Computational Mechanics

, Volume 46, Issue 1, pp 17–29 | Cite as

Multiscale sequentially-coupled arterial FSI technique

  • Tayfun E. Tezduyar
  • Kenji Takizawa
  • Creighton Moorman
  • Samuel Wright
  • Jason Christopher
Original Paper

Abstract

Multiscale versions of the Sequentially-Coupled Arterial Fluid–Structure Interaction (SCAFSI) technique are presented. The SCAFSI technique was introduced as an approximate FSI approach in arterial fluid mechanics. It is based on the assumption that the arterial deformation during a cardiac cycle is driven mostly by the blood pressure. First we compute a “reference” arterial deformation as a function of time, driven only by the blood pressure profile of the cardiac cycle. Then we compute a sequence of updates involving mesh motion, fluid dynamics calculations, and recomputing the arterial deformation. The SCAFSI technique was developed and tested in conjunction with the stabilized space–time FSI (SSTFSI) technique. Beyond providing a computationally more economical alternative to the fully coupled arterial FSI approach, the SCAFSI technique brings additional flexibility, such as being able to carry out the computations in a spatially or temporally multiscale fashion. In the test computations reported here for the spatially multiscale versions of the SCAFSI technique, we focus on a patient-specific middle cerebral artery segment with aneurysm, where the arterial geometry is based on computed tomography images. The arterial structure is modeled with the continuum element made of hyperelastic (Fung) material.

Keywords

Cardiovascular fluid mechanics Fluid–structure interaction Sequentially-coupled arterial FSI Multiscale Space–time methods Cerebral aneurysm Hyperelastic material Patient-specific data 

