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Auction Dynamics

This page illustrates the method for computing multiphase volume constrained curvature motion proposed in: Auction Dynamics: a volume constrained MBO scheme [1].

Summary

Suppose that $\mathbf{\Sigma}=(\Sigma_1, \ldots, \Sigma_N)$ is a partition of the torus $\mathbb{T}^d$ into $N$ disjoint phases. Let $\Gamma_{ij}$ denote the interface between phase $i$ and phase $j$. We then define the surface energy $E$ of the partition as \begin{equation} E(\mathbf{\Sigma})=\sum_{i\neq j} \sigma_{ij} H^{d-1}(\Gamma_{ij})\end{equation} where $\sigma_{ij}=\sigma_{ji}>0$ is a surface tension constant that represents the relative affinity between the phases and $H^{d-1}$ is the $d-1$ dimensional Hausdorff measure. Volume preserving mean curvature flow is the gradient flow of $E$ (for a singular metric concentrated on the interfaces) with the additional constraint that the volumes of each phase must remain fixed i.e. $m(\Sigma_i)=V_i$ for all time.

Videos

The following videos illustrate multiphase volume preserving curvature motion on a periodic domain