Adaptive and anisotropic mesh strategy for thin shell problems. Case of inhibited parabolic shells

The aim of this paper is to show the reliability of an adaptive and anisotropic mesh procedure for thin shell problems. We consider singular perturbation problems only for parabolic shells whose behavior is described by the Koiter model. The corresponding system of equations, which depends on the relative thickness ε of the shell, is elliptic except at the limit for ε = 0 where it is parabolic. In a first part of this paper, we study theoretically the phenomena of internal layers appearing during the singular perturbation process, when the loading is somewhat singular. These layers have very different structures either they are along or across the asymptotic lines of the middle surface of the shell. In a second part, numerical computations are performed using a finite element software coupled with an adaptive anisotropic mesh generator. This technique enables to approach accurately the singularities and the layers predicted by the theory especially for very small values of the thickness. The efficiency of such a procedure in comparison with uniform meshes is put in a prominent position.