Computational Mechanics

, Volume 54, Issue 4, pp 1055–1071 | Cite as

Fluid–structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation

  • Ming-Chen HsuEmail author
  • David Kamensky
  • Yuri Bazilevs
  • Michael S. Sacks
  • Thomas J. R. Hughes
Original Paper


We propose a framework that combines variational immersed-boundary and arbitrary Lagrangian–Eulerian methods for fluid–structure interaction (FSI) simulation of a bioprosthetic heart valve implanted in an artery that is allowed to deform in the model. We find that the variational immersed-boundary method for FSI remains robust and effective for heart valve analysis when the background fluid mesh undergoes deformations corresponding to the expansion and contraction of the elastic artery. Furthermore, the computations presented in this work show that the arterial wall deformation contributes significantly to the realism of the simulation results, leading to flow rates and valve motions that more closely resemble those observed in practice.


Fluid–structure interaction Bioprosthetic heart valve Variational immersed-boundary method Arbitrary Lagrangian–Eulerian formulation Isogeometric analysis Arterial wall deformation 



Y. Bazilevs was supported by the NSF CAREER Award No. 1055091. T. J. R. Hughes was supported by grants from the Office of Naval Research (N00014-08-1-0992), the National Science Foundation (CMMI-01101007), and SINTEF (UTA10-000374) with the University of Texas at Austin. M. S. Sacks was supported by NIH/NHLBI grants R01 HL108330 and HL119297, and FDA contract HHSF223201111595P. D. Kamensky was partially supported by the CSEM Graduate Fellowship. We thank the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing HPC resources that have contributed to the research results reported in this paper.


  1. 1.
    Schoen FJ, Levy RJ (2005) Calcification of tissue heart valve substitutes: progress toward understanding and prevention. Ann Thorac Surg 79(3):1072–1080CrossRefGoogle Scholar
  2. 2.
    Pibarot P, Dumesnil JG (2009) Prosthetic heart valves: selection of the optimal prosthesis and long-term management. Circulation 119(7):1034–1048CrossRefGoogle Scholar
  3. 3.
    Siddiqui RF, Abraham JR, Butany J (2009) Bioprosthetic heart valves: modes of failure. Histopathology 55:135–144CrossRefGoogle Scholar
  4. 4.
    Hales S (1733) Statical essays: containing haemastaticks: or, an account of some hydraulick and hydrostatical experiments made on the blood and blood-vessels of animals. W. Innys and R. Manby; T. Woodward, LondonGoogle Scholar
  5. 5.
    Frank O (1899) Die grundform des arteriellen pulses. Erste abhandlung. Mathematische analyse. Zeitschrift für Biologie 37:485–526Google Scholar
  6. 6.
    Sagawa K, Lie RK, Schaefer J (1990) Translation of otto Frank’s paper “Die Grundform des arteriellen pulses” Zeitschrift für Biologie 37: 483–526 (1899). J Mol Cell Cardiol 22(3):253–254Google Scholar
  7. 7.
    Frank O (1990) The basic shape of the arterial pulse. first treatise: mathematical analysis. J Mol Cell Cardiol 22(3):255–277CrossRefGoogle Scholar
  8. 8.
    Westerhof N, Lankhaar J-W, Westerhof BE (2009) The arterial Windkessel. Med Biol Eng Comput 47(2):131–141CrossRefGoogle Scholar
  9. 9.
    Westerhof N, Bosman F, De Vries CJ, Noordergraaf A (1969) Analog studies of the human systemic arterial tree. J Biomech 2(2):121–143CrossRefGoogle Scholar
  10. 10.
    Stergiopulos N, Westerhof BE, Westerhof N (1999) Total arterial inertance as the fourth element of the windkessel model. Am J Physiol 276:H81–88Google Scholar
  11. 11.
    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 Methods Appl Mech Eng 195:3776–3796zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Kamensky D, Hsu M-C, Schillinger D, Evans JA, Aggarwal A, Bazilevs Y, Sacks MS, Hughes TJR (2014) A variational immersed boundary framework for fluid-structure interaction: Isogeometric implementation and application to bioprosthetic heart valves. Comput Methods Appl Mech Eng. In review. Also appeared as ICES REPORT 14–12, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, May 2014Google Scholar
  13. 13.
    Sotiropoulos F, Yang X (2014) Immersed boundary methods for simulating fluid-structure interaction. Prog Aerosp Sci 65:1–21CrossRefGoogle Scholar
  14. 14.
    Mittal R, Iaccarino G (2005) Immersed boundary methods. Ann Rev Fluid Mech 37:239–261MathSciNetCrossRefGoogle Scholar
  15. 15.
