An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure

We develop a loosely coupled fluid-structure interaction finite element solver based on the Lie operator splitting scheme. The scheme is applied to the interaction between an incompressible, viscous, Newtonian fluid, and a multilayered structure, which consists of a thin elastic layer and a thick poroelastic material. The thin layer is modeled using the linearly elastic Koiter membrane model, while the thick poroelastic layer is modeled as a Biot system. We prove a conditional stability of the scheme and derive error estimates. Theoretical results are supported with numerical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1054–1100, 2015

[1]  Weidong Zhao,et al.  Finite Element Approximations for Stokes–darcy Flow with Beavers–joseph Interface Conditions * , 2022 .

[2]  Fabio Nobile,et al.  Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems , 2009 .

[3]  John A. Hudson,et al.  Comprehensive rock engineering : principles, practice, and projects , 1993 .

[4]  Ivan Yotov,et al.  Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach , 2014, 1403.5707.

[5]  Luca Gerardo-Giorda,et al.  Analysis and Optimization of Robin-Robin Partitioned Procedures in Fluid-Structure Interaction Problems , 2010, SIAM J. Numer. Anal..

[6]  Gerhard A Holzapfel,et al.  Constitutive modelling of passive myocardium: a structurally based framework for material characterization , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[8]  R. Schreiber.  Numerical Methods for Partial Differential Equations , 1999 .

[9]  Miguel Angel Fernández,et al.  Displacement-velocity correction schemes for incompressible fluid-structure interaction , 2011 .

[10]  R. Glowinski.  Finite element methods for incompressible viscous flow , 2003 .

[11]  E. Miglio,et al.  Mathematical and numerical models for coupling surface and groundwater flows , 2002 .

[12]  W. Layton,et al.  A decoupling method with different subdomain time steps for the nonstationary stokes–darcy model , 2013 .

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

[14]  Annalisa Quaini,et al.  Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction , 2009, J. Comput. Phys..

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

[16]  G. C. Lee.,et al.  Numerical simulation for the propagation of nonlinear pulsatile waves in arteries. , 1992, Journal of biomechanical engineering.

[17]  Jinchao Xu,et al.  A Two-Grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow , 2007, SIAM J. Numer. Anal..

[18]  Miguel Angel Fernández,et al.  Incremental displacement-correction schemes for the explicit coupling of a thin structure with an incompressible fluid , 2011 .

[19]  W A Wall,et al.  Material model of lung parenchyma based on living precision-cut lung slice testing. , 2011, Journal of the mechanical behavior of biomedical materials.

[20]  A. Quarteroni,et al.  Fluid–structure algorithms based on Steklov–Poincaré operators , 2006 .

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

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

[23]  K Perktold,et al.  Mathematical and numerical models for transfer of low-density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow. , 2005, Journal of biomechanics.

[24]  J D Humphrey,et al.  Mechanics of the arterial wall: review and directions. , 1995, Critical reviews in biomedical engineering.

[25]  N. Koshiba,et al.  Multiphysics simulation of blood flow and LDL transport in a porohyperelastic arterial wall model. , 2007, Journal of biomechanical engineering.

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

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

[28]  C. Peskin,et al.  A computational fluid dynamics of `clap and fling' in the smallest insects , 2005, Journal of Experimental Biology.

[29]  VIVETTE GIRAULT,et al.  DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition , 2009, SIAM J. Numer. Anal..

[30]  Alfio Quarteroni,et al.  Robin-Robin Domain Decomposition Methods for the Stokes-Darcy Coupling , 2007, SIAM J. Numer. Anal..

[31]  E. Ramm,et al.  Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows , 2007 .

[32]  A. Cheng,et al.  Fundamentals of Poroelasticity , 1993 .

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

[34]  R. Vito,et al.  Blood vessel constitutive models-1995-2002. , 2003, Annual review of biomedical engineering.

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

[36]  G. Holzapfel,et al.  Anisotropic mechanical properties of tissue components in human atherosclerotic plaques. , 2004, Journal of biomechanical engineering.

[37]  Annalisa Quaini,et al.  Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement , 2012, J. Comput. Phys..

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

[39]  G. Saidel,et al.  Permeability change of arterial endothelium is an age-dependent function of lesion size in apolipoprotein E-null mice. , 2008, American journal of physiology. Heart and circulatory physiology.

[40]  Miguel Angel Fern.  Incremental displacement-correction schemes for incompressible uid-structure interaction: stability and convergence analysis , 2013 .

[41]  Xiao-Chuan Cai,et al.  Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling , 2010, J. Comput. Phys..

[42]  Xiaohong Zhu,et al.  Decoupled schemes for a non-stationary mixed Stokes-Darcy model , 2009, Math. Comput..

