An implicit time integrator for Cosserat rods based on the spherical Bézier interpolation

The updating procedure of an implicit time scheme integrator is based on linear operations of vectors defined at different instants. When the procedure involves rotational coordinates these quantities belong to different tangent spaces to SO(3). This issue makes the process formally not-consistent with the manifold. At the same time, a correct implementation should transport and project incremental rotation variables, or spin vectors, as well as angular velocities, and accelerations onto the same tangent space. Furthermore, spatial discretization of spin vectors should also be consistent with the interpolation of rotations in SO(3). The spherical Bézier interpolation of Cosserat rods proposed by the same authors in [1] solves these problems and provides a generalization to higher order interpolation of the Crisfield's generalized basis functions consistent with SO(3). Spin is interpolated through local basis functions updated during the motion, changing the configuration. This feature allows the use of an Eulerian approach for the update of local spin vectors. In this paper, the high-order spherical linear interpolation is further extended to the angular velocity and acceleration fields utilizing De Casteljau's algorithm. These outcomes are used to implement the dynamics of non-linear space Cosserat rods based on an Eulerian implicit time integrator and multiplicative update of rotations. Finally, three numerical applications demonstrate the feasibility of the proposed method and accordance with the results reported in the literature.