Graphics Reference
In-Depth Information
-12-
Shallow Water
We'll finish off the topic with two chapters on special cases of water simula-
tion that allow much faster and simpler algorithms. In both cases, we will
use the simplifying assumption that the water surface can be represented
as a
height field
y
=
h
(
x, z
): the water region is all of the points where
y<h
(
x, z
), excluding solids. The most important solid is of course the
bottom, which we also represent as a height field
y
=
b
(
x, z
), giving a water
region defined by
b
(
x, z
)
<y<h
(
x, z
)
,
and thus the water depth is
d
(
x, z
)=
h
(
x, z
)
b
(
x, z
). We will actually use
depth
d
as a primary simulation variable, reconstructing the height
h
=
b
+
d
as needed. This geometric simplification rules out many interesting effects
such as convincing splashes, breaking waves, droplets or sprays, but still
allows many interesting wave motions.
1
For the purposes of the topic,
we'll also restrict the bottom to be stationary—
b
(
x, z
) remains constant—
though allowing it to move is a fairly easy generalization if you follow the
modeling steps in this chapter.
For the height field assumption to remain a good approximation for the
water throughout the simulation, we also need to restrict our attention to
height fields that aren't too steep and velocities which aren't too extreme:
mainly we will be looking at fairly calm waves. For example, a tall column
of water can be represented with a height field in the first frame, but when
it starts to collapse it is almost bound to start splashing around in more
general ways that will rule it out.
While you can of course use the height field representation to track the
water surface in conjunction with a full three-dimensional solver as detailed
earlier in the topic—see Foster and Metaxas [Foster and Metaxas 96]—
we'll make some further approximations to reduce the complexity of the
equations. In this chapter we'll look at the case where the water is shallow,
−
1
Many authors have worked out ways to bring back some of these features, usually
by way of adding a particle system for the extra effects. For example, see the articles by
O'Brien and Hodgins [O'Brien and Hodgins 95] and Thurey et al. [Thuerey et al. 07].
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