## Preprint 2010-13

## Lourenco Beirao-Da-Veiga, David Mora, Rodolfo Rodríguez:

### Numerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner-Mindlin plate model

### Abstract:

This paper deals with the approximation of the bending of a clamped plate modeled by Reissner-Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a new mixed formulation in terms of the bending moments, shear stress, rotations and transversal displacement. To prove that the resulting variational formulation is well posed, we use standard Babuska-Brezzi theory with appropriate t-dependent norms. The problem is discretized by standard finite elements and error estimates are proved with constants independent of the plate thickness. Moreover, these constants depend on norms of the solution that can be a priori bounded independently of the plate thickness, which leads to the conclusion that the method is locking-free. A local post-processing leading to H1-approximations of transversal displacement and rotations is introduced. Finally, we report numerical experiments confirming the theoretical results.

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

**Lourenco BEIRAO-DA-VEIGA, David MORA, Rodolfo RODRíGUEZ: ***Numerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner-Mindlin plate model*. Numerical Methods for Partial Differential Equations, vol. 29, 1, pp. 40-63, (2013).