Pressure boundary conditions for blood flows

Simulations of blood flows in arteries require numerical solutions of fluidstructure interactions involving Navier-Stokes equations coupled with large displacement visco-elasticity for the vessels.Among the various simplifications which have been proposed, the surface pressure model leads to a hierarchy of simpler models including one that involves only the pressure. The model exhibits fundamental frequencies which can be computed and compared with the pulse. Yet unconditionally stable time discretizations can be constructed by combining implicit time schemes with Galerkin-characteristic discretization of the convection terms in the Navier-Stokes equations. Such problems with prescribed pressure on the walls will be shown to be efficient and accurate as an approximation of the full fluid structure interaction problem.

[1]  Miguel A. Fernández,et al.  Incremental displacement-correction schemes for incompressible fluid-structure interaction , 2012, Numerische Mathematik.

[2]  F P T Baaijens,et al.  A three-dimensional computational analysis of fluid-structure interaction in the aortic valve. , 2003, Journal of biomechanics.

[3]  O. Pironneau,et al.  Conditions aux limites sur la pression pour les équations de Stokes et de Navier-Stokes , 1986 .

[4]  G. Cottet,et al.  EULERIAN FORMULATION AND LEVEL SET MODELS FOR INCOMPRESSIBLE FLUID-STRUCTURE INTERACTION , 2008 .

[5]  P. Tallec,et al.  Fluid structure interaction with large structural displacements , 2001 .

[6]  M. Costabel,et al.  Singularities of Electromagnetic Fields¶in Polyhedral Domains , 2000 .

[7]  Rolf Rannacher.  Incompressible Viscous Flows , 2004 .

[8]  I. Marshall.  Computational simulations and experimental studies of 3D phase‐contrast imaging of fluid flow in carotid bifurcation geometries , 2010, Journal of magnetic resonance imaging : JMRI.

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

[10]  J. P. Benque,et al.  A finite element method for Navier-Stokes equations , 1980 .

[11]  Jacques Rappaz,et al.  Conservation schemes for convection-diffusion equations with Robin boundary conditions , 2013 .

[12]  Zhiyong Si.  Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems , 2011, Numerical Algorithms.

[13]  Mate Kosor,et al.  Mechanical behavior of fully expanded commercially available endovascular coronary stents. , 2011, Texas Heart Institute journal.

[14]  Charles A. Taylor,et al.  Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries , 2006 .

[15]  Oscar Gonzalez,et al.  Exact energy and momentum conserving algorithms for general models in nonlinear elasticity , 2000 .

[16]  Miguel A. Fernández,et al.  ACCELERATION OF A FIXED POINT ALGORITHM FOR FLUID-STRUCTURE INTERACTION USING TRANSPIRATION CONDITIONS , 2003 .

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

[18]  Marc Thiriet.  Control of Cell Fate in the Circulatory and Ventilatory Systems , 2011 .

[19]  Olivier Pironneau,et al.  Finite Element Analysis of Multi-Component Assemblies: CAD-Based Domain Decomposition , 2014 .

[20]  M. Costabel,et al.  Maxwell and Lamé eigenvalues on polyhedra , 1999 .

[21]  D. Holdsworth,et al.  Flow Patterns at the Stenosed Carotid Bifurcation: Effect of Concentric versus Eccentric Stenosis , 2000, Annals of Biomedical Engineering.

[22]  Fang Q. Hu,et al.  Absorbing boundary conditions for nonlinear Euler and Navier-Stokes equations based on the perfectly matched layer technique , 2008, J. Comput. Phys..

[23]  L. Heltai,et al.  A finite element approach to the immersed boundary method , 2003 .

[24]  J. Boyle,et al.  Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches , 2008 .

[25]  J. C. Simo,et al.  On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry , 1996 .

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

[27]  O. Pironneau.  Finite Element Methods for Fluids , 1990 .

[28]  C. Peskin,et al.  A three-dimensional computational method for blood flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid , 1989 .

[29]  John B. Bell,et al.  Staggered Schemes for Fluctuating Hydrodynamics , 2012, Multiscale Model. Simul..

[30]  Masahisa Tabata,et al.  Stability and convergence of a Galerkin‐characteristics finite element scheme of lumped mass type , 2009 .

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

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

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

[34]  Clive A. J. Fletcher,et al.  Incompressible Viscous Flow , 1991 .

[35]  Paolo Crosetto,et al.  Parallel Algorithms for Fluid-Structure Interaction Problems in Haemodynamics , 2011, SIAM J. Sci. Comput..

[36]  Bertrand Maury,et al.  Moving meshes with freefem++ , 2012, J. Num. Math..

[37]  Olivier Pironneau,et al.  A nouveau sur les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression , 1987 .