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

SUMMARY In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM 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. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[2]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[3]  Peter Hansbo,et al.  A hierarchical NXFEM for fictitious domain simulations , 2011 .

[4]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[5]  Wolfgang A. Wall,et al.  An XFEM‐based embedding mesh technique for incompressible viscous flows , 2011 .

[6]  Bertrand Maury.  Numerical Analysis of a Finite Element/Volume Penalty Method , 2009, SIAM J. Numer. Anal..

[7]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[8]  Michel Salaün,et al.  High‐order extended finite element method for cracked domains , 2005 .

[9]  G. Golub,et al.  Structured inverse eigenvalue problems , 2002, Acta Numerica.

[10]  H. Meijer,et al.  An extended finite element method for the simulation of particulate viscoelastic flows , 2010 .

[11]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[12]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[13]  G. Legendre,et al.  Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation , 2008 .

[14]  André Massing,et al.  A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem , 2012, J. Sci. Comput..

[15]  Isabelle Ramière,et al.  Convergence analysis of the Q1‐finite element method for elliptic problems with non‐boundary‐fitted meshes , 2008 .

[16]  D. Chopp,et al.  Modelling crack growth by level sets , 2013 .

[17]  T. Belytschko,et al.  The extended/generalized finite element method: An overview of the method and its applications , 2010 .

[18]  Jean-François Remacle,et al.  Imposing Dirichlet boundary conditions in the eXtended Finite Element Method , 2011 .

[19]  Patrick Hild,et al.  A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics , 2010, Numerische Mathematik.

[20]  Patrick Laborde,et al.  Crack tip enrichment in the XFEM method using a cut-off function , 2008 .

[21]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[22]  Marius Tucsnak,et al.  Convergence of the Lagrange-Galerkin Method for the Equations Modelling the Motion of a Fluid-Rigid System , 2005, SIAM J. Numer. Anal..

[23]  Ted Belytschko,et al.  Modelling crack growth by level sets in the extended finite element method , 2001 .

[24]  Jaroslav Haslinger,et al.  A New Fictitious Domain Approach Inspired by the Extended Finite Element Method , 2009, SIAM J. Numer. Anal..

[25]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

[26]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[27]  Anders Klarbring,et al.  Fictitious domain/mixed finite element approach for a class of optimal shape design problems , 1995 .

[28]  Ted Belytschko,et al.  An extended finite element method with higher-order elements for curved cracks , 2003 .

[29]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .

[30]  Marius Tucsnak,et al.  An Initial and Boundary Value Problem Modeling of Fish-like Swimming , 2008 .

[31]  Ted Belytschko,et al.  The extended finite element method for rigid particles in Stokes flow , 2001 .

[32]  M. Gunzburger,et al.  Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses , 1992 .

[33]  Juhani Pitkäranta,et al.  Local stability conditions for the Babuška method of Lagrange multipliers , 1980 .

[34]  R. Glowinski,et al.  Error analysis of a fictitious domain method applied to a Dirichlet problem , 1995 .

[35]  Nicolas Moës,et al.  A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method , 2009 .

[36]  Nicolas Moës,et al.  Imposing Dirichlet boundary conditions in the extended finite element method , 2006 .

[37]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[38]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[39]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[40]  A. Quarteroni.  Numerical Models for Differential Problems , 2009 .

[41]  T. Belytschko,et al.  Non‐planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model , 2002 .

[42]  P. Angot,et al.  A Fictitious domain approach with spread interface for elliptic problems with general boundary conditions , 2007 .

[43]  G. Hou,et al.  Numerical Methods for Fluid-Structure Interaction — A Review , 2012 .

[44]  Helio J. C. Barbosa,et al.  Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms , 1992 .

[45]  Loredana Smaranda,et al.  Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time , 2009 .

[46]  Erik Burman,et al.  Stabilized finite element methods for the generalized Oseen problem , 2007 .

[47]  T. Belytschko,et al.  Extended finite element method for three-dimensional crack modelling , 2000 .

[48]  Helio J. C. Barbosa,et al.  The finite element method with Lagrange multiplier on the boundary: circumventing the Babuscka-Brezzi condition , 1991 .

[49]  Patrick Laborde,et al.  Crack tip enrichment in the XFEM using a cutoff function , 2008 .

[50]  R. Glowinski,et al.  A distributed Lagrange multiplier/fictitious domain method for particulate flows , 1999 .