Abstract In this work, we propose a position-based finite element formulation for incompressible Newtonian flows under total Lagrangian description. Such formulation is different from the traditional finite element approach used in fluid dynamics by using current nodal positions as main variable instead of nodal velocities. The variational form of the governing equations is derived by applying the stationary total mechanical energy principle written in terms of current positions. To introduce full incompressibility, we use equal-order mixed finite elements combined to a pressure stabilizing Petrov-Galerkin technique, circumventing Ladyzhenskaya-Babuska-Brezzi conditions. This leads to a method directly applicable to finite strain incompressible flow problems that, if combined to re-meshing techniques, is capable of simulating more complex problems, resulting into a suitable tool for free-surface flows analysis in general. The efficiency and robustness of the proposed approach is demonstrated with numerical studies and comparison of obtained results to numerical and experimental data from literature.

中文翻译：

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自由表面不可压缩流的基于拉格朗日位置的总有限元公式

摘要 在这项工作中，我们提出了一种基于位置的有限元公式，用于在总拉格朗日描述下不可压缩的牛顿流。这种公式不同于流体动力学中使用的传统有限元方法，它使用当前节点位置而不是节点速度作为主要变量。控制方程的变分形式是通过应用根据当前位置编写的固定总机械能原理推导出来的。为了引入完全不可压缩性，我们将等阶混合有限元与压力稳定 Petrov-Galerkin 技术相结合，绕过 Ladyzhenskaya-Babuska-Brezzi 条件。这导致了一种直接适用于有限应变不可压缩流动问题的方法，如果结合重新划分网格技术，能够模拟更复杂的问题，最终形成了一个适用于一般自由表面流动分析的工具。通过数值研究以及将获得的结果与文献中的数值和实验数据进行比较，证明了所提出方法的效率和稳健性。