ALE or Arbitrary Lagrangian Eulerian formulation is used to model the interaction between fluids and solids; in particular,
the fluid loading on structures. It can also be used to model fluid-like behavior, as seen in plastic deformation of materials.

Smooth Particle Hydrodynamics (SPH) is a meshless numerical method based on interpolation theory. It allows any function
to be expressed in terms of its values at a set of disordered point's so-called particles.

Smooth Particle Hydrodynamics (SPH) is a meshless numerical method based on interpolation theory. It allows any function
to be expressed in terms of its values at a set of disordered point's so-called particles.

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:(1)

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

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 $2h$ 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).

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
$\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 $2h$.