Abstract
Simulation divergence due to backflow is a common, but not fully addressed, problem in three-dimensional simulations of blood flow in the large vessels. Because backflow is a naturally occurring physiologic phenomenon, careful treatment is necessary to realistically model backflow without artificially altering the local flow dynamics. In this study, we quantitatively compare three available methods for treatment of outlets to prevent backflow divergence in finite element Navier–Stokes solvers. The methods examined are (1) adding a stabilization term to the boundary nodes formulation, (2) constraining the velocity to be normal to the outlet, and (3) using Lagrange multipliers to constrain the velocity profile at all or some of the outlets. A modification to the stabilization method is also discussed. Three model problems, a short and long cylinder with an expansion, a right-angle bend, and a patient-specific aorta model, are used to evaluate and quantitatively compare these methods. Detailed comparisons are made to evaluate robustness, stability characteristics, impact on local and global flow physics, computational cost, implementation effort, and ease-of-use. The results show that the stabilization method offers a promising alternative to previous methods, with reduced effect on both local and global hemodynamics, improved stability, little-to-no increase in computational cost, and elimination of the need for tunable parameters.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bove EL, Migliavacca F, de Leval MR, Balossino R, Pennati G, Lloyd TR, Khambadkone S, Hsia TY, Dubini G (2008) Use of mathematic modeling to compare and predict hemodynamic effects of the modified blalock-taussig and right ventricle-pulmonary artery shunts for hypoplastic left heart syndrome. J Thorac Cardiovasc Surg 136(2): 312–320.e2
Migliavacca F, Balossino R, Pennati G, Dubini G, Hsia TY, de Leval MR, Bove EL (2006) Multiscale modelling in biofluidynamics: application to reconstructive paediatric cardiac surgery. J Biomech 39(6): 1010–1020
Lagana K, Dubini G, Migliavacca F, Pietrabissa R, Pennati G, Veneziani A, Quarteroni A (2002) Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39: 359–364
Urquiza SA, Blanco PJ, Vnere MJ, Feijo RA (2006) Multidimensional modelling for the carotid artery blood flow. Comput Meth Appl Mech Eng 195(33–36): 4002–4017
Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA (2006) Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput Meth Appl Mech Eng 195(29–32): 3776–3796
Blanco PJ, Feijo RA, Urquiza SA (2007) A unified variational approach for coupling 3d-1d models and its blood flow applications. Comput Meth Appl Mech Eng 196(41–44): 4391–4410
Heywood JG, Rannacher R, Turek S (1996) Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 22(5): 325–352
Formaggia L, Gerbeau JF, Nobile F, Quarteroni A (2002) Numerical treatment of defective boundary conditions for the Navier–Stokes equations. SIAM J Numer Anal 40: 376–401
Formaggia L, Veneziani A, Vergara C (2008) A new approach to numerical solution of defective boundary valve problems in incompressible fluid dynamics. SIAM J Numer Anal 46(6): 2769–2794
Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of blood flow in arteries. Comput Methods Appl Mech Eng 158(1–2): 155–196
Taylor CA, Cheng CP, Espinosa LA, Tang BT, Parker D, Herfkens RJ (2002) In vivo quantification of blood flow and wall shear stress in the human abdominal aorta during lower limb exercise. Ann Biomed Eng 30: 402–408
Lagan K, Balossino R, Migliavacca F, Pennati G, Bove EL, de Leval MR, Dubini G (2005) Multiscale modeling of the cardiovascular system: application to the study of pulmonary and coronary perfusions in the univentricular circulation. J Biomech 38(5): 1129–1141
Marsden AL, Vignon-Clementel IE, Chan F, Feinstein JA, Taylor CA (2007) Effects of exercise and respiration on hemodynamic efficiency in CFD simulations of the total cavopulmonary connection. Ann Biomed Eng 35: 250–263
Pekkan K, Dasi LP, Nourparvar P, Yerneni S, Tobita K, Fogel MA, Keller B, Yoganathan A (2008) In vitro hemodynamic investigation of the embryonic aortic arch at late gestation. J Biomech 41(8): 1697–1706
Pekkan K, Dur O, Sundareswaran K, Kanter K, Fogel M, Yoganathan A, Undar A (2008) Neonatal aortic arch hemodynamics and perfusion during cardiopulmonary bypass. J Biomech Eng 130(6): 061012
Tezduyar TE, Ramakrishnan S, Sathe S (2008) Stabilized formulations for incompressible flows with thermal coupling. Int J Numer Methods Fluids 57: 1189–1209
Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Space-time finite element computation of complex fluid-structure interactions. Int J Numer Methods Fluids 64: 1201–1218
de Zelicourt D, Ge L, Wang C, Sotiropoulos F, Gilmanov A, Yoganathan A (2009) Flow simulations in arbitrarily complex cardiovascular anatomies - an unstructured cartesian grid approach. Comput Fluids 38(9): 1749–1762
