Skip to main content
SpringerLink
Log in

Finite element analysis of pressure formulation of the elastoacoustic problem

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we analyze the non symmetric pressure/displacement formulation of the elastoacoustic vibration problem and show its equivalence with the (symmetric) stiffness coupling formulation. We introduce discretizations for these problems based on Lagrangian finite elements. We show that both formulations are also equivalent at discrete level and prove optimal error estimates for eigenfunctions and eigenvalues. Both formulations are rewritten such as to be solved with a standard Matlab eigensolver. We report numerical results comparing the efficiency of the methods over some test examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babuška, I., Osborn, J.: Eigenvalue problems. Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet y J.L. Lions, eds., North Holland, Amsterdam, 1991

  2. Bermúdez, A., Durán, R., Muschietti, M.A., Rodríguez, R., Solomin, J.: Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal. 32, 1280–1295 (1995)

    Google Scholar 

  3. Bermúdez, A., Rodríguez, R.: Finite element computation of the vibration modes of a fluid-solid system. Comput. Methods Appl. Mech. Eng. 119, 355–370 (1994)

    Article  Google Scholar 

  4. Bermúdez, A., Rodríguez, R.: Analysis of a finite element method for pressure/ potential formulation of elastoacoustic spectral problems. Math. Comp. 71, 537–552 (2002)

    Article  Google Scholar 

  5. Conca, C., Planchard, J., Vanninathan, M.: Fluids and Periodic Structures. Masson, Paris, 1995

  6. Dauge, M.: Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Lecture Notes in Mathematics, 1341, Springer, Berlin, 1988

  7. Dauge, M.: Problèmes de Neumann et de Dirichlet sur un polyèdre dans ℝ3: régularité dans des spaces de Sobolev L p . C. R. Acad. Sci. Paris, Série I 307, 27–32 (1988)

    Google Scholar 

  8. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985

  9. Hamdi, M., Ousset, Y., Verchery, G.: A displacement method for the analysis of vibrations of coupled fluid-structure systems. Int. J. Numer. Methods Eng. 13, 139–150 (1978)

    MATH  Google Scholar 

  10. Kolata, W.G.: Approximation in variationally posed eigenvalues problems. Numer. Math. 29, 159–171 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Morand, H.J-P., Ohayon, R.: Interactions Fluides-Structures. Recherches en Mathématiques Appliquées 23, Masson, Paris, 1992

  12. Ohayon, R., Soize, C.: Structural Acoustics and Vibration. New York: Academic Press, 1998

  13. Olson, L.G., Bathe, K.J.: A study of displacement-based fluid finite elements for calculating frequencies of fluid and fluid-structure systems. Nucl. Eng. Design 76, 137–151 (1983)

    Article  Google Scholar 

  14. von Petersdorff, T.: Boundary value problems of elasticity in polyhedra: singularities and approximation with boundary element methods. PhD Thesis, Technical University Darmstadt, Darmstadt, Germany, 1989

  15. von Petersdorff, T., Stephan, E.P.: Regularity of mixed boundary value problems in ℝ3 and boundary integral methods on graded meshes. Math. Methods Appl. Sci. 12, 229–249 (1990)

    MATH  Google Scholar 

  16. Petyt, M.: Introduction to Finite Element Vibration Analysis. Cambridge-New York: Cambridge University Press, 1990

  17. Sandberg, G.: A new strategy for solving fluid-structure problems. Int. J. Numer. Methods Eng. 38, 357–370 (1995)

    MathSciNet  MATH  Google Scholar 

  18. Wandinger, J.: Analysis of small vibrations of coupled fluid-structure systems. Z. angew. Math. Mech. 74, 37–42 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Zienkiewicz, O.C., Newton, R.E.: Coupled vibration of a structure submerged in a compressible fluid. Proc. Int. Symp. on Finite Element Techniques, Stuttgart, 1–15 (1969)

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    A. Bermúdez

  2. Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    P. Gamallo

  3. Departamento de Matemáticas, Universidade da Coruña, 15071, A Coruña, Spain

    L. Hervella-Nieto

  4. GI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile

    R. Rodríguez

Authors
  1. A. Bermúdez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Gamallo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Hervella-Nieto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Bermúdez.

Additional information

Partially supported by Xunta de Galicia (Spain) through grant No. PGIDT00-PXI20701PR and by MCYT Research Project DPI2001-1613-C02-02

Partially supported by Xunta de Galicia (Spain) through grant No. PGIDT00-PXI20701PR and by MCYT Research Project DPI2001-1613-C02-02

Partially supported by FONDAP in Applied Mathematics (Chile) and by MCYT Research Projects DPI2001-1613-C02-02 and BFM2001-3261-C02-02

Partially supported by FONDECYT (Chile) through grant No. 1.990.346 and FONDAP in Applied Mathematics (Chile)

Mathematics Subject Classification (1991): 65N30

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bermúdez, A., Gamallo, P., Hervella-Nieto, L. et al. Finite element analysis of pressure formulation of the elastoacoustic problem. Numer. Math. 95, 29–51 (2003). https://doi.org/10.1007/s00211-002-0414-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-002-0414-0

Keywords

Search

Navigation