Energy-dissipative momentum-conserving time-stepping algorithms for the dynamics 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. RID="†" ID="†" Our motivation behind the developments presented in this paper started from a number of very instructive conversations with Professor M.A. Crisfield. His insight in the numerical treatment of the structural problems considered here was unique. It was for us a great privilege to interact with him and enjoy of his friendship. These interactions were always very rewarding, given especially how contagious his enthusiasm for his work was. We would like to dedicate this modest contribution to his memory. Currently at: E.T.S.I.C.C.P., Universidad Politécnica de Madrid, Spain Dedicated to the memory of Prof. mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years. Financial support for this research was provided by the AFOSR under contract no. F49620-00-1-0360 with UC Berkeley. This support is gratefully acknowledged.