    Peskin CS (2002) The immersed boundary method. Acta Numerica 11:479–517zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    De Hart J, Peters GWM, Schreurs PJG, Baaijens FPT (2003) A three-dimensional computational analysis of fluid-structure interaction in the aortic valve. J Biomech 36:103–112CrossRefGoogle Scholar
  17. 17.
    De Hart J, Baaijens FPT, Peters GWM, Schreurs PJG (2003) A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve. J Biomech 36:699–712CrossRefGoogle Scholar
  18. 18.
    Astorino M, Gerbeau J-F, Pantz O, Traoré K-F (2009) Fluid-structure interaction and multi-body contact: application to aortic valves. Comput Methods Appl Mech Eng 198:3603–3612zbMATHCrossRefGoogle Scholar
  19. 19.
    Astorino M, Hamers J, Shadden SC, Gerbeau J-F (2012) A robust and efficient valve model based on resistive immersed surfaces. Int J Numer Method Biomed Eng 28(9):937–959MathSciNetCrossRefGoogle Scholar
  20. 20.
    Griffith BE (2012) Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. Int J Numer Method Biomed Eng 28:317–345zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Borazjani I (2013) Fluid-structure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves. Comput Methods Appl Mech Eng 257(0):103–116Google Scholar
  22. 22.
    Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29:329–349zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Donea J, Giuliani S, Halleux JP (1982) An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput Methods Appl Mech Eng 33(1–3):689–723zbMATHCrossRefGoogle Scholar
  24. 24.
    Donea J, Huerta A, Ponthot J-P, Rodriguez-Ferran A (2004) Arbitrary Lagrangian-Eulerian methods. In encyclopedia of computational mechanics, Vol 3 Fluids, chapter 14. WileyGoogle Scholar
  25. 25.
    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(3):339–351zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    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(3):353–371zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Takizawa K, Tezduyar TE (2011) Multiscale space-time fluid-structure interaction techniques. Comput Mech 48:247–267zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26(10):27–36CrossRefGoogle Scholar
  29. 29.
    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 119:73–94zbMATHCrossRefGoogle Scholar
  30. 30.
    Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid-structure interactions with large displacements. J Appl Mech 70:58–63zbMATHCrossRefGoogle Scholar
  31. 31.
    Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193:2019–2032zbMATHCrossRefGoogle Scholar
  32. 32.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43:3–37zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    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– 1218Google Scholar
  34. 34.
    Tezduyar TE (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch Comput Methods Eng 8:83–130zbMATHCrossRefGoogle Scholar
  35. 35.
    Akin JE, Tezduyar TE, Ungor M (2007) Computation of flow problems with the mixed interface-tracking/interface-capturing technique (MITICT). Comput Fluids 36:2–11zbMATHCrossRefGoogle Scholar
  36. 36.
    Cruchaga MA, Celentano DJ, Tezduyar TE (2007) A numerical model based on the mixed interface-tracking/interface-capturing technique (MITICT) for flows with fluid-solid and fluid-fluid interfaces. Int J Numer Methods Fluids 54:1021–1030zbMATHCrossRefGoogle Scholar
  37. 37.
    Akkerman I, Bazilevs Y, Benson DJ, Farthing MW, Kees CE ( 2011) Free-surface flow and fluid-object interaction modeling with emphasis on ship hydrodynamics. J Appl Mech, Accepted for publicationGoogle Scholar
  38. 38.
    Wick T (2013) Coupling of fully Eulerian and arbitrary Lagrangian–Eulerian methods for fluid-structure interaction computations. Comput Mech, 52(5)Google Scholar
  39. 39.
    Wick T (2014) Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity. Comput Mech 53(1):29–43zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Bazilevs Y, Hsu M-C, Scott MA (2012) Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput Methods Appl Mech Eng 249–252:28–41MathSciNetCrossRefGoogle Scholar
  41. 41.
    Formaggia L, Gerbeau JF, Nobile F, Quarteroni A (2001) On the coupling of 3D and 1D Navier-stokes equations for flow problems in compliant vessels. Comput Methods Appl Mech Eng 191:561–582zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Gerbeau J-F, Vidrascu M, Frey P (2005) Fluid-structure interaction in blood flows on geometries based on medical imaging. Comput Struct 83:155–165CrossRefGoogle Scholar
  43. 43.
