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.

本文使用傅里叶分析技术从理论上证明了静压问题的SPH积分公式解的唯一性，以及这种解对精确线性压力场的收敛性。当在自由表面上施加压力的Dirichlet边界条件（BC）时，此问题涉及内核的截断。假设某些假设，最重要的是压力场的变化发生在平滑长度h的长度尺度上。理论分析辅以数值测试。除了自由表面的BC之外，数值解还要求截断其下方的无限子域，从而对压力施加Neumann BC。与以前的研究（基于流动扩展的类别专门对其进行评估）相反，本文采用全局方法寻求截断方程与这些BC的数值解的一致性和收敛性。在这些数值测试中，与理论结果一致，对于离散化，数值显示了精确解的收敛性，其中粒子间距离与h的比率为1阶。但是，当该比率变为零时，则h也趋于零，表明长度小于h的长度尺度出现在解决方案中，并且会聚消失。结论对于SPH从业者很重要，因为将比率设置为一阶是减少计算时间的标准做法。