SeMA Journal

, Volume 55, Issue 1, pp 59–108 | Cite as

Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit

Article
First Online:

Abstract

Over the last decade, the numerical simulation of incompressible fluid-structure interaction has been a very active research field and the subject of numerous works. This is due, in particular, to the increasing interest of the research community in the simulation of blood flows in large arteries. In this context, the fluid equations have to be solved in a moving domain and the incompressibility constraint makes the coupling sensitive to the added-mass effect. As a result, the solution procedure has to be designed carefully in order to guarantee efficiency without compromising numerical stability. In this paper, we review some of the coupling schemes recently proposed in the literature. Some numerical results that show the effectiveness of the novel approaches are also presented.

Keywords

Discontinuous Galerkin Coupling Scheme Couple Problem GMRES Iteration Explicit Coupling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Allaire. Conception optimale de structures, volume 58 of Mathématiques & Applications. Springer, 2007.Google Scholar
  2. [2]
    D.N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742–760, 1982.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    D.N. Arnold, F. Brezzi, B. Cockburn, and D. Marini. Discontinuous Galerkin methods for elliptic problems. In Discontinuous Galerkin methods (Newport, RI, 1999), volume 11 of Lect. Notes Comput. Sci. Eng., pages 89–101. Springer, 2000.Google Scholar
  4. [4]
    M. Astorino. Interaction fluide-structure dans le système cardiovasculaire. Analyse numérique et simulation. PhD thesis, Université Paris VI, France, 2010.Google Scholar
  5. [5]
    M. Astorino, F. Chouly, and M. A. Fernández. Robin based semi-implicit coupling in fluid-structure interaction: Stability analysis and numerics. SIAM J. Sci. Comput., 31(6):4041–4065, 2009.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    M. Astorino, J.-F. Gerbeau, O. Pantz, and K.-F. Traoré. Fluid-structure interaction and multi-body contact: application to aortic valves. Comput. Methods Appl. Mech. Engrg., 198(45–46):3603–3612, 2009.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    M. Astorino and C. Grandmont. Convergence analysis of a projection semi-implicit coupling scheme for fluid-structure interaction problems. Numer. Math., 116:721–767, 2010.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    S. Badia and R. Codina. On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity. Internat. J. Numer. Methods Engrg., 72(1):46–71, 2007.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    S. Badia, F. Nobile, and C. Vergara. Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comp. Phys., 227:7027–7051, 2008.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    S. Badia, F. Nobile, and C. Vergara. Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems. Comput. Methods Appl. Mech. Engrg., 198(33–36):2768–2784, 2009.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    S. Badia, A. Quaini, and A. Quarteroni. Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effect. Comput. Methods Appl. Mech. Engrg., 197(49–50):4216–4232, 2008.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    S. Badia, A. Quaini, and A. Quarteroni. Splitting methods based on algebraic factorization for fluid-structure interaction. SIAM J. Sci. Comput., 30(4):1778–1805, 2008.MathSciNetCrossRefGoogle Scholar
  13. [13]
    G.A. Baker. Finite element methods for elliptic equations using nonconforming elements. Math. Comp., 31(137):45–59, 1977.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    K.J. Bathe and H. Zhang. Finite element developments for general fluid flows with structural interactions. Int. J. Num. Meth. Engng., 2004.Google Scholar