    Nobile F, Vergara C (2008) An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. SIAM J Sci Comput 30:731–763zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Bazilevs Y, Hsu M-C, Zhang Y, Wang W, Kvamsdal T, Hentschel S, Isaksen J (2010) Computational fluid-structure interaction: methods and application to cerebral aneurysms. Biomech Model Mechanobiol 9:481–498CrossRefGoogle Scholar
  45. 45.
    Perego M, Veneziani A, Vergara C (2011) A variational approach for estimating the compliance of the cardiovascular tissue: an inverse fluid-structure interaction problem. SIAM J Sci Comput 33:1181–1211zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Takizawa K, Bazilevs Y, Tezduyar TE (2012) Space-time and ALE-VMS techniques for patient-specific cardiovascular fluid-structure interaction modeling. Arch Comput Methods Eng 19:171–225MathSciNetCrossRefGoogle Scholar
  47. 47.
    Nitsche J (1971) Uber ein variationsprinzip zur losung von Dirichlet-problemen bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind. Abh Math Univ Hamburg 36:9–15zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Hsu M-C, Bazilevs Y (2012) Fluid-structure interaction modeling of wind turbines: simulating the full machine. Comput Mech 50:821–833zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput Mech 38:310–322zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    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 198:3534–3550zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Bazilevs Y, Hsu M-C, Benson D, Sankaran S, Marsden A (2009) Computational fluid-structure interaction: methods and application to a total cavopulmonary connection. Comput Mech 45:77– 89Google Scholar
  52. 52.
    Bazilevs Y, Hsu M-C, Zhang Y, Wang W, Liang X, Kvamsdal T, Brekken R, Isaksen J (2010) A fully-coupled fluid-structure interaction simulation of cerebral aneurysms. Comput Mech 46:3– 16Google Scholar
  53. 53.
    Zhang Y, Wang W, Liang X, Bazilevs Y, Hsu M-C, Kvamsdal T, Brekken R, Isaksen JG (2009) High-fidelity tetrahedral mesh generation from medical imaging data for fluid-structure interaction analysis of cerebral aneurysms. Comput Model Eng Sci 42:131–150Google Scholar
  54. 54.
    Hsu M-C, Bazilevs Y (2011) Blood vessel tissue prestress modeling for vascular fluid-structure interaction simulations. Finite Elem Anal Des 47:593–599MathSciNetCrossRefGoogle Scholar
  55. 55.
    Bazilevs Y, Hughes TJR (2007) Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput Fluids 36:12–26zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Bazilevs Y, Michler C, Calo VM, Hughes TJR (2007) Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput Methods Appl Mech Eng 196:4853–4862zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Bazilevs Y, Michler C, Calo VM, Hughes TJR (2010) Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Comput Methods Appl Mech Eng 199:780–790zbMATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Höllig KH (2003) Finite element methods with B-splines. SIAM, PhiladelphiazbMATHCrossRefGoogle Scholar
  59. 59.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, ChichesterCrossRefGoogle Scholar
  61. 61.
    Bazilevs Y, Hsu M-C, Takizawa K, Tezduyar TE (2012) ALE-VMS and ST-VMS methods for computer modeling of wind-turbine rotor aerodynamics and fluid-structure interaction. Math Models Methods Appl Sci 22:1230002CrossRefGoogle Scholar
  62. 62.
    Takizawa K, Bazilevs Y, Tezduyar TE, Hsu M-C, Øiseth O, Mathisen KM, Kostov N, McIntyre S (2014) Engineering analysis and design with ALE-VMS and Space-Time methods. Arch Comput Methods Eng. doi: 10.1007/s11831-014-9113-0
  63. 63.
    Bazilevs Y, Takizawa K, Tezduyar TE, Hsu M-C, Kostov N, McIntyre S (2014) Aerodynamic and FSI analysis of wind turbines with the ALE-VMS and ST-VMS methods. Arch Comput Methods Eng. doi: 10.1007/s11831-014-9119-7
  64. 64.
    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 32:199–259zbMATHMathSciNetCrossRefGoogle Scholar
  65. 65.
    Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44zbMATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190:411–430zbMATHCrossRefGoogle Scholar
  67. 67.
    Hughes TJR, Mazzei L, Oberai AA, Wray A (2001) The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence. Phys Fluids 13:505–512CrossRefGoogle Scholar
  68. 68.
    Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43:555–575zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Hughes TJR, Scovazzi G, Franca LP (2004) Multiscale and stabilized methods. In E. Stein, R. de Borst, and TJR Hughes (eds), Encyclopedia of Computational Mechanics, vol 3 Fluids, chapter 2. WileyGoogle Scholar
  70. 70.
    Bazilevs Y, Calo VM, Cottrel 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 197:173–201zbMATHCrossRefGoogle Scholar
  71. 71.
