Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-15T04:29:59.428Z Has data issue: false hasContentIssue false

A Parallel Domain Decomposition Algorithm for Simulating Blood Flow with Incompressible Navier-Stokes Equations with Resistive Boundary Condition

Published online by Cambridge University Press:  20 August 2015

Yuqi Wu*
Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309, USA
Xiao-Chuan Cai*
Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309, USA
Get access


We introduce and study a parallel domain decomposition algorithm for the simulation of blood flow in compliant arteries using a fully-coupled system of nonlinear partial differential equations consisting of a linear elasticity equation and the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition plays an interesting role in the accuracy of the blood flow simulation and we provide a numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition method for solving the fully coupled nonlinear system on a supercomputer with a few hundred processors.

Research Article
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Badia, S., Quaini, A. and Quarteroni, A., Splitting methods based on algebraic factorization for fluid-structure interaction, SIAM J. Sci. Comput., 30 (2008), 17781805.Google Scholar
[2]Balay, S., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. and Zhang, H., PETSc users manual, Technical report, Argonne National Laboratory, 2010.Google Scholar
[3]Barker, A. T., Monolithic Fluid-Structure Interaction Algorithms for Parallel Computing with Application to Blood Flow, PhD thesis, University of Colorado at Boulder, 2009.Google Scholar
[4]Barker, A. T. and Cai, X.-C., Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling, J. Comput. Phys., 229 (2010), 642659.Google Scholar
[5]Bazilevs, Y., Calo, V., Zhang, Y. and Hughes, T., Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Comput. Mech., 38 (2006), 310322.Google Scholar
[6]Bazilevs, Y., Calo, V., Hughes, T. and Zhang, Y., Isogeometric fluid-structure interaction: theory, algorithms and computations, Comput. Mech., 43 (2008), 337.Google Scholar
[7]Cai, X.-C., Gropp, W. D., Keyes, D. E., Melvin, R. G. and Young, D. P., Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput., 19 (1998), 246265.Google Scholar
[8]Cai, X.-C. and Sarkis, M., A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21 (1999), 792797.CrossRefGoogle Scholar
[9]Causin, P., Gerbeau, J. F. and Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Eng., 194 (2005), 45064527.Google Scholar
[10]Dennis, J. E. Jr. and Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1996.Google Scholar
[11]Donea, J., Giuliani, S. and Halleux, J. P., An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl. Mech. Eng., 33 (1982), 689723.CrossRefGoogle Scholar
[12]Eisenstat, S. C. and Walker, H. F., Globally convergent inexact Newton method, SIAM J. Optim., 4 (1994), 393422.Google Scholar
[13]Eisenstat, S. C. and Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), 1632.CrossRefGoogle Scholar
[14]Farhat, C. and Geuzaine, P., Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids, Comput. Methods Appl. Mech. Eng., 193 (2004), 40734095.CrossRefGoogle Scholar
[15]Farhat, C., Geuzaine, P. and Grandmont, C., The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids, J. Comput. Phys., 174 (2001), 669694.Google Scholar
[16]Figueroa, C. A., Vignon-Clementel, I. E., Jansen, K. E., Hughes, T. J. R. and Taylor, C. A., A coupled momentum method for modeling blood flow in three-dimensional deformable arteries, Comput. Methods Appl. Mech. Eng., 195 (2006), 56855706.CrossRefGoogle Scholar
[17]Formaggia, L., Gerbeau, J. F., Nobile, F. and Quarteroni, A., On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comput. Methods Appl. Mech. Eng., 191 (2001), 561582.CrossRefGoogle Scholar
[18]Fung, Y. C., Biomechanics: Circulation, 2nd edition, Springer-Verlag, New York, 1997.Google Scholar
[19]Heil, M., An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems, Comput. Methods Appl. Mech. Eng., 193 (2004), 123.Google Scholar
[20]Hughes, T. J. R., Liu, W. K. and Zimmermann, T. K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 29 (1981), 329349.CrossRefGoogle Scholar
[21]Hwang, F.-N. and Cai, X.-C., A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations, J. Comput. Phys., 204 (2005), 666691.Google Scholar
[22]Karypis, G., Aggarwal, R., Schloegel, K., Kumar, V. and Shekhar, S., METIS/ParMETIS web page, University of Minnesota, 2010, Scholar
[23]Le, P. Tallec and Mouro, J., Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Eng., 190 (2001), 30393067.Google Scholar
[24]Michler, C., van Brummelen, E. H., Hulshoff, S. J. and de Borst, R., The relevance of conservation for stability and accuracyof numerical methods for fluid-structure interaction, Comput. Methods Appl. Mech. Eng., 192 (2003), 41954215.CrossRefGoogle Scholar
[25]Nichols, W. W. and O’Rourke, M. F., McDonald’s Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles, Oxford University Press, New York, 1998.Google Scholar
[26]Nobile, F., Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics, PhD thesis, Ecole Polytechnique Federade Lausanne, 2001.Google Scholar
[27]Owen, S. J. and Shepherd, J. F., CUBIT project web page, Sandia National Laboratories, 2010, Scholar
[28]Piperno, S. and Farhat, C., Partitioned procedures for the transient solution of coupled aeroe-lastic problems-part II: energy transfer analysis and three-dimensional applications, Com-put. Methods Appl. Mech. Eng., 190 (2001), 31473170.Google Scholar
[29]Saad, Y. and Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsysmetric linear system, SIAM J. Sci. Stat. Comput., 7 (1986), 856869.Google Scholar
[30]Taylor, C. A. and Draney, M. T., Experimental and computational methods in cardiovascular fluid mechanics, Ann. Rev. Fluid Mech., 36 (2004), 197231.Google Scholar
[31]Taylor, C. A. and Humphrey, J. D., Open problems in computational vascular biomechanics: hemodynamics and arterial wall mechanics, Comput. Methods Appl. Mech. Eng., 198 (2009), 35143523.Google Scholar
[32]Vignon, I. E. and Taylor, C. A., Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries, Wave Motion, 39 (2004), 361374.Google Scholar
[33]Vignon-Clementel, I. E., Figueroa, C. A., Jansen, K. E. and Taylor, C. A., Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries, Comput. Methods Appl. Mech. Eng., 195 (2006), 37763796.Google Scholar