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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
2/2013

Threshold Dynamics for Networks with Arbitrary Surface Tensions

Selim Esedoglu and Felix Otto

Abstract

We present and study a new algorithm for simulating the $N$-phase mean curvature motion for an arbitrary set of (isotropic) $\frac{N(N-1)}{2}$ surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two-phase case.

An new energetic interpretation of this algorithm allows to extend it in a natural way to the case of $N$ phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in sense that at every time step, it decreases an energy functional that is an approximation (in the sense of $\Gamma$-convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties.

An important application is the motion of grain boundaries in polycrystalline materials:
It is also established that the above-mentioned large class of surface tensions contains the Read-Shockley surface tensions, both in the 2D and 3D settings.

Received:
08.01.13
Published:
08.01.13

Related publications

inJournal
2015 Repository Open Access
Selim Esedoglu and Felix Otto

Threshold dynamics for networks with arbitrary surface tensions

In: Communications on pure and applied mathematics, 68 (2015) 5, pp. 808-864