Energy-dissipative momentum-conserving time-stepping algorithms of nonlinear Cosserat rods

This paper presents the development of energy-dissipative momentum-conserving algorithms for the numerical integration of the dynamics of nonlinear Cosserat rods. The proposed numerical schemes exhibit a non-negative energy dissipation, controllable through the appropriate algorithmic parameters including an energy-conserving scheme as a particular case. These conservation/dissipation properties are proven rigorously in the general nonlinear setting, accounting specifically for the finite element implementation of the rotational degrees of freedom associated to the motion of the rod's cross-sections. In particular, we consider a direct parameterization of the director fields defining these sections, hence leading to frame-indifferent approximations of the strain measures defining the rod's mechanical response. The robustness added by these considerations when comparing the proposed numerical schemes with existing conserving schemes is illustrated with several representative numerical simulations.