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Computational Mechanics

, Volume 48, Issue 3, pp 277–291 | Cite as

A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations

  • Mahdi Esmaily Moghadam
  • Yuri BazilevsEmail author
  • Tain-Yen Hsia
  • Irene E. Vignon-Clementel
  • Alison L. Marsden
  • Modeling of Congenital Hearts Alliance (MOCHA)
Original Paper

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.

Keywords

Neumann boundary conditions Outflow stabilization Lagrange multipliers Normal velocity constraint Patient-specific blood flow Flow reversal Navier–Stokes FEM solver Cardiovascular simulation 

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References

  1. 1.
    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.e2CrossRefGoogle Scholar
  2. 2.
    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): 1010–1020CrossRefGoogle Scholar
  3. 3.
    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: 359–364Google Scholar
  4. 4.
    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): 4002–4017zbMATHCrossRefGoogle Scholar
  5. 5.
    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): 3776–3796MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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): 4391–4410zbMATHCrossRefGoogle Scholar
  7. 7.
    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): 325–352MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    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: 376–401MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    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): 2769–2794MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of blood flow in arteries. Comput Methods Appl Mech Eng 158(1–2): 155–196MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    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: 402–408CrossRefGoogle Scholar
  12. 12.
    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): 1129–1141CrossRefGoogle Scholar
  13. 13.
    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: 250–263CrossRefGoogle Scholar
  14. 14.
    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): 1697–1706CrossRefGoogle Scholar
  15. 15.
    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): 061012CrossRefGoogle Scholar
  16. 16.
    Tezduyar TE, Ramakrishnan S, Sathe S (2008) Stabilized formulations for incompressible flows with thermal coupling. Int J Numer Methods Fluids 57: 1189–1209MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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: 1201–1218zbMATHCrossRefGoogle Scholar
  18. 18.
    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): 1749–1762zbMATHCrossRefGoogle Scholar
  19. 19.
    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): 1890–1905MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    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: 326–344CrossRefGoogle Scholar
  21. 21.
    Vignon-Clementel IE (2006) A Coupled Multidomain Method for Computational Modeling of Blood Flow. PhD thesis, StanfordGoogle Scholar
  22. 22.
    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): 3534–3550MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    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): 3551–3566MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    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): 199–259MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43(5): 555–575MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    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): 173–201MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    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): 93–116zbMATHCrossRefGoogle Scholar
  28. 28.
    Franca LP, Frey SL (1992) Stabilized finite element methods: II. the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 99(2–3): 209–233MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    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: 221–242zbMATHCrossRefGoogle Scholar
  30. 30.
    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): 305–319MathSciNetzbMATHGoogle Scholar
  31. 31.
    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): 415–456MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Gresho PM, Sani RL (2000) Incompressible flow and the finite element method, vol 2. WileyGoogle Scholar
  33. 33.
    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 1266–1280Google Scholar
  34. 34.
    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): 3–13CrossRefGoogle Scholar
  35. 35.
    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): 625–640CrossRefGoogle Scholar
  36. 36.
    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–H2086Google Scholar
  37. 37.
    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: 3195–3209CrossRefGoogle Scholar
  38. 38.
    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): 5634–5655MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput Mech 38: 310–322MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43: 3–37MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    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: 308–323zbMATHCrossRefGoogle Scholar
  42. 42.
    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

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Mahdi Esmaily Moghadam
    • 1
  • Yuri Bazilevs
    • 2
    Email author
  • Tain-Yen Hsia
    • 3
  • Irene E. Vignon-Clementel
    • 4
  • Alison L. Marsden
    • 1
  • Modeling of Congenital Hearts Alliance (MOCHA)
  1. 1.Mechanical and Aerospace Engineering DepartmentUniversity of CaliforniaSan DiegoUSA
  2. 2.Structural Engineering DepartmentUniversity of CaliforniaSan DiegoUSA
  3. 3.Cardiac Unit, Institute of Child HealthGreat Ormond Street Hospital for ChildrenLondonUK
  4. 4.INRIA Paris-RocquencourtRocquencourtFrance

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