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Extended Finite Elements Method for Fluid-Structure Interaction with an Immersed Thick Non-linear Structure

  • Christian Vergara
  • Stefano ZoncaEmail author
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 16)

Abstract

We consider an Extended Finite Element method to solve fluid-structure interaction problems in the case of an immersed thick structure described by non-linear finite elasticity. This method, that belongs to the family of the Cut Finite Element methods, allows to consider unfitted meshes for the fluid and solid domains by maintaining the fluid mesh fixed in time as the solid moves. We review the state of the art about the numerical methods for fluid-structure interaction problems and we present an overview of the Cut Finite Element methods. We describe the numerical discretization proposed here to handle the case of a thick immersed structure with size comparable or smaller than the fluid mesh element size in the case of non-linear finite elasticity. Finally, we present some three-dimensional numerical results of the proposed method.

Notes

Acknowledgements

The authors gratefully acknowledge the financial support of the Italian MIUR by the grant PRIN12, number 201289A4LX, “Mathematical and numerical models of the cardiovascular system, and their clinical applications”. C. Vergara has been partially supported by the H2020-MSCA-ITN-2017, EU project 765374 “ROMSOC—Reduced Order Modelling, Simulation and Optimization of Coupled systems”.

References

  1. 1.
    Alauzet, F., Fabrèges, B., Fernández, M.A., Landajuela, M.: Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures. Comput. Methods Appl. Mech. Eng. 301, 300–335 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aletti, M., Gerbeau, J.-F., Lombardi, D.: Modeling autoregulation in three-dimensional simulations of retinal hemodynamics. J. Model. Ophthalmol. 1, 88–115 (2015)Google Scholar
  3. 3.
    Annavarapu, C., Hautefeuille, M., Dolbow, J.E.: A robust Nitsche’s formulation for interface problems. Comput. Methods Appl. Mech. Eng. 225–228, 44–54 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arciero, J., Harris, A., Siesky, B., Amireskandari, A., Gershuny, V., Pickrell, A., Guidoboni, G.: Theoretical analysis of vascular regulatory mechanisms contributing to retinal blood flow autoregulation mechanisms contributing to retinal autoregulation. Invest. Ophthalmol. Vis. Sci. 54(8), 5584–5593 (2013)CrossRefGoogle Scholar
  5. 5.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Astorino, M., Gerbeau, J.-F., Pantz, O., Traoré, K.-F.: Fluid-structure interaction and multi-body contact: application to the aortic valves. Comput. Methods Appl. Mech. Eng. 198, 3603–3612 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Basting, S., Quaini, A., Čanić, S., Glowinski, R.: Extended ALE Method for fluid–structure interaction problems with large structural displacements. J. Comput. Phys. 331, 312–336 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bazilevs, Y., Hsu, M.-C., Kiendl, J., Wüchner, R., Bletzinger, K.-U.: 3D simulation of wind turbine rotors at full scale. Part II: fluid–structure interaction modeling with composite blades. Int. J. Numer. Methods Fluids 65(1–3), 236–253 (2011)zbMATHGoogle Scholar
  9. 9.
    Becker, R., Burman, E., Hansbo, A.: A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198(41–44), 3352–3360 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Beckert, A., Wendland, H.: Multivariate interpolation for fluid-structure-interaction problems using radial basis functions. Aerosol Sci. Technol. 5(2), 125–134 (2001)CrossRefGoogle Scholar
  11. 11.
    Belytschko, T., Moës, N., Usui, S., Parimi, C.: Arbitrary discontinuities in finite elements. Int. J. Numer. Methods Eng. 50(4), 993–1013 (2001)CrossRefGoogle Scholar
  12. 12.
    Benedettini, F., Rega, G., Alaggio, R.: Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions. J. Sound Vib. 182(5), 775–798 (1995)CrossRefGoogle Scholar
