Computational Mechanics

, Volume 38, Issue 4–5, pp 417–429

A Solution for the Incompressibility Dilemma in Partitioned Fluid–Structure Interaction with Pure Dirichlet Fluid Domains

  • Ulrich Küttler
  • Christiane Förster
  • Wolfgang A. Wall
Original Paper

Abstract

In a subset of fluid–structure interaction (FSI) problems of incompressible flow and highly deformable structures all popular partitioned approaches fail to work. This also holds for recently quite popular strong coupling approaches based on Dirichlet–Neumann substructuring. This subset can be described as the special case where the fluid domain is entirely enclosed by Dirichlet boundary conditions, i.e. prescribed velocities. A vivid simple example would be a balloon with prescribed inflow rate. In such cases the incompressibility of the fluid cannot be satisfied during standard alternating FSI iterations as the deformation of the coupling surface is determined by the structural displacement that usually does not know about the current constraint on the fluid field. By analyzing this deficiency of the partitioned algorithm a small augmentation is proposed which allows to overcome the dilemma of incompressibility and fixed boundary velocities by introducing the volume constraint on the structural system of equations. In contrast to the original accelerated strong coupling partitioned method, the relaxation which ensures convergence of the iteration over the different fields has now to be performed on the coupling forces rather than on the displacements. In addition, two alternative approaches are discussed for the solution of the dilemma. The capability of the proposed method to deal with largely changing volumes of enclosed fluid is demonstrated by means of numerical examples.

Keywords

Incompressible fluid Fluid–structure interaction Pure Dirichlet domain Augmented Dirichlet–Neumann approach 

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References

  1. 1.
    Braess H, Wriggers P (2000) Arbitrary Lagrangian Eulerian finite element analysis of free surface flow. Comput Meth Appl Mech Eng 190:95–109CrossRefMATHGoogle Scholar
  2. 2.
    Causin P, Gerbeau J-F, Nobile F (2005) Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Comput Meth Appl Mech Eng 194:4506–4527CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Chorin AJ (1997) A numerical method for solving incompressible viscous flow problems. J Comput Phys 135:118–125CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation; the generalized-α method. J Appl Math 60:371–375MathSciNetMATHGoogle Scholar
  5. 5.
    Deparis S (2004) Numerical analysis of axisymmetric flows and methods for fluid–structure interaction arising in blood flow simulation. PhD thesis, EPFLGoogle Scholar
  6. 6.
    Deparis S, Discacciati M, Fourestey G, Quarteroni A (2006) Fluid–structure algorithms based on Steklov–Poincaré operators. Comput Meth Appl Mech Eng DOI 10.1016/j.cma.2005.09.029Google Scholar
  7. 7.
    Fernández MÁ, Moubachir M (2005) A Newton method using exact Jacobians for solving fluid–structure coupling. Comput Struct 83(2–3):127–142CrossRefGoogle Scholar
  8. 8.
    Förster Ch, Wall WA, Ramm E (2006) Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible flow, Preprint SFB 404, University of StuttgartGoogle Scholar
  9. 9.
    Förster Ch, Wall WA, Ramm E (2006) On the geometric conservation law in transient flow calculations on deforming domains. Int J Numer Meth Fluids 50(12):1369–1379CrossRefMATHGoogle Scholar
  10. 10.
    Gerbeau J-F, Vidrascu M (2003) A quasi-Newton algorithm based on a reduced model for fluid–structure interaction problems in blood flows. Math Model and Numer Anal 37(4):631–647CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Gerbeau J-F, Vidrascu M, Frey P (2005) Fluid–structure interaction in blood flows on geometries coming from medical imaging. Comput Struct 83:155–165CrossRefGoogle Scholar
  12. 12.
    Heil M (2004) An efficient solver for the fully coupled solution of large-displacement fluid–structure interaction problems. Comput Meth Appl Mech Eng 193:1–23CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Hübner B, Walhorn E, Dinkler D (2004) A monolithic approach to fluid–structure interaction using space–time finite elements. Comput Meth Appl Mech Eng 193:2087–2104CrossRefMATHGoogle Scholar
  14. 14.
    Le Tallec P, Mouro J (2001) Fluid structure interaction with large structural displacements. Comput Meth Appl Mech Eng 190:3039–3067CrossRefMATHGoogle Scholar
  15. 15.
    Matthies HG, Steindorf J (2003) Partitioned strong coupling algorithms for fluid–structure interaction. Comput Struct 81:805–812CrossRefGoogle Scholar
  16. 16.
    Mok DP (2001) Partitionierte Lösungsansätze in der Strukturdynamik und der Fluid–Struktur-Interaktion. PhD thesis, Institut für Baustatik, Universität StuttgartGoogle Scholar
  17. 17.
    Mok DP, Wall WA (2001) Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures. In: Wall WA, Bletzinger K-U, Schweitzerhof K (eds) Trends in computational structural mechanicsGoogle Scholar
  18. 18.
    Piperno S (1997) Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2d inviscid aeroelastic simulations. Int J Numer Meth Fluids 25:1207–1226CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Raback P, Ruokolainen J, Lyly M, Järvinen E (2001) Fluid–structure interaction boundary conditions by artificial compressibility. ECCOMAS Comp Fluid Dynamcs Conference 2001, Swansea, Wales, UKGoogle Scholar
  20. 20.
    Rogers SE (1995) A comparison of implicit schemes for the incompressible Navier–Stokes equations and artificial compressibility. AIAA J 33(11):2066–2072CrossRefMATHGoogle Scholar
  21. 21.
    Tezduyar TE (2004) Finite element methods for fluid dynamics with moving boundaries and interfaces. In: Stein E, De Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3, chapt 17. Wiley, New York, pp 1–1Google Scholar
  22. 22.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2004) Space–time techniques for finite element computation of flows with moving boundaries and interfaces. In: Proceedings of the III international on numerical methods in engineering and applied sciences, Monterrey, Mexico, January 2004Google Scholar
  23. 23.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2006) Space–time finite element techniques for computation of fluid–structure interactions. Comp Meth Appl Mech Engng 195:2002–2027CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Vierendeels J (2005) Implicit coupling of partitioned fluid-structure interaction solvers using a reduced order model. In: 35th AIAA fluid dynamics conference and exhibit. Toronto, Ontario, June 6–9 2005. AIAAGoogle Scholar
  25. 25.
    Wall WA (1999) Fluid–Struktur-Interaktion mit stabilisierten Finiten Elementen. PhD thesis, Institut für Baustatik, Universität StuttgartGoogle Scholar
  26. 26.
    Wall WA, Mok DP, Ramm E (1999) Partitioned analysis approach of the transient coupled response of viscous fluids and flexible structures. In: Wunderlich W (ed), Solids, structures and coupled problems in engineering, proceedings of the European conference on computational mechanics ECCM ’99, MunichGoogle Scholar
  27. 27.
    Wall WA, Genkinger S, Ramm E (2006) A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Computers & Fluids, DOI 10.1016/j.comfluid.2005.08.007Google Scholar

Copyright information

© Springer Verlag 2006

Authors and Affiliations

  • Ulrich Küttler
    • 1
  • Christiane Förster
    • 2
  • Wolfgang A. Wall
    • 1
  1. 1.Chair of Computational MechanicsTU MunichGarchingGermany
  2. 2.Institute of Structural MechanicsUniversity of Stuttgart Pfaffenwaldring 7StuttgartGermany

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