Graphics Reference
In-Depth Information
says the values stored on the particles shouldn't change, and thus there is
no filtering and no information loss. In some sense particles are perfect for
advection. In fact, we've already seen some uses for marker particles in
Chapter 5 for rendering smoke and Chapter 6 for tracking liquids.
Many people have in fact constructed fluid simulators that only use
particles with no grids at all, except perhaps as acceleration structures for
quickly finding which particles are close together. Just within graphics
this includes the seminal work of Miller and Pearce [Miller and Pearce 89],
the introduction of smoothed particle hydrodynamics (SPH) to graphics
by Desbrun and Cani [Desbrun and Cani 96], further development of SPH
for animation for example by Muller et al. [Muller et al. 03], and mesh-free
vortex methods by Angelidis and Neyret [Angelidis and Neyret 05] or Park
and Kim [Park and Kim 05]. However, while particles are excellent for
advection, grids are extremely hard to beat when it comes to eciently
and accurately enforcing incompressibility. Thus we will instead focus on
hybrid methods that use particles and grids together, trying to exploit the
best of both.
It should also be pointed out, before we get into details, that these
particle methods really are only worthwhile for fields with zero diffusion
(or viscosity, or conduction, or whatever other name is appropriate for
the quantity in question). If there is significant physical diffusion, strong
enough to show up on the grid length scale, then using an Eulerian advec-
tion scheme should work perfectly well without need for particles. We will
in fact take a look at incorporating small amounts of diffusion into particle
methods, but the emphasis is on small: for large diffusion you're probably
best off sticking to Eulerian methods.
10.1
Particle Advection
The details of a particle-grid method lie in how information is passed back
and forth between particles and the grid—which we'll explore for several
examples in the rest of this chapter. Underlying all of them, however, is a
particle advection stage, where we update particle positions based on the
MAC grid velocity field. As we discussed back in Chapter 3, the simplest
time integration scheme, forward Euler, really is insucient. In fact, the
requirements for particle advection are probably a bit stricter than for
tracing trajectories in semi-Lagrangian advection, since errors in particle
advection are accumulated over many time steps instead of being reset each
time step as in the semi-Lagrangian method.
