SPH Cells Distribution

The particles should be distributed through a hexagonal compact or a cubic net.

Hexagonal Compact Net

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

The nominal value h₀ of h for the hexagonal compact net is the distance between any particle and its closest neighbor.

The mass of the particle which will be considered is the mass m_p defined in the property which is associated to the part that the particle belongs to.

The mass of the particles m_p should relate to the density of the material

ρ

and to the size h₀ of the hexagonal compact net, with respect to Equation 1:(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} = \frac{h_{0}^{3}}{\sqrt{2}}

Notice that choosing h₀ for the smoothing length insures naturally consistency up to order one, if the previous equation is satisfied.

Weight functions vanish at distance 2h where, h is the smoothing length. In a hexagonal compact net with size h₀, each particle has exactly 54 neighbors within the distance 2h₀:

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
	6	18
	24	42
2`h`₀	12	54
	24	78

If SPH correction is used, notice that a small value for the smoothing length will increase the instability of the SPH method in case of tension behavior.

So, if using a hexagonal compact net and SPH corrections, it is recommended to set the smoothing length lower than the inter-particles minimum distance, only when there is no tension in the physical problem.

In the case of tension behavior, the value of the smoothing length can be set somewhat larger than the inter-particles distance, in order to limit the tension instability of SPH method. But a large value will result in excessive computational time.

Cubic Net

Let, c the side length of each elementary cube into the net.

The mass of the particles m_p should relate to the density of the material

ρ

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

m_{p} = c^{3} ρ

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
	12	18
	8	26
2c	6	32
	24	56
	24	80
	12	92
3c	6	98

From experience, a larger number of neighbors should be taken into account than in the hexagonal compact net, in order to solve the tension instability. A value of the smoothing length between 1.25c and 1.5c seems to be well-suited.

If choosing the smoothing length h=1.5c, each particle has 98 neighbors within the distance 2h.