Mean-Curvature Flow of Voronoi Diagrams

Abstract

We study the evolution of grain boundary networks by the mean-curvature flow under the restriction that the networks are Voronoi diagrams for a set of points. For such evolution we prove a rigorous universal upper bound on the coarsening rate. The rate agrees with the rate predicted for the evolution by mean-curvature flow of the general grain boundary networks, namely that the typical grain area grows linearly in time. We perform a numerical simulation which provides evidence that the dynamics achieves the rate of coarsening that agrees with the upper bound in terms of scaling.

Appendix: Derivation of Coordinates for \(v_{ijk}\)

Equation (10) may be derived as follows: The vertex $v_{ijk}$ must satisfy

$$\begin{aligned} (x_i-v_{ijk}) \cdot (x_i-v_{ijk}) = (x_j-v_{ijk}) \cdot (x_j-v_{ijk}) = (x_k-v_{ijk}) \cdot (x_k-v_{ijk}), \end{aligned}$$

from which we subtract $v_{ijk} \cdot v_{ijk}$ to obtain

$$\begin{aligned} x_i \cdot x_i-2x_i \cdot v_{ijk} = x_j \cdot x_j-2x_j \cdot v_{ijk} = x_k \cdot x_k-2x_k \cdot v_{ijk}, \end{aligned}$$

which can be combined to obtain

$$\begin{aligned} 2(x_j-x_i) \cdot v_{ijk} = x_j \cdot x_j-x_i \cdot x_i \end{aligned}$$

and

$$\begin{aligned} 2(x_k-x_j) \cdot v_{ijk} = x_k \cdot x_k-x_j \cdot x_j. \end{aligned}$$

In matrix form, this can be expressed as

$$\begin{aligned} \left( \begin{array}{c} (x_j-x_i)^T \\ (x_k-x_j)^T \end{array} \right) v_{ijk} = \frac{1}{2} \left( \begin{array}{c} x_j \cdot x_j-x_i \cdot x_i \\ x_k \cdot x_k-x_j \cdot x_j \end{array} \right) . \end{aligned}$$

(23)

We compute

$$\begin{aligned} \left( \begin{array}{c} (x_j-x_i)^T \\ (x_k-x_j)^T \end{array} \right) ^{-1} = \frac{\left( \begin{array}{cc} (x_j-x_k)^\perp&(x_j-x_i)^\perp \end{array} \right) }{(x_j-x_i)\cdot (x_k-x_j)^\perp }. \end{aligned}$$

(24)

(10) arises by left-multiplying (23) by (24), observing that $a \cdot b^\perp = -b \cdot a^\perp$, and collecting like terms.