8 Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

For the elastodynamic simulation of a geometrically exact beam [1], an asynchronous variational integrator (AVI) [2] is derived from a PDE viewpoint. Variational integrators are symplectic and conserve discrete momentum maps and since the presented integrator 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]. The discrete Euler-Lagrange equations are derived in a general manner and then applied to the beam. 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 behavior, i.e. there is no drift of the total energy for conservative systems. 8.