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Isogeometric fluid-structure interaction: theory, algorithms, and computations

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Abstract

We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-α time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.

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References

  1. Abedi P, Patracovici B, Haber RB (2006) A spacetime discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput Methods Appl Mech Eng 195: 3247–3273

    Article  MATH  Google Scholar 

  2. Akkerman I, Bazilevs Y, Calo VM, Hughes TJR, Hulshoff S (2008) The role of continuity in residual-based variational multiscale modeling of turbulence. Comput Mech 41: 371–378

    Article  MathSciNet  Google Scholar 

  3. Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput Mech 38: 310–322

    Article  MathSciNet  Google Scholar 

  4. Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197: 173–201

    Article  MathSciNet  Google Scholar 

  5. Bazilevs Y, Beiraoda Veiga L, Cottrell JA, Hughes TJR, Sangalli G (2006) Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math Models Methods Appl Sci 16: 1031–1090

    Article  MATH  MathSciNet  Google Scholar 

  6. Bazilevs Y, Hughes TJR (2007) Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput Fluids 36: 12–26

    Article  MATH  MathSciNet  Google Scholar 

  7. Bazilevs Y, Michler C, Calo VM, Hughes TJR (2007) Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput Methods Appl Mech Eng 196: 4853–4862

    Article  MathSciNet  Google Scholar 

  8. Bishop R, Goldberg S (1980) Tensor analysis on manifolds. Dover, New York

    Google Scholar 

  9. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, Berlin

    MATH  Google Scholar 

  10. 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: 199–259

    Article  MATH  MathSciNet  Google Scholar 

  11. Calo VM (2004) Residual-based multiscale turbulence modeling: finite volume simulation of bypass transistion. Ph.D. Thesis, Department of Civil and Environmental Engineering, Stanford University

  12. Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. J Appl Mech 60: 371–75

    Article  MATH  MathSciNet  Google Scholar 

  13. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195: 5257–5297

    Article  MATH  MathSciNet  Google Scholar 

  14. Cottrell JA, Reali A, Hughes TJR (2007) Studies of refinement and continuity in isogeometric structural analysis. Comput Methods Appl Mech Eng 196: 4160–4183

    Article  MathSciNet  Google Scholar 

  15. Dettmer W, Perić D (2006) A computational framework for fluid-structure interaction: finite element formulation and applications. Comput Methods Appl Mech Eng 195: 5754–5779

    Article  MATH  Google Scholar 

  16. Elguedj T, Bazilevs Y, Calo VM, Hughes TJR (2008) B-bar and F-bar projection methods for nearly incompressible linear and nonlinear elasticity and plasticity using higher-order NURBS elements. Comput Methods Appl Mech Eng 197: 2732–2762

    Article  Google Scholar 

  17. Farhat C, Geuzaine P, Grandmont C (2001) The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids. J Comput Phys 174(2): 669–694

    Article  MATH  MathSciNet  Google Scholar 

  18. Farhat C, van der Zee K, Geuzaine P (2006) Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity. Comput Methods Appl Mech Eng 195: 1973–2001

    Article  MATH  Google Scholar 

  19. Fernandez MA, Moubachir M (2005) A Newton method using exact jacobians for solving fluid-structure coupling. Comput Struct 83: 127–142

    Article  Google Scholar 

  20. Figueroa A, Vignon-Clementel IE, Jansen KE, Hughes TJR, Taylor CA (2006) A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput Methods Appl Mech Eng 195: 5685–5706

    Article  MATH  MathSciNet  Google Scholar 

  21. Flanders H (1963) Differential forms with applications to the physical sciences. Academic Press, London

    MATH  Google Scholar 

  22. Formaggia L, Nobile F (2005) Stability analysis of second-order time accurate schemes for ALE-FEM. Comput Methods Appl Mech Eng 193: 4097–4116

    Article  MathSciNet  Google Scholar 

  23. Guillermin V, Pollack A (1974) Differential topology. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  24. Heil M, Hazel A, Boyle J (2008) Solvers for large-displacement fluid-structure interaction problems: segregated vs. monolithic approaches. Comput Mech. doi:10.1007/s00466-008-0270-6

  25. Heywood JG, Rannacher R, Turek S (1996) Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int J Numer Methods Fluids 22: 325–352

