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8.1.3 Smoothed Particle Hydrodynamics (SPH)
 PIC technique uses both Eulerian and Lagrangian elements.
Average density within a cell = number of point particles
in that cell.
 Can we represent the local fluid density without a grid?
Lucy [LUCY 77] and Gingold and Monaghan [GINGOLD 77,MONAGHAN 92]:
load each particle with a spatially extended
interpolation kernel
Average local density
= sum over the individual contributions.
Let
denote the interpolation kernel centered
around
. Then the local density at
is

(8.33) 
Generally, a property
is represented by
its ``smoothed particle estimate''

(8.34) 
Form of the interpolation kernel: Gaussian or polynomial
Example:

(8.35) 
with a width chosen such that the number of particles within
is in 2 dimensions and for 3 dimensions.
Now rewrite the Lagrangian equations of motion
8.8, 8.9 and 8.11.
in smoothed particle form.
Note:
In the momentum equation
,
interpolating and directly would not conserve
linear and angular momentum [MONAGHAN 92].
Use the identity

(8.36) 
and the SPH expressions for
and
to find

(8.37) 
with
.
If is Gaussian, this equation describes the motion of particle
under the influence of central pair forces

(8.38) 
The SPH equivalents of the other Lagrangian flow equations are

(8.39) 
where
, and

(8.40) 
Note: The density equation need not be integrated; just update all
positions , then invoke
equ. 8.33 to find
.
To update
the obvious relation

(8.41) 
might be used; a better way is

(8.42) 
with
. This relation
maintains angular and linear momentum conservation,
with the additional advantage that nearby particles will have similar
velocities [MONAGHAN 89].
To solve equs. 8.39, 8.37, 8.41 and
8.40, use any suitable algorithm (see Chapter 4).
Popular schemes are the leapfrog algorithm, predictorcorrector and
RungeKutta methods.
Example: Variant of the halfstep technique ([MONAGHAN 89]):
Given all particle positions at time , the local density
at
is computed from
8.33. Writing equs. 8.37 and 8.40 as

(8.43) 
compute the predictors

(8.44) 
and

(8.45) 
Determine midpoint values of
,
and according to

(8.46) 
etc. From these, compute midpoint values of ,
and and insert these in correctors of the type

(8.47) 
Here is an overview of the method:
Smoothed particle hydrodynamics (SPH)
Note: SPH is not restricted to compressible inviscid flow.
Incompressibility may be introduced by using an equation of state
that keeps compressibility below a few percent [MONAGHAN 92],
and viscosity is added by an additional term in the
equations of motions for momentum and energy, equs. 8.37
and 8.40:
The artificial viscosity term is modeled in the following way:

(8.50) 
where is the speed of sound, is defined by

(8.51) 
and
and
etc.
This form of introduces the effects of shear and bulk viscosity.
The parameters and should be near
and for best results [MONAGHAN 92]. The quantity
prevents singularities for
. It should be
chosen such that
.
Additional features:
Thermal conduction may be included. See [MONAGHAN 89].
Interfaces: Introduce dummy particles on the far side of
the boundary. By picking the properties of these particles appropriately
one can mimick a free surface or a ``sticky'' solid boundary.
See Nugent and Posch [NUGENT 00] for
free surfaces, and Ivanov [IVANOV 00] for rough interfaces
Sample application:
``RayleighBénard'' convection
A fluid layer is heated carefully from below and cooled from above.
Formation of stable convective rolls transporting heat from
the bottom to the top. See the Figure for a match
between SPH and an Eulertype calculation [HOOVER 99]:
 Computing times comparable for both calculations
 Results are in good agreement
 Fluctuations in SPH (like in any particletype calculation), none
in Euler
 SPH code is quite simple  similar to an MD program; Euler code
very massive
Comparison of Smoothed Particle Hydrodynamics with an
Eulerian finitedifference calculation. The density (above) and
temperature (below) contours for a stationary RayleighBénard
flow are shown. Left: SPH; right: Euler.
(From [HOOVER 99], with kind permission by the author)
Next: 8.2 Incompressible Flow with
Up: 8.1 Compressible Flow without
Previous: 8.1.2 ParticleinCell Method (PIC)
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001