[43]  Ivan Yotov,et al.  Effects of poroelasticity on fluid-structure interaction in arteries: A computational sensitivity study , 2015 .

[44]  Hoang Tran,et al.  Long time stability of four methods for splitting the evolutionary Stokes-Darcy problem into Stokes and Darcy subproblems , 2012, J. Comput. Appl. Math..

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

[46]  Miguel A. Fernández,et al.  An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes , 2014 .

[47]  R. Showalter.  Poro-plastic filtration coupled to Stokes flow , 2005 .

[48]  J-F Gerbeau,et al.  External tissue support and fluid–structure simulation in blood flows , 2012, Biomechanics and modeling in mechanobiology.

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

[50]  Abimael F. D. Loula,et al.  Micromechanical computational modeling of secondary consolidation and hereditary creep in soils , 2001 .

[51]  R. Armentano,et al.  Arterial wall mechanics in conscious dogs. Assessment of viscous, inertial, and elastic moduli to characterize aortic wall behavior. , 1995, Circulation research.

[52]  Fabio Nobile,et al.  Numerical approximation of fluid-structure interaction problems with application to haemodynamics , 2001 .

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

[54]  Roland Glowinski,et al.  Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow , 2009, J. Comput. Phys..

[55]  Mária Lukáčová-Medvid’ová,et al.  Kinematic splitting algorithm for fluid–structure interaction in hemodynamics , 2013 .

[56]  Fabio Nobile,et al.  Time accurate partitioned algorithms for the solution of fluid–structure interaction problems in haemodynamics , 2013 .

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

[58]  Kambiz Vafai,et al.  Effect of the fluid-structure interactions on low-density lipoprotein transport within a multi-layered arterial wall. , 2012, Journal of biomechanics.

[59]  Yiannis Ventikos,et al.  Coupling Poroelasticity and CFD for Cerebrospinal Fluid Hydrodynamics , 2009, IEEE Transactions on Biomedical Engineering.

[60]  Ivan Yotov,et al.  Domain Decomposition for Stokes-Darcy Flows with Curved Interfaces , 2013, ICCS.

[61]  Dalin Tang,et al.  Multi-Physics MRI-Based Two-Layer Fluid-Structure Interaction Anisotropic Models of Human Right and Left Ventricles with Different Patch Materials: Cardiac Function Assessment and Mechanical Stress Analysis. , 2011, Computers & structures.

[62]  oris,et al.  Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible , viscous fluid in a cylinder with deformable walls , 2012 .

[63]  Matteo Lesinigo.  Lumped Mathematical Models for Intracranial Dynamics , 2013 .

[64]  Ivan Yotov,et al.  Coupling Fluid Flow with Porous Media Flow , 2002, SIAM J. Numer. Anal..

[65]  Chung Bang Yun,et al.  Fluid-structure-soil interaction analysis of cylindrical liquid storage tanks subjected to horizontal earthquake loading , 2002 .

[66]  Suncica Canic,et al.  Existence of a solution to a fluid-multi-layered-structure interaction problem , 2013, 1305.5310.

[67]  Suncica Canic,et al.  A partitioned scheme for fluid-composite structure interaction problems , 2015, J. Comput. Phys..

[68]  Ionel Michael Navon,et al.  VARIATM—A FORTRAN program for objective analysis of pseudostress wind fields using large-scale conjugate-gradient minimization , 1991 .

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

[70]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[71]  Anne M. Robertson,et al.  Structurally motivated damage models for arterial walls. Theory and application , 2012 .

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

[73]  Mikel Landajuela,et al.  A fully decoupled scheme for the interaction of a thin-walled structure with an incompressible fluid☆ , 2013 .

[74]  Nikolai D. Botkin,et al.  Dispersion relations for acoustic waves in heterogeneous multi-layered structures contacting with fluids , 2007, J. Frankl. Inst..

[75]  Béatrice Rivière,et al.  Locally Conservative Coupling of Stokes and Darcy Flows , 2005 .

[76]  Cornel Marius Murea,et al.  A fast method for solving fluid–structure interaction problems numerically , 2009 .

[77]  Andro Mikelić,et al.  Convergence of iterative coupling for coupled flow and geomechanics , 2013, Computational Geosciences.

[78]  Y C Fung,et al.  The degree of nonlinearity and anisotropy of blood vessel elasticity. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[79]  R. D. Bauer,et al.  Separate determination of the pulsatile elastic and viscous forces developed in the arterial wall in vivo , 2004, Pflügers Archiv.

[80]  Suncica Canic,et al.  Modeling Viscoelastic Behavior of Arterial Walls and Their Interaction with Pulsatile Blood Flow , 2006, SIAM J. Appl. Math..