Approximation of the vibration modes of a plate by Reissner-Mindlin equations
HTML articles powered by AMS MathViewer
- by R. G. Durán, L. Hervella-Nieto, E. Liberman, L. Hervella-Nieto and J. Solomin PDF
- Math. Comp. 68 (1999), 1447-1463 Request permission
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.References
- B.S. Al Janabi and E. Hinton, A study of the free vibrations of square plates with various edge conditions, in Numerical Methods and Software for Dynamic Analysis of Plates and Shells, E. Hinton, ed., Pineridge Press, Swansea, 1987.
- Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. MR 1025088, DOI 10.1137/0726074
- P. G. Ciarlet and J.-L. Lions (eds.), Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR 1115235
- K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982.
- 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, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 491–503. MR 811058
- K.J. Bathe and E.N. Dvorkin, A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation, Internat. J. Numer. Methods Eng. 21 (1985) 367–383.
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Franco Brezzi, Michel Fortin, and Rolf Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 125–151. MR 1115287, DOI 10.1142/S0218202591000083
- Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
- Ricardo Durán and Elsa Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58 (1992), no. 198, 561–573. MR 1106965, DOI 10.1090/S0025-5718-1992-1106965-0
- D. J. Dawe and O. L Roufaeil, Rayleigh-Ritz vibration analysis of Mindlin plates, J. Sound. Vib., 12 (1980) 345–359.
- H.C. Huang and E. Hinton, A nine node Lagrangian Mindlin plate element with enhanced shear interpolation, Eng. Comput., 1 (1984) 369–379.
- Thomas J. R. Hughes, The finite element method, Prentice Hall, Inc., Englewood Cliffs, NJ, 1987. Linear static and dynamic finite element analysis; With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. MR 1008473
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- P. Peisker and D. Braess, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, RAIRO Modél. Math. Anal. Numér. 26 (1992), no. 5, 557–574 (English, with English and French summaries). MR 1177387, DOI 10.1051/m2an/1992260505571
- Juhani Pitkäranta and Manil Suri, Design principles and error analysis for reduced-shear plate-bending finite elements, Numer. Math. 75 (1996), no. 2, 223–266. MR 1421988, DOI 10.1007/s002110050238
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
- L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
- O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 2, McGraw-Hill, 1989.
Additional Information
- R. G. Durán
- Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
- ORCID: 0000-0003-1349-3708
- 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
- 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). - © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 1447-1463
- MSC (1991): Primary 65N25, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-99-01094-7
- MathSciNet review: 1648387