## 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
Full Text:

### References:

 [1] S. Altschuler and M. A. Grayson: Shortening space curves and flow through singularities. J. Differential Geom. 35 (1992), 283-298. · Zbl 0782.53001 [2] J. W. Barrett, H. Garcke, and R. Nurnberg: On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comp. 29 (2007), 1006-1041. · Zbl 1148.65074 [3] M. Beneš: Phase field model of microstructure growth in solidification of pure substances. Acta Math. Univ. Comenian. 70 (2001), 123-151. · Zbl 0990.80006 [4] M. Beneš: Mathematical analysis of phase-field equations with numerically efficient coupling terms. Interfaces and Free Boundaries 3 (2001), 201-221. · Zbl 0986.35116 [5] M. Beneš, K. Mikula, T. Oberhuber, and D. Ševčovič: Comparison study for level set and direct Lagrangian methods for computing Willmore flow of closed planar curves. Computing and Visualization in Science 12 (2009), No. 6, 307-317. · Zbl 1213.35033 [6] K. Deckelnick and G. Dziuk: Mean curvature flow and related topics. Frontiers in Numerical Analysis (2002), 63-108. · Zbl 1027.65134 [7] G. Dziuk, A. Schmidt, A. Brillard, and C. Bandle: Course on Mean Curvature Flow. Manuscript 75 pp., Freiburg 1994. [8] F. Kroupa: Long-range elastic field of semi-infinite dislocation dipole and of dislocation jog. Phys. Status Solidi 9 (1965), 27-32. [9] K. Mikula and D. Ševčovič: Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61 (2001), 5, 1473-1501. · Zbl 0980.35078 [10] K. Mikula and D. Ševčovič: Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Visualization Sci. 6 (2004), 4, 211-225. [11] V. Minárik and J. Kratochvíl: Dislocation dynamics - analytical description of the interaction force between dipolar loops. Kybernetika 43 (2007), 841-854. · Zbl 1135.74013 [12] V. Minárik, J. Kratochvíl, and K. Mikula: Numerical Simulation of dislocation dynamics by means of parametric approach. Proc. Czech-Japanese Seminar in Applied Mathematics (M. Beneš, J. Mikyška, and T. Oberhuber, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague 2005, pp. 128-138. · Zbl 1312.74022 [13] V. Minárik, J. Kratochvíl, K. Mikula, and M. Beneš: Numerical simulation of dislocation dynamics. Numerical Mathematics and Advanced Applications - ENUMATH 2003 (M. Feistauer, V. Dolejší, P. Knobloch, and K. Najzar, Springer-Verlag, New York 2004, pp. 631-641. · Zbl 1163.65309 [14] T. Mura: Micromechanics of Defects in Solids. Springer-Verlag, Berlin 1987. · Zbl 0652.73010 [15] T. Oberhuber: Finite difference scheme for the Willmore flow of graphs. Kybernetika 43 (2007), 855-867. · Zbl 1140.53032 [16] S. Osher and R. P. Fedkiw: Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, New York 2003. · Zbl 1026.76001 [17] P. Pauš: Numerical simulation of dislocation dynamics. Proceedings of Slovak-Austrian Congress, Magia (M. Vajsáblová and P. Struk, Bratislava, pp. 45-52. [18] P. Pauš and M. Beneš: Topological changes for parametric mean curvature flow. Proc. Algoritmy Conference (A. Handlovičová, P. Frolkovič, K. Mikula, and D. