In order to keep an almost constant number of neighbors contributing at each particle, use
smoothing length varying in time and in space.
Consider
the smoothing length related to particle
;
(1)
and
if kernel correction
or
(2)
and
without kernel correction
At each time step, density is updated for each particle
, according to:
(3)
with
(4)
Where,
-
- Mass of a particle
-
- Density
-
- Velocity
Strain tensor is obtained by the same way when non pure hydrodynamic laws are used or in
the other words when law uses deviatoric terms of the strain tensor:
(5)
Next the constitutive law is integrated for each particle. Then Forces are computed
according to:
(6)
Where
and
are pressures at particles
and
, and
is a term for artificial viscosity. The expression is more
complex for non pure hydrodynamic laws.
注: The previous equation reduces to the following
one when there is no kernel correction:
(7)
since
Then, in order particles to keep almost a constant number of neighbors into their kernels (
is kept constant), search distances are updated according
to:
(8)