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In-Depth Information
vorticity confinement term (see Chapter 9). Using some smooth Gaussian-
like kernel function
ξ
around each particle, with support spanning at least
a few grid cells, the acceleration from each particle applied to nearby grid-
velocity values is taken as
f
p
(
x
)=
x
p
−
ω
p
.
x
×
x
p
−
x
The confinement parameter
again can be tuned to get more or less influ-
ence from the spin particles.
10.4
Particle-in-Cell Methods
A major part of the numerical viscosity we have battled is due to the
Eulerian advection. Techniques such as vorticity confinement and spin
particles can partially mitigate this, but as it turns out, we can do even
better. In this section, we will replace the velocity advection step
Du/Dt
=
0 itself with particle advection.
Simply storing velocity vectors on particles and advecting them around
isn't enough, of course: pressure projection to keep the velocity field
divergence-free globally couples all the velocities together. Somehow we
need to account for particle-particle interactions on top of advection. We'll
now take a look at a general class of methods that can eciently treat
particle-particle interaction by way of a grid.
The particle-in-cell (PIC) approach was pioneered at Los Alamos Na-
tional Laboratory in the early 1950s, though Harlow's 1963 paper [Har-
low 63] is perhaps the first journal article describing it.
1
The basic PIC
method begins with all quantities—velocities included—stored on particles
that sample the entire fluid region. For graphics work, eight particles per
grid cell, initialized as usual from a jittered grid, seems to be about right.
In each time step, we first transfer the quantities such as velocity from
the particles to the grid—perhaps just as we did with smoke concentration
earlier in this chapter. All the non-advection terms, such as acceleration
due to gravity, pressure projection and resolution of boundary conditions,
viscosity, etc., are integrated on the grid, just as in a fully Eulerian solver.
1
Harlow and other team members in the T-3 Division at Los Alamos also created
the MAC grid and marker particle method, and pioneered the use of the vorticity-
streamfunction formulation used in the previous section, to name just a few of their
many contributions to computational fluid dynamics; see the review article by Har-
low [Harlow 04] for more on the history of this group.
