas, est celui de Newton. La difficulté principale de cette méthode vient du calcul de la matrice Jacobienne, qui requiert l'évaluation des dérivées des termes nonlinéaires. En particulier, la nonlinéarité dueà la dépendance du domaine fluide du déplacement de la la structure fait intervenir des dérivées de forme dans le Jacobien du système. Le calcul analytique et l'assemblage de ces dérivées ne sont pas triviaux (ces termes sont souvent négligés ou approximés dans la littérature), et sont décrits dans le troisième chapitre. A la fin de ce chapitre on décrit l'implémentation de cette partie dans un code auxéléments finis, ce qui constitue une contribution originale de ce travail. Dans le quatrième chapitre onétudie des méthodes de résolution du système linéaire (Ja-cobien). Une méthode efficace normalement utilisée pour ce type de systèmes (grands, creux et nonlinéaires) est GMRES. Cette méthode est utilisée normalement dans les cas pratiques avec un préconditionneur. Après avoir résumé des méthodes utilisées en FSI pour résoudre le système linéairisé, un nouveau type de préconditionneurs est proposé. Ces preconditionnerus permettent de traiter séparément les blocs qui correspondent aux problèmes différents (comme fluide et structure). Une analyse proposée pour ce type de préconditionneurs montre que le conditionnement du système global préconditionné ne dépend que du conditionnement des problèmes découplés. Dans la troisième partie, les méthodes décrites dans les chapitres précédents sont appliquéesà des cas cliniques. Plusieurs battements cardiaques consécutifs sont simulé dans le cas de l'aorte thoracique d'un sujet sain, ainsi que dans le cas d'un pontage fémoro-poplité. Le WSS est calculé et différentes méthodes sont comparées (modèle 1D, paroi rigide, FSI), ainsi que différentes discrétisations spatiales et temporelles. Enfin dans le dernier chapitre, les résultats de scalabilité forte et faible sont montrés, sur des maillages différents, avec différentes méthodes. Les simulations de ce chapitre ontété réalisées sur des clustersà haute performance (Cray XT5, Cray XT6, Blue Gene/P).

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