Numerical algorithms for mass conservation applications

This dissertation focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier-Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart-Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. Quadrilateral and hexahedral finite element spaces are considered to investigate the error convergence of different variables. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart-Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier-Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the dissertation considers an interface advection problem using marker approximation, in the context of multiphase flow simulations. A C++ library is presented, where the implemented algorithm is able to initialize, advect, and reconstruct a marker cloud performing a best-fit quadratic interpolation. Several numerical tests, equipped with an analytical velocity field for the surface advection, are presented to demonstrate the robustness of the algorithm. Lastly, a comparison with an interpolated velocity by using Raviart-Thomas basis functions is shown, with the aim of maintaining zero divergence, mitigating the classical issue of finite element mass loss.