  13. 13.
    Bertrand, F., Tanguy, P.A., Thibault, F.: A three-dimensional fictitious domain method for incompressible fluid flow problems. Int. J. Numer. Methods Fluids 25(6), 719–736 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Boffi, D., Gastaldi, L: A finite element approach for the immersed boundary method. Comput. Struct. 81(8–11), 491–501 (2003). K.J Bathe 60th Anniversary IssueMathSciNetCrossRefGoogle Scholar
  15. 15.
    Boffi, D., Gastaldi, L., Heltai, L.: Numerical stability of the finite element immersed boundary method. Math. Models Methods Appl. Sci. 17(10), 1479–1505 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Boffi, D., Gastaldi, L., Heltai, L., Peskin, C.: On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Eng. 197(25–28), 2210–2231 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Borazjani, I., Ge, L., Sotiropoulos, F.: Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies. J. Comput. Phys. 227(16), 7587–7620 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Borazjani, I., Ge, L., Sotiropoulos, F.: High-resolution fluid–structure interaction simulations of flow through a bi-leaflet mechanical heart valve in an anatomic aorta. Ann. Biomed. Eng. 38(2), 326–344 (2010)CrossRefGoogle Scholar
  19. 19.
    Braun, A.L., Awruch, A.M.: Finite element simulation of the wind action over bridge sectional models: Application to the Guamá river bridge (Pará State, Brazil). Finite Elem. Anal. Des. 44(3), 105–122 (2008)CrossRefGoogle Scholar
  20. 20.
    Burman, E.: Ghost penalty. C. R. Math. Acad. Sci. Paris 348(21–22), 1217–1220 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Burman, E., Fernández, M.A.: Stabilized explicit coupling for fluid-structure interaction using Nitsche’s method. C. R. Acad. Sci. Paris Sér. I Math. 345, 467–472 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Burman, E., Fernández, M.A.: An unfitted Nitsche method for incompressible fluid-structure interaction using overlapping meshes. Comput. Methods Appl. Mech. Eng. 279, 497–514 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for Oseen’s equations. SIAM J. Numer. Anal. 44(3), 1248–1274 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104(7), 472–501 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    De Hart, J., Baaijens, F.P.T., Peters, G.W.M., Schreurs, P.J.G.: A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve. J. Biomech. 36(5), 699–712 (2003). Cardiovascular BiomechanicsGoogle Scholar
  26. 26.
    De Hart, J., Peters, G.W.M., Schreurs, P.J.G., Baaijens, F.P.T.: A three-dimensional computational analysis of fluid–structure interaction in the aortic valve. J. Biomech. 36(1), 103–112 (2003)CrossRefGoogle Scholar
  27. 27.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69. Springer, Berlin (2012)Google Scholar
  28. 28.
    Donea, J.: An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interaction. Comput. Methods Appl. Mech. Eng. 33, 689–723 (1982)CrossRefGoogle Scholar
  29. 29.
    Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, New York (2003)CrossRefGoogle Scholar
  30. 30.
    Douglas, J., Dupont, T.: Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, pp. 207–216. Springer, Berlin (1976)Google Scholar
  31. 31.
    Dowell, E.H., Hall, K.C.: Modeling of fluid-structure interaction. Ann. Rev. Fluid Mech. 33(1), 445–490 (2001)CrossRefGoogle Scholar
  32. 32.
