Numerical integration of the stiff dynamics of geometrically exact shells: an energy-dissipative momentum-conserving scheme

This paper presents a new family of time-stepping algorithms for the integration of the dynamics of non-linear shells. We consider the geometrically exact shell theory involving an inextensible director eld (the so-called ve-parameter shell model). The main characteristic of this model is the presence of the group of nite rotations in the con guration manifold describing the deformation of the solid. In this context, we develop time-stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system, and allow the introduction of a controllable non-negative energy dissipation to handle the high numerical sti ness characteristic of these problems. A series of algorithmic parameters for the di erent components of the deformation of the shell (i.e. membrane, bending and transverse shear) fully control this numerical dissipation, recovering existing energy-momentum schemes as a particular choice of these algorithmic parameters. We present rigorous proofs of the numerical properties of the resulting algorithms in the full non-linear range. Furthermore, it is argued that the numerical dissipation is introduced in the high-frequency range by considering the proposed algorithm in the context of a linear problem. The nite element implementation of the resulting methods is described in detail as well as considered in the nal arguments proving the aforementioned conservation=dissipation properties. We present several representative numerical simulations illustrating the performance of the newly proposed methods. The robustness gained over existing methods in these sti problems is con rmed in particular