A total Lagrangian position-based finite element formulation for free-surface incompressible flows

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-Babuška-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.