On the truncated integral SPH solution of the hydrostatic problem

Uniqueness of solutions to the SPH integral formulation of the hydrostatic problem, and the convergence of such solution to the exact linear pressure field, are theoretically demonstrated in this paper using Fourier analytical techniques. This problem involves the truncation of the kernel when the Dirichlet boundary condition (BC) on the pressure is imposed at the free surface. Certain hypotheses are assumed, the most important being that the variations of the pressure field occur in length scales of the order of the smoothing length, h. The theoretical analysis is complemented with numerical tests. In addition to the BC at the free surface, the numerical solution requires truncating the infinite subdomain below it, imposing a Neumann BC for the pressure. The consistency and convergence of the numerical solution of the truncated equation with these BCs are sought herein with a global approach, as opposed to previous studies which exclusively assessed it based on the class of the flow extensions. In these numerical tests, and consistently with the theoretical results, the convergence to the exact solution is shown numerically for discretizations with an inter-particle distance to h ratio of order one. However, when this ratio goes to zero as h also goes to zero, it is shown that length scales shorter than h appear in the solution, and that convergence is lost. The conclusions are important for SPH practitioners as setting that ratio to be of order one is a standard practice to lower the computational time.