Graphics Reference
In-Depth Information
-10-
Hybrid Particle Methods
Advection is one of the central themes of fluid simulation in this topic.
We've already struggled with the excessive numerical dissipation produced
by semi-Lagrangian advection with linear interpolation, and improved it
significantly with a Catmull-Rom interpolant. Even sharper grid-based
methods have been developed—however, all Eulerian schemes have a fun-
damental limit.
One way of looking at a time step of any Eulerian advection schemes is
as follows:
Begin with the field sampled on a grid.
Reconstruct the field as a continuous function from the grid samples.
Advect the reconstructed field.
Resample the advected field on the grid.
Technically speaking, some Eulerian schemes might use more than just the
values sampled on the grid—e.g., multistep time integration will also use
several past values of the field on the grid—but these details don't change
the gist: the key is the resampling at the end.
The problem is that a general incompressible velocity field, though it
preserves volumes, may introduce significant distortions: at any point in
space, the advected field may be stretched out along some axes and squished
together along others. From a rendering perspective, you might think of
this as a local magnification (stretching out) along some axes and a local
minification (squishing together) along the others. And just as in render-
ing, while resampling a stretched out or magnified field doesn't lose any
information, resampling a shrunk or minified field can lead to information
loss. If the advected field has details varying at the grid scale Δ x ,assoon
as those get shrunk in advection, resampling will at best destroy them (or
worse, if care is not taken in the numerical method, cause them to alias as
spurious lower-frequency artifacts—again, just as in rendering).
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