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 (Figure 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

ρ

and to the size

h_{0}

of the hexagonal compact net, with respect to:(1)

m_{p} \approx \frac{h_{0}^{3}}{\sqrt{2}} ρ

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 $m_{P} = \frac{ρ V}{n}$ .

Where,

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

Note: Choosing

h_{0}

for the smoothing length insures naturally consistency up to order 1 if the previous equation is satisfied.

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

Table 1. Number of Neighbors in a Hexagonal Compact Net
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

ρ

and to the size

c

of the net, with respect to the following equation:(3)

m_{p} \approx c^{3} ρ

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

2 h

.

Table 2. Number of Neighbors in a Cubic Net
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