Numerical modeling of fluid–structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains

The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid–structure interactions and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid–structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid–structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple – but nonsymmetric – curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid–structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often‐used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F‐bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second‐order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non‐rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid–structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  Alfio Quarteroni,et al.  A Comparison of Preconditioners for the Steklov–Poincaré Formulation of the Fluid-Structure Coupling in Hemodynamics , 2015 .

[2]  Andreas Fischle,et al.  A parallel Newton-Krylov-FETI-DP Solver based on FEAP: Large-scale applications and scalability for problems in the mechanics of soft biological tissues in arterial wall structures , 2015 .

[3]  Huidong Yang,et al.  Partitioned solution algorithms for fluid-structure interaction problems with hyperelastic models , 2015, J. Comput. Appl. Math..

[4]  Alfio Quarteroni,et al.  Fluid-structure interaction simulations of cerebral arteries modeled by isotropic and anisotropic constitutive laws , 2015, Computational Mechanics.

[5]  Matthias Mayr,et al.  A Temporal Consistent Monolithic Approach to Fluid-Structure Interaction Enabling Single Field Predictors , 2015, SIAM J. Sci. Comput..

[6]  Jiyuan Tu,et al.  Computational Fluid Structure Interaction , 2015 .

[7]  Alfio Quarteroni,et al.  A Rescaled Localized Radial Basis Function Interpolation on Non-Cartesian and Nonconforming Grids , 2014, SIAM J. Sci. Comput..

[8]  Xiao-Chuan Cai,et al.  A fully implicit domain decomposition based ALE framework for three-dimensional fluid-structure interaction with application in blood flow computation , 2014, J. Comput. Phys..

[9]  Daniel Balzani,et al.  Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis , 2014 .

[10]  A. Quarteroni,et al.  Aspects of Arterial Wall Simulations: Nonlinear Anisotropic Material Models and Fluid Structure Interaction , 2014 .

[11]  A. Klawonn,et al.  Augmented Lagrange methods for quasi‐incompressible materials—Applications to soft biological tissue , 2013, International journal for numerical methods in biomedical engineering.

[12]  Yuri Bazilevs,et al.  Computational Fluid-Structure Interaction: Methods and Applications , 2013 .

[13]  Thomas Richter,et al.  A Fully Eulerian formulation for fluid-structure-interaction problems , 2013, J. Comput. Phys..

[14]  Raimund Erbel,et al.  A Simultaneous Augmented Lagrange Approach for the Simulation of Soft Biological Tissue , 2013, Domain Decomposition Methods in Science and Engineering XX.

[15]  Ulrich Langer,et al.  Domain Decomposition Solvers for Some Fluid‐Structure Interaction Problems , 2012 .

[16]  Raimund Erbel,et al.  Parallel simulation of patient‐specific atherosclerotic arteries for the enhancement of intravascular ultrasound diagnostics , 2012 .

[17]  Stefan Turek,et al.  FEM multigrid techniques for fluid-structure interaction with application to hemodynamics , 2012 .

[18]  Jan Vierendeels,et al.  Multi-level quasi-Newton coupling algorithms for the partitioned simulation of fluid-structure interaction , 2012 .

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

[20]  M. Benzi,et al.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2010) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2267 Modified augmented Lagrangian preconditioners for the incompressible Navier , 2022 .

[21]  A. Quarteroni,et al.  Fluid―structure interaction simulation of aortic blood flow , 2011 .

[22]  W. Wall,et al.  Truly monolithic algebraic multigrid for fluid–structure interaction , 2011 .

[23]  Rolf Rannacher,et al.  An Adaptive Finite Element Method for Fluid-Structure Interaction Problems Based on a Fully Eulerian Formulation , 2011 .

[24]  Wolfgang A. Wall,et al.  An XFEM Based Fixed-Grid Approach for 3D Fluid-Structure Interaction , 2011 .

[25]  Stefan Turek,et al.  Numerical Benchmarking of Fluid-Structure Interaction: A Comparison of Different Discretization and Solution Approaches , 2011 .

[26]  W. Zulehner,et al.  A Newton Based Fluid–Structure Interaction Solver with Algebraic Multigrid Methods on Hybrid Meshes , 2011 .

[27]  Robert L. Taylor,et al.  FEAP - - A Finite Element Analysis Program , 2011 .

[28]  A. Klawonn,et al.  On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies , 2010 .

[29]  Wolfgang A. Wall,et al.  Coupling strategies for biomedical fluid–structure interaction problems , 2010 .

[30]  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..

[31]  A. Klawonn,et al.  Highly scalable parallel domain decomposition methods with an application to biomechanics , 2010 .

[32]  J. Pereira,et al.  BLOCK PRECONDITIONING WITH SCHUR COMPLEMENTS FOR MONOLITHIC FLUID-STRUCTURE INTERACTIONS , 2010 .

[33]  Stefan Turek,et al.  Numerical Simulation of Laminar Incompressible Fluid-Structure Interaction for Elastic Material with Point Constraints , 2010 .

