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

Abstract

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.

A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems

“…The geometry of the model domain is described independently by the means of a separate geometry model, e.g., a CAD or level-set based description, or even another, independently generated tessellation of the geometry boundary. Combined with suitable techniques to properly impose boundary or interface conditions, for instance by means of Lagrange multipliers or Nitsche-type methods [16][17][18][19], complex geometries can be simply embedded into an easy-to-generate background. As the embedded geometry can cut arbitrarily through the background mesh, a main challenge is to devise unfitted finite element methods that are geometrical robustness in the sense that standard a priori error and conditioning number estimates also hold in the unfitted case with constants independent of the particular cut configuration.…”

Numerical Simulation of Dynamic Fluid-Structure Interaction with Elastic Structure–Rigid Obstacle Contact

“…The densities, respectively, the viscosities of fluid and visco-elastic materials were not the same. For a rigid thick body immersed in an incompressible fluid, the convergence of a penalization method was presented in [7] and the extended finite element method (XFEM) was used in [8]. Nitsche-XFEM was used in [9] for a thin elastic structure immersed in an incompressible fluid.…”

Three-Dimensional Simulation of Fluid–Structure Interaction Problems Using Monolithic Semi-Implicit Algorithm

“…In some methods such as the Arbitrary Lagrangian Eulerian (ALE) framework, the fluid equations are written over a moving mesh which follows the structure displacement (see [18,19]). Other methods use a fixed mesh for fluid domain: immersed boundary method [20], distributed Lagrange multiplier [21,22], penalization [23,24], extended finite element method (XFEM) [25,26], and Nitsche-XFEM [27]. Distributed Lagrange multiplier strategy with remeshing is used in [28] and a monolithic fictitious domain without Lagrange multiplier is employed in [29,30].…”