Corrected ALE-ISPH with novel Neumann boundary condition and density-based particle shifting technique

It is well-known in the Smoothed Particle Hydrodynamics (SPH) community that correction in the gradient and Laplacian operators have the potential to drastically increase the accuracy of the method at the expense of computational stability. This paper proposes a stable implementation of such corrections in all derivative operators to the Arbitrary Lagrangian Eulerian incompressible SPH (ALE-ISPH) method, in addition to a novel Neumann boundary condition (BC) applied directly on the velocity (as opposed to traditional BCs where the constraint is applied on the acceleration). In this way, the pressure is solved for both water and wall particles simultaneously, leading to a pressure field that obeys non-penetration BC and divergence-free at the same time. Furthermore, to stabilize the method, we have developed a novel density-based particle shifting technique (PST), specifically designed to deal with incompressible fluids. In this formulation, the numerical density is given as one of the most critical constraint variables. As a result, the proposed density-based PST can maintain the fluid's overall volume for the whole simulation. In addition, it also provides numerical stability as it prevents particle clustering and leads the fluid domain to an isotropic composition. First, we verified the proposed corrected formulation with the novel Neumann BC for both non-penetration and non-slip conditions with the simulation of hydrostatic pressure and Poisenuille flow, respectively. Then, we tested the proposed density-based PST with the rotating square patch problem with results comparable to previous studies. Lastly, we verified the proposed method for the dam break with an obstacle test, a highly dynamic problem.