Objective energy–momentum conserving integration for the constrained dynamics of geometrically exact beams

In this paper the results in [S. Leyendecker, P. Betsch, P. Steinmann, Energy-conserving integration of constrained Hamiltonian systems—a comparison of approaches, Comput. Mech. 33 (2004) 174–185] are extended to geometrically exact beams. The finite element formulation for nonlinear beams in terms of directors, providing a framework for the objective description of their dynamics, is considered. Geometrically exact beams are analysed as Hamiltonian systems subject to holonomic constraints with a Hamiltonian being invariant under the action of SO(3). The reparametrisation of the Hamiltonian in terms of the invariants of SO(3) is perfectly suited for a temporal discretisation which leads to energy–momentum conserving integration. In this connection the influence of alternative procedures, the Lagrange multiplier method, the Penalty method and the augmented Lagrange method, for the treatment of the constraints is investigated for the example of a beam with concentrated masses.