Kybernetika 45 no. 4, 591-604, 2009

Direct Approach to Mean-curvature Flow with Topological Changes

Petr Pauš and Michal Beneš


This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves $ \Gamma (t) : S \rightarrow \mathbb{R} ^2 $, $ t \geqq 0 $. The curves are driven by the normal velocity $v$ which is the function of curvature $\kappa$ and the position. The evolution law reads as: $v = -\kappa + F$. The motion law is treated using direct approach numerically solved by two schemes, i. e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved by tangential redistribution of curve points which allows long time computations and better accuracy. The results of dislocation dynamics simulation are presented (e. g., dislocations in channel or Frank-Read source). We also introduce an algorithm for treatment of topological changes in the evolving curve.


dislocation dynamics, mean curvature flow, parametric approach


35L65, 76M12, 80A20