1. Introduction

Large shell transformations are characterized by large movements of the mean surface and large rotations of the fibers that are initially normal to this surface. The transformation is therefore exactly represented, at least in the continuous problem. The derivation of the finite element objects associated with the linearized system of equations resulting from the principle of virtual work is carried out without any simplifying hypothesis on displacements or rotations. In addition, a new selective numerical integration scheme is presented in order to solve the problem of membrane blocking and transverse shear.

The degrees of freedom of rotation selected are the components of the spatial iterative rotation vector. Between two iterations, it is the vector of the infinitesimal rotation superimposed on the deformed configuration. This choice leads to a tangent stiffness matrix that is not symmetric. This is due to the non-vectorial nature of the large rotations which actually belong to the differential manifold \(\mathit{SO}(3)\). The rotations must remain less than \(2\pi\) because of the choice to update large rotations implemented in Code_Aster, for which there is no bijection between the total rotation vector and the orthogonal rotation matrix.

An important difference compared to linear analysis should be noted. Finite element objects are directly constructed in the global coordinate system; nodal movements and rotations are measured in the global coordinate system.