An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

本文介绍了一种用于任意弯曲和变形二维材料和界面的拉格朗日-欧拉（ALE）有限元方法。ALE理论是通过在曲面上赋予面内速度而不依赖于面内材料速度的网格开发的，可以任意指定。提出了该理论的有限元实现方式，并将其应用于具有平面内不可压缩流的弯曲和变形表面。通过将表面张力局部投影到分段线性函数的不连续空间上，可以消除与平面内不可压缩性相关的数值不稳定性。基于具有曲线坐标的任意表面参数化的通用等参有限元方法，已针对多个数值基准进行了测试和验证。通过将ALE显影技术应用于圆柱状流体膜，获得了新的物理见解，在计算和分析上，它们对非轴对称扰动是稳定的，而当其长度超过其圆周时，对于长波长轴对称扰动则不稳定。拉格朗日方案是ALE公式的特例。尽管无法对具有持续剪切流的流体膜进行建模，但拉格朗日方案已通过重现圆柱不稳定性来验证。但是，相对于ALE结果，发现拉格朗日模拟具有节点数少，因此误差较大的空间未解析区域。当长轴对称扰动的长度超过其圆周时，则不稳定。拉格朗日方案是ALE公式的特例。尽管无法对具有持续剪切流的流体膜进行建模，但拉格朗日方案已通过重现圆柱不稳定性来验证。但是，相对于ALE结果，发现拉格朗日模拟具有节点数少，因此误差较大的空间未解析区域。当长轴对称扰动的长度超过其圆周时，则不稳定。拉格朗日方案是ALE公式的特例。尽管无法对具有持续剪切流的流体膜进行建模，但拉格朗日方案已通过重现圆柱不稳定性来验证。但是，相对于ALE结果，发现拉格朗日模拟具有节点数少，因此误差较大的空间未解析区域。