    Tezduyar TE, Sathe S (2007) Modelling of fluid-structure interactions with the space-time finite elements: solution techniques. Int J Numer Methods Fluids 54(6–8):855–900zbMATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    Hsu M-C, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2010) Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comput Methods Appl Mech Eng 199:828–840zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Johnson C (1987) Numerical solution of partial differential equations by the finite element method. Cambridge University Press, SwedenzbMATHGoogle Scholar
  74. 74.
    Brenner SC, Scott LR (2002) The mathematical theory of finite element methods, 2nd edn. Springer, BerlinzbMATHCrossRefGoogle Scholar
  75. 75.
    Ern A, Guermond JL (2004) Theory and practice of finite elements. Springer, BerlinzbMATHCrossRefGoogle Scholar
  76. 76.
    Evans JA, Hughes TJR (2013) Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Comput Methods Appl Mech Eng 123:259–290zbMATHMathSciNetGoogle Scholar
  77. 77.
    Hsu M-C, Akkerman I, Bazilevs Y (2014) Finite element simulation of wind turbine aerodynamics: validation study using NREL Phase VI experiment. Wind Energy 17:461–481CrossRefGoogle Scholar
  78. 78.
    Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of blood flow in arteries. Comput Methods Appl Mech Eng 158:155–196zbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    Rispoli F, Corsini A, Tezduyar TE (2007) Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Comput Fluids 36:121–126zbMATHCrossRefGoogle Scholar
  80. 80.
    Tezduyar TE (2007) Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. Comput Fluids 36:191–206zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    Tezduyar TE, Senga M (2007) SUPG finite element computation of inviscid supersonic flows with YZ \(\beta \) shock-capturing. Comput Fluids 36:147–159zbMATHCrossRefGoogle Scholar
  82. 82.
    Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2007) YZ \(\beta \) discontinuity-capturing for advection-dominated processes with application to arterial drug delivery. Int J Numer Methods Fluids 54:593–608zbMATHMathSciNetCrossRefGoogle Scholar
  83. 83.
    Catabriga L, de Souza DAF, Coutinho ALGA, Tezduyar TE (2009) Three-dimensional edge-based SUPG computation of inviscid compressible flows with YZ \(\beta \) shock-capturing. J Appl Mech 76:021208CrossRefGoogle Scholar
  84. 84.
    Bazilevs Y, Akkerman I (2010) Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method. J Computat Phys 229:3402–3414zbMATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Hsu M-C, Akkerman I, Bazilevs Y (2012) Wind turbine aerodynamics using ALE-VMS: validation and the role of weakly enforced boundary conditions. Comput Mech 50:499–511zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Esmaily-Moghadam M, Bazilevs Y, Hsia T-Y, Vignon-Clementel IE, Marsden AL (2011) Modeling of congenital hearts alliance (MOCHA). A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Comput Mech 48:277–291Google Scholar
  87. 87.
    Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  88. 88.
    Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199:357–373zbMATHCrossRefGoogle Scholar
  89. 89.
    Trefethen LN (2012) Gibbs phenomenon. In Approximation theory and approximation practice, chapter 9. SIAM, Philadelphia, Pennsylvania, USAGoogle Scholar
  90. 90.
    Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems. J Comput Phys 135(2):118–125MathSciNetCrossRefGoogle Scholar
  91. 91.
    Kiendl J, Bletzinger K-U, Linhard J, Wüchner R (2009) Isogeometric shell analysis with Kirchhoff-Love elements. Comput Methods Appl Mech Eng 198:3902–3914zbMATHCrossRefGoogle Scholar
  92. 92.
    Kiendl J, Bazilevs Y, Hsu M-C, Wüchner R, Bletzinger K-U (2010) The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches. Comput Methods Appl Mech Eng 199:2403–2416zbMATHCrossRefGoogle Scholar
  93. 93.
    Kiendl J (2011) Isogeometric analysis and shape optimal design of shell structures. PhD thesis, Lehrstuhl für Statik, Technische Universität MünchenGoogle Scholar
  94. 94.
    Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, ChichesterGoogle Scholar
  95. 95.
    Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-\(\alpha \) method. J Appl Mech 60:371–375zbMATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    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 190:305–319Google Scholar
  97. 97.
    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 195:5743–5753zbMATHMathSciNetCrossRefGoogle Scholar
  98. 98.
    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 195:2002–2027zbMATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    Tezduyar TE, Sathe S (2007) Modeling of fluid-structure interactions with the space-time finite elements: solution techniques. Int J Numer Methods Fluids 54:855–900zbMATHMathSciNetCrossRefGoogle Scholar
  100. 100.