  15. [15]
    Y. Bazilevs, V.M. Calo, T.J.R. Hughes, and Y. Zhang. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech., 43(1):3–37, 2008.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    R. Becker, P. Hansbo, and R. Stenberg. A finite element method for domain decomposition with non-matching grids. M2AN Math. Model. Numer. Anal., 37(2):209–225, 2003.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    C. Bernardi, Y. Maday, and A. T. Patera. Domain decomposition by the mortar element method. In Asymptotic and numerical methods for partial differential equations with critical parameters (Beaune, 1992), volume 384 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 269–286. Kluwer Acad. Publ., Dordrecht, 1993.Google Scholar
  18. [18]
    C. Bertoglio, D. Chapelle, M.A. Fernández, J.-F. Gerbeau, and P. Moireau. Filtering-based data assimilation in vascular fluid-structure interaction through displacement measurements at the interface. In IV European Congress on Computational Mechanics (ECCM IV): Solids, Structures and Coupled Problems in Engineering, 2010.Google Scholar
  19. [19]
    P. N. Brown and Y. Saad. Convergence theory of nonlinear Newton-Krylov algorithms. SIAM J. Optim., 4(2):297–330, 1994.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    E. Burman and M.A. Fernández. Robin-based explicit coupling schemes in incompressible fluid-structure interaction. In preparation.Google Scholar
  21. [21]
    E. Burman and M.A. Fernández. Stabilized explicit coupling for fluid-structure interaction using Nitsche’s method. C. R. Math. Acad. Sci. Paris, 345(8):467–472, 2007.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    E. Burman and M.A. Fernández. Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility. Comput. Methods Appl.Mech. Engrg., 198(5–8):766–784, 2009.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    E. Burman and P. Hansbo. A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math., 198:35–51, 2007.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    E. Burman and P. Zunino. A domain decomposition method based on interior penalties for advection-diffusion-reaction problems. Siam Jour. Num. Anal., 44:1612–1638, 2006.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    P. Causin, J.-F. Gerbeau, and F. Nobile. Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg., 194 (42–44):4506–4527, 2005.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    D. Chapelle and K.J. Bathe. The Finite Element Analysis of Shells —Fundamentals. Springer, 2003.Google Scholar
  27. [27]
    D. Chapelle, P. Moireau, and P. Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete Contin. Dyn. Syst., 23(1–2):65–84, 2009.MathSciNetMATHGoogle Scholar
  28. [28]
    Y. Cheng, H. Oertel, and T. Schenkel. Fluid-structure coupled CFD simulation of the left ventricular flow during filling phase. Ann. Biomed. Eng., 33(5):567–576, 2005.CrossRefGoogle Scholar
  29. [29]
    A.J. Chorin. On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comp., 23:341–353, 1969.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    P.G. Ciarlet. Mathematical elasticity. Vol. I, volume 20 of Studies in Mathematics and its Applications. North-Holland, 1988.Google Scholar
  31. [31]
    J. Degroote, S. Annerel, and J. Vierendeels. Stability analysis of Gauss-Seidel iterations in a partitioned simulation of fluid-structure interaction. Comp. & Struct., 88(5–6):263–271, 2010.CrossRefGoogle Scholar
  32. [32]
    J. Degroote, K.-J. Bathe, and J. Vierendeels. Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction. Comp.& Struct., 87(11–12):793–801, 2009.CrossRefGoogle Scholar
  33. [33]
