Graphics Reference
In-Depth Information
son 06] have established that these hurdles can be overcome using robust
collision methods developed for cloth animation, but at present the imple-
mentation diculty associated with the approach makes other techniques
more attractive.
6.2
Level Set Methods
Instead of building an implicit function around particles, then sampling
it on a grid for rendering, we can dispense with particles and work with
the grid directly. This is the core idea of
level set methods
. Their chief
advantage over marker particles is the elimination of blobby artifacts: level
sets can easily give you beautifully smooth water surfaces.
Here we'll touch on just the basics we need; readers might look at the
topic by Osher and Fedkiw [Osher and Fedkiw 02], for example, for a more
detailed study of the numerics and applications of level sets.
Define the implicit surface function
φ
i,j,k
at the centers of simulation
grid cells (i.e., in the same locations as pressure, etc.). Tri- or bilinear
interpolation can be used to estimate
φ
(
x
) in between cell centers. The
surface is taken to be the points where
φ
(
x
) = 0; by convention we'll
identify the region where
φ
(
x
)
<
0 to be the water, or more generally the
inside of the surface, and the region where
φ
(
x
)
>
0 to be the air, or the
outside.
This gives us a lot of freedom to choose
φ
: infinitely many functions
have the same zero level set. Several arguments can be made that
signed
distance
is the most convenient function.
6.2.1 Signed Distance
Given any closed set
S
of points, the
distance function
for the set is
distance
S
(
x
)=min
p∈S
x
−
p
;
x
(
s
); using the flow map Φ
t
which takes initial positions to their positions after being
advected up to time
t
, this curve is advected to Φ
t
(
x
(
s
)), which is also smooth if Φ
t
is smooth. In reality, the smoothness underlying Navier-Stokes breaks down as we ap-
proach molecular scales, at which point a continuum velocity field makes little sense.
Topology changes are ultimately outside of the domain of continuum mechanics. How-
ever, numerically we tend not to worry about this issue and simply make the assumption
that if a connecting tendril of water gets thinner than a grid cell it breaks apart.
