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

DOI:
https://doi.org/10.1090/S0025-5718-2011-02456-7

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.

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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

Email:
rduran@dm.uba.ar

**Rodolfo Rodríguez**

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

Email:
rodolfo@ing-mat.udec.cl

**Frank Sanhueza**

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

Email:
fsanhueza@unab.cl

DOI:
https://doi.org/10.1090/S0025-5718-2011-02456-7

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.