    Farhat, C., Lesoinne, M., Le Tallec, P.: 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. Eng. 157(1–2), 95–114 (1998)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Formaggia, L., Miglio, E., Mola, A., Parolini, N.: Fluid–structure interaction problems in free surface flows: application to boat dynamics. Int. J. Numer. Methods Fluids 56(8), 965–978 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Formaggia, L., Miglio, E., Mola, A., Montano, A.: A model for the dynamics of rowing boats. Int. J. Numer. Methods Fluids 61(2), 119–143 (2009)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Formaggia, L., Vergara, C., Zonca, S.: Unfitted extended finite elements for composite grids. Comput. Math. Appl. 76(4), 893–904 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ge, L., Sotiropoulos, F.: A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries. J. Comput. Phys. 225(2), 1782–1809 (2007)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gerstenberger, A.: An XFEM based fixed-grid approach to fluid-structure interaction. PhD thesis, Technical University of Munich (2010)Google Scholar
  38. 38.
    Gerstenberger, A., Wall, W.A.: An extended finite element method based approach for large deformation fluid-structure interaction. In: Wesseling, P., Onate, E., Periaux, J. (eds.) Proceedings of the European Conference on Computational Fluid Dynamics (2006)Google Scholar
  39. 39.
    Gerstenberger, A., Wall, W.A.: An extended finite element method/Lagrange multiplier based approach for fluid–structure interaction. Comput. Methods Appl. Mech. Eng. 197(19), 1699–1714 (2008)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Gerstenberger, A., Wall, W.A.: An embedded Dirichlet formulation for 3D continua. Int. J. Numer. Methods Eng. 82(5), 537–563 (2010)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Glowinski, R., Pan, T.-W., Periaux, J.: A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111(3–4), 283–303 (1994)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Glowinski, R., Pan, T.-W., Periaux, J.: A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 112(1), 133–148 (1994)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Glowinski, R., Pan, T.-W., Periaux, J.: A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies: (I) case where the rigid body motions are known a priori. C. R. Acad. Sci. Ser. I-Math. 324(3), 361–369 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Glowinski, R., Pan, T.-W., Hesla, T.I., Joseph, D.D.: A distributed lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 25(5), 755–794 (1999)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Glowinski, R., Pan, T.-W., Hesla, T.I., Joseph, D.D., Periaux, J.: A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169(2), 363–426 (2001)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Griffith, B.E.: Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. Int. J. Numer. Methods Biomed. Eng. 28(3), 317–345 (2012)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Griffith, B.E., Hornung, R.D., McQueen, D.M., Peskin, C.S.: An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys. 223(1), 10–49 (2007)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Griffith, B.E., Luo, X., McQueen, D.M., Peskin, C.S.: Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method. Int. J. Appl. Mech. 1, 137–176 (2009)CrossRefGoogle Scholar
  49. 49.
    Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193(33–35), 3523–3540 (2004)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Hansbo, A., Hansbo, P., Larson, M.G.: A finite element method on composite grids based on Nitsche’s method. ESAIM: Math. Model. Numer. Anal. 37(3), 495–514 (2003)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Hansbo, P., Larson, M.G., Zahedi, S.: Characteristic cut finite element methods for convection–diffusion problems on time dependent surfaces. Comput. Methods Appl. Mech. Eng. 293, 431–461 (2015)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Harun, Z., Reda, E., Abdullah, S.: Large eddy simulation of the wind flow over skyscrapers. Recent Adv. Mech. Mech. Eng. 15, 72–79 (2015)Google Scholar
  55. 55.
    Hirt, C.W., Amsden, A.A., Cook, J.L.: An arbitrary lagrangian-eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974)CrossRefGoogle Scholar
  56. 56.
