#
Computational Results

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1. Interface

The goal of the linear stability analysis through eigenvalue calulation
is to study the change of the temperature field and the geometry of the
solid-liquid interface during the solidification process.
The formulation of the eigenvalue problem couples temperature field U with
the interfacial boundary N. Therefore the computed eigenvector
consists of the numerical values for both variables U and N. In particular,
if the discretization is applied to a rectangular grid with
nx cells in the x-direction, ny cells in the y-direction,
the last ny+1 entries of an eigenvector provides a pattern
of the moving boundary up to a constant scalar.
Several change of variables are used to map the parabolically shaped
interface onto a horizontal line in an infinite domain, then from
an infinite domain to a rectangular box. (See technical report
for details.) The "raw" eigenvectors computed correspond to variables
defined in a finite rectangular box. Four typical interface excitation
modes corresponds to four different exictation frequencies (eigenvalues) are
shown in the following figure.

We may use the inverse of the coordinate tranform to plot the
interface on the (truncated) infinite domain. Notice that
the interface change is localized near the original of the
domain. This is more so when the frequency of the growth
is higher.

One more reverse transformation allows us to plot the
interfacial change on top of the parabolic needle (dotted
curve). We can view the dynamic behavious by letting the
interface propagate for a few time steps. The shape of the
following plots depends on the curvature of the
parabola defined in the original problem. The time-dependent
behavior of the interface also depends on the thermal diffusivity.

##
2. Temperature

The following figure shows the temperature profile of the solid
corresponding to one of the excited modes. The frequency of
the mode is 0.102. It easy to observe, after comparing
with the interface plot, that the growth of the solid corresponds to
a more dramatric temperature change near the tip.