Asynchronous variational Lie group integration for geometrically exact beam dynamics

Abstract

For the elastodynamic simulation of a spatially discretized beam, asynchronous variational integrators (AVI) offer the possibility to use different time steps for every element [1]. They are symplectic and conserve discrete momentum maps and since the presented integrator for geometrically exact beam dynamics [2] is derived in the Lie group setting (SO(3) for the representation of rotational degrees of freedom), it intrinsically preserves the group structure without the need for constraints [3]. A decrease of computational cost is to be expected in situations, where the time steps have to be very low in certain parts of the beam but not everywhere, e.g. if some regions of the beam are moving faster than others. The implementation allows synchronous as well as asynchronous time stepping and shows very good energy behaviour, i.e. there is no drift of the total energy for conservative systems.