\section{A Step-by-Step Example}

On the following pages we illustrate the partitioning, placement, and routing
of an unstructured finite element mesh 
with roughly 5000 nodes on an $8 \times 8$ torus.
The mesh was designed to predict stresses in a
cracked plate, and because of the singularity in stress at the crack tip,
is hundreds of times more dense at the center 
than near the edges.  The purpose of this small example is 
to show the behavior of our partitioning method on an unstructured mesh
with a large variation in resolution.

Figure 2 illustrates the original unstructured mesh $M$ for the 
problem.  Note that the mesh is so refined near the center that the figure is
completely black.  Figure 3 is the underlying graph $G$ of $M$,
after having been vertex-partitioned into 64
blocks.  Due to the fineness of the original mesh, many of the
partition blocks are indistinguishable.  Figure 4 illustrates the
64-vertex minor graph $G'$ derived by collapsing all the vertices in each 
block of
the partition to a single supervertex.  Finally, Figure 5 shows a
placement and routing of $G'$ on an $8 \times 8$ torus.  Edges extending
past the boundary of the grid wrap around to the other side.