    J. Degroote, A. Swillens, P. Bruggeman, R. Haelterman, P. Segers, and J. Vierendeels. Simulation of fluid-structure interaction with the interface artificial compressibility method. Int. J. Numer. Meth. Biomed. Engng., 26(3–4):276–289, 2010.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    P. Degroote, J. Bruggeman, R. Haelterman, and J. Vierendeels. Stability of a coupling technique for partitioned solvers in FSI applications. Comp. & Struct., 86(23–24):2224–2234, 2008.CrossRefGoogle Scholar
  35. [35]
    S. Deparis. Numerical Analysis of Axisymmetric Flows and Methods for Fluid-Structure Interaction Arising in Blood Flow Simulation. PhD thesis, EPFL, Switzerland, 2004.Google Scholar
  36. [36]
    S. Deparis, M. Discacciati, G. Fourestey, and A. Quarteroni. Fluid-structure algorithms based on Steklov-Poincaré operators. Comput. Methods Appl. Mech. Engrg., 195(41–43):5797–5812, 2006.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    S. Deparis, M.A. Fernández, and L. Formaggia. Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions. M2AN Math. Model. Numer. Anal., 37(4):601–616, 2003.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    W. Dettmer and D. Perić. A computational framework for fluid-rigid body interaction: finite element formulation and applications. Comput. Methods Appl. Mech. Engrg., 195(13–16):1633–1666, 2006.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    N. Diniz dos Santos, J.-F. Gerbeau, and J.-F. Bourgat. A partitioned fluid-structure algorithm for elastic thin valves with contact. Comput. Methods Appl. Mech. Engrg., 197(19–20):1750–1761, 2008.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    J. Donéa, S. Giuliani, and J. P. Halleux. An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comp. Meth. Appl. Mech. Engng., pages 689–723, 1982.Google Scholar
  41. [41]
    C. Farhat, M. Lesoinne, and P. Le Tallec. Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput. Methods Appl. Mech. Engrg., 157:95–114, 1998.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    C. Farhat, K. van der Zee, and Ph. Geuzaine. Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear aeroelasticity. Comput. Methods Appl. Mech. Engrg., 195(17–18):1973–2001, 2006.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    C.A. Felippa, K.C. Park, and C. Farhat. Partitioned analysis of coupled mechanical systems. Comput. Methods Appl. Mech. Engrg., 190(24–25):3247–3270, 2001.MATHCrossRefGoogle Scholar
  44. [44]
    M.A. Fernández. Incremental displacement-correction schemes for the explicit coupling of a thin structure with an incompressible fluid. C. R. Acad. Sci. Paris Sér. I Math., 2011. In press. DOI: 10.1016/j.crma.2011.03.001.Google Scholar
  45. [45]
    M.A. Fernández, L. Formaggia, J.-F. Gerbeau, and A. Quarteroni. The derivation of the equations for fluids and structures. In Cardiovascular mathematics, volume 1 of MS&A. Model. Simul. Appl., pages 77–121. Springer, 2009.Google Scholar
  46. [46]
    M.A. Fernández and J.-F. Gerbeau. Algorithms for fluid-structure interaction problems. In Cardiovascular mathematics, volume 1 of MS&A. Model. Simul. Appl., pages 307–346. Springer, 2009.Google Scholar
  47. [47]
    M.A. Fernández, J.-F. Gerbeau, and C. Grandmont. A projection algorithm for fluid-structure interaction problems with strong added-mass effect. C. R. Math. Acad. Sci. Paris, 342(4):279–284, 2006.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    M.A. Fernández, J.F. Gerbeau, and C. Grandmont. A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Num. Meth. Engrg., 69(4):794–821, 2007.MATHCrossRefGoogle Scholar
  49. [49]
    M.A. Fernández and M. Moubachir. A Newton method using exact Jacobians for solving fluid-structure coupling. Comp. & Struct., 83:127–142, 2005.CrossRefGoogle Scholar
  50. [50]
    M.A. Fernández and J. Mullaert. Displacement-velocity correction schemes for incompressible fluid-structure interaction. Submitted to C. R. Acad. Sci. Paris Sér. I Math., http://hal.inria.fr/inria-00583126/en/.
