SPH Cell Distribution

It is recommended to distribute the particles through a hexagonal compact or a cubic net.

Hexagonal Compact Net

A cubic centered faces net realizes a hexagonal compact distribution and this can be useful to build the net (図 1). The nominal value ${h}_{0}$ is the distance between any particle and its closest neighbor. The mass of the particle ${m}_{p}$ may be related to the density of the material $\rho$ and to the size ${h}_{0}$ of the hexagonal compact net, with respect to:(1)
${m}_{p}\approx \frac{{h}_{0}^{3}}{\sqrt{2}}\rho$
Since the space can be partitioned into polyhedras surrounding each particle of the net, each one with a volume:(2)
${V}_{p}\approx \frac{{h}_{0}^{3}}{\sqrt{2}}$

But, due to discretization error at the frontiers of the domain, mass consistency better corresponds to ${m}_{P}=\frac{\rho V}{n}$ .

Where,
$V$
Total volume of the domain and $n$ the number of particles distributed in the domain

Weight functions vanish at distance $2h$ where $h$ is the smoothing length. In an hexagonal compact net with size ${h}_{0}$ , each particle has exactly 54 neighbors within the distance ${\mathrm{2h}}_{0}$ (表 1).

Distance d Number of Particles at Distance d Number of Particles within Distance d
${h}_{0}$ 12 12
$\sqrt{2}{h}_{0}$ 6 18
$\sqrt{3}{h}_{0}$ 24 42
$2{h}_{0}$ 12 54
$\sqrt{5}{h}_{0}$ 24 78

Cubic Net

Let $c$ the side length of each elementary cube into the net. The mass of the particles ${m}_{p}$ should be related to the density of the material $\rho$ and to the size $c$ of the net, with respect to the following equation:(3)
${m}_{p}\approx {c}^{3}\rho$
By experience, a larger number of neighbors must be taken into account with the hexagonal compact net, in order to solve the tension instability as explained in following sections. A value of the smoothing length between 1.25c and 1.5c seems to be suitable. In the case of smoothing length h=1.5c, each particle has 98 neighbors within the distance $2h$ .

Distance d Number of Particles at Distance d Number of Particles within Distance d
$c$ 6 6
$\sqrt{2}c$ 12 18
$\sqrt{3}c$ 8 26
2c 6 32
$\sqrt{5}c$ 24 56
$\sqrt{6}c$ 24 80
2 $\sqrt{2}c$ 12 92
3c 6 98