Computational Results

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.