Approximation of shell geometry for nonlinear analysis

Philippe Destuynder

doi:10.1016/S0045-7825(97)00202-8

Approximation of shell geometry for nonlinear analysis

1998-04-01

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Philippe Destuynder *

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Michel Salaün

DOI：10.1016/S0045-7825(97)00202-8

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A major difficulty in the approximation of shell structures is the definition of the medium surface. Except for the particular case where it is known from analytical formula, one has only reduced information, like the positions of vertices and the normal vector to the medium surface at these points. The situation is worse for geometrically nonlinear problems, like arbitrary Euler-Lagrange formulations, where the geometry is continuously updated. The goal of this paper is then to define a then to define a mathematical analysis of shell model approximations based on an updating of the geometry. The first step is to define an approximation of all the quantities necessary for a shell formulations. For instance, it is possible to construct an approximation of the curvature tensor based on a linear interpolation of the unit normal on each plane element. Then, the derivatives are piecewise constant. One can guess that this is an improvement if the method is compared to the classical flat element method. The second step is to define a finite element scheme where the shell geometry is approximated by the method mentioned previously. As a matter of fact, there is a major difficulty because the third-order derivatives of the mapping which defines the medium surface of the shell, are necessary in a consistent shell model like the Koiter one (derivatives of the curvature operator which are not approximated in the method). Using a mixed finite element technique enables one to overcome this problem. But, obviously, the convergence when the mesh size tends to zero requires that these third-order derivatives are bounded. (C) 1998 Elsevier Science S.A.

Approximation of shell geometry for nonlinear analysis

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