  51. [51]
    C.A. Figueroa, I.E. Vignon-Clementel, K.E. Jansen, T.J.R. Hughes, and C.A. Taylor. A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput. Methods Appl. Mech. Engrg., 195(41–43):5685–5706, 2006.MathSciNetMATHCrossRefGoogle Scholar
  52. [52]
    L. Formaggia, J.-F. Gerbeau, F. Nobile, and A. Quarteroni. On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comp. Meth. Appl. Mech. Engrg., 191(6–7):561–582, 2001.MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    L. Formaggia, K. Perktold, and A. Quarteroni. Basic mathematical models and motivations. In Cardiovascular mathematics, volume 1 of MS&A. Model. Simul. Appl., pages 47–75. Springer, 2009.Google Scholar
  54. [54]
    C. Förster, W.A. Wall, and E. Ramm. Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Engrg., 196(7):1278–1293, 2007.MathSciNetMATHCrossRefGoogle Scholar
  55. [55]
    G. Fourestey and S. Piperno. A second-order time-accurate ALE Lagrange-Galerkin method applied to wind engineering and control of bridge profiles. Comput. Methods Appl. Mech. Engrg., 193(39–41):4117–4137, 2004.MATHCrossRefGoogle Scholar
  56. [56]
    Y.C. Fung. Biomechanics, Mechanical properties of living tissues. Springer, 1993.Google Scholar
  57. [57]
    M.J. Gander and L. Halpern. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal., 45(2):666–697, 2007.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    M.W. Gee, U. Küttler, and W. Wall. Truly monolithic algebraic multigrid for fluid-structure interaction. Int. J. Numer. Meth. Engng., 2010. To appear.Google Scholar
  59. [59]
    J.-F. Gerbeau and M. Vidrascu. A quasi-Newton algorithm based on a reduced model for fluid-structure interactions problems in blood flows. Math. Model. Num. Anal., 37(4):631–648, 2003.MathSciNetMATHCrossRefGoogle Scholar
  60. [60]
    J.-F. Gerbeau, M. Vidrascu, and P. Frey. Fluid-structure interaction in blood flows on geometries based on medical imaging. Comp. & Struct., 83(2–3):155–165, 2005.CrossRefGoogle Scholar
  61. [61]
    C. Grandmont and Y. Maday. Nonconforming grids for the simulation of fluid-structure interaction. In Domain decomposition methods, 10 (Boulder, CO, 1997), volume 218 of Contemp. Math., pages 262–270. Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  62. [62]
    J.-L. Guermond. Some implementation of projection methods for Navier-Stokes equations. M2AN Math. Model. Numer. Anal., 30:637–667, 1996.MathSciNetMATHGoogle Scholar
  63. [63]
    J. L. Guermond, P. Minev, and J. Shen. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg., 195(44–47):6011–6045, 2006.MathSciNetMATHCrossRefGoogle Scholar
  64. [64]
    G. Guidoboni, R. Glowinski, N. Cavallini, and S. Canic. Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow. J. Comp. Phys., 228(18):6916–6937, 2009.MathSciNetMATHCrossRefGoogle Scholar
  65. [65]
    G. Guidoboni, R. Glowinski, N. Cavallini, S. Canic, and S. Lapin. A kinematically coupled time-splitting scheme for fluid-structure interaction in blood flow. Appl. Math. Lett., 22(5):684–688, 2009.MathSciNetMATHCrossRefGoogle Scholar
  66. [66]
    M.E. Gurtin. An introduction to continuum mechanics, volume 158 of Mathematics in Science and Engineering. Academic Press Inc., 1981.Google Scholar
  67. [67]
    P. Hansbo. Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt., 28(2):183–206, 2005.MathSciNetMATHGoogle Scholar
  68. [68]
    P. Hansbo and J. Hermansson. Nitsche’s method for coupling non-matching meshes in fluid-structure vibration problems. Computational Mechanics, 32(1–2):134–139, 2003.MATHCrossRefGoogle Scholar
  69. [69]
