Graphics Reference
In-Depth Information
and if relevant, the viscous stress tensor times the triangle normal. The
torque can similarly be approximated. However, in other circumstances
(e.g., objects tesselated at a very different scale, or thin curves such as
rope) these surface integrals can be inconvenient. We can then transform
the surface integral to volume integrals, which can be approximated directly
on the grid. For example, for pressure the net force is
S
∇
F
=
−
pn
=
−
p
(11.2)
∂S
and the net torque is
x
C
)
pn
=
T
=
−
(
x
−
S
∇×
[(
x
−
x
X
)
p
]
,
(11.3)
∂S
where
x
C
is the center of mass. The volume integrals can be broken up
into sums over the appropriate grid cells, using the appropriate volume
fractions (which are approximated for the pressure solve anyhow).
This general approach has met with success in many graphics papers
(e.g., [Takahashi et al. 02, Guendelman et al. 05]) and is quite attractive
from a software architecture point of view—the internal dynamics of fluids
and solids remain cleanly separated, with new code only for integrating
fluid forces applied to solids—but does suffer from a few problems that
may necessitate smaller than desirable time steps. For example, if we
start with a floating solid initially resting at equilibrium: after adding
acceleration due to gravity all velocities are Δ
tg
, the fluid pressure solve
treats this downward velocity at the solid surface as a constraint and thus
leads to non-zero fluid velocities, and finally the pressure field (perturbed
from hydrostatic equilibrium) doesn't quite cancel the velocity of the solid;
the solid sinks to some extent, and the water starts moving. These errors
are proportional to the time-step size and thus of course can be reduced,
but at greater expense.
11.3 The Immersed Boundary Method
A somewhat stronger coupling scheme is epitomized by the immersed
boundary method (the classic reference is the review article by Peskin
[Peskin 02]). Here we give the fluid pressure solve leeway to change the
solid velocity by, in effect, momentarily pretending the solid is also fluid
(just of a different density). In particular, rather than impose the solid
velocity as boundary conditions for the fluid pressure solve, we add the
