With recent advances in variable-length structures for use in soft actuation, energy harvesting, energy dissipation and metamaterials, the mathematical modelling and numerical simulation of physical systems with time-varying domains is becoming increasingly important. The planar nonlinear dynamics of one-dimensional elastic structures with variable domain is formulated from a Lagrangian approach by using a non-material reference frame. An Arbitrary Lagrangian-Eulerian (ALE) scheme is proposed where the domain is reparametrized based on a priori unknown configuration parameters. Based on this formulation, a Finite Element (FE) method is developed for theoretically predicting the evolution of a rod constrained at its ends by one or two sliding-sleeves, whose position and inclination can be varied in time, and under external loadings. Finally, case studies and instability problems are investigated to assess the reliability of the proposed formulation against others available and to demonstrate its effectiveness. With respect to previously developed methods for this type of structural problems, the present ALE-FE approach shows a strong theoretical and implementation simplicity, maintaining an efficient and fast convergence according to the cases analyzed. An open source code realized for the present ALE-FE model is made available for solving the nonlinear dynamics of planar systems constrained by one or two independent sliding sleeves. The present research paves the way for further extensions to easily implement solvers for the three-dimensional dynamics of flexible one- and two-dimensional structural systems with moving boundary conditions.

中文翻译：

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一种简单而有效的变长挠性棒非线性平面动力学 ALE-FE 方法


随着用于软驱动、能量收集、能量耗散和超材料等的可变长度结构的最新进展，具有时变域的物理系统的数学建模和数值模拟变得越来越重要。具有可变域的一维弹性结构的平面非线性动力学是使用非材料参考系从拉格朗日方法公式化的。提出了一种任意拉格朗日-欧拉 （ALE） 方案，其中域根据先验未知配置参数重新参数化。基于这个公式，开发了一种有限元 （FE） 方法，用于从理论上预测杆的末端受一个或两个滑动套筒约束的杆的演变，其位置和倾斜度可以随时间变化，并在外部载荷下变化。最后，调查案例研究和不稳定性问题，以评估所提出的公式与其他可用公式的可靠性并证明其有效性。对于以前开发的此类结构问题的方法，目前的 ALE-FE 方法显示出很强的理论和实施简单性，根据分析的案例保持了高效和快速的收敛。为本 ALE-FE 模型实现的开源代码可用于求解受一个或两个独立滑套约束的平面系统的非线性动力学。本研究为进一步的扩展铺平了道路，以便轻松实现具有移动边界条件的柔性一维和二维结构系统的三维动力学求解器。