    Hsu, M.-C., Kamensky, D., Bazilevs, Y., Sacks, M.S., Hughes, T.J.R.: Fluid–structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation. Comput. Mech. 54(4), 1055–1071 (2014)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Hughes, T.J.R., Hulbert, G.M.: Space-time finite element methods for elastodynamics: formulations and error estimates. Comput. Methods Appl. Mech. Eng. 66(3), 339–363 (1988)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Hughes, T.J.R., Liu, W.K., Zimmermann, T.K.: Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45(1–3), 285–312 (1984)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Jonsson, T., Larson, M.G., Larsson, K.: Cut finite element methods for elliptic problems on multipatch parametric surfaces. Comput. Methods Appl. Mech. Eng. 324, 366–394 (2017)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Kamakoti, R., Shyy, W.: Fluid–structure interaction for aeroelastic applications. Prog. Aerosp. Sci. 40(8), 535–558 (2004)CrossRefGoogle Scholar
  62. 62.
    Kamensky, D., Hsu, M.-C., Schillinger, D., Evans, J.A., Aggarwal, A., Bazilevs, A., Sacks, M.S., Hughes, T.J.R.: An immersogeometric variational framework for fluidstructure interaction: application to bioprosthetic heart valves. Comput. Methods Appl. Mech. Eng. 284, 1005–1053 (2015)CrossRefGoogle Scholar
  63. 63.
    Katayama, S., Umetani, N., Sugiura, S., Hisada, T.: The sinus of Valsalva relieves abnormal stress on aortic valve leaflets by facilitating smooth closure. J. Thorac. Cardiovasc. Surg. 136(6), 1528–1535 (2008)CrossRefGoogle Scholar
  64. 64.
    Lai, Y.G., Chandran, K.B., Lemmon, J.: A numerical simulation of mechanical heart valve closure fluid dynamics. J. Biomech. 35(7), 881–892 (2002)CrossRefGoogle Scholar
  65. 65.
    Le, T.B., Sotiropoulos, F.: Fluid–structure interaction of an aortic heart valve prosthesis driven by an animated anatomic left ventricle. J. Comput. Phys. 244, 41–62 (2013)MathSciNetCrossRefGoogle Scholar
  66. 66.
    LifeV.: The parallel finite element library for the solution of PDEs (2018). http://www.lifev.org
  67. 67.
    Liu, W.K., Liu, Y., Farrell, D., Zhang, L.T., Wang, X.S., Fukui, Y., Patankar, N., Zhang, Y., Bajaj, C., Lee, J., Hong, J., Chen, X., Hsu, H.: Immersed finite element method and its applications to biological systems. Comput. Methods Appl. Mech. Eng. 195(13), 1722–1749 (2006)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Marom, G.: Numerical methods for fluid–structure interaction models of aortic valves. Arch. Comput. Meth. Eng. 22(4), 595–620 (2015)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Massing, A., Larson, M.G., Logg, A.: Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions. SIAM J. Sci. Comput. 35(1), C23–C47 (2013)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche overlapping mesh method for the Stokes problem. Numer. Math. 128(1), 73–101 (2014)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A Nitsche-based cut finite element method for a fluid-structure interaction problem. Commun. Appl. Math. Comput. Sci. 10(2), 97–120 (2015)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Mayer, U.M., Popp, A., Gerstenberger, A., Wall, W.A.: 3D fluid-structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach. Computat. Mech. 46(1), 53–67 (2010)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37(1), 239–261 (2005)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)CrossRefGoogle Scholar
  75. 75.
    Morgenthal, G.: Fluid Structure Interaction in Bluff-body Aerodynamics and Long-span Bridge Design: Phenomena and Methods. University of Cambridge, Department of Engineering Cambridge (2000)Google Scholar
  76. 76.
    Morsi, Y.S., Yang, W.W., Wong, C.S., Das, S.: Transient fluid–structure coupling for simulation of a trileaflet heart valve using weak coupling. J. Artif. Organs 10(2), 96–103 (2007)CrossRefGoogle Scholar
  77. 77.
    Nguyen, H., Reynen, J.: A space-time least-square finite element scheme for advection-diffusion equations. Comput. Methods Appl. Mech. Eng., 42(3), 331–342 (1984)CrossRefGoogle Scholar
  78. 78.