    P. Hansbo, J. Hermansson, and T. Svedberg. Nitsche’s method combined with space-time finite elements for ALE fluid-structure interaction problems. Comput. Methods Appl. Mech. Engrg., 193:4195–4206, 2004.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    M. Heil. An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems. Comput. Methods Appl. Mech. Engrg., 193(1–2):1–23, 2004.MathSciNetMATHCrossRefGoogle Scholar
  71. [71]
    M. Heil, A.L. Hazel, and J. Boyle. Solvers for large-displacement fluid-structure interaction problems: Segregated vs. monolithic approaches. Comp. Mech., 43:91–101, 2008.MATHCrossRefGoogle Scholar
  72. [72]
    B. Hübner, E. Walhorn, and D. Dinkle. A monolithic approach to fluid-structure interaction using space-time finite elements. Comp. Meth. Appl. Mech. Engng., 193:2087–2104, 2004.MATHCrossRefGoogle Scholar
  73. [73]
    E. Järvinen, P. Rȧback, M. Lyly, and J.-P. Salenius. A method for partitioned fluid-structure interaction computation of flow in arteries. Medical Engineering & Physics, 30(7):917–923, 2008.CrossRefGoogle Scholar
  74. [74]
    M.M. Joosten, W.G. Dettmer, and D. Perić. Analysis of the block Gauss-Seidel solution procedure for a strongly coupled model problem with reference to fluid-structure interaction. Internat. J. Numer. Methods Engrg., 78(7):757–778, 2009.MathSciNetMATHCrossRefGoogle Scholar
  75. [75]
    M. Juntunen and R. Stenberg. Nitsche’s method for general boundary conditions. Math. Comp., 78(267):1353–1374, 2009.MathSciNetMATHCrossRefGoogle Scholar
  76. [76]
    S.A. Kock, J.V. Nygaard, N. Eldrup, E.-T. Fründ, A. Klaerke, W.P. Paaske, E. Falk, and W. Yong Kim. Mechanical stresses in carotid plaques using mri-based fluid-structure interaction models. J. Biomech., 41(8):1651–1658, 2008.CrossRefGoogle Scholar
  77. [77]
    U. Küttler, M.W. Gee, C. Förster, A. Comerford, and W.A. Wall. Coupling strategies for biomedical fluid-structure interaction problems. Int. J. Numer. Meth. Biomed. Engng., 26(3–4):305–321, 2009.Google Scholar
  78. [78]
    U. Küttler and W.A. Wall. Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comp. Mech., 43(1):61–72, 2008.MATHCrossRefGoogle Scholar
  79. [79]
    U. Küttler and W.A. Wall. Vector extrapolation for strong coupling fluid-structure interaction solvers. J. App. Mech., 76(2):021205, 2009.CrossRefGoogle Scholar
  80. [80]
    L. Lanoye, J. Vierendeels, P. Segers, and P. Verdonck. Vascular fluid-structure-interaction using Fluent and Abaqus software. J. Biomech., 39 (Supplement 1):S440–S440, 2006.CrossRefGoogle Scholar
  81. [81]
    W. Layton, H. K. Lee, and J. Peterson. A defect-correction method for the incompressible Navier-Stokes equations. Appl. Math. Comput., 129(1):1–19, 2002.MathSciNetMATHCrossRefGoogle Scholar
  82. [82]
    P. Le Tallec. Numerical methods for nonlinear three-dimensional elasticity. In Handbook of numerical analysis, Vol. III, pages 465–622. North-Holland, 1994.CrossRefGoogle Scholar
  83. [83]
    P. Le Tallec and P. Hauret. Energy conservation in fluid structure interactions. In Numerical methods for scientific computing. Variational problems and applications, pages 94–107. Internat. Center Numer. Methods Eng. (CIMNE), Barcelona, 2003.Google Scholar
  84. [84]
    P. Le Tallec and S. Mani. Numerical analysis of a linearised fluid-structure interaction problem. Numer. Math., 87(2):317–354, 2000.MathSciNetMATHCrossRefGoogle Scholar
  85. [85]
    P. Le Tallec and J. Mouro. Fluid structure interaction with large structural displacements. Comput. Meth. Appl. Mech. Engrg., 190:3039–3067, 2001.MATHCrossRefGoogle Scholar
  86. [86]
