Robin Based Semi-Implicit Coupling in Fluid-Structure Interaction: Stability Analysis and Numerics

In this paper, we propose a semi-implicit coupling scheme for the numerical simulation of fluid-structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespective of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective nonlinearities through the use of the Chorin-Temam projection scheme within the fluid. Stability comes from the implicit pressure-solid coupling and a specific Robin treatment of the explicit viscous-solid coupling, derived from Nitsche's method.

[1]  Alexandre J. Chorin,et al.  On the Convergence of Discrete Approximations to the Navier-Stokes Equations , 1969 .

[2]  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.

[3]  Jeanine Weekes Schroer,et al.  The Finite String Newsletter Abstracts of Current Literature Glisp User's Manual , 2022 .

[4]  Yvon Maday,et al.  Nonconforming Grids for the Simulation of Fluid-Structure Interaction , 1998 .

[5]  S. R. Parks,et al.  Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms , 2006, Journal of Fluid Mechanics.

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

[7]  Matteo Astorino,et al.  An added-mass free semi-implicit coupling scheme for fluid–structure interaction , 2009 .

[8]  Ekkehard Ramm,et al.  Accelerated iterative substructuring schemes for instationary fluid-structure interaction , 2001 .

[9]  R. Temam.  Une méthode d'approximation de la solution des équations de Navier-Stokes , 1968 .

[10]  A. Quarteroni,et al.  On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels , 2001 .

[11]  Erik Burman,et al.  Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility , 2009 .

[12]  Miguel Angel Fernández,et al.  A Newton method using exact jacobians for solving fluid-structure coupling , 2005 .

[13]  Roland Glowinski,et al.  A kinematically coupled time-splitting scheme for fluid-structure interaction in blood flow , 2009, Appl. Math. Lett..

[14]  Matteo Astorino,et al.  Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems , 2010, Numerische Mathematik.

[15]  Fabio Nobile,et al.  Fluid-structure partitioned procedures based on Robin transmission conditions , 2008, J. Comput. Phys..

[16]  Lutz Tobiska,et al.  Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations , 1996 .

[17]  Cornel Marius Murea,et al.  A stable time advancing scheme for solving fluid-structure interaction problem at small structural displacements , 2008 .

[18]  F. NOBILE,et al.  An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions , 2008, SIAM J. Sci. Comput..

[19]  Jean-Luc Guermond,et al.  International Journal for Numerical Methods in Fluids on Stability and Convergence of Projection Methods Based on Pressure Poisson Equation , 2022 .

[20]  Erik Burman,et al.  Stabilized explicit coupling for fluid-structure interaction using Nitsche s method , 2007 .

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

[22]  Alfio Quarteroni,et al.  Cardiovascular mathematics : modeling and simulation of the circulatory system , 2009 .

[23]  Annalisa Quaini,et al.  Splitting Methods Based on Algebraic Factorization for Fluid-Structure Interaction , 2008, SIAM J. Sci. Comput..

[24]  P. Tallec,et al.  Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity , 1998 .

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

[26]  Patrick Le Tallec,et al.  Effects of the flexibility of the arterial wall on the wall shear stresses and wall tension in Abdominal Aortic Aneurysms. , 2005 .

[27]  P. Hansbo,et al.  Nitsche's method combined with space–time finite elements for ALE fluid–structure interaction problems☆ , 2004 .

[28]  Miguel A. Fernández,et al.  A projection algorithm for fluid–structure interaction problems with strong added-mass effect , 2006 .

[29]  Fabio Nobile,et al.  Added-mass effect in the design of partitioned algorithms for fluid-structure problems , 2005 .

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