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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 can be obtained if the contribution of transverse shear is neglected. Here is the thickness of the shell, the mesh spacing, and a smoothness parameter. In the bending-dominated case, the uniformly optimal bound is achievable but requires that membrane and transverse shear strains are of order as . In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound .

**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

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65N30,
73K15

Retrieve articles in all journals with MSC (2000): 65N30, 73K15

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