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Locking-free finite element methods for shells

Authors: Douglas N. Arnold and Franco Brezzi
Journal: Math. Comp. 66 (1997), 1-14
MSC (1991): Primary 65N30, 73K15, 73V05
MathSciNet review: 1370847
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Abstract: We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer $k$. The methods are based on a nonstandard mixed formulation, and the $k$th method employs triangular Lagrange finite elements of degree $k+2$ augmented by bubble functions of degree $k+3$ for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree $k$ for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the $H^{1}$ norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order $k+1$ uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.

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Additional Information

Douglas N. Arnold
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
MR Author ID: 27240

Franco Brezzi
Affiliation: Istituto di Analisi Numerica del C.N.R., Università di Pavia, 27100 Pavia, Italy

Keywords: Shell, locking, finite element
Received by editor(s): December 2, 1993
Received by editor(s) in revised form: April 3, 1995, and November 13, 1995
Additional Notes: The work of the first author was supported by National Science Foundation grants DMS-9205300 and DMS-9500672. The work of the second author was partially supported by the HCM Program on Shells, contract number ERBCHRXCT 940536.
Article copyright: © Copyright 1997 American Mathematical Society