    Bazilevs Y, Takizawa K, Tezduyar TE (2013) Computational fluid-structure interaction: methods and applications. Wiley, ChichesterCrossRefGoogle Scholar
  101. 101.
    Wriggers P (1995) Finite element algorithms for contact problems. Arch Comput Methods Eng 2:1–49MathSciNetCrossRefGoogle Scholar
  102. 102.
    Wriggers P (2006) Computational contact mechanics, 2nd edn. Springer, BerlinzbMATHCrossRefGoogle Scholar
  103. 103.
    Laursen TA (2003) Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, BerlinCrossRefGoogle Scholar
  104. 104.
    De Lorenzis L, Temizer İ, Wriggers P, Zavarise G (2011) A large deformation frictional contact formulation using NURBS-based isogeometric analysis. Int J Numer Methods Fluids 87:1278–1300zbMATHGoogle Scholar
  105. 105.
    Dimitri R, De Lorenzis L, Scott MA, Wriggers P, Taylor RL, Zavarise G (2014) Isogeometric large deformation frictionless contact using T-splines. Comput Methods Appl Mech Eng 269:394–414CrossRefGoogle Scholar
  106. 106.
    Bellhouse BJ, Bellhouse FH (1968) Mechanism of closure of the aortic valve. Nature 217(5123):86–87CrossRefGoogle Scholar
  107. 107.
    Sun W, Abad A, Sacks MS (2005) Simulated bioprosthetic heart valve deformation under quasi-static loading. J Biomech Eng 127(6):905–914CrossRefGoogle Scholar
  108. 108.
    Yap CH, Saikrishnan N, Tamilselvan G, Yoganathan AP (2011) Experimental technique of measuring dynamic fluid shear stress on the aortic surface of the aortic valve leaflet. J Biomech Eng 133(6):061007CrossRefGoogle Scholar
  109. 109.
    Felner JM (1990) The second heart sound. In Clinical methods: the history, physical, and laboratory, 3rd edn, chapter 23. Butterworths, Boston, USAGoogle Scholar
  110. 110.
    Sabbah HN, Stein PD (1978) Relation of the second sound to diastolic vibration of the closed aortic valve. Am J Physiol Heart Circ Physiol 234(6):H696–H700Google Scholar
  111. 111.
    Kendall ME, Rembert JC, Greenfield JC Jr (1973) Pressure-flow studies in man: the nature of the aortic flow pattern in both valvular mitral insufficiency and the prolapsing mitral valve syndrome. Am Heart J 86(3):359–365CrossRefGoogle Scholar
  112. 112.
    Uther JB, Peterson KL, Shabetai R, Braunwald E (1973) Measurement of ascending aortic flow patterns in man. J Appl Physiol 34(4):513–518Google Scholar
  113. 113.
    Kim H, Lu J, Sacks MS, Chandran KB (2008) Dynamic simulation of bioprosthetic heart valves using a stress resultant shell model. Ann Biomed Eng 36(2):262–275CrossRefGoogle Scholar
  114. 114.
    Nollert G, Miksch J, Kreuzer E, Reichart B (2003) Risk factors for atherosclerosis and the degeneration of pericardial valves after aortic valve replacement. J Thorac Cardiovasc Surg 126(4):965–968Google Scholar
  115. 115.
    Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer, New YorkCrossRefGoogle Scholar
  116. 116.
    Tong P, Fung Y-C (1976) The stress-strain relationship for the skin. J Biomech 9(10):649–657CrossRefGoogle Scholar
  117. 117.
    Fung YC (1993) Biomechanics: mechanical properties of living tissues, second edition edn. Springer, New YorkGoogle Scholar
  118. 118.
    Iyengar AKS, Sugimoto H, Smith DB, Sacks MS (2001) Dynamic in vitro quantification of bioprosthetic heart valve leaflet motion using structured light projection. Ann Biomed Eng 29(11):963–973Google Scholar
  119. 119.
    Sugimoto H, Sacks MS (2013) Effects of leaflet stiffness on in vitro dynamic bioprosthetic heart valve leaflet shape. Cardiovasc Eng Tech 4(1):2–15CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ming-Chen Hsu
    • 1
    Email author
  • David Kamensky
    • 2
  • Yuri Bazilevs
    • 3
  • Michael S. Sacks
    • 2
  • Thomas J. R. Hughes
    • 2
  1. 1.Department of Mechanical EngineeringIowa State UniversityAmesUSA
  2. 2.Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  3. 3.Department of Structural Engineering University of California, San DiegoLa JollaUSA

Personalised recommendations