Isogeometric collocation for implicit dynamics of three-dimensional beams undergoing finite motions

We propose a novel approach to the implicit dynamics of shear-deformable geometrically exact beams, based on the isogeometric collocation method combined with the Newmark time integration scheme extended to the rotation group SO(3). The proposed formulation is fully consistent with the underlying geometric structure of the configuration manifold. The method is highly efficient, stable, and does not suffer from any singularity problem due to the (material) incremental rotation vector employed to describe the evolution of finite rotations. Consistent linearization of the governing equations, variables initialization and update procedures are the most critical issues which are discussed in detail in the paper. Numerical applications involving very large motions and different boundary conditions demonstrate the capabilities of the method and reveal the critical role that the high-order approximation in space may have in improving the accuracy of the solution