Nested and parallel sparse algorithms for arterial fluid mechanics computations with boundary layer mesh refinement

Arterial fluid-structure interaction (FSI) computations involve a number of numerical challenges. Because blood flow is incompressible, iterative solution of the fluid mechanics part of the linear equation system at every nonlinear iteration of each time step is one of those challenges, especially for computations over slender domains and in the presence of boundary layer mesh refinement. In this paper we address that challenge. As test cases, we use equation systems from stabilized finite element computation of a bifurcating middle cerebral artery segment with aneurysm, with thin layers of elements near the arterial wall. We show how the preconditioning techniques, we propose for solving these large sparse nonsymmetric systems, perform at different time steps of the computation over a cardiac cycle. We also present a new hybrid parallel sparse linear system solver 'DD-Spike' and demonstrate its scalability.

[1]  Tayfun E. Tezduyar,et al.  Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions , 2010 .

[2]  Tayfun E. Tezduyar,et al.  Space-time finite element techniques for computation of fluid-structure interactions , 2005 .

[3]  Ahmed H. Sameh,et al.  A parallel hybrid banded system solver: the SPIKE algorithm , 2006, Parallel Comput..

[4]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[5]  Tayfun E. Tezduyar,et al.  Sequentially-Coupled Arterial Fluid-Structure Interaction (SCAFSI) technique , 2009 .

[6]  T. Tezduyar.  Computation of moving boundaries and interfaces and stabilization parameters , 2003 .

[7]  Tayfun E. Tezduyar,et al.  Modeling of Fluid-Structure Interactions with the Space-Time Techniques , 2006 .

[8]  Tayfun E. Tezduyar,et al.  PARALLEL COMPUTATION OF INCOMPRESSIBLE FLOWS WITH COMPLEX GEOMETRIES , 1997 .

[9]  A. Sameh,et al.  Preconditioning Techniques for Nonsymmetric Linear Systems in the Computation of Incompressible Flows , 2009 .

[10]  Tayfun E. Tezduyar,et al.  Modelling of fluid–structure interactions with the space–time finite elements: Solution techniques , 2007 .

[11]  Hans-Joachim Bungartz,et al.  Fluid-Structure Interaction on Cartesian Grids: Flow Simulation and Coupling Environment , 2006 .

[12]  Wulf G. Dettmer,et al.  On the coupling between fluid flow and mesh motion in the modelling of fluid–structure interaction , 2008 .

[13]  Tayfun E. Tezduyar,et al.  Multiscale sequentially-coupled arterial FSI technique , 2010 .

[14]  David J. Kuck,et al.  On Stable Parallel Linear System Solvers , 1978, JACM.

[15]  David J. Kuck,et al.  Practical Parallel Band Triangular System Solvers , 1978, TOMS.

[16]  Toshio Kobayashi,et al.  Influence of wall thickness on fluid–structure interaction computations of cerebral aneurysms , 2010 .

[17]  Tayfun E. Tezduyar,et al.  Modelling of fluid–structure interactions with the space–time finite elements: Arterial fluid mechanics , 2007 .

[18]  René de Borst,et al.  On the Nonnormality of Subiteration for a Fluid-Structure-Interaction Problem , 2005, SIAM J. Sci. Comput..

[19]  Arif Masud,et al.  An adaptive mesh rezoning scheme for moving boundary flows and fluid-structure interaction , 2007 .

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

[21]  D. Peric,et al.  A computational framework for fluid–structure interaction: Finite element formulation and applications , 2006 .

[22]  Murat Manguoglu,et al.  Solution of linear systems in arterial fluid mechanics computations with boundary layer mesh refinement , 2010 .

[23]  Pascal Frey,et al.  Fluid-structure interaction in blood flows on geometries based on medical imaging , 2005 .

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

[25]  Yuri Bazilevs,et al.  Computational fluid–structure interaction: methods and application to a total cavopulmonary connection , 2009 .

[26]  W. Wall,et al.  A Solution for the Incompressibility Dilemma in Partitioned Fluid–Structure Interaction with Pure Dirichlet Fluid Domains , 2006 .

[27]  Rainald Löhner,et al.  Extending the Range and Applicability of the Loose Coupling Approach for FSI Simulations , 2006 .

[28]  Duncan H. Lawrie,et al.  The computation and communication complexity of a parallel banded system solver , 1984, TOMS.

[29]  Michael L. Accorsi,et al.  Parachute fluid-structure interactions: 3-D computation , 2000 .

[30]  T. Tezduyar,et al.  A parallel 3D computational method for fluid-structure interactions in parachute systems , 2000 .

[31]  Toshiaki Hisada,et al.  Fluid–structure interaction analysis of the two-dimensional flag-in-wind problem by an interface-tracking ALE finite element method , 2007 .