    Nicaise, S., Renard, Y., Chahine, E.: Optimal convergence analysis for the extended finite element method. Int. J. Numer. Methods Eng. 86(4–5), 528–548 (2011)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Parolini, N., Quarteroni, A.: Mathematical models and numerical simulations for the America’s cup. Comput. Methods Appl. Mech. Eng. 194(9), 1001–1026 (2005)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Peskin, C.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10(2), 252–271 (1972)MathSciNetCrossRefGoogle Scholar
  81. 81.
    Pettigrew, M.J., Taylor, C.E.: Vibration analysis of shell-and-tube heat exchangers: an overview - Part 1: flow, damping, fluid elastic instability. J. Fluids Struct. 18(5), 469–483 (2003)CrossRefGoogle Scholar
  82. 82.
    Rega, G.: Nonlinear vibrations of suspended cables–Part I: modeling and analysis. Appl. Mech. Rev. 57(6), 443–478 (2004)CrossRefGoogle Scholar
  83. 83.
    Schott, B., Wall, W.A.: A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 276, 233–265 (2014)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Stijnen, J.M.A., De Hart, J., Bovendeerd, P.H.M., van de Vosse, F.N.: Evaluation of a fictitious domain method for predicting dynamic response of mechanical heart valves. J. Fluids Struct. 19(6), 835–850 (2004)CrossRefGoogle Scholar
  85. 85.
    Temam, R.: Navier-Stokes Equations. North-Holland Publishing Company, Amsterdam (1977)zbMATHGoogle Scholar
  86. 86.
    Trivedi, C., Cervantes, M.J.: Fluid-structure interactions in francis turbines: a perspective review. Renew. Sust. Energ. Rev. 68, 87–101 (2017)CrossRefGoogle Scholar
  87. 87.
    van Loon, R.: A 3D method for modelling the fluid-structure interaction of heart valves. PhD thesis, Technische Universiteit Eindhoven (2005)Google Scholar
  88. 88.
    van Loon, R., Anderson, P.D., van de Vosse, F.N.: A fluid–structure interaction method with solid-rigid contact for heart valve dynamics. J. Comput. Phys. 217(2), 806–823 (2006)MathSciNetCrossRefGoogle Scholar
  89. 89.
    van Loon, R., Anderson, P.D., van de Vosse, F.N., Sherwin, S.J.: Comparison of various fluid–structure interaction methods for deformable bodies. Comput. Struct. 85(11–14), 833–843 (2007). Fourth MIT Conference on Computational Fluid and Solid MechanicsGoogle Scholar
  90. 90.
    Votta, E., Le, T.B., Stevanella, M., Fusini, F., Caiani, E.G., Redaelli, A., Sotiropoulos, F.: Toward patient-specific simulations of cardiac valves: State-of-the-art and future directions. J. Biomech. 46(2), 217–228 (2013). Special Issue: Biofluid Mechanics.Google Scholar
  91. 91.
    Wang, X., Liu, W.K.: Extended immersed boundary method using FEM and RKPM. Comput. Methods Appl. Mech. Eng. 193(12), 1305–1321 (2004)MathSciNetCrossRefGoogle Scholar
  92. 92.
    Weinberg, E.J., Mack, P.J., Schoen, F.J., García-Cardeña, G., Mofrad, M.R.K.: Hemodynamic environments from opposing sides of human aortic valve leaflets evoke distinct endothelial phenotypes in vitro. Cardiovasc. Eng. 10(1), 5–11 (2010)CrossRefGoogle Scholar
  93. 93.
    Zhang, L.T., Gay, M.: Immersed finite element method for fluid-structure interactions. J. Fluids Struct. 23(6), 839–857 (2007)CrossRefGoogle Scholar
  94. 94.
    Zhang, L.T., Gerstenberger, A., Wang, X., Liu, W.K.: Immersed finite element method. Comput. Methods Appl. Mech. Eng. 193(21), 2051–2067 (2004)MathSciNetCrossRefGoogle Scholar
  95. 95.
    Zhang, H., Liu, L., Dong, M., Sun, H.: Analysis of wind-induced vibration of fluid–structure interaction system for isolated aqueduct bridge. Eng. Struct. 46, 28–37 (2013)CrossRefGoogle Scholar
  96. 96.
    Zonca, S., Vergara, C., Formaggia, L.: An unfitted formulation for the interaction of an incompressible fluid with a thick structure via an XFEM/DG approach. SIAM J. Sci. Comput. 40(1), B59–B84 (2018)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.MOX, Dipartimento di Matematica, Politecnico di MilanoMilanoItaly

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