[34]  Wolfgang A. Wall,et al.  A computational strategy for prestressing patient‐specific biomechanical problems under finite deformation , 2010 .

[35]  A. E. Ehret,et al.  A Universal Model for the Elastic, Inelastic and Active Behaviour of Soft Biological Tissues , 2009 .

[36]  Oliver Rheinbach,et al.  Parallel Iterative Substructuring in Structural Mechanics , 2009 .

[37]  K. Bathe,et al.  Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction , 2009 .

[38]  Xiao-Chuan Cai,et al.  Parallel monolithic fluid-structure interaction algorithms with application to blood flow simulation , 2009 .

[39]  A. Quarteroni,et al.  The derivation of the equations for fluids and structure , 2009 .

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

[41]  A. Klawonn,et al.  Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy , 2008, Computer methods in biomechanics and biomedical engineering.

[42]  C. Vuik,et al.  A comparison of preconditioners for incompressible Navier–Stokes solvers , 2008 .

[43]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

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

[45]  Wing Kam Liu,et al.  The immersed/fictitious element method for fluid–structure interaction: Volumetric consistency, compressibility and thin members , 2008 .

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

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

[48]  Daniel Balzani,et al.  Analysis of thin shells using anisotropic polyconvex energy densities , 2008 .

[49]  Carlo Sansour,et al.  On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy , 2008 .

[50]  Jean-Frédéric Gerbeau,et al.  Domain decomposition based Newton methods for fluid-structure interaction problems , 2008 .

[51]  John N. Shadid,et al.  A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations , 2008, J. Comput. Phys..

[52]  A. Klawonn,et al.  Large‐scale simulation of arterial walls: mechanical modeling , 2007 .

[53]  John N. Shadid,et al.  Least Squares Preconditioners for Stabilized Discretizations of the Navier-Stokes Equations , 2007, SIAM J. Sci. Comput..

[54]  Miguel A. Fernández,et al.  Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence , 2007, Numerische Mathematik.

[55]  SIMULATION OF ARTERIAL WALLS: AN ALGEBRAIC INTERFACE TO ITERATIVE SUBSTRUCTURING , 2007 .

[56]  Thomas Dunne,et al.  Adaptive Finite Element Approximation of Fluid-Structure Interaction Based on Eulerian and Arbitrary Lagrangian-Eulerian Variational Formulations , 2007 .

[57]  Maxim A. Olshanskii,et al.  An Augmented Lagrangian-Based Approach to the Oseen Problem , 2006, SIAM J. Sci. Comput..

[58]  D Balzani,et al.  Simulation of discontinuous damage incorporating residual stresses in circumferentially overstretched atherosclerotic arteries. , 2006, Acta biomaterialia.

[59]  G. Holzapfel,et al.  A polyconvex framework for soft biological tissues. Adjustment to experimental data , 2006 .

[60]  Tayfun E. Tezduyar,et al.  Solution techniques for the fully discretized equations in computation of fluid–structure interactions with the space–time formulations , 2006 .

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

[62]  John N. Shadid,et al.  Block Preconditioners Based on Approximate Commutators , 2005, SIAM J. Sci. Comput..

[63]  Stefan Turek,et al.  A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics , 2006 .

[64]  D. Balzani.  Polyconvex anisotropic energies and modeling of damage applied to arterial walls , 2006 .

[65]  Giovanna Guidoboni,et al.  Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics, and Experimental Validation , 2006, Annals of Biomedical Engineering.

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

[67]  P. Neff,et al.  A variational approach for materially stable anisotropic hyperelasticity , 2005 .

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

[69]  D. Dinkler,et al.  A monolithic approach to fluid–structure interaction using space–time finite elements , 2004 .

[70]  Jean-Frédéric Gerbeau,et al.  A Quasi-Newton Algorithm Based on a Reduced Model for Fluid-Structure Interaction Problems in Blood Flows , 2003 .

[71]  Miguel Angel Fernández,et al.  An exact Block–Newton algorithm for solving fluid–structure interaction problems , 2003 .

[72]  P. Neff,et al.  Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions , 2003 .

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

[74]  Gerhard A. Holzapfel,et al.  A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications , 2001 .

[75]  A. Quarteroni,et al.  Factorization methods for the numerical approximation of Navier-Stokes equations , 2000 .

[76]  R. Ogden,et al.  A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models , 2000 .

[77]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[78]  J. C. Simo,et al.  Numerical analysis and simulation of plasticity , 1998 .

[79]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[80]  Rekha Ranjana Rao,et al.  A Newton-Raphson Pseudo-Solid Domain Mapping Technique for Free and Moving Boundary Problems , 1996 .

[81]  J. C. Simo,et al.  On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects , 1987 .

[82]  J. P. Boehler,et al.  Introduction to the Invariant Formulation of Anisotropic Constitutive Equations , 1987 .

[83]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[84]  D. Spalding,et al.  A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows , 1972 .