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Numerics of Fluid-Structure Interaction

  • Sebastian Bönisch
  • Thomas Dunne
  • Rolf Rannacher
Part of the Oberwolfach Seminars book series (OWS, volume 37)

Abstract

This chapter describes numerical methods for simulating the interaction of viscous liquids with rigid or elastic bodies.

General examples of fluid-solid/structure interaction (FSI) problems are flow transporting rigid or elastic particles (particulate flow), flow around elastic structures (airplanes, submarines) and flow in elastic structures (hemodynamics, transport of fluids in closed containers). In all these settings the dilemma in modeling the coupled dynamics is that the fluid model is normally based on an Eulerian perspective in contrast to the usual Lagrangian formulation of the solid model. This makes the setup of a common variational description difficult. However, such a variational formulation of FSI is needed as the basis of a consistent Galerkin discretization with a posteriori error control and mesh adaptation, as well as the solution of optimal control problems based on the Euler-Lagrange approach.

Keywords

Posteriori Error Free Fall Error Representation Eulerian Approach Mesh Adaptation 
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.

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References

  1. [1]
    E. Bänsch and W. Dörfler, Adaptive finite elements for exterior domain problems. Numer. Math. 80 (1998), 497–523.CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    W. Bangerth and R. Rannacher, Adaptive Finite-Element Methods for Differential Equations. Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel 2003.Google Scholar
  3. [3]
    R. Becker and M. Braack, A finite-element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001), 173–199.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    R. Becker, M. Braack, and D. Meidner: Gascoigne: A C++ numerics library for scientific computing. Institute of Applied Mathematics, University of Heidelberg, URL http://www.gascoigne.uni-hd.de/.
  5. [5]
    R. Becker and Th. Dunne, VisuSimple: An interactive visualization utility for scientific computing. Abschlußband SFB 359, Reactive Flows, Diffusion and Transport (W. Jäger et al., eds.), Springer, Berlin-Heidelberg New York, 2007.Google Scholar
  6. [6]
    R. Becker, V. Heuveline, and R. Rannacher, An optimal control approach to adaptivity in computational fluid mechanics. Int. J. Numer. Meth. Fluids. 40 (2002), 105–120.CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    R. Becker and R. Rannacher, An optimal control approach to error estimation and mesh adaptation in finite-element methods. Acta Numerica 2000 (A. Iserles, ed.), pp. 1–102, Cambridge University Press, 2001.Google Scholar
  8. [8]
    A. Belmonte, H. Eisenberg, and E. Moses, From flutter to tumble, Inertial drag and froude similarity in falling paper. Phys. Rev. Lett. 81 (1998), 345–348.CrossRefGoogle Scholar
  9. [9]
    S. Bönisch, Adaptive Finite-Element Methods for Rigid Particulate Flow Problems. Doctoral thesis, Institute of Applied Mathematics, University of Heidelberg, 2006.Google Scholar
  10. [10]
    S. Bönisch and V. Heuveline, On the numerical simulation of the instationary free fall of a solid in a fluid. I. The Newtonian case. Computer & Fluids 36 (2007), 1434–1445.CrossRefGoogle Scholar
  11. [11]
    S. Bönisch and V. Heuveline, On the numerical simulation of the free fall of a solid in a fluid. II. The viscoelastic case. SFB Preprint 2004-32, University of Heidelberg, 2004.Google Scholar
  12. [12]
    S. Bönisch and V. Heuveline, Advanced flow visualization with HiVision. Abschlußband SFB 359, Reactive Flows, Diffusion and Transport (W. Jäger et al., eds.), Springer, Berlin-Heidelberg New York, 2007.Google Scholar
  13. [13]
    S. Bönisch, V. Heuveline, R. Rannacher, Numerical simulation of the free fall problem. Proc. Int. Conf. on High Performance Scientific Computing (HPSCHanoi 2003), Hanoi, March 2003, (H.G. Bock, et al., eds.), Springer, Berlin-Heidelberg, 2005.Google Scholar
  14. [14]
    S. Bönisch, V. Heuveline und P. Wittwer: Adaptive boundary conditions for exterior flow problems, J. Math. Fluid Mech. 7 (2005), 85–107.CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    S. Bönisch, V. Heuveline und P. Wittwer: Second order adaptive boundary conditions for exterior flow problems: non-symmetric stationary flows in two dimensions, J. Math. Fluid Mech. 8 (2006), 1–26.CrossRefMathSciNetGoogle Scholar