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References

  1. 1.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2004) Influence of wall elasticity on image-based blood flow simulation. Jpn Soc Mech Eng J Ser A Jpn 70: 1224–1231Google Scholar
  2. 2.
    Gerbeau JF, Vidrascu M, Frey P (2005) Fluid–structure interaction in blood flow on geometries based on medical images. Comput Struct 83: 155–165CrossRefGoogle Scholar
  3. 3.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2006) Computer modeling of cardiovascular fluid–structure interactions with the Deforming-Spatial-Domain/Stabilized Space–Time formulation. Comput Methods Appl Mech Eng 195: 1885–1895MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2006) Fluid–structure interaction modeling of aneurysmal conditions with high and normal blood pressures. Comput Mech 38: 482–490MATHCrossRefGoogle Scholar
  5. 5.
    Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38: 310–322MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2007) Influence of wall elasticity in patient-specific hemodynamic simulations. Comput Fluids 36: 160–168MATHCrossRefGoogle Scholar
  7. 7.
    Tezduyar TE, Sathe S, Cragin T, Nanna B, Conklin BS, Pausewang J, Schwaab M (2007) Modeling of fluid–structure interactions with the space-time finite elements: arterial fluid mechanics. Int J Numer Methods Fluids 54: 901–922MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2007) Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm—Dependence of the effect on the aneurysm shape. Int J Numer Methods Fluids 54: 995–1009MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tezduyar TE, Sathe S, Schwaab M, Conklin BS (2008) Arterial fluid mechanics modeling with the stabilized space–time fluid-structure interaction technique. Int J Numer Methods Fluids 57: 601–629MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43: 3–37MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2008) Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: Influence of structural modeling. Comput Mech 43: 151–159MATHCrossRefGoogle Scholar
  12. 12.
    Tezduyar TE, Schwaab M, Sathe S (2008) Sequentially-Coupled Arterial Fluid–Structure Interaction (SCAFSI) technique. Comput Methods Appl Mech Eng published online, doi:10.1016/j.cma.2008.05.024
  13. 13.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2008) Fluid–structure interaction modeling of blood flow and cerebral aneurysm: Significance of artery and aneurysm shapes. Comput Methods Appl Mech Eng published online doi:10.1016/j.cma.2008.08.020
  14. 14.
    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 published online doi:10.1016/j.cma.2009.04.015
  15. 15.
    Takizawa K, Christopher J, Tezduyar TE, Sathe S (2009) Space–time finite element computation of arterial fluid–structure interactions with patient-specific data. Commun Numer Methods Eng published online, doi:10.1002/cnm.1241
  16. 16.
    Tezduyar TE, Takizawa K, Christopher J (2009) Multiscale sequentially-coupled arterial fluid–structure interaction (SCAFSI) technique. In: Hartmann S, Meister A, Schaefer M, Turek S (eds) International workshop on fluid-structure interaction—theory, numerics and applications. Kassel University PressGoogle Scholar
  17. 17.
    Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2009) Influence of wall thickness on fluid–structure interaction computations of cerebral aneurysms. Commun Numer Methods Eng published online. doi:10.1002/cnm.1289
  18. 18.
    Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26: 27–36CrossRefGoogle Scholar
  19. 19.
    Tezduyar TE, Aliabadi SK, Behr M, Mittal S (1994) Massively parallel finite element simulation of compressible and incompressible flows. Comput Methods Appl Mech Eng 119: 157–177MATHCrossRefGoogle Scholar
  20. 20.
    Mittal S, Tezduyar TE (1994) Massively parallel finite element computation of incompressible flows involving fluid-body interactions. Comput Methods Appl Mech Eng 112: 253–282MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Mittal S, Tezduyar TE (1995) Parallel finite element simulation of 3D incompressible flows—Fluid-structure interactions. Int J Numer Methods Fluids 21: 933–953MATHCrossRefGoogle Scholar
  22. 22.
    Johnson AA, Tezduyar TE (1997) Parallel computation of incompressible flows with complex geometries. Int J Numer Methods Fluids 24: 1321–1340MATHCrossRefGoogle Scholar
  23. 23.
    Johnson AA, Tezduyar TE (1999) Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23: 130–143MATHCrossRefGoogle Scholar
  24. 24.
    Kalro V, Tezduyar TE (2000) A parallel 3D computational method for fluid–structure interactions in parachute systems. Comput Methods Appl Mech Eng 190: 321–332MATHCrossRefGoogle Scholar
  25. 25.
    Stein K, Benney R, Kalro V, Tezduyar TE, Leonard J, Accorsi M (2000) Parachute fluid–structure interactions: 3-D computation. Comput Methods Appl Mech Eng 190: 373–386MATHCrossRefGoogle Scholar
  26. 26.
    Tezduyar T, Osawa Y (2001) Fluid–structure interactions of a parachute crossing the far wake of an aircraft. Comput Methods Appl Mech Eng 191: 717–726MATHCrossRefGoogle Scholar
  27. 27.
    Ohayon R (2001) Reduced symmetric models for modal analysis of internal structural-acoustic and hydroelastic-sloshing systems. Comput Methods Appl Mech Eng 190: 3009–3019MATHCrossRefGoogle Scholar
  28. 28.
    Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid–structure interactions with large displacements. J Appl Mech 70: 58–63MATHCrossRefGoogle Scholar
  29. 29.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2004) Space–time techniques for finite element computation of flows with moving boundaries and interfaces. In: Gallegos S, Herrera I, Botello S, Zarate F, Ayala G (eds) Proceedings of the III international congress on numerical methods in engineering and applied science, CD-ROM, Monterrey, MexicoGoogle Scholar
  30. 30.
    Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193: 2019–2032MATHCrossRefGoogle Scholar
  31. 31.
    van Brummelen EH, de Borst R (2005) On the nonnormality of subiteration for a fluid-structure interaction problem. SIAM J Scient Comput 27: 599–621MATHCrossRefGoogle Scholar
  32. 32.
    Michler C, van Brummelen EH, de Borst R (2005) An interface Newton–Krylov solver for fluid–structure interaction. Int J Numer Methods Fluids 47: 1189–1195MATHCrossRefGoogle Scholar
