## Preprint 2009-13

## Ricardo Durán, Rodolfo Rodríguez, Frank Sanhueza:

### Computation of the vibration modes of a Reissner-Mindlin laminated plate

### 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.

This preprint gave rise to the following definitive publication(s):

**Ricardo DURáN, Rodolfo RODRíGUEZ, Frank SANHUEZA: ***Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate*. Mathematics of Computation, vol. 80, 275, pp. 1239-1264, (2011).