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.

中文翻译：

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柔性电缆任意拉格朗日-欧拉公式的变分积分方法


变分积分方法有利于长时间模拟，因为它们具有显著的对称和动量守恒特性，以及具有有限能量误差的近乎能量守恒的特性。然而，这些工作都没有被引入任意的拉格朗日-欧拉 （ALE） 公式中，这些公式对于部署系留卫星和卷筒系统等应用至关重要。在本文中，首次开发了一种新的柔性电缆 ALE 公式的变分方法。ALE 公式中的网格节点不固定在特定的材料点上，这使得经典的变分方案无效。D'Alembert 原理不是直接采用汉密尔顿原理的非物质体积，而是以积分的形式等价地推导出来。此外，引入等参坐标来解决动网格引起的时空耦合。虚拟作品被整合到时空域中，从而为变分积分方案产生了优雅而简化的推导和简洁的方程。对变长电缆或可变网格的几个基准进行了仿真和分析，验证了当前变分积分方法对 ALE 电缆元件的有效性。