APPROXIMATION OF SHELL GEOMETRY FOR NONLINEAR-ANALYSIS

A major difficulty in the approximation of shell structures is the def inition of the medium surface. Except for the particular case where it is known from analytical formula, one has only reduced information, l ike the positions of vertices and the normal vector to the medium surf ace at these points. The situation is worse for geometrically nonlinea r problems, like arbitrary Euler-Lagrange formulations, where the geom etry is continuously updated. The goal of this paper is then to define a then to define a mathematical analysis of shell model approximation s based on an updating of the geometry. The first step is to define an approximation of all the quantities necessary for a shell formulation s. For instance, it is possible to construct an approximation of the c urvature 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 techniqu e enables one to overcome this problem. But, obviously, the convergenc e when the mesh size tends to zero requires that these third-order der ivatives are bounded. (C) 1998 Elsevier Science S.A.