Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations

Authors:
Ville Havu and Juhani Pitkäranta

Journal:
Math. Comp. **71** (2002), 923-943

MSC (2000):
Primary 65N30; Secondary 74K25

DOI:
https://doi.org/10.1090/S0025-5718-01-01376-X

Published electronically:
November 20, 2001

MathSciNet review:
1898740

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Abstract: We consider a bilinear reduced-strain finite element formulation for a shallow shell model of Reissner-Naghdi type. The formulation is closely related to the facet models used in engineering practice. We estimate the error of this scheme when approximating an inextensional displacement field. We make the strong assumptions that the domain and the finite element mesh are rectangular and that the boundary conditions are periodic and the mesh uniform in one of the coordinate directions. We prove then that for sufficiently smooth fields, the convergence rate in the energy norm is of optimal order uniformly with respect to the shell thickness. In case of elliptic shell geometry the error bound is furthermore quasioptimal, whereas in parabolic and hyperbolic geometries slightly enhanced smoothness is required, except for the degenerate cases where the characteristic lines are parallel with the mesh lines. The error bound is shown to be sharp.

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Additional Information

**Ville Havu**

Affiliation:
Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 Helsinki Univ. of Tech., Finland

Email:
Ville.Havu@hut.fi

**Juhani Pitkäranta**

Affiliation:
Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 Helsinki Univ. of Tech., Finland

Email:
Juhani.Pitkaranta@hut.fi

DOI:
https://doi.org/10.1090/S0025-5718-01-01376-X

Keywords:
Finite elements,
locking,
shells

Received by editor(s):
April 12, 2000

Received by editor(s) in revised form:
September 28, 2000

Published electronically:
November 20, 2001

Article copyright:
© Copyright 2001
American Mathematical Society