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.
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.