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visually pleasing. Because it is more complex, it is often used along simple,
regular
grid
discretizations [Stam 99]. For liquids, several intermediate steps are
required for the surface to behave adequately [Enright et al. 02]. Implementing
these steps using rasterization APIs is challenging. For instance, [Crane et al. 07]
only partially implements them and the fluid behaves more like a single phase
fluid. Furthermore, the strong formulation requires a surface representation like a
level-set
density field, which requires its own set of specificities (re-initialization).
Again, in [Crane et al. 07] the level-set method is only partially implemented
and had to be hacked into staying at a certain height; preventing them from
generating such scenarios as the water jet shown in Figure 4.3.
The weak formulation on the other hand, requires only a simple soft con-
straint to keep the fluid from compressing. It is much simpler, but also less
accurate. It is often used along particle discretizations and
mesh-free
numeri-
cal schemes like
Smooth Particles Hydrodynamics
(SPH) [Desbrun and Cani 96].
The advantage of the weak formulation along particles is really for liquids. This
combination allowed reproducing the behavior of liquids without computing any
surface boundary conditions similar to [Enright et al. 02]. Additionally, the par-
ticles can be used directly to render the liquid surface and there is no need for a
level-set. The drawback however, is that particles require finding their neighbors,
in order to compute forces ensuring they keep at a minimal distance. Typically
buckets or other spacial sorting algorithms are used to cluster the particles into
groups [Amada et al. 04,Rozen et al. 08,Bayraktar et al. 08], which can be cum-
bersome to implement using rasterization APIs.
Instead of sorting the particles, our method makes use of rasterization ca-
pabilities. First, we rasterize, or splat, the particle density onto a grid. Then,
we use simple finite difference to compute all the interacting forces—including
the soft incompressibility constraint—on the grid, in a single pass. Some have
considered splatting before [Kolb and Cuntz 05] but had to splat for each force
(
pressure and viscosity
), while we only need to splat once—for all the forces. Fi-
nally, the particles are corrected and moved by sampling the grid—which in turn
can also be used
directly
to render the liquid surface by ray casting the splatted
density
. Overall, our method allows treating each particle independently while
making sure they automatically repulse one another and avoid rigid obstacles in
the domain.
4.3 Simple Liquid Model
Our liquid model is best described as follows: particles are allowed to move
freely in the domain while their mass and kinetic energy remains conserved. For
instance, if we add gravity it should translate into velocity
u
. To keep the particles
from interpenetrating, we add a soft incompressibility constraint
P
derived from