  16. [16]
    M. Braack and E. Burman: Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM J. Numer. Anal. 43 (2006), 2544–2566.MathSciNetMATHGoogle Scholar
  17. [17]
    R. Bürger, R. Liu, and W.L. Wendland, Existence and stability for mathematical models of sedimentation-consolidation processes in several space dimensions. J. Math. Anal. Appl. 264 (2001), 288–310.CrossRefMathSciNetMATHGoogle Scholar
  18. [18]
    C. Conca, J.S. Martin, and M. Tucsnak, Existence of the solutions for the equations modelling the motion of rigid body in a viscous fluid. Commun. Partial Differ. Equations 25 (2000), 1019–1042.CrossRefMATHGoogle Scholar
  19. [19]
    B. Desjardin and M. Esteban, Existence of weak solutions for the motion of rigid bodies in viscous fluids. Arch. Rational Mech. Anal. 46 (1999), 59–71.CrossRefGoogle Scholar
  20. [20]
    Th. Dunne, Adaptive Finite-Element Simulation of Fluid Structure Interaction Based on an Eulerian Formulation. Doctoral thesis, Institute of Applied Mathematics, University of Heidelberg, 2007.Google Scholar
  21. [21]
    Th. Dunne and R. Rannacher, Adaptive finite-element approximation of fluidstructure interaction based on an Eulerian variational formulation. In ‘FluidStructure Interaction: Modelling, Simulation, Optimisation’ (H.-J. Bungartz and M. Schäfer, eds.), Springer’s LNCSE-Series, 2006.Google Scholar
  22. [22]
    S.B. Field, M. Klaus, M.G. Moore, and F. Nori Chaotic dynamics of falling disks. Nature 388 (1997), 252–254.CrossRefGoogle Scholar
  23. [23]
    G.P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rational Mech. Anal. 148 (1999), 53–88.CrossRefMathSciNetMATHGoogle Scholar
  24. [24]
    G.P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications. Handbook of Mathematical Fluid Mechanics (S. Friedlander and D. Serre, eds.), Elsevier, 2001.Google Scholar
  25. [25]
    G.P. Galdi and A. Vaidya, Translational steady fall of symmetric bodies in a NavierStokes liquid, with application to particle sedimentation. J. Math. Fluid Mech. 3 (2001), 183–211.Google Scholar
  26. [26]
    G.P. Galdi, A. Vaidya, M. Pokorny, D.D. Joseph, J. Feng, Orientation of bodies sedimenting in a second-order liquid at non-zero Reynolds number. Math. Models Methods Appl. Sci. 12 (2002),1653–1690.CrossRefMathSciNetMATHGoogle Scholar
  27. [27]
    V. Girault and P.-A. Raviart, Finite-Element Methods for the Navier-Stokes Equations. Springer: Berlin-Heidelberg-New York, 1986.MATHGoogle Scholar
  28. [28]
    R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, and J. Periaux, A distributed Lagrange multiplier/fictitious domain method for flow around moving rigid bodies: Application to particulate flow. Int. J. Numer. Meth. Fluids 30 (1999), 1043–1066.CrossRefMATHGoogle Scholar
  29. [29]
    M.D. Gunzburger, H.C. Lee, and G.A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2 (2000), 219–266.CrossRefMathSciNetMATHGoogle Scholar
  30. [30]
    V. Heuveline and R. Rannacher, Adaptive FEM for eigenvalue problems with application in hydrodynamic stability analysis. Proc. Int. Conf. ‘Advances in Numerical Mathematics’, Moscow, Sept. 16-17, 2005 (W. Fitzgibbon et al., eds.), pp. 109–140, Institute of Numerical Mathematics RAS, Moscow, 2006.Google Scholar
  31. [31]
    V. Heuveline, HiFlow: A multi-purpose finite-element package, Rechenzentrum, Universität Karlsruhe, URL http://hiflow.de/.
  32. [32]
    V. Heuveline, HiVision: A visualization platform, Rechenzentrum, Universität Karlsruhe, URL http://hiflow.de/.
  33. [33]
    K.-H. Hoffmann and V.N. Starovoitov, Zur Bewegung einer Kugel in einer zähen Flüssigkeit. Doc. Math. 5 (2000), 15–21.MathSciNetMATHGoogle Scholar
  34. [34]
    J. Hron and S. Turek, Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow. In ‘Fluid-Structure Interaction: Modelling, Simulation, Optimisation’ (H.-J. Bungartz and M. Schäfer, eds.), in Springer’s LNCSE-Series, 2006.Google Scholar
  35. [35]
    J. Hron and S. Turek, Fluid-structure interaction with applications in biomechanics. In ‘Fluid-Structure Interaction: Modelling, Simulation, Optimisation’ (H.-J. Bungartz and M. Schäfer, eds.), in Springer’s LNCSE-Series, 2006.Google Scholar
  36. [36]
    H.H. Hu, Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996), 335–352.CrossRefMATHGoogle Scholar
  37. [37]
    H.H. Hu, D.D. Joseph, and M.J. Crochet, Direct simulation of fluid particle motions. Theor. Comp. Fluid Dyn. 3 (1992), 285–306.CrossRefMATHGoogle Scholar
  38. [38]
