Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation

Legendre, Guillaume; Takahashi, Takéo

doi:10.1051/m2an:2008020

Cited by 8 publications

(10 citation statements)

References 29 publications

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“…Using similar ideas as in [21], we can define the weak formulation of the problem (8)- (12) and (16)- (17). The first equation of (8) and (11)- (12) correspond to the following integral identity:…”

Section: The Weak Formulation and The Main Resultsmentioning

confidence: 99%

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Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid

Nečasová¹,

Takahashi

Tucsnak

2011

Acta Appl Math

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We consider the three-dimensional motion of a self-propelled deformable structure into a viscous incompressible fluid. The deformation of the solid is given whereas its position is unknown. Such a system could model the propulsion of fish-like swimmers. The equations of motion of the fluid are the Navier-Stokes equations and the equations for the structure are deduced from Newton's laws. The corresponding system is a free boundary problem and the main result of the paper is the existence of weak solutions for this problem.

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Section: The Weak Formulation and The Main Resultsmentioning

confidence: 99%

“…4 for more details) for a fluid-rigid body system. However, there are no numerical analysis of the convergence of a numerical scheme based on penalization method as in [20] or [17]. Such an analysis could be performed by combining the methods developed in these two articles with the analysis performed here.…”

mentioning

confidence: 99%

Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid

Nečasová¹,

Takahashi

Tucsnak

2011

Acta Appl Math

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“…Let us give the functional spaces we use for the continuous problem (1)- (4). For the velocity u, we consider the following spaces…”

Section: Remarkmentioning

confidence: 99%

A fictitious domain approach for the Stokes problem based on the extended finite element method

Court

Fournié

Lozinski

2013

Numerical Methods in Fluids

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SUMMARYIn the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by eXtended Finite Element Method [31] and studied for the Poisson problem in [22]. The method allows computations in domains whose boundaries do not match. A mixed finite element method is used for the fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given and several numerical tests are performed to illustrate the capabilities of the method.

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“…The development of efficient numerical methods for the simulation of fluid-structure interaction problems is a great challenge. On one hand some methods are based on meshes which fit to the geometry of the computational domain, like in [11,18,19] for instance. In that case re-meshing is necessary when the geometry is led to evolve, which can be greedy in resources, especially for complex problems.…”

Section: Introductionmentioning

confidence: 99%

A fictitious domain approach for Fluid-Structure Interactions based on the eXtended Finite Element Method.

2014

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In this work we develop a fictitious domain method for the Stokes problem which allows computations in domains whose boundaries do not depend on the mesh. The method is based on the ideas of Xfem and has been first introduced for the Poisson problem. The fluid part is treated by a mixed finite element method, and a Dirichlet condition is imposed by a Lagrange multiplier on an immersed structure localized by a level-set function. A stabilization technique is carried out in order to get the convergence for this multiplier. The latter represents the forces that the fluid applies on the structure. The aim is to perform fluid-structure simulations for which these forces have a central role. We illustrate the capacities of the method by extending it to the incompressible Navier-Stokes equations coupled with a moving rigid solid.Résumé. Dans ce travail nous développons une méthode de type domaines fictifs pour le problème de Stokes permettant des calculs dans des domaines indépendants du maillage. La méthode est basée sur les idées de Xfem et a d'abordété introduite pour le problème de Poisson. La partie fluide est traitée avec une méthodeéléments finis mixte, et une condition de Dirichlet est imposée par un multiplicateur de Lagrange sur une structure immergée localisée par une fonction level-set. Une technique de stabilisation est mise en oeuvre pour obtenir la convergence de ce multiplicateur. Ce dernier représente les forces que le fluide exerce sur la structure. L'objectif est de réaliser des simulations fluide-structure pour lesquelles ces forces ont un rôle central. Nous illustrons les capacités de la méthode en l'étendant auxéquations de Navier-Stokes incompressibles couplées avec un solide rigide en mouvement. * This work is partially supported by the ANR-project CISIFS: 09-BLAN-0213-03, and the foundation STAE in the context of the RTRA platform SMARTWING. It is in particular based on the use of the Getfem++ library [17], and thus on collaborative efforts with Yves Renard.

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Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation

Cited by 8 publications

References 29 publications

Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid

Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid

A fictitious domain approach for the Stokes problem based on the extended finite element method

A fictitious domain approach for Fluid-Structure Interactions based on the eXtended Finite Element Method.

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