Graphics Reference
In-Depth Information
11.2 Weak Coupling
For objects large or heavy enough to significantly affect the fluid flow,
but light enough to be affected in turn by the fluid, we need methods for
simulating both in tandem. One common approach to implementing this
two-way coupling is sometimes termed
weak coupling
. In this scheme, we
interleave the solid- and fluid-simulation steps. At its simplest, we get the
following algorithm for each time step:
•
Advect the fluid, and update the solid positions (and orientations if
relevant).
•
Integrate non-coupled forces into all velocities (e.g., gravity, internal
elasticity forces).
•
Solve for the pressure to make the fluid incompressible, enforcing the
solid-wall boundary condition with the current solid velocities held
fixed.
•
Update the solid velocities from forces due to the new fluid pressure
and from contact/collision.
More complicated schemes are of course possible, e.g., with repeated alter-
nations between fluid and solid or with substeps to get higher accuracy for
the internal elastic forces, but the essence of weak coupling remains: one
pressure solve for fluid treats the solid velocities as fixed, and one update
to solid velocities treats the fluid pressure as fixed.
In terms of implementation, we have already covered the fluid aspect
of this problem since, from the point of view of the fluid solver, the solid
is always treated as fixed as before.
1
All that needs to be added is the
fluid-to-solid stage, where fluid forces are applied to the solid.
For a rigid object, the fluid-to-solid coupling amounts to finding the net
force and torque due to the fluid, which we have seen in surface integral form
in the previous section. If the geometry of the solid objects is tesselated
finely enough (i.e., on a scale comparable to the grid spacing Δ
x
)these
surface integrals can be directly approximated with numerical quadrature.
For example, if the object surface is represented by a triangle mesh, the
force could be as simple as summing over the triangles the product of
triangle area with pressure interpolated at the centroid of the triangle,
1
Though as Guendelman et al. [Guendelman et al. 05] point out, if the solids are
thin, care must be taken in advection—in the semi-Lagrangian approach, if a particle
trajectory is traced back
through
a solid, the fluid velocity at the interfering solid wall
should be used, interpolated in a one-sided way only from the correct side of the solid;
for particle methods, collision detection should be used to ensure particles don't pass
through solid walls.