    T.J.R. Hughes and A.N. Brooks, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation. Comput. Meth. Appl. Mech. Engrg. 32 (1982), 199–259.CrossRefMathSciNetMATHGoogle Scholar
  39. [39]
    T.J.R. Hughes, L.P. Franc, and M. Balestra, A new finite-element formulation for computational fluid mechanics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation. Comput. Meth. Appl. Mech. Engrg. 59 (1986), 85–99.CrossRefMATHGoogle Scholar
  40. [40]
    C. Liu and N.J. Walkington, An Eulerian description of fluids containing visco-elastic particles. Arch. Rat. Mech. Anal. 159 (2001), 229–252.CrossRefMathSciNetMATHGoogle Scholar
  41. [41]
    N.A. Patankar, A formulation for fast computations of rigid particulate flows. Center Turbul. Res., Ann. Res. Briefs, 185–196 (2001).Google Scholar
  42. [42]
    N.A. Patankar, Physical interpretation and mathematical properties of the stress-DLM formulation for rigid particulate flows. Int. J. Comp. Meth. Engrg Sci. Mech. 6 (2005), 137–143.CrossRefGoogle Scholar
  43. [43]
    N.A. Patankar and D.D. Joseph, Modeling and numerical simulation of particulate flows by the Eulerian-Lagrangian approach. Int. J. Multiphase Flow 27 (2001), 1659–1684.Google Scholar
  44. [44]
    N.A. Patankar and D.D. Joseph, Lagrangian numerical simulation of particulate flows. Int. J. Multiphase Flow 27 (2001), 1685–1706.CrossRefMATHGoogle Scholar
  45. [45]
    N.A. Patankar, P. Singh, D.D. Joseph, R. Glowinski, and T.-W. Pan, A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 26 (2000), 1509–1524.CrossRefMATHGoogle Scholar
  46. [46]
    R. Rannacher, Finite-element methods for the incompressible Navier-Stokes equations. In ‘Fundamental Directions in Mathematical Fluid Mechanics’ (G.P. Galdi et al., eds.), pp. 191–293, Birkhäuser, Basel, 2000.Google Scholar
  47. [47]
    R. Rannacher, Methods for numerical flow simulation. In this volume.Google Scholar
  48. [48]
    R. Rannacher and F.-T. Suttmeier, Error estimation and adaptive mesh design for FE models in elasto-plasticity. In Error-Controlled Adaptive FEMs in Solid Mechanics (E. Stein, ed.), pp. 5–52, John Wiley, Chichster, 2002.Google Scholar
  49. [49]
    N. Sharma and N.A. Patankar, A fast computation technique for the direct numerical sim,ulation of rigid particulate flows. J. Comp. Phys. 205 (2005), 439–457.CrossRefMATHGoogle Scholar
  50. [50]
    T.E. Tezduyar, M. Behr, and J. Liou, A new strategy for finite-element flow computations involving moving boundaries and interfaces-the deforming-spatialdomain/space-time procedures: I. The concept and preliminary tests, II. Computation of free-surface ows, two-liquid ows and ows with drifting cylinders. Computer Methods in Applied Mechanics and Engineering, 1992.Google Scholar
  51. [51]
    S.V. Tsynkov, Numerical solution of problems on unbounded domains. a review. Appl. Numer. Math. 27 (1998), 465–532.CrossRefMathSciNetMATHGoogle Scholar
  52. [52]
    S.V. Tsynkov, External boundary conditions for three-dimensional problems of computational aerodynamics. SIAM J. Sci. Comput. 21 (1999), 166–206.CrossRefMathSciNetMATHGoogle Scholar
  53. [53]
    S. Turek, Efficient solvers for incompressible flow problems: an algorithmic and computational approach. Springer, Heidelberg-Berlin-New York, 1999.MATHGoogle Scholar
  54. [54]
    S.O. Unverdi and G. Tryggvason, Computations of multi-fluid flows. Physica D 60 (1992), 70–83.Google Scholar
  55. [55]
    VisuSimple, VisuSimple: An open source interactive visualization utility for scientific computing. Institute of Applied Mathematics, University of Heidelberg, URL http://www.visusimple.uni-hd.de/.
  56. [56]
    D. Wan and S. Turek, Direct numerical simulstion of particulate flow via multigrid FEM techniques and the fictitious boundary method. Int. J. Numer. Meth. Fluids 51 (2006), 531–566.CrossRefMathSciNetMATHGoogle Scholar
  57. [57]
    J. Wang, R. Bai, C. Lewandowski, G.P. Galdi, and D.D. Joseph, Sedimentation of cylindrical particles in a viscoelastic liquid: shape-tilting. China Particuology 2 (2004), 13–18.CrossRefGoogle Scholar
  58. [58]
    H.F. Weinberger, On the steady fall of a body in a Navier-stokes fluid. Proc. Symp. Pure Mathematics 23 (1973), 421–440.Google Scholar
  59. [59]
    P. Wittwer, On the structure of stationary solutions of the Navier-Stokes equations. Commun. Math. Phys. 226 (2002), 455–474.CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Sebastian Bönisch
    • 1
  • Thomas Dunne
    • 1
  • Rolf Rannacher
    • 1
  1. 1.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany

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