A low-cost space-time finite element for free-surface flows under total Lagrangian description

Resumo

In this work, we propose a position-based space-time finite element formulation for incompressible Newtonian flows under a total Lagrangian description. This formulation is suggested within the context of finite strain free-surface flows and differs from the traditional finite element approach for fluid dynamics by utilizing current nodal positions as the main variables instead of nodal velocities. In contrast to time-marching methods, space-time formulations involve applying the finite element technique not only to the spatial domain but also to the temporal domain. The proposed approach employs a finite element discretization that can be unstructured or structured in space, but is always structured in time. Thus, the space-time shape functions are expressed as a tensor product of linear shape functions in the spatial direction and specially designed quadratic shape functions in the temporal direction. Consequently, the space-time domain is divided into time slabs that are solved progressively, with the final velocities and positions from the previous time slab serving as initial conditions for the current one, thereby reducing the dimension of the discrete system of equations. The proposed shape functions in the temporal direction yield a system of equations of the same size as standard time-marching methods, but with advantages stemming from the space-time discretization: different stability and high-frequency dissipation can be achieved based on the selection of the time test functions. To solve the incompressible problem stably, we employ mixed equal-order position-pressure finite elements with Petrov-Galerkin/pressure stabilization (PSPG). This formulation possesses several significant features that justify its development: 1) the space-time formulation facilitates dynamic re-meshing by permitting the spatial discretization at the end of one time slab to differ entirely from the spatial discretization at its beginning, thus extending its applicability to flows with undefined distortion and topological changes; 2) by considering positions as variational parameters, it becomes straightforward to couple with Lagrangian hyper-elastic solid solvers, which may also be formulated in terms of positions or displacements in a monolithic way. Numerical examples conducted to validate the formulation demonstrate its robustness and efficiency for finite strain free surface flows, including phenomena such as dam breaks with smooth surfaces and sloshing.