In this paper we introduce a new procedure to solve free-surface problems based on applying the Newmark family of time integration schemes to non-Eulerian formulations of the problem (i.e., in non-current domains). The methods obtained within this framework present important advantages over other methods in literature, as for instance, that the computational domain is independent of the current unknowns, the convective term disappears, and modelling and tracking of the free surface is straightforward. Moreover, the Newmark algorithm is convenient to obtain accurate and stable methods for solving continuum mechanics models which can be written in terms of either displacement, velocity or acceleration. We consider a viscous Newtonian fluid in a time dependent domain which may undergo large deformations along the time but not topological changes of interfaces. A unified formulation providing the general framework of the proposed schemes is stated in this context. They are combined with finite elements methods for space discretization. In particular, the Newmark family of pure Lagrange–Galerkin methods is stated. The non-Eulerian formulations are advantageous to obtain linear methods. More precisely, linearized versions of the standard (non-linear) Newmark schemes that present good approximation properties are also proposed. Moreover, we deal with problems associated with high mesh distortion by proposing a reinitialization method to be applied in the non-Eulerian Newmark framework that preserves the order of convergence. In order to assess the performance of the proposed numerical methods, we solve different problems in two space dimensions.

在本文中，我们基于将Newmark系列时间积分方案应用于问题的非欧拉公式（即在非当前域中），介绍了一种解决自由表面问题的新方法。在此框架内获得的方法相对于文献中的其他方法具有重要的优势，例如，计算域独立于当前的未知数，对流项消失，自由表面的建模和跟踪非常简单。此外，Newmark算法方便地获得用于求解连续力学模型的准确而稳定的方法，该模型可以用位移，速度或加速度来表示。我们考虑在时间相关域中的粘性牛顿流体，该流体可能会随时间经历较大的变形，但不会发生界面的拓扑变化。在这种情况下，提出了提供拟议方案总体框架的统一表述。它们与有限元方法结合用于空间离散化。尤其指出了Newmark系列的纯Lagrange-Galerkin方法。非欧拉公式有利于获得线性方法。更准确地说，还提出了具有良好近似特性的标准（非线性）Newmark方案的线性化版本。此外，我们通过提出一种重新初始化方法来解决与高网格变形有关的问题，该方法将在保留收敛顺序的非欧拉纽马克框架中应用。