Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model |
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Authors: | Lourenço Beirão da Veiga David Mora Rodolfo Rodríguez |
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Affiliation: | 1. Dipartimento di Matematica “F. Enriques”, Università Degli Studi di Milano, Via Saldini 50, Milano 20133, Italy;2. Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Casilla 5‐C, Concepción, Chile;3. Centro de Investigación en Ingeniería Matemática (CI 2 MA), Universidad de Concepción, Concepción, Chile;4. CI 2 MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160‐C, Concepción, Chile |
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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 Babu?ka‐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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 |
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Keywords: | bending moment formulation error analysis locking‐free finite elements Reissner‐Mindlin |
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