    M. Lesoinne and C. Farhat. Higher-order subiteration-free staggered algorithm for nonlinear transient aeroelastic problems. AIAA Journal, 36(9):1754–1757, 1998.CrossRefGoogle Scholar
  87. [87]
    Z. Li and C. Kleinstreuer. Computational analysis of type II endoleaks in a stented abdominal aortic aneurysm model. J. Biomech., 39(14):2573–2582, 2006.CrossRefGoogle Scholar
  88. [88]
    Y. Maday. Analysis of coupled models for fluid-structure interaction of internal flows. In Cardiovascular mathematics, volume 1 of MS&A. Model. Simul. Appl., pages 279–306. Springer, 2009.Google Scholar
  89. [89]
    H.G. Matthies and J. Steindorf. Partitioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. Comp. & Struct., 80(27–30):1991–1999, 2002.CrossRefGoogle Scholar
  90. [90]
    D.P. Mok and W.A. Wall. Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures. In W.A. Wall, K.U. Bletzinger, and K. Schweizerhof, editors, Trends in computational structural mechanics, Barcelona, 2001. CIMNE.Google Scholar
  91. [91]
    D.P. Mok, W.A. Wall, and E. Ramm. Partitioned analysis approach for the transient, coupled response of viscous fluids and flexible structures. In W. Wunderlich, editor, Proceedings of the European Conference n Computational Mechanics. ECCM’99, TU Munich, 1999.Google Scholar
  92. [92]
    D.P. Mok, W.A. Wall, and E. Ramm. Accelerated iterative substructuring schemes for instationary fluid-structure interaction. In Bathe. K.J., editor, Computational Fluid and Solid Mechanics, pages 1325–1328. Elsevier, 2001.CrossRefGoogle Scholar
  93. [93]
    H. Morand and R. Ohayon. Fluid-Structure Interaction: Applied Numerical Methods. John Wiley & Sons, 1995.Google Scholar
  94. [94]
    J. Mouro. Interactions fluide structure en grands déplacements. Résolution numérique et application aux composants hydrauliques automobiles. PhD thesis, Ecole Polytechnique, France, 1996.Google Scholar
  95. [95]
    J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36:9–15, 1971.MathSciNetMATHCrossRefGoogle Scholar
  96. [96]
    F. Nobile. Numerical approximation of fluid-structure interaction problems with application to haemodynamics. PhD thesis, EPFL, Switzerland, 2001.Google Scholar
  97. [97]
    F. Nobile and C. Vergara. An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. SIAM J. Sci. Comput., 30(2):731–763, 2008.MathSciNetMATHCrossRefGoogle Scholar
  98. [98]
    T. Nomura and T.J.R. Hughes. An arbitray Lagrangian-Eulerian finite element method for interaction of fluid and rigid body. Comput. Methods Appl. Mech. Eng., 95(1):115–138, 1992.MATHCrossRefGoogle Scholar
  99. [99]
    S. Piperno. Simulation numérique de phénomènes d’interaction fluide-structure. PhD thesis, Ecole Nationale des Ponts et Chaussées, France, 1995.Google Scholar
  100. [100]
    S. Piperno. Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations. Internat. J. Numer. Methods Fluids, 25(10):1207–1226, 1997.MathSciNetMATHCrossRefGoogle Scholar
  101. [101]
    S. Piperno and P.E. Bournet. Numerical simulations of wind effects on flexible civil engineering structures. Rev. Eur. Élém. Finis, 8(5–6):659–687, 2001.Google Scholar
  102. [102]
    S. Piperno and C. Farhat. Design of efficient partitioned procedures for the transient solution of aeroelastic problems. In Fluid-structure interaction, Innov. Tech. Ser., pages 23–49. Kogan Page Sci., 2003.Google Scholar
  103. [103]
    A. Quaini and A. Quarteroni. A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method. Math. Models Methods Appl. Sci., 17(6):957–983, 2007.MathSciNetMATHCrossRefGoogle Scholar
  104. [104]
    K. Riemslagh, J. Vierendeels, and E. Dick. An efficient coupling procedure for flexible wall fluid-structure interaction. In ECCOMAS 2000, Barcelona, 2000.Google Scholar