  33. 33.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2006) Space–time finite element techniques for computation of fluid–structure interactions. Comput Methods Appl Mech Eng 195: 2002–2027MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Tezduyar TE, Sathe S, Stein K (2006) Solution techniques for the fully-discretized equations in computation of fluid–structure interactions with the space–time formulations. Comput Methods Appl Mech Eng 195: 5743–5753MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Tezduyar TE, Sathe S, Stein K, Aureli L (2006) Modeling of fluid–structure interactions with the space–time techniques. In: Bungartz H-J, Schafer M (eds) Fluid–structure interaction. Lecture notes in computational science and engineering, vol 53. Springer, Heidelberg, pp 50–81Google Scholar
  36. 36.
    Dettmer W, Peric D (2006) A computational framework for fluid-structure interaction: finite element formulation and applications. Comput Methods Appl Mech Eng 195: 5754–5779MATHCrossRefGoogle Scholar
  37. 37.
    Khurram RA, Masud A (2006) A multiscale/stabilized formulation of the incompressible Navier–Stokes equations for moving boundary flows and fluid–structure interaction. Comput Mech 38: 403–416MATHCrossRefGoogle Scholar
  38. 38.
    Kuttler U, Forster C, Wall WA (2006) A solution for the incompressibility dilemma in partitioned fluid–structure interaction with pure Dirichlet fluid domains. Comput Mech 38: 417–429CrossRefGoogle Scholar
  39. 39.
    Lohner R, Cebral JR, Yang C, Baum JD, Mestreau EL, Soto O (2006) Extending the range of applicability of the loose coupling approach for FSI simulations. In: Bungartz H-J, Schafer M (eds) Fluid–structure interaction. Lecture notes in computational science and engineering, vol 53. Springer, Heidelberg, pp 82–100Google Scholar
  40. 40.
    Bletzinger K-U, Wuchner R, and Kupzok A (2006) Algorithmic treatment of shells and free form-membranes in FSI. In: Bungartz H-J, Schafer M (eds) Fluid–Structure Interaction. Lecture notes in computational science and engineering, vol 53. Springer, Heidelberg, pp 336–355Google Scholar
  41. 41.
    Masud A, Bhanabhagvanwala M, Khurram RA (2007) An adaptive mesh rezoning scheme for moving boundary flows and fluid– structure interaction. Comput Fluids 36: 77–91MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Sawada T, Hisada T (2007) Fuid–structure interaction analysis of the two dimensional flag-in-wind problem by an interface tracking ALE finite element method. Comput Fluids 36: 136–146MATHCrossRefGoogle Scholar
  43. 43.
    Wall WA, Genkinger S, Ramm E (2007) A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Comput Fluids 36: 169–183MATHCrossRefGoogle Scholar
  44. 44.
    Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space–time finite elements: Solution techniques. Int J Numer Methods Fluids 54: 855–900MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Kuttler U, Wall WA (2008) Fixed-point fluid–structure interaction solvers with dynamic relaxation. Comput Mech 43: 61–72CrossRefGoogle Scholar
  46. 46.
    Dettmer WG, Peric D (2008) On the coupling between fluid flow and mesh motion in the modelling of fluid–structure interaction. Comput Mech 43: 81–90MATHCrossRefGoogle Scholar
  47. 47.
    Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28: 1–44MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Tezduyar TE, Behr M, Liou J (1992) 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. Comput Methods Appl Mech Eng 94: 339–351MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Tezduyar TE, Behr M, Mittal S, Liou J (1992) 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. Comput Methods Appl Mech Eng 94: 353–371MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43: 555–575MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Hughes TJR, Brooks AN (1979) A multi-dimensional upwind scheme with no crosswind diffusion. In: Hughes TJR (ed) Finite element methods for convection dominated flows, AMD-vol 34. ASME, New York, pp 19–35Google Scholar
  52. 52.
    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: 199–259MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    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: 221–242MATHCrossRefGoogle Scholar
  54. 54.
    Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška–Brezzi condition: A stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59: 85–99MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Tezduyar TE, Behr M, Mittal S, Johnson AA (1992) Computation of unsteady incompressible flows with the finite element methods—space–time formulations, iterative strategies and massively parallel implementations. In: New methods in transient analysis, PVP-Vol 246/AMD-Vol 143. ASME, New York, pp 7–24Google Scholar
  56. 56.
    Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput Methods Appl Mech Eng 119: 73–94MATHCrossRefGoogle Scholar
  57. 57.
    Tezduyar TE (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch Comput Methods Eng 8: 83–130MATHCrossRefGoogle Scholar
  58. 58.
    Tezduyar TE (2004) Finite element methods for fluid dynamics with moving boundaries and interfaces. In: Stein E, De Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3: Fluids, Chapter 17. Wiley, LondonGoogle Scholar
  59. 59.
    Tezduyar TE, Cragin T, Sathe S, Nanna B (2007) FSI computations in arterial fluid mechanics with estimated zero-pressure arterial geometry. In: Onate E, Garcia J, Bergan P, Kvamsdal T (eds) Marine CIMNE. Barcelona, SpainGoogle Scholar
  60. 60.
    Tezduyar TE, Schwaab M, Sathe S (2007) Arterial fluid mechanics with the sequentially-coupled arterial FSI technique. In: Onate E, Papadrakakis M, Schrefler B (eds) Coupled problems 2007, CIMNE. Barcelona, SpainGoogle Scholar
  61. 61.
    Saad Y, Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Scient Stat Comput 7: 856–869MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Huang H, Virmani R, Younis H, Burke AP, Kamm RD, Lee RT (2001) The impact of calcification on the biomechanical stability of atherosclerotic plaques. Circulation 103: 1051–1056Google Scholar
  63. 63.
    Frank O (1899) Die grundform des arteriellen pulses. Zeitung fur Biologie 37: 483–586Google Scholar
  64. 64.
    Takizawa K, Moorman C, Wright S, Christopher J, Tezduyar TE (2009) Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions. Comput Mech (submitted to)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Tayfun E. Tezduyar
    • 1
  • Kenji Takizawa
    • 1
  • Creighton Moorman
    • 1
  • Samuel Wright
    • 1
  • Jason Christopher
    • 1
  1. 1.Mechanical EngineeringRice University, MS 321HoustonUSA

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