    Article  MATH  MathSciNet  Google Scholar 

  26. Holzapfel GA (2004) Computational biomechanics of soft biological tissue. In: Stein E, De Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, Solids and structures, chap s18, vol 2. Wiley, London

  27. Holzapfel GA (2000) Nonlinear solid mechanics, a continuum approach for engineering. Wiley, Chichester

    MATH  Google Scholar 

  28. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola

    Google Scholar 

  29. Hughes TJR, Calo VM, Scovazzi G (2004) Variational and multiscale methods in turbulence. In: Gutkowski W, Kowalewski TA (eds) In Proceedings of the XXI International Congress of Theoretical and Applied Mechanics (IUTAM), Kluwer, pp 153–163

  30. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl Mech Eng 194: 4135–4195

    Article  MATH  MathSciNet  Google Scholar 

  31. Hughes TJR, Hulbert GM (1988) Space-time finite element methods for elastodynamics: formulations and error estimates. Comput Methods Appl Mech Eng 66: 339–363

    Article  MATH  MathSciNet  Google Scholar 

  32. Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29: 329–349

    Article  MATH  MathSciNet  Google Scholar 

  33. Hughes TJR, Wells GN (2005) Conservation properties for the Galerkin and stabilised forms of the advection-diffusion and incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 194: 1141–1159

    Article  MATH  MathSciNet  Google Scholar 

  34. Hulbert GM, Hughes TJR (1990) Space-time finite element methods for second order hyperbolic equations. Comput Methods Appl Mech Eng 84: 327–348

    Article  MATH  MathSciNet  Google Scholar 

  35. Idelsohn SR, Oñate E, Del Pin F, Calvo N (2006) Fluid-structure interaction using the particle finite element method. Comput Methods Appl Mech Eng 195: 2100–2123

    Article  MATH  Google Scholar 

  36. Jansen KE, Whiting CH, Hulbert GM (1999) A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190: 305–319

    Article  MathSciNet  Google Scholar 

  37. 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: 73–94

    Article  MATH  Google Scholar 

  38. Johnson AA, Tezduyar TE (1997) Parallel computation of incompressible flows with complex geometries. Int J Numer Methods Fluids 24: 1321–1340

    Article  MATH  Google Scholar 

  39. Johnson AA, Tezduyar TE (1999) Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23: 130–141

    Article  MATH  Google Scholar 

  40. Johnson C (1987) Numerical solution of partial differential equations by the finite element method. Cambridge University Press, Sweden

    MATH  Google Scholar 

  41. Johnson C, Nävert U, Pitkäranta J (1984) Finite element methods for linear hyperbolic problems. Comput Methods Appl Mech Eng 45: 285–312

    Article  MATH  Google Scholar 

  42. Küttler U, Förster C, Wall WA (2006) A solution for the incompressibility dilemma in partitioned fluid-structure interaction with pure Dirichlet fluid domains. Comput Mech 38: 417–429

    Article  Google Scholar 

  43. Kuhl E, Hulshoff S, de Borst R (2003) An arbitrary Lagrangian Eulerian finite element approach for fluid-structure interaction phenomena. Int J Numer Methods Eng 57: 117–142

    Article  MATH  Google Scholar 

  44. Lang S (1972) Differential manifolds. Addison-Wesley, Reading

    MATH  Google Scholar 

  45. Lang S (1995) Differential and riemannian manifolds (graduate texts in mathematics, vol. 160). Springer, Heidelberg

    Google Scholar 

  46. Le Tallec P, Mouro J (2001) Fluid structure interaction with large structural displacements. Comput Methods Appl Mech Eng 190:3039–3068

    Article  MATH  Google Scholar 

  47. Marsden JE, Hughes TJR (1993) Mathematical foundations of elasticity. Dover Publications Inc., New York

    Google Scholar 

  48. Masud A, Hughes TJR (1997) A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems. Comput Methods Appl Mech Eng 148: 91–126

    Article  MathSciNet  Google Scholar 

  49. Michler C, van Brummelen EH, de Borst R (2006) Error-amplification analysis of subiteration-preconditioned GMRES for fluid-structure interaction. Comput Methods Appl Mech Eng 195: 2124–2148

