Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-09T07:57:13.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal error estimates for FEM approximations of dynamic nonlinear shallow shells

Published online by Cambridge University Press:  15 April 2002

Irena Lasiecka  and
Rich Marchand
Irena Lasiecka
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA.
Rich Marchand
Affiliation:
Department of Mathematics Sciences, United States Military Academy, West Point, NY 10996, USA.
Get access

Abstract

Finite element semidiscrete approximations on nonlinear dynamic shallow shell models in considered. It is shown that the algorithm leads to global, optimal rates of convergence. The result presented in the paper improves upon the existing literature where the rates of convergence were derived for small initial data only [19].

Keywords

Finite elementsnonlinear dynamic shellsoptimal error estimatesglobal existence and uniqueness.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 63 - 84
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Bernadou, Méthodes d'Éléments Finis pour les Problèmes de Coques Minces. Masson, Paris-Milan-Barcelone (1994).
M. Bernadou and P.G. Ciarlet, Sur l'ellipticité du modèle linéaire de coques de W.T. Koiter, in Computing Methods in Applied Sciences and Engineering (Lecture Notes in Economics and Mathematical Systems), Springer-Verlag (1976) 89-136.
M. Bernadou and B. Lalanne, On the approximations of free vibration modes of a general thin shell, application to turbine blades, in The Third European Conference on Mathematics in Industry, J. Manley et al. Eds., Kluwer Academic Publishers and B.G. Teubner Stuttgart (1990) 257-264.
M. Bernadou, P.G. Ciarlet and B. Miara, Existence theorems for two-dimensional linear shell theories. Technical Report 1771, Unité de Recherche INRIA-Rocquencourt (1992).
Bernadou, M. and An, J.T. Oden existence theorem for a class of nonlinear shallow shell problems. J. Math. Pures Appl. 60 (1981) 285-308.
P.G. Ciarlet, The Finite Element Method For Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
Delfour, M.C. and Zolésio, J.P., Tangential differential equations for dynamical thin/shallow shells. J. Differential Equations 128 (1995) 125-167. CrossRef
W. Flügge, Tensor Analysis and Continuum Mechanics. Springer-Verlag (1972).
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Verlag, New York (1984).
Glowinski, R. and Pironneau, O., Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev. 21 (1979) 167-212. CrossRef
R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in Domain Decomposition Methods for Partial Differential Equations, SIAM (1988) 144-172.
A.E. Green and W. Zerna, Theoretical Elasticity. Oxford University Press, 2nd. edn. (1968).
W.T. Koiter, On the nonlinear theory of thin elastic shells, in Proc. Kon. Ned. Akad. Wetensch., Vol. B (1966) 1-54.
J.E. Lagnese, Boundary Stabilization of Thin Plates. SIAM, Philadelphia, Pennsylvania (1989).
Lasiecka, I., Uniform stabilization of a full von Karman system with nonlinear boundary feedback. SIAM J. Control 36 (1998) 1376-1422. CrossRef
Lasiecka, I., Weak, classical and intermediate solutions to full von Karman system of dynamic nonlinear elasticity. Applicable Anal. 68 (1998) 123-145.
J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969).
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1. Springer Verlag (1972).
Mansfield, L., Analysis of finite element methods for the nonlinear dynamic analysis of shells. Numerische Mathematik 42 (1983) 213-235. CrossRef
R. Marchand, Finite element approximations of control problems arising in nonlinear shell theory. Ph.D. thesis, University of Virginia (1996).
V.G. Mazya and T.V. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions. Pitman (1985).
A. Raoult, Analyse mathématique de quelques modèles de plaques et de poutres élastiques ou élasto-plastiques. Doctoral Dissertation, Université Pierre et Marie Curie, Paris (1988).
H.L. Royden, Real Analysis. Macmillan Publishing Company, 3rd edn. (1988).
V.I. Sedenko, The uniqueness of generalized solutions of initial boundary value problem for Marguerre-Vlasov equation in the nonlinear oscillation theory of shallow shells. Izwestia Vysshyh Uchebnych Zavedenij (1994) 1-2.
V. Thomee, Galerkin Finite Element Methods for Parabolic Problems. Springer Verlag (1984).