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Direct approach to mean-curvature flow with topological changes. (English) Zbl 1192.65128

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

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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