Free access article

ESAIM: Proc., September 2005, Vol. 14, pp. 115-123
DOI: 10.1051/proc:2005010

A penalty method for the simulation of fluid - rigid body interaction

João Janela¹, Aline Lefebvre² and Bertrand Maury<sup>2

1</sup>  CEMAT, Instituto Superior Técnico and Departamento de Matemática, ISEG, Lisbon, Portugal;
²  Laboratoire de Mathématiques, Université Paris Sud, Orsay, France;

jjanela@iseg.utl.pt
Aline.Lefebvre@math.u-psud.fr
Bertrand.Maury@math.u-psud.fr

Abstract
We present here a method to simulate the motion of a rigid body in a fluid. The method is based on a variational formulation on the whole fluid/solid domain, with some constraints on the unknown and the test functions. These constraints are relaxed by introducing a penalty term, which leads to a minimization problem over unconstrained functional spaces. This makes the method straightforward to implement from any finite element Stokes/Navier-Stokes solver. It is shown that, as the penalty parameter goes to infinity, we recover the coupled fluid-solid equations. We apply this approach to a simplified 2D model of the aortic valve.

Résumé

Nous présentons ici une méthode de simulation du mouvement d'un corps rigide dans un fluide. Cette méthode est basée sur une formulation variationnelle sur tout le domaine fluide/solide, avec des contraintes sur l'inconnue et sur les fonctions test. On relaxe ces contraintes en introduisant un terme de pénalisation, qui conduit à un problème de minimisation sur des espaces fonctionnels non contraints. Ainsi, tout solveur éléments finis pour Stokes/Navier Stokes permet de programmer aisément cette méthode. On montre que, quand le paramètre de pénalisation tend vers l'infini, on retrouve le système d'équations couplées fluide/solide. Cette approche est appliquée à un modèle 2D simplifié de la valve aortique.

Mathematics Subject Classification. 35Q30, 65M25, 65M60

Key words: Navier-Stokes equation, rigid motion, finite element, penalty method, valve


© EDP Sciences, ESAIM 2005

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