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Approximation of the vibration modes
of a plate by Reissner-Mindlin equations


Authors: R. G. Durán, L. Hervella-Nieto, E. Liberman, L. Hervella-Nieto and J. Solomin
Journal: Math. Comp. 68 (1999), 1447-1463
MSC (1991): Primary 65N25, 65N30
DOI: https://doi.org/10.1090/S0025-5718-99-01094-7
Published electronically: May 19, 1999
MathSciNet review: 1648387
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family. Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding $L^2$-estimate for a load problem which is proven here. This optimal order $L^2$-estimate is of interest in itself. Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory.


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Additional Information

R. G. Durán
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: rduran@dm.uba.ar

L. Hervella-Nieto
Affiliation: Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain
Email: luisher@zmat.usc.es

E. Liberman
Affiliation: Comisión de Investigaciones Científicas de la Provincia de Buenos Aires and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 172., 1900 La Plata, Argentina
Email: elsali@mate.unlp.edu.ar

L. Hervella-Nieto
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 4009, Concepción, Chile
Email: rodolfo@ing-mat.udec.cl

J. Solomin
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 172., 1900 La Plata, Argentina
Email: solo@mate.unlp.edu.ar

DOI: https://doi.org/10.1090/S0025-5718-99-01094-7
Keywords: Mixed methods, Reissner-Mindlin, plates, eigenvalues
Received by editor(s): November 13, 1997
Published electronically: May 19, 1999
Additional Notes: The first author was partially supported by UBA through grant EX-071. Member of CONICET (Argentina).
The fourth author was partially supported by FONDECYT (Chile) through grant No. 1.960.615 and FONDAP-CONICYT (Chile) through Program A on Numerical Analysis.
The fifth author was partially supported by SECYT through grant PIP-292. Member of CONICET (Argentina).
Article copyright: © Copyright 1999 American Mathematical Society

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