Although most of the homeworks in CPS 615 involve solving a field equation over a square grid, real world topologies are rarely so simple. This fact makes the successful use of HPF more difficult, as the proper data distribution methods are now less clear than the common (BLOCK,BLOCK) used in the square grid. While this distribution method has the best edge over area ratio, it will often cause a crippling load imbalance in the new topologies.
A topology (or simple variations of it) that is often used in CFD simulations is illustrated in Figure 1. This generic grid has an ``inlet'' that narrows down into a channel, which continues on until an ``outlet'' slopes back upward. This geometry is seen in vents, air ducts, wind tunnels, diffusers, nozzles, lock sluices and other fluid flow apparati and as such it is important in CFD simulations. The geometry shown below is actually modeled on an air duct that I saw in Link Hall. At any rate, this topology and most of its variations exhibit symmetry in addition to the qualities distributed above. This characteristic is not trivial as it aids in determining the proper distribution to be used.
Clearly, a simple (BLOCK,BLOCK) distribution leads to
a bad load imbalance for the air duct simulation. In an
effort to find a more suitable method, a motivating example
concerning the duct in Figure 1 was created.
The duct in this example starts with an
inlet that slopes downward at a 45 degree angle
until one quarter of its length. A channel
continues for one half of the length, after
which the boundary slopes upward at a 45 degree angle
for the final quarter. It is thus desired to visualize
flow through this duct using potential flow theory.
The vent was modeled using a 
grid that
was shaped using a logical variable. Finding the streamlines
across this grid requires finding the values of the stream function 
at each grid point; however, this is the classic solution
of Laplace's equation, 
. Since
lines of constant 
are streamlines, this exercise
will yield the flow streamlines, the desired result.
Figure 1: The stream function values were set at arbitrary constants
at the upper and lower edges (solid boundaries). Along the
inlet and outlet the 
values vary linearly between the
top and bottom fixed values.