# SPH Cells Distribution

The particles should be created in a hexagonal compact, face centered cubic or a cubic net packing.

The hexagonal compact net and face centered cubic are recommended for use in Radioss and give similar results. A face center cubic particle distribution can be created using the HyperMesh SPH panel. A HyperMesh Tcl macro to generate hexagonal compact net is available from the Altair Connect website.

## Hexagonal Compact Net

A cubic centered faces net realizes a hexagonal compact distribution, which can be useful to build the net.

The hexagonal compact net distribution can be created in HyperMesh using a Tcl script available by searching in Altair Connect for the Hexagonal Compact Net Tcl script. When using this script, the pitch or distance between any particle and its closest neighbor is entered as h0.

The mass of the particle mp is defined in the property /PROP/SPH.

The SPH particle mass relates to the material density $\rho$ and the pitch h0 of the hexagonal compact net. This particle mass can be calculated as:(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)
${V}_{p}=\frac{{h}_{0}^{3}}{\sqrt{2}}$
Due to discretization differences in the volume, the mass can be more accurately represented by:(3)
${m}_{P}=\frac{\rho V}{n}$
Where,
$V$
Total volume filled by the particles.
$n$
Total number of particles distributed in the volume.

For hexagonal compact net, the recommended smoothing length h in /PROP/SPH is the pitch h0 which is the smallest distance between the particles. A smoothing length smaller than is this can only be used when there is no tension physical problems material. If the material does include tensile behavior, then a smoothing length larger than h0 can be used to increase stability but there will be an increase in the computational cost.

The following table shows the number of neighbors with when different smoothing lengths h in /PROP/SPH are used. The accuracy and computational cost of the simulation improves as the smooth length increases.
Table 1. Number of Neighbors in a Hexagonal Compact Net
Distance d Number of particles at distance d Number of particles within distance d
h0 12 12
$\sqrt{2}{h}_{0}$ 6 18
$\sqrt{3}{h}_{0}$ 24 42
2h0 12 54
$\sqrt{5}{h}_{0}$ 24 78

## Face Centered Cubic

The face centered cubic arranges the particles in groups of 14, forming the corners and the center of each face of a cube.
Similar to hexagonal compact net, each particle has 12 neighbors and the mass of a particle is:(4)
${m}_{P}=\frac{\rho V}{n}$

For face centered cubic the recommended smoothing length h in /PROP/SPH is the pitch entered when created the sph mesh in HyperMesh. The pitch h0 is the smallest distance between the particles. A smoothing length smaller than h0 can only be used when there is no tension physical problems material. If the material does include tensile behavior, then a smoothing length larger than h0 can be used to increase stability but there will be an increase in the computational cost.

## Simple Cubic Net

Let, c the side length of each elementary cube into the net.
The mass of the particles mp should relate to the density of the material $\rho$ and to the size c of the net, with respect to the following equation:(5)
${m}_{p}={c}^{3}\rho$
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
$\sqrt{2}c$ 12 92
3c 6 98

From experience using cubic net, a higher smoothing length compared to face centered cubic or hexagonal compact net is needed to solve the tension instability. This higher smoothing length increases the computational cost since more neighbor particles have to be included in the calculation for each particle.

For cubic net, a smoothing length h between 1.25c and 1.5c is recommended in /PROP/SPH.