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Approximation of the vibration modes of a plate coupled with afluid by low-order isoparametric finiteelements

Published online by Cambridge University Press:  15 December 2004

Erwin Hernández
Erwin Hernández*
Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaiso, Chile. [email protected].
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Abstract

We analyze an isoparametric finite element method to compute the vibration modes of a plate, modeled by Reissner-Mindlin equations, in contact with a compressible fluid, described in terms of displacement variables. To avoid locking in the plate, we consider a low-order method of the so called MITC (Mixed Interpolation of Tensorial Component) family on quadrilateral meshes. To avoid spurious modes in the fluid, we use a low-order hexahedral Raviart-Thomas elements and a non conforming coupling is used on the fluid-structure interface. Applying a general approximation theory for spectral problems, under mild assumptions, we obtain optimal order error estimates for the computed eigenfunctions, as well as a double order for the eigenvalues. These estimates are valid with constants independent of the plate thickness. Finally, we report several numerical experiments showing the behavior of the methods.

Keywords

Reissner-MindlinMITC methodsfluid-structure interaction.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 6 , November 2004 , pp. 1055 - 1070
Copyright
© EDP Sciences, SMAI, 2004

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References

S.M. Alessandrini, D.N. Arnold, R.S. Falk and A.L. Madureira, Derivation and justification of plate models by variational methods, in Plates and Shells, M. Fortin Ed., AMS, Providence, CRM Proc. Lect. Notes Ser. 21 (1999) 1–20.
Arnold, D.N. and Falk, R.S., A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26 (1989) 1276–1290. CrossRef
K.J. Bathe and F. Brezzi, On the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation in Mathematics of Finite Elements an Applications V, J.R. Whiteman Ed., Academic Press, London (1985) 491–503.
Bathe, K.J. and Dvorkin, E.N., A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int. J. Numer. Methods Eng. 21 (1985) 367–383. CrossRef
Bermúdez, A. and Rodríguez, R., Finite element computation of the vibration modes of a fluid-solid system. Comp. Methods Appl. Mech. Eng. 119 (1994) 355–370. CrossRef
Bermúdez, A., Gamallo, P. and Rodríguez, R., An hexahedral face element for elastoacoustic vibration problems. J. Comp. Acoust. 119 (1994) 355–370.
Bermúdez, A., Durán, R., Muschietti, M.A., Rodríguez, R. and Solomin, J., Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal. 32 (1995) 1280–1295. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Brezzi, F., Fortin, M. and Stenberg, R., Quasi-optimal error bounds for approximation of shear-stresses in Mindlin-Reissner plate models. Math. Models Methods Appl. Sci. 1 (1991) 125–151. CrossRef
Durán, R. and Liberman, E., On mixed finite element methods for the Reissner-Mindlin plate model. Math. Comp. 58 (1992) 561–573. CrossRef
Durán, R., Hervella-Nieto, L., Liberman, E., Rodríguez, R. and Solomin, J., Approximation of the vibration modes of a plate by Reissner-Mindlin equations. Math. Comp. 68 (1999) 1447–1463. CrossRef
Durán, R., Hervella-Nieto, L., Liberman, E., Rodríguez, R. and Solomin, J., Finite element analysis of the vibration problem of a plate coupled with a fluid. Numer. Math. 86 (2000) 591–616. CrossRef
Durán, R., Hernández, E., Hervella-Nieto, L., Liberman, E. and Rodríguez, R., Computation of the vibration modes of plates and shells by low-order MITC quadrilateral finite elements. SIAM J. Numer. Anal. 41 (2003) 1751–1772. CrossRef
P. Gamallo, Métodos numéricos de elementos finitos en problemas de interacción fluido-estructura. Ph.D. Thesis, U. de Santiago de Compostela, Spain (2002).
V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1986).
T.J.R. Hughes, The Finite Element Method: Linear Static and Dinamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ (1987).
H.J.-P. Morand and R. Ohayon, Fluid-structure interactions. John Wiley & Sons, New York (1995).
Rodríguez, R. and Solomin, J., The order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems. Math. Comp. 65 (1996) 1463–1475. CrossRef
Raviart, P.A. and Thomas, J.M., A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, Springer-Verlag, Berlin, Heidelberg, New York. Lect. Notes Math. 606 (1977) 292–315. CrossRef
Stenberg, R. and Suri, M., An hp error analysis of MITC plate elements. SIAM J. Numer. Anal. 34 (1997) 544–568. CrossRef
J.M. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes. Thèse de Doctorat d'Etat, Université Pierre et Marie Curie, Paris 6, France (1977).