Projection Schemes for Fluid Flows through a Porous Interface

This paper presents a numerical method to simulate an incompressible fluid through an immersed porous interface. The interface is modeled by a surface measure term in the Navier-Stokes equations and it is characterized by a resistance parameter. This approach can be used, for example, to model valves or to simulate blood flood through an immersed stent. Starting from a monolithic formulation proposed recently, a fractional step algorithm is derived. The difficult point is that this formulation is singular when the resistance vanishes, which can be a serious issue in some applications. We show that an appropriate Nitsche treatment of the interface condition fixes this problem and ensures uniform energy stability in time for any nonnegative value of the resistance. The theoretical stability and convergence results are illustrated with numerical experiments.

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

[2]  Matteo Astorino,et al.  Fluid-Structure Interaction in the Cardiovascular System. Numerical Analysis and Simulation , 2010 .

[3]  Frans N. van de Vosse,et al.  An approximate projec-tion scheme for incompressible ow using spectral elements , 1996 .

[4]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) , 1969 .

[5]  Miguel A. Fernández,et al.  Numerical simulation of blood flows through a porous interface , 2008 .

[6]  Miguel A. Fernández,et al.  A projection semi‐implicit scheme for the coupling of an elastic structure with an incompressible fluid , 2007 .

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

[8]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[9]  Miguel A. Fernández,et al.  Robin Based Semi-Implicit Coupling in Fluid-Structure Interaction: Stability Analysis and Numerics , 2009, SIAM J. Sci. Comput..

[10]  Grégoire Allaire,et al.  Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes , 1991 .

[11]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[12]  Rolf Stenberg,et al.  Nitsche's method for general boundary conditions , 2009, Math. Comput..

[13]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[14]  Santiago Badia,et al.  Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition , 2007, Numerische Mathematik.

[15]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[16]  Peter Hansbo,et al.  Nitsche's method for interface problems in computa‐tional mechanics , 2005 .

[17]  P. Hansbo,et al.  A finite element method for domain decomposition with non-matching grids , 2003 .

[18]  K. Goda,et al.  A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows , 1979 .

[19]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[20]  A. Quarteroni,et al.  A SEMI-IMPLICIT APPROACH FOR FLUID-STRUCTURE INTERACTION BASED ON AN ALGEBRAIC FRACTIONAL STEP METHOD , 2007 .