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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  • 1. K.J. Bathe, E.N. Dvorkin, A formulation of general shell elements - the use of mixed interpolation of tensorial components, Int. J. Numer. Methods Engrg. 22 (1986) 697-722.
  • 2. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
  • 3. V. Havu, J. Pitkäranta, Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations, Math. Comp. 71 (2002) 923-943.
  • 4. M. Malinen, On the classical shell model underlying bilinear degenerated shell finite elements, Int. J. Numer. Methods Engrg. 52 (2001) 389-416.
  • 5. J. Pitkäranta, Y. Leino, O. Ovaskainen, and J. Piila, Shell deformation states and the finite element method: a benchmark study of cylindrical shells, Comput. Methods Appl. Mech. Engrg. 128 (1995), no. 1-2, 81–121. MR 1376906, 10.1016/0045-7825(95)00870-X
  • 6. Juhani Pitkäranta, The first locking-free plane-elastic finite element: historia mathematica, Comput. Methods Appl. Mech. Engrg. 190 (2000), no. 11-12, 1323–1366. MR 1807008, 10.1016/S0045-7825(00)00163-8
  • 7. Juhani Pitkäranta, The problem of membrane locking in finite element analysis of cylindrical shells, Numer. Math. 61 (1992), no. 4, 523–542. MR 1155337, 10.1007/BF01385524

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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
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