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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Analysis of a bilinear finite element for shallow shells. II: Consistency error


Authors: Ville Havu and Juhani Pitkäranta
Journal: Math. Comp. 72 (2003), 1635-1653
MSC (2000): Primary 65N30; Secondary 73K15
Published electronically: March 4, 2003
MathSciNet review: 1986797
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Abstract: We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound $O(h + t^{-1}h^{1+s})$ can be obtained if the contribution of transverse shear is neglected. Here $t$ is the thickness of the shell, $h$ the mesh spacing, and $s$ a smoothness parameter. In the bending-dominated case, the uniformly optimal bound $O(h)$ is achievable but requires that membrane and transverse shear strains are of order $O(t^2)$ as $t \rightarrow 0$. In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound $O(h)$.


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Additional Information

Ville Havu
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 Helsinki University of Technology, Finland
Email: Ville.Havu@hut.fi

Juhani Pitkäranta
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 Helsinki University of Technology, Finland
Email: Juhani.Pitkaranta@hut.fi

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01508-4
PII: S 0025-5718(03)01508-4
Keywords: Finite elements, locking, shells
Received by editor(s): January 18, 2001
Received by editor(s) in revised form: February 7, 2002
Published electronically: March 4, 2003
Article copyright: © Copyright 2003 American Mathematical Society