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Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate

Authors: Ricardo G. Durán, Rodolfo Rodríguez and Frank Sanhueza
Journal: Math. Comp. 80 (2011), 1239-1264
MSC (2010): Primary 65N25, 74K10, 65N30
Published electronically: February 7, 2011
MathSciNet review: 2785457
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Abstract: This paper deals with the finite element approximation of the vibration modes of a laminated plate modeled by the Reissner-Mindlin equations; DL3 elements are used for the bending terms and standard piecewise linear continuous elements for the in-plane displacements. An a priori estimate of the regularity of the solution, independent of the plate thickness, is proved for the corresponding load problem. This allows using the abstract approximation theory for spectral problems to study the convergence of the proposed finite element method. Thus, optimal order error estimates including a double order for the vibration frequencies are obtained under appropriate assumptions. These estimates are independent of the plate thickness, which leads to the conclusion that the method is locking-free. Numerical tests are reported to assess the performance of the method.

References [Enhancements On Off] (What's this?)

  • 1. D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26 (1989) 1276-1290. MR 1025088 (91c:65068)
  • 2. F. Auricchio, C. Lovadina, and E. Sacco, Analysis of mixed finite elements for laminated composite plates, Comput. Methods Appl. Mech. Engrg., 190 (2000) 4767-4783. MR 1840799 (2002d:74062)
  • 3. F. Auricchio and E. Sacco, A mixed-enhanced finite element for the analysis of laminated composite plates, Internat. J. Numer. Methods Engrg., 44 (1999) 1481-1504.
  • 4. I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, 1991, pp. 641-787. MR 1115240
  • 5. 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 Engrg., 21 (1985) 367-383.
  • 6. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 7. D. Chenais and J.-C. Paumier, On the locking phenomenon for a class of elliptic problems, Numer. Math., 67 (1994) 427-440. MR 1274440 (95g:65147)
  • 8. R. Durán, E. Hernández, L. Hervella-Nieto, E. Liberman, and R. Rodríguez, Error estimates for low-order isoparametric quadrilateral finite elements for plates, SIAM J. Numer. Anal., 41 (2003) 1751-1772. MR 2035005 (2004m:65192)
  • 9. R. Durán, L. Hervella-Nieto, E. Liberman, R. Rodríguez, and J. Solomin, Approximation of the vibration modes of a plate by Reissner-Mindlin equations, Math. Comp., 68 (1999) 1447-1463. MR 1648387 (99m:73045)
  • 10. R. Durán and E. Liberman, On mixed finite element methods for Reissner Mindlin Plate model, Math. Comp., 58 (1992) 561-573. MR 1106965 (92f:65135)
  • 11. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. MR 1335452 (96a:47025)
  • 12. V. Kozlov, V. Maz'ya, and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs 85, AMS, Providence, RI, 2001. MR 1788991 (2001i:35069)
  • 13. O.O. Ochoa and J.N. Reddy, Finite element analysis of composite laminates, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
  • 14. P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics 606, Springer, Berlin, 1977, pp. 292-315. MR 0483555 (58:3547)
  • 15. J.N. Reddy, Energy and Variational Methods in Applied Mechanics, Wiley, New York, 1984.
  • 16. J.N. Reddy, Mechanics of Laminated Composite Plates - Theory and Analysis, CRC Press, Boca Raton, 1997.

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Additional Information

Ricardo G. Durán
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina

Rodolfo Rodríguez
Affiliation: CI$^{2}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

Frank Sanhueza
Affiliation: Escuela de Obras Civiles, Universidad Andres Bello, Autopista 7100, Concepción, Chile.

Keywords: Reissner-Mindlin, laminated plates, spectral problems.
Received by editor(s): August 24, 2009
Received by editor(s) in revised form: June 9, 2010
Published electronically: February 7, 2011
Additional Notes: The first author was partially supported by Universidad de Buenos Aires under grant X070. Member of CONICET (Argentina).
The second author was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile (Chile).
The third author was supported by a CONICYT fellowship (Chile).
All authors were partially supported by ANPCyT through grant PICT RAÍCES 2006, No. 1307 (Argentina).
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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