Geometrically Exact Kirchhoff Beam Theory: Application to Cable Dynamics

In this article, the finite element simulation of cables is investigated for future applications to robotics and hydrodynamics. The solution is based on the geometrically exact approach of Cosserat beams in finite transformations, as initiated by Simo in the 1980s. However, the internal basic kinematics of the beam theory is not those of Reissner–Timoshenko but rather those of Kirchhoff. Based on these kinematics, the dynamic model adopted is a nonlinear extension of the so-called linear model of twisted and stretched Euler–Bernoulli beams. In agreement with the investigated applications, one or both of the ends of the cable are submitted to predefined motions. This model is also implemented into a computational fluid dynamics code, which solves the Reynolds-averaged Navier–Stokes equations. Regarding this last point, an implicit/iterative algorithm including a conservative load transfer for the variable hydrodynamic forces exerted all along the beam length has been used to reach a stable coupling. The relevance of the approach is tested through three advanced examples. The first is related to the prediction of cable motion in robotics. Then, the two last illustrations deal with fluid-structure interaction (FSI). A 2D classical benchmark in FSI is first investigated, and, at last, a computation illustrates the procedure in a 3D case.