Non-Eulerian Newmark Methods: A Powerful Tool for Free-Boundary Continuum Mechanics Problems

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.