Energy preserving/decaying schemes for non-linear beam dynamics using the helicoidal approximation

We develop two unconditionally stable displacement based time stepping schemes for the non-linear dynamic response of beams. The first algorithm guarantees the exact discrete conservation of energy and momentum. The second is associated with an energy decay inequality that achieves control of the unresolved frequencies by means of a numerical dissipation mechanism. Both schemes emanate from a weak form of the equations of dynamic equilibrium referred to a fixed pole. Space and time discretizations are based on the exponential parameterization of motion. This implies that the beam reference line and the trajectories of the beam nodes are helicoids in space. The exponential mapping approach allows a unified treatment of translations and rotations, greatly simplifying the derivation of the algorithms and their analysis. The capabilities and performance of the new schemes are demonstrated and discussed with the aid of numerical simulations.