Variational integration approach for arbitrary Lagrangian-Eulerian formulation of flexible cables

Variational integration approaches are favorable for long-time simulations, due to their remarkable symplectic and momentum conservation properties, as well as the nearly energy-preserving feature with the bounded energy error. However, none of the work has been introduced into arbitrary Lagrangian-Eulerian (ALE) formulations, which are crucial to applications such as the deployment of tether satellites and reeving systems. In this paper, a novel variational approach for the ALE formulation of flexible cables is developed for the first time. The fact that the mesh nodes in ALE formulations are not fixed at specific material points makes the classical variational schemes ineffective. Instead of directly adopting Hamilton's principle for non-material volume, D'Alembert's principle in the form of integrals is deduced equivalently. Moreover, isoparametric coordinates are introduced to resolve the spacetime coupling caused by the moving mesh. The virtual works are integrated within the spacetime domain, resulting in elegant and simplified derivations and concise equations for a variational integration scheme. Several benchmarks with either variable-length cables or variable grids are simulated and analyzed, verifying the effectiveness of the present variational integration approach for ALE cable elements.