  105. [105]
    S. Rugonyi and K.J. Bathe. On finite element analysis of fluid flows coupled with structural interaction. CMES — Comp. Modeling Eng. Sci., 2(2):195–212, 2001.Google Scholar
  106. [106]
    A.-V. Salsac, S.R. Sparks, J.M. Chomaz, and J.C. Lasheras. Evolution of the wall shear stresses during the progressive enlargement of symmetric abdominal aortic aneurysms. J. Fluid Mech., 550:19–51, 2006.CrossRefGoogle Scholar
  107. [107]
    M. Schäfer, M. Heck, and S. Yigit. An implicit partitioned method for the numerical simulation of fluid-structure interaction. In Fluid-structure interaction, volume 53 of Lect. Notes Comput. Sci. Eng., pages 171–194. Springer, 2006.Google Scholar
  108. [108]
    J. Sokolowski and J.-P. Zolésio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer, 1992.Google Scholar
  109. [109]
    H.J. Stetter. The defect correction principle and discretization methods. Numer. Math, 29:425–443, 1978.MathSciNetMATHCrossRefGoogle Scholar
  110. [110]
    E. W. Swim and P. Seshaiyer. A nonconforming finite element method for fluid-structure interaction problems. Comput. Methods Appl. Mech. Engrg., 195(17–18):2088–2099, 2006.MathSciNetMATHCrossRefGoogle Scholar
  111. [111]
    S. Sy and C.M. Murea. A stable time advancing scheme for solving fluid-structure interaction problem at small structural displacements. Comput. Methods Appl. Mech. Engrg., 198(2):210–222, 2008.MathSciNetMATHCrossRefGoogle Scholar
  112. [112]
    R. Temam. Une methode d’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France, 96:115–152, 1968.MathSciNetMATHGoogle Scholar
  113. [113]
    T.E Tezduyar. Finite element methods for fluid dynamics with moving boundaries and interfaces. Arch. Comput. Methods Engrg., 8:83–130, 2001.MATHCrossRefGoogle Scholar
  114. [114]
    T.E. Tezduyar and S. Sathe. Modelling of fluid-structure interactions with the space-time finite elements: solution techniques. Internat. J. Numer. Methods Fluids, 54(6–8):855–900, 2007.MathSciNetMATHCrossRefGoogle Scholar
  115. [115]
    M. Thiriet. Biology and Mechanics of Blood Flows. Part II: Mechanics and Medical Aspects. CRM Series in Mathematical Physics. Springer, 2008.Google Scholar
  116. [116]
    J. Vierendeels, Lanoye. L., J. Degroote, and P. Verdonck. Implicit coupling of partitioned fluid-structure interaction problems with reduced order models. Comp. & Struct., 85(11–14):970–976, 2007.CrossRefGoogle Scholar
  117. [117]
    W.A. Wall, S. Genkinger, and E. Ramm. A strong coupling partitioned approach for fluid-structure interaction with free surfaces. Comput. & Fluids, 36(1):169–183, 2007.MATHCrossRefGoogle Scholar
  118. [118]
    H. Watanabe, S. Sugiura, H. Kafuku, and T. Hisada. Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method. Biophysical Journal, 87(3):2074–2085, 2004.CrossRefGoogle Scholar
  119. [119]
    M.F. Wheeler. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal., 15(1):152–161, 1978.MathSciNetMATHCrossRefGoogle Scholar
  120. [120]
    L. Wiechert, T. Rabczuk, A. Comerford, R. Metzke, and W.A. Wall. Towards stresses and strains in the respiratory system. In Mathematical and numerical modelling of the human lung, volume 23 of ESAIM Proc., pages 98–113. EDP Sci., 2008.Google Scholar
  121. [121]
    X.L. Yang, Y. Liu, and J.M. Yang. Fluid-structure interaction in a pulmonary arterial bifurcation. J. Biomech., 40(12):2694–2699, 2007.CrossRefGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2011

Authors and Affiliations

  1. 1.INRIA, RocquencourtLe Chesnay CedexFrance

Personalised recommendations