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Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model
Author(s) -
da Veiga Lourenço Beirão,
Mora David,
Rodríguez Rodolfo
Publication year - 2013
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21698
Subject(s) - mathematics , finite element method , bending moment , mathematical analysis , discretization , bending of plates , partial derivative , displacement (psychology) , reduction (mathematics) , bending , geometry , physics , psychology , psychotherapist , thermodynamics
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 H 1 ‐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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013