root/trunk/matml/transport/problems/castabox/castabox.tex

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\item Cast-a-Box

The {\tt castabox} software required for this problem simulates transient
three-dimensional heat conduction with phase change and complex boundary
conditions in a right regular hexahedron using finite differences on a uniform
grid.  You will use {\tt castabox} to modify a casting design such that the
last place to freeze is not the inside, but the top surface, preventing the
formation of a large shrinkage cavity in the ``casting''.

Note that by ignoring fluid flow and convective heat transport, and not
actually modeling the change in geometry due to shrinkage, this program
enormously simplifies actual casting processes and is not likely to be terribly
accurate.  But it does illustrate the principle in a simple, fast parallel C
program which is easy to read and modify if you are interested in doing so.

\begin{enumerate}
\item Run {\tt castabox} with its default geometry and boundary conditions, and
  capture (and submit) a couple of temperature plots and their associated
  times, particularly one when the top surface is just fully frozen.  (You may
  find the program {\tt xv} useful for grabbing window snapshots.)  Document
  the temperatures represented by each contour surface.

\item If the top surface freezes with liquid trapped beneath it (as you should
  find under the default conditions), solidification shrinkage will lead to the
  formation of a {\em shrinkage cavity}.  (Ice, silicon, and certain other
  materials have the opposite problem: expansion during solidification of
  trapped liquid can fracture the casting.)

  Propose a design change which will produce a fully-solid box casting of the
  same geometry and material ({\em i.e.} same properties).  Show with a {\tt
    castabox} simulation that your design change is likely to work, by again
  capturing a couple of temperature plots, particularly when the top surface is
  just fully frozen.
\end{enumerate}
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