Isogeometric Collocation Methods for the Nonlinear Dynamics of Three-Dimensional Timoshenko Beams

In this paper the isogeometric collocation (IGA-C) method is used to solve the dynamic problem of geometrically exact beams. The kinematics of a spatial Timoshenko beam undergoing finite displacements and rotations involves the Lie group ({mathrm{SO(3)}}). Most of the computational complexities originate from the presence of such a non-additive and non-commutative rotation group. By employing the incremental rotation vector to describe the evolution of finite rotations, we discuss how the IGA-C method can efficiently be used in both explicit and implicit Newmark-based schemes.