International audienceWe address the numerical simulation of fluid-structure systems involving an incompressible viscous fluid. This issue is particularly difficult to face when the fluid added-mass acting on the structure is strong, as it happens in hemodynamics for example. Indeed, several works have shown that, in such situations, implicit coupling seems to be necessary in order to avoid numerical instabilities. Although significant improvements have been achieved during the last years, solving implicit coupling often exhibits a prohibitive computational cost. In this work, we introduce a semi-implicit coupling scheme which remains stable for a reasonable range of the discretization parameters. The first idea consists in treating implicitly the added-mass effect, whereas the other contributions (geometrical non-linearities, viscous and convective effects) are treated explicitly. The second idea, relies on the fact that this kind of explicit-implicit splitting can be naturally performed using a Chorin-Temam projection scheme in the fluid. We prove (conditional) stability of the scheme for a fully discrete formulation. Several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches

This paper aims at introducing a partitioned Newton based method for solving nonlinear coupled systems arising in the numerical approximation of fluid-structure interaction problems. We provide a method which characteristic lies in the use of exact cross jacobians evaluation involving the shape derivative of the fluid state with respect to solid motion perturbations. Numerical tests based on an implementation inside a 3D fluid-structure interaction code show how the exactness of the cross jacobians computation guarantee the overall convergence of the Newton's loop.

In this work we propose a stabilized explicit coupling scheme for fluid-structure interaction problems involving a viscous incompressible fluid. The coupled discrete formulation is based on Nitsche's method with a time penalty term giving L 2 -control on the fluid pressure variations at the interface. The scheme is stable, in the energy norm, irrespectively of the fluid-structure density ratio. Numerical experiments, in two and three dimensions, show that optimal time accuracy can be obtained by performing a few defect-correction iterations.Key-words: Fluid-structure interaction, Nitsche's method, fluid incompressibility, time discretization, explicit coupling, loosely coupled schemes, defectcorrection method

International audienceIn this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207–216] to Oseen's equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pres-sure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L 2-control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results

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Schémas avec correction de déplacement incrémentale pour l'interaction fluide-structure: stabilité et convergence Résumé : Dans cet article nous introduisons une classe de schémas avec correction de déplacement incrémentale pour le couplage explicite d'une structure mince et d'un fluide incompressible. Ces méthodes imposent un traitement spécifique Robin-Neumann explicite du couplage à l'interface. Nous proposons une analyse générale de stabilité et de convergence, qui traite à la fois les variantes incrémentales et non-incrémentale. Leurs propriétés de stabilité sont indépendants de l'effet de masse ajoutée. La précision supérieure des schémas incrémentaux (par rapport à la variante originale non-incrémentale) est mise en évidence par des estimations d'erreur a priori, puis confirmée par des expériences numériques dans un cas test connu.

This paper deals with the numerical simulation of electrocardiograms (ECG). Our aim is to devise a mathematical model, based on partial differential equations, which is able to provide realistic 12-lead ECGs. The main ingredients of this model are classical: the bidomain equations coupled to a phenomenological ionic model in the heart, and a generalized Laplace equation in the torso. The obtention of realistic ECGs relies on other important features--including heart-torso transmission conditions, anisotropy, cell heterogeneity and His bundle modeling--that are discussed in detail. The numerical implementation is based on state-of-the-art numerical methods: domain decomposition techniques and second order semi-implicit time marching schemes, offering a good compromise between accuracy, stability and efficiency. The numerical ECGs obtained with this approach show correct amplitudes, shapes and polarities, in all the 12 standard leads. The relevance of every modeling choice is carefully discussed and the numerical ECG sensitivity to the model parameters investigated.

Abstract:We introduce a class of explicit coupling schemes for the numerical solution of fluid-structure interaction problems involving a viscous incompressible fluid and a general thinwalled structure (e.g., including damping and non-linear behavior). The fundamental ingredient in these methods is the (parameter free) explicit Robin interface condition for the fluid, which enables the fluid-solid splitting through appropriate extrapolations of the solid velocity and fluid stress on the interface. The resulting solution procedures are genuinely partitioned. Stability and error estimates are provided for all the variants (depending on the extrapolations), using energy arguments within a representative linear setting. In particular, we show that one of them yields added-mass free unconditional stability and optimal (first-order) time accuracy. A comprehensive numerical study, involving different examples from the literature, supports the theory.Key-words: fluid-structure interaction, incompressible fluid, non-linear shell, visco-elasticity, time-discretization, explicit coupling scheme, loosely coupled scheme, Robin-Neumann scheme, partitioned procedure, finite element method, error estimate. Schémas Robin-Neumann explicites pour le couplage d'un fluide incompressible avec une structure mince Résumé :Cet article présente une famille de schémas numériques explicites qui permet de résoudre un problème d'interaction fluide-structure faisant intervenir un fluide visqueux incompressible et une structure mince (qui peut être amortie ou non linéaire). L'outil principal de ces méthodes réside dans une condition aux limites de type Robin pour le fluide (sans paramètre à ajuster), qui permet le découplage entre les deux sous-problèmes et fait intervenir une extrapolation de la vitesse du solide et des efforts du fluide. Le schéma qui en résulte est partitionné. L'article fournit des estimations d'énergie et d'erreur a priori qui permettent de montrer la stabilité et la convergence du schéma pour tous les ordres d'extrapolation lorsque le problème est linéaire. Plus précisément, on montre qu'une des variantes est à la fois inconditionnellement stable, insensible à l'effet de masse ajoutée et possède une précision en temps optimale (c'est-à-dire d'ordre un). Ces résultats sont illustrés par de nombreux exemples numériques tirés de la littérature. Mots-clés :interaction fluide-structure, fluide incompressible, coque non-linéaire, viscoélasticité, discrétisation en temps, schéma de couplage explicite, schéma faiblement couplé, schéma Robin-Neumann, algorithme partitionné, méthode des éléments finis, estimation d'erreur.Explicit Robin-Neumann schemes 3