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Examples

These various sources of error can all substantially reduce the confidence one may have in the reliability of an individual result of this method applied to discretized interfaces. The strength of the method, however, is in accurately reflecting the spatial distribution of curvature throughout a complex system composed of one or more interfaces. Where the method has been applied to sintering simulations [3,4] it has provided quite realistic predictions of the direction of mass transport throughout complex 2D and 3D microstructures. Further illustrations of this strength are provided by the following two examples, one in three dimensions and the other in two.

The first example is that of computing the mean curvature of the surface of a digitized sphere. Fig. 5 shows the computed mean curvature of a sphere plotted against its true mean curvature, R⁻¹. The template sphere diameter used was 9 , and Eq. 32 was used to relate the pixel count to the mean curvature (no averaging over neighboring surface pixels was performed). The values shown are averages of the values computed for every surface pixel in one octant, and error bars represent plus and minus 1 standard deviation. The average value of mean curvature for each sphere is very accurate, although the precision is low because no averaging was performed over neighboring surface sites. This example is nothing more than a verification of the calibrated relation between pixel count and curvature, Eq. 32. But it does point out that the method should provide a good measurement of, for instance, the average driving force for mass transfer between the various particles comprising a sintering microstructure, although precision error would cause fluctuations about that average value.

The second example is that of computing the curvature along a sinusoidal arc in two dimensions, like that analyzed by Mullins [7]. The arc is described by

The curvature of an arc described by Eq. 34 can be calculated analytically according to the expression

where f_x and f_xx are the first and second derivatives of f with respect to x. Figure 6 shows both the computed and analytical values of the curvature at a number of points along one wavelength (2 π k⁻¹ ) of the surface. Each individual computed value represents an average over neighboring surface pixels that share either a common edge or corner. Extremal curvature values are predicted quite well by the method (to within 3%), and somewhat greater deviations from the true curvature occur at lower curvature magnitudes.

Figure 6: Computed and analytical values of the curvature for a sinusoidally-perturbed surface (y = A sin(kx)) plotted along one wavelength λ (2 π k⁻¹ ) of the perturbation. In this graph, A = 30 , λ = 100 , and the diameter of the template circle is 15 .