This article 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 mixed formulation based on the Hellinger-Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babuška-Brezzi theory with appropriate t -dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking-free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H1 -type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method.

Numerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner-Mindlin plate model / L. Beirão da Veiga, D. Mora, R. Rodríguez. - In: NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0749-159X. - 29:1(2013 Jan), pp. 40-63. [10.1002/num.21698]

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

#### Abstract

This article 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 mixed formulation based on the Hellinger-Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babuška-Brezzi theory with appropriate t -dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking-free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H1 -type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method.
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bending moment formulation; error analysis; locking-free finite elements; Reissner-Mindlin
Settore MAT/08 - Analisi Numerica
gen-2013
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/176732`