# 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 $\rho $ and to the size ${h}_{0}$ of the hexagonal compact net, with respect to:

Since the space can be partitioned into polyhedras surrounding each particle of the net, each one with a volume:

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

- $V$
- Total volume of the domain
- $n$
- 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 ${\mathrm{2h}}_{0}$ (Table 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:

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 |