    Article  MATH  Google Scholar 

  50. Michler C, van Brummelen EH, Hulshoff SJ, de Borst R (2003) The relevance of conservation for stability and accuracy of numerical methods for fluid-structure interaction. Comput Methods Appl Mech Eng 192: 4195–4215

    Article  MATH  Google Scholar 

  51. Nobile F (2001) Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics. Ph.D. Thesis, EPFL

  52. Piperno S, Farhat C (2001) Partitined procedures for the transient solution of coupled aeroelastic problems. Part II: Energy transfer analysis and three-dimensional applications. Comput Methods Appl Mech Eng 190: 3147–3170

    Article  MATH  Google Scholar 

  53. Piperno S, Farhat C, Larrouturou B (1995) Partitined procedures for the transient solution of coupled aeroelastic problems. Part I: Model problem, theory and two-dimensional application. Comput Methods Appl Mech Eng 124: 79–112

    Article  MATH  MathSciNet  Google Scholar 

  54. Saad Y, Schultz MH (1986) GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7: 856–869

    Article  MATH  MathSciNet  Google Scholar 

  55. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  56. Babuška I (1973) The finite element method with Lagrange multipliers. Numer Math 20: 179–192

    Article  MATH  Google Scholar 

  57. Spivak M (1965) Calculus on manifolds. Benjamin, New York

    MATH  Google Scholar 

  58. Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid-structure interactions with large displacements. J Appl Mech 70: 58–63

    Article  MATH  Google Scholar 

  59. Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193: 2019–2032

    Article  MATH  Google Scholar 

  60. Texas Advanced Computing Center (TACC). http://www.tacc.utexas.edu

  61. Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of blood flow in arteries. Comput Methods Appl Mech Eng 158: 155–196

    Article  MATH  MathSciNet  Google Scholar 

  62. Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: relevance to atherosclerosis. Ann Biomed Eng 26: 975–987

    Article  Google Scholar 

  63. Taylor CA, Hughes TJR, Zarins CK (1999) Effect of exercise on hemodynamic conditions in the abdominal aorta. J Vasc Surg 29: 1077–1089

    Article  Google Scholar 

  64. 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: 339–351

    Article  MATH  MathSciNet  Google Scholar 

  65. 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: 353–371

    Article  MATH  MathSciNet  Google Scholar 

  66. Tezduyar TE, Sathe S (2007) Modelling of fluid-structure interactions with the space-time finite elements: solution techniques. Int J Numer Methods Fluids 54: 855–900

    Article  MATH  MathSciNet  Google Scholar 

  67. Tezduyar TE, Sathe S, Cragin T, Nanna B, Conklin BS, Pausewang J, Schwaab M (2007) Modelling of fluid-structure interactions with the space-time finite elements: Arterial fluid mechanics. Int J Numer Methods Fluids 54: 901–922

    Article  MATH  MathSciNet  Google Scholar 

  68. 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: 601–629

    Article  MATH  MathSciNet  Google Scholar 

  69. Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43: 555–575

    Article  MATH  MathSciNet  Google Scholar 

  70. Tezduyar TE, Behr M, Mittal S, Johnson AA (1992) Computation of unsteady incompressible flows with the stabilized 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 7–24

  71. 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: 2002–2027

    Article  MATH  MathSciNet  Google Scholar 

  72. 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: 3776–3796

    Article  MATH  MathSciNet  Google Scholar 

  73. Wall W (1999) Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen. Ph.D. Thesis, Institut für Baustatik, Universität Stuttgart

  74. Zhang Y, Bazilevs Y, Goswami S, Bajaj C, Hughes TJR (2007) Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput Methods Appl Mech Eng 196: 2943–2959

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Structural Engineering, University of California, San Diego 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Y. Bazilevs

  2. Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, 1 University Station C0200, Austin, TX, 78712, USA

    V. M. Calo & T. J. R. Hughes

  3. Mechanical Engineering, Carnegie Mellon University, Scalfe Hall 303, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    Y. Zhang

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  1. Y. Bazilevs
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  2. V. M. Calo
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  3. T. J. R. Hughes
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  4. Y. Zhang
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Correspondence to Y. Bazilevs.

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Bazilevs, Y., Calo, V.M., Hughes, T.J.R. et al. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43, 3–37 (2008). https://doi.org/10.1007/s00466-008-0315-x

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