[32]  Arif Masud,et al.  A Multiscale/stabilized Formulation of the Incompressible Navier–Stokes Equations for Moving Boundary Flows and Fluid–structure Interaction , 2006 .

[33]  Yuri Bazilevs,et al.  Determination of Wall Tension in Cerebral Artery Aneurysms by Numerical Simulation , 2008, Stroke.

[34]  T. Tezduyar,et al.  Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling , 2008 .

[35]  Ekkehard Ramm,et al.  A strong coupling partitioned approach for fluid–structure interaction with free surfaces , 2007 .

[36]  Tayfun E. Tezduyar,et al.  Massively parallel finite element simulation Of compressible and incompressible flows , 1994 .

[37]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[38]  S. Mittal,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II: Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders , 1992 .

[39]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[40]  T. Tezduyar,et al.  Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm—Dependence of the effect on the aneurysm shape , 2007 .

[41]  Tayfun E. Tezduyar,et al.  Space–time finite element computation of arterial fluid–structure interactions with patient‐specific data , 2010 .

[42]  Toshio Kobayashi,et al.  Influence of wall elasticity on image-based blood flow simulations , 2004 .

[43]  W. Wall,et al.  Fixed-point fluid–structure interaction solvers with dynamic relaxation , 2008 .

[44]  Toshio Kobayashi,et al.  Computer modeling of cardiovascular fluid-structure interactions with the deforming-spatial-domain/stabilized space-time formulation , 2006 .

[45]  Tayfun E. Tezduyar,et al.  Automatic mesh update with the solid-extension mesh moving technique , 2004 .

[46]  Michael W. Berry,et al.  Multiprocessor Schemes for Solving Block Tridiagonal Linear Systems , 1988 .

[47]  T. Tezduyar,et al.  Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements , 2003 .

[48]  van Eh Harald Brummelen,et al.  An interface Newton–Krylov solver for fluid–structure interaction , 2005 .

[49]  Ryo Torii,et al.  Role of 0D peripheral vasculature model in fluid–structure interaction modeling of aneurysms , 2010 .

[50]  A. Sameh,et al.  A nested iterative scheme for computation of incompressible flows in long domains , 2008 .

[51]  Marek Behr,et al.  Parallel finite-element computation of 3D flows , 1993, Computer.

[52]  Tayfun E. Tezduyar,et al.  Fluid-structure interactions of a parachute crossing the far wake of an aircraft , 2001 .

[53]  Yuri Bazilevs,et al.  A fully-coupled fluid-structure interaction simulation of cerebral aneurysms , 2010 .

[54]  Roger Ohayon,et al.  Reduced symmetric models for modal analysis of internal structural-acoustic and hydroelastic-sloshing systems , 2001 .

[55]  S. Mittal,et al.  Massively parallel finite element computation of incompressible flows involving fluid-body interactions , 1994 .

[56]  Tayfun E. Tezduyar,et al.  PARALLEL FINITE ELEMENT SIMULATION OF 3D INCOMPRESSIBLE FLOWS: FLUID-STRUCTURE INTERACTIONS , 1995 .

[57]  T. Tezduyar,et al.  Arterial fluid mechanics modeling with the stabilized space–time fluid–structure interaction technique , 2008 .

[58]  Thomas J. R. Hughes,et al.  Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device , 2009 .

[59]  Ahmed Sameh,et al.  SPIKE: A parallel environment for solving banded linear systems , 2007 .

[60]  Ming-Chen Hsu,et al.  Computational vascular fluid–structure interaction: methodology and application to cerebral aneurysms , 2010, Biomechanics and modeling in mechanobiology.

[61]  Tayfun E. Tezduyar,et al.  Advanced mesh generation and update methods for 3D flow simulations , 1999 .

[62]  Toshio Kobayashi,et al.  Fluid-structure interaction modeling of blood flow and cerebral aneurysm: Significance of artery and aneurysm shapes , 2009 .

[63]  Tayfun E. Tezduyar,et al.  Interface projection techniques for fluid–structure interaction modeling with moving-mesh methods , 2008 .

[64]  Toshio Kobayashi,et al.  Influence of wall elasticity in patient-specific hemodynamic simulations , 2007 .

[65]  Jack J. Dongarra,et al.  On some parallel banded system solvers , 1984, Parallel Comput..

[66]  Tayfan E. Tezduyar,et al.  Stabilized Finite Element Formulations for Incompressible Flow Computations , 1991 .

[67]  Ahmed Sameh,et al.  A nested iterative scheme for indefinite linear systems in particulate flows , 2004 .

[68]  Roland Wüchner,et al.  Algorithmic treatment of shells and free form-membranes in FSI , 2006 .

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

[70]  T. Tezduyar,et al.  Fluid–structure Interaction Modeling of Aneurysmal Conditions with High and Normal Blood Pressures , 2006 .