, Volume 95, Supplement 1, pp 343–361 | Cite as

Numerical simulation of fluid–structure interaction of compressible flow and elastic structure

  • Jaroslava Hasnedlová
  • Miloslav Feistauer
  • Jaromír Horáček
  • Adam Kosík
  • Václav Kučera


The paper is concerned with fluid–structure interaction problem of compressible flow and elastic structure in 2D domains with a special interest in medical applications to airflow in human vocal folds. The viscous flow in a time dependent domain is described by the Navier–Stokes equations written with the aid of the Arbitrary Lagrangian–Eulerian (ALE) method. The equations of motion for elastic deformations of the human vocal folds are coupled with the equations for the fluid flow using either loose or strong coupling. The space discretization of the flow problem is carried out by the discontinuous Galerkin finite element method. For the time discretization we use a semi-implicit scheme. In order to derive the space-time discretization of the elastic body problem, we apply the finite element method using continuous piecewise linear elements. For the time discretization we use the Newmark scheme. Results of numerical experiments are presented.


Fluid–structure interaction Compressible flow ALE method Discontinuous Galerkin finite element method Coupling algorithms 

Mathematics Subject Classification (2000)

74F10 65M60 65M12 



This work was supported by the grants No. P101/11/0207 (J. Horáček) and 201/08/0012 (M. Feistauer, V. Kučera) of the Czech Science Foundation, and by the grants SVV-2012-265316 and GAChU 549912 financed by the Charles University in Prague (J. Hasnedlová-Prokopová and A. Kosík).


  1. 1.
    Fung YC (1969) An introduction to the theory of aeroelaticity. Dover Publications, New YorkGoogle Scholar
  2. 2.
    Dowell EH (1974) Aeroelasticity of plates and shells. Kluwer, DordrechtGoogle Scholar
  3. 3.
    Naudasher E, Rockwell D (1994) Flow-induced vibrations. A. A. Balkema, RotterdamGoogle Scholar
  4. 4.
    Dowell EH (1995) A modern course in aeroelaticity. Kluwer, DordrechtGoogle Scholar
  5. 5.
    Bisplinghoff RL, Ashley H, Halfman RL (1996) Aeroelaticity. Dover, New YorkGoogle Scholar
  6. 6.
    Paidoussis MP (1998) Fluid–structure interactions. Slender structures and axial flow, vol I. Academic Press, San DiegoGoogle Scholar
  7. 7.
    Paidoussis MP (2004) Fluid–structure interactions. Slender structures and axial flow, vol II. Academic Press, LondonGoogle Scholar
  8. 8.
    Hoffman KH, Starovoitov VN (1999) On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv Math Sci Appl 9:633–648MathSciNetGoogle Scholar
  9. 9.
    Grandmont C (2008) Existence of a weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J Math Sci 40:716–737MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Guidorzi M, Padula M, Plotnikov PI (2008) Hopf solutions to a fluid-elastic interaction model. Math Models Methods Appl Sci 18:215–269MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Neustupa J (2009) Existence of a weak solution to the Navier–Stokes equation in a general time-varying domain by the Rothe method. Math Methods Appl Sci 32:653–683MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Titze IR (2000) Principles of voice production. National Center for Voice and Speech, Iowa CityGoogle Scholar
  13. 13.
    Alipour F, Brücker Ch, Cook DD, Gömmel A, Kaltenbacher M, Mattheus W, Mongeau L, Nauman E, Schwarze R, Tokuda I, Zörner S (2011) Mathematical models and numerical schemes for simulation of human phonation. Curr Bioinf 6:323–343CrossRefGoogle Scholar
  14. 14.
    Horáček J, Šidlof P, Švec JG (2005) Numerical simulation of self-oscillations of human vocal folds with Hertz model of impact forces. J Fluids Struct 20:853–869CrossRefGoogle Scholar
  15. 15.
    Alipour F, Titze IR (1996) Combined simulation of two-dimensional airflow and vocal fold vibration. In: Davis PJ, Fletcher NH (eds) Vocal fold physiology, controlling complexity and chaos. San DiegoGoogle Scholar
  16. 16.
    De Vries MP, Schutte HK, Veldman AEP, Verkerke GJ (2002) Glottal flow through a two-mass model: comparison of Navier–Stokes solutions with simplified models. J Acoust Soc Am 111:1847–1853CrossRefGoogle Scholar
  17. 17.
    Titze IR (2006) The myoelastic aerodynamic theory of phonation. National Center for Voice and Speech, DenverGoogle Scholar
  18. 18.
    Punčochářová-Pořízková P, Kozel K, Horáček J (2011) Simulation of unsteady compressible flow in a channel with vibrating walls—influence of frequency. Comput Fluids 46:404–410MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Horáček J, Švec JG (2002) Aeroelastic model of vocal-fold-shaped vibrating element for studying the phonation threshold. J Fluids Struct 16:931–955CrossRefGoogle Scholar
  20. 20.
    Zhang Z, Neubauer J, Berry DA (2007) Physical mechanisms of phonation onset: a linear stability analysis of an aeroelastic continuum model of phonation. J Acoust Soc Am 122:2279–2295CrossRefGoogle Scholar
  21. 21.
    Nomura T, Hughes TJR (1992) An arbitrary Lagrangian–Eulerian finite element method for interaction of fluid and a rigid body. Comput Methods Appl Mech Eng 95:115–138zbMATHCrossRefGoogle Scholar
  22. 22.
    Feistauer M, Horáček J, Kučera V, Prokopová J (2012) On numerical solution of compressible flow in time-dependent domains. Math Bohemica 137: 1–16Google Scholar
  23. 23.
    Badia S, Codina R (2007) On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity. Int J Numer Meth Eng 72:46–71MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  • Jaroslava Hasnedlová
    • 1
  • Miloslav Feistauer
    • 1
  • Jaromír Horáček
    • 2
  • Adam Kosík
    • 1
  • Václav Kučera
    • 1
  1. 1.Department of Numerical Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  2. 2.Institute of ThermomechanicsThe Academy of Sciences of the Czech RepublicPraha 8Czech Republic

Personalised recommendations