Marsden AL, Feinstein JA, Taylor CA (2008) A computational framework for derivative-free optimization of cardiovascular geometries. Comput Methods Appl Mech Eng 197(21–24): 1890–1905
Borazjani I, Ge L, Sotiropoulos F (2010) High-resolution fluid-structure interaction simulations of flow through a bi-leaflet mechanical heart valve in an anatomic aorta. Ann Biomed Eng 38: 326–344
Vignon-Clementel IE (2006) A Coupled Multidomain Method for Computational Modeling of Blood Flow. PhD thesis, Stanford
Bazilevs Y, Gohean JR, Hughes TJR, Moser RD, Zhang Y (2009) Patient-specific isogeometric fluid-structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Comput Methods Appl Mech Eng 198(45–46): 3534–3550
Kim HJ, Figueroa CA, Hughes TJR, Jansen KE, Taylor CA (2009) Augmented lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite element simulations of blood flow. Comput Meth Appl Mech Eng 198(45–46): 3551–3566
Brooks AN, Hughes TJR (1982) Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32(1–3): 199–259
Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43(5): 555–575
Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197(1–4): 173–201
Whiting CH, Jansen KE (2001) A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis. Int J Numer Methods Fluids 35(1): 93–116
Franca LP, Frey SL (1992) Stabilized finite element methods: II. the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 99(2–3): 209–233
Tezduyar TE, Mittal S, Ray SE, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity–pressure elements. Comput Methods Appl Mech Eng 95: 221–242
Jansen KE, Whiting CH, Hulbert GM (2000) A generalized-[alpha] method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190(3–4): 305–319
Shakib F, Hughes TJR, Johan Z (1989) A multi-element group preconditioned gmres algorithm for nonsymmetric systems arising in finite element analysis. Comput Methods Appl Mech Eng 75(1–3): 415–456
Gresho PM, Sani RL (2000) Incompressible flow and the finite element method, vol 2. Wiley
Schmidt JP, Delp SL, Sherman MA, Taylor CA, Pande VS, Altman RB (2008) The Simbios National Center: systems biology in motion. In: Proceedings of the IEEE, vol 96, issue 8, pp 1266–1280
Vignon-Clementel IE, Marsden AL, Feinstein JA (2010) A primer on computational simulation in congenital heart disease for the clinician. Progress Pediatr Cardiol 30(1–2): 3–13
Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA (2010) Outflow boundary conditions for three-dimensional simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput Methods Biomech Biomed Eng 13(5): 625–640
Migliavacca F, Pennati G, Dubini G, Fumero R, Pietrabissa R, Urcelay G, Bove EL, Hsia TY, De Leval MR (2001) Modeling of the norwood circulation: effects of shunt size, vascular resistances, and heart rate. Am J Physiol Heart Circ Physiol 280: H2076–H2086
Kim H, Vignon-Clementel IE, Coogan J, Figueroa C, Jansen KE, Taylor CA (2010) Patient-specific modeling of blood flow and pressure in human coronary arteries. Ann Biomed Eng 38: 3195–3209
Sahni O, Muller J, Jansen KE, Shephard MS, Taylor CA (2006) Efficient anisotropic adaptive discretization of the cardiovascular system. Comput Methods Appl Mech Eng 195(41–43): 5634–5655
Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput Mech 38: 310–322
Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43: 3–37
Takizawa K, Moorman C, Wright S, Purdue J, McPhail T, Chen PR, Warren J, Tezduyar TE (2011) Patient-specific arterial fluid-structure interaction modeling of cerebral aneurysms. Int J Numer Methods Fluids 65: 308–323
Tezduyar TE, Takizawa K, Brummer T, Chen PR (2011) Space–time fluid–structure interaction modeling of patient-specific cerebral aneurysms. Int J Numer Methods Biomed Eng 27. doi:10.1002/cnm.1433
Author information
Authors and Affiliations
Consortia
Corresponding author
Additional information
MOCHA Investigators: Edward Bove MD and Adam Dorfman MD (University of Michigan, USA); Andrew Taylor MD, Alessandro Giardini MD, Sachin Khambadkone MD, Marc de Leval MD, Silvia Schievano PhD, and T-Y Hsia MD (Institute of Child Health, UK); G. Hamilton Baker MD and Anthony Hlavacek (Medical University of South Carolina, USA); Francesco Migliavacca PhD, Giancarlo Pennati PhD, and Gabriele Dubini PhD (Politecnico di Milano, Italy); Richard Figliola PhD and John McGregor PhD (Clemson University, USA); Alison Marsden PhD (University of California, San Diego, USA); Irene Vignon-Clementel (National Institute of Research in Informatics and Automation, France).
Rights and permissions
About this article
Cite this article
Esmaily Moghadam, M., Bazilevs, Y., Hsia, TY. et al. A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Comput Mech 48, 277–291 (2011). https://doi.org/10.1007/s00466-011-0599-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-011-0599-0