On the truncated integral SPH solution of the hydrostatic problem

Abstract

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.

Appendices

Transforming an integral equation into a (full) convolution equation

In this appendix, a generalization of the processes followed in Sects. 5 and 6 to obtain full convolution equations [(23) and (27)] from their preceding truncated integral equations [(1) and (15), respectively,] is provided. Note that a convolution equation is of the form (28). This is not the case of Eqs. (1) and (15), which are instead of the form

$$\begin{aligned} \displaystyle \int _{-\infty }^0 W_h(x-y) f_h(y) \mathrm{d}y = g(x), \quad x \in (-\infty ,0), \end{aligned}$$

(41)

where $f_h(z)$ and g(z) are functions defined only for $z < 0$.

The process consists of two main steps:

• 1. In the first step, the integral is extended to the whole real line by introducing the extended function

  $$\begin{aligned} f_h^-(z) := {\left\{ \begin{array}{ll} f_h(z), &{}z < 0, \\ 0, &{}z \ge 0. \end{array}\right. } \end{aligned}$$

  (42)

  Applying this extension to (41) leads to an integral in which the limits of integration have been extended to the whole space ${\mathbb {R}}$:

  $$\begin{aligned} \displaystyle \int _{-\infty }^{\infty } W_h(x-y) f^-_h(y) \mathrm{d}y = g(x), \quad x \in (-\infty ,0). \end{aligned}$$

  (43)

• 2. In a second step, a full convolution equation is obtained by requiring that (43) holds for every value of x in the whole real line. The left-hand side term becomes then $(W_h *f^-_h)(x)$. This convolution may take values different from zero for some $x > 0$. To take this into account, an extra unknown, $g^+_h(x)$, has to be introduced in the right-hand side. This new function has to be identically zero for $x \in (-\infty , 0)$⁴ so the equation is unchanged in the region $x<0$. The function g(x) is extended by zero for $x > 0$. All these considerations finally lead to

  $$\begin{aligned} \displaystyle \int _{-\infty }^{\infty } W_h(x-y) f^-_h(y) \mathrm{d}y = g^-(x) + g^+_h(x), \quad x \in {\mathbb {R}}. \end{aligned}$$

  (44)

Deduction of a full convolution equation for the error $e_h$

Let us now particularize the process described in A to deduce Eq. (27). Consider Eq. (15):

• In the first step, the function $p_h(y)$ is extended by zero to ${\mathbb {R}}$, getting

  $$\begin{aligned} \displaystyle \int _{-\infty }^{\infty } W_h(x-y) p^-_h(y) \mathrm{d}y = p(x), \quad x \in (-\infty ,0), \end{aligned}$$

  (45)

  with p(x) the exact solution, $p(x) = -x$, defined for $x \in (-\infty ,0)$.

• To extend the previous equation to the whole real line, the new unknown $g^+_h(x)$ is introduced, leading to

  $$\begin{aligned} (W_h *p^-_h )(x) = p^-(x) + g^+_h(x), \quad x \in {\mathbb {R}}. \end{aligned}$$

  (46)

  Finally, substituting $p_h^-(y)$ by $p^-(y) + e^-_h(y)$ in (46) and rearranging the resulting different terms one recovers equation (23):

  $$\begin{aligned} (W_h *e^-_h )(x) = p^-(x) - (W_h *p^-)(x) + g^+_h(x), \quad x \in {\mathbb {R}}. \end{aligned}$$

  (47)