Classical and all‐floating FETI methods for the simulation of arterial tissues

High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example we choose the artery which is - as most other biological tissues - characterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison to classical FETI, the all-floating approach has not only advantages concerning the implementation but in many cases also concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations we will also discuss some limitations concerning the dependence on material parameters.

[1]  J. Mandel,et al.  Convergence of a substructuring method with Lagrange multipliers , 1994 .

[2]  M. Fortin,et al.  Finite Elements for the Stokes Problem , 2008 .

[3]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[4]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[5]  Olaf Steinbach,et al.  Boundary Element Tearing and Interconnecting Methods , 2003, Computing.

[6]  Rudolf A. Römer,et al.  On large‐scale diagonalization techniques for the Anderson model of localization , 2005, SIAM J. Sci. Comput..

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

[8]  Gerhard A Holzapfel,et al.  Comparison of a multi-layer structural model for arterial walls with a fung-type model, and issues of material stability. , 2004, Journal of biomechanical engineering.

[9]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[10]  Olaf Steinbach,et al.  Classical and All-floating FETI Methods with Applications to Biomechanical Models , 2015 .

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

[12]  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.

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

[14]  J. Remacle,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[15]  Petra Pustejovská,et al.  Existence Analysis for a Model Describing Flow of an Incompressible Chemically Reacting Non-Newtonian Fluid , 2014, SIAM J. Math. Anal..

[16]  H. W. Weizsäcker,et al.  Biomechanical behavior of the arterial wall and its numerical characterization , 1998, Comput. Biol. Medicine.

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

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

[19]  R. Ogden,et al.  Biomechanics of Soft Tissue in Cardiovascular Systems , 2003 .

[20]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[21]  Clemens Pechstein,et al.  Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems , 2012, Lecture Notes in Computational Science and Engineering.

[22]  Anthony T. Patera,et al.  Nonconforming mortar element methods: Application to spectral discretizations , 1988 .

[23]  Olof B. Widlund,et al.  A Domain Decomposition Method with Lagrange Multipliers and Inexact Solvers for Linear Elasticity , 2000, SIAM J. Sci. Comput..

[24]  Miroslav Bulíček,et al.  On Existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index , 2013 .

[25]  Y. Maday,et al.  Optimal convergence properties of the FETI domain decomposition method , 2007 .

[26]  O. Widlund,et al.  FETI and Neumann--Neumann Iterative Substructuring Methods: Connections and New Results , 1999 .

[27]  Olaf Steinbach,et al.  FETI Methods for the Simulation of Biological Tissues , 2013, Domain Decomposition Methods in Science and Engineering XX.

[28]  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.

[29]  Olaf Schenk,et al.  Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization , 2007, Comput. Optim. Appl..

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

[31]  G. Plank,et al.  A Novel Rule-Based Algorithm for Assigning Myocardial Fiber Orientation to Computational Heart Models , 2012, Annals of Biomedical Engineering.

[32]  R. Ogden,et al.  Constitutive modelling of arteries , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[33]  Olaf Steinbach,et al.  The all-floating boundary element tearing and interconnecting method , 2009, J. Num. Math..

[34]  Olaf Steinbach,et al.  Stability Estimates for Hybrid Coupled Domain Decomposition Methods , 2003 .

[35]  J D Humphrey,et al.  Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms. , 2012, Journal of biomechanics.

[36]  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.

[37]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[38]  P. Wriggers.  Nonlinear Finite Element Methods , 2008 .

[39]  R. Rivlin,et al.  Stress-Deformation Relations for Isotropic Materials , 1955 .

[40]  Annie Raoult,et al.  Symmetry groups in nonlinear elasticity: an exercise in vintage mathematics , 2008 .

[41]  Gerhard A. Holzapfel,et al.  Collagen in Arterial Walls: Biomechanical Aspects , 2008 .

[42]  D. Rixen,et al.  FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method , 2001 .

[43]  P. Flory,et al.  Thermodynamic relations for high elastic materials , 1961 .

[44]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[45]  Olof B. Widlund,et al.  Selecting Constraints in Dual-Primal FETI Methods for Elasticity in Three Dimensions , 2005 .

[46]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[47]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[48]  P. Deuflhard.  Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms , 2011 .

[49]  Gerhard A. Holzapfel,et al.  Modelling the layer-specific three-dimensional residual stresses in arteries, with an application to the human aorta , 2010, Journal of The Royal Society Interface.

[50]  Barbara I. Wohlmuth,et al.  A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier , 2000, SIAM J. Numer. Anal..

[51]  Gerhard A. Holzapfel,et al.  Structural and Numerical Models for the (Visco)elastic Response of Arterial Walls with Residual Stresses , 2003 .

[52]  A. Ibrahimbegovic.  Nonlinear Solid Mechanics , 2009 .

[53]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

[54]  Z. Dostál,et al.  Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure , 2011 .

[55]  Antje Baer,et al.  Direct Methods In The Calculus Of Variations , 2016 .

[56]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[57]  Z. Dostál,et al.  Total FETI—an easier implementable variant of the FETI method for numerical solution of elliptic PDE , 2006 .

[58]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

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

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

[61]  R. Ogden,et al.  Hyperelastic modelling of arterial layers with distributed collagen fibre orientations , 2006, Journal of The Royal Society Interface.

[62]  David A. Patterson,et al.  Computer Architecture: A Quantitative Approach , 1969 .

[63]  Olaf Steinbach,et al.  On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems , 2014, Numerische Mathematik.