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 (図 1). The nominal value is the distance between any particle and its closest neighbor.
The mass of the particle may be related to the density of the material
and to the size 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)
But, due to discretization error at the frontiers of the domain, mass consistency better
corresponds to .
Where,
Total volume of the domain and the number of particles distributed in the domain
注: Choosing for the smoothing length insures naturally consistency up to
order 1 if the previous equation is satisfied.
Weight functions vanish at distance where is the smoothing length. In an hexagonal compact net with size , each particle has exactly 54 neighbors within the distance (表 1).
表 1. Number of Neighbors in a Hexagonal Compact Net
Distance d
Number of Particles at Distance
d
Number of Particles within Distance
d
12
12
6
18
24
42
12
54
24
78
Cubic Net
Let the side length of each elementary cube into the net. The mass
of the particles should be related to the density of the material
and to the size of the net, with respect to the following
equation:(3)
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 .