Coupling Fluids and Solids - Fluid Simulation for Computer Graphics - page 153

mass and velocity of the solid to the fluid grid and then solve for pres-
sure throughout the whole domain. The usual fluid fractions are used as
weights in determining the average density and average velocity in each u -,
v -, and w -cell, and then the fractions actually used in determining the pres-
sure equations are full. (Incidentally, this approach was in fact combined
with the approach of the previous section in the paper of Guendelman et
al. [Guendelman et al. 05], where this pressure is used instead of the less ac-
curate pressure of the classic voxelized pressure solve to update the solid's
velocity.)
A related method, the rigid fluid approach of Carlson et al. [Carlson
et al. 04], simplifies the solve somewhat by moreover assuming the density
of the solid to be the same as the fluid and adding a corrective buoyancy
force as a separate step, recovering a rigid body's velocity directly from
averaging the velocity on the grid after the pressure solve (i.e., finding
the average translational and angular velocity of the grid cells the solid
occupies) rather than integrating pressure forces over the surface of the
body. This can work extremely well if the ratio of densities isn't too large.
For inviscid flow, simply averaging the solid and fluid velocities in mixed
cells as is typically done in the immersed boundary method may lead to
excessive numerical dissipation. Recall that the tangential velocity of the
solid is not coupled to the tangential velocity of the fluid: only the nor-
mal components are connected for inviscid flows. When averaging the full
velocities together we are, in essence, constraining the fluid to the viscous
boundary condition u = u solid . Therefore it is recommended if possible
to extrapolate the tangential component of fluid velocity into the cells oc-
cupied by the solid and only average the normal component of the solid's
velocity onto the grid. For very thin solids, such as cloth, this is partic-
ularly simple since extrapolation isn't required—just a separation of the
solid velocities into normal and tangential components.
This approach helps reduce some of the artifacts of the previous weak-
coupling method, but it doesn't succeed in all cases. For example, starting
a simulation with a floating object resting at equilibrium still ends up
creating false motion, since in the pressure solve the solid object appears
to be an odd-shaped wave on the fluid surface.
11.4 General Sparse Matrices
Before getting into strong coupling, where we compute fluid and solid forces
simultaneously, we need to take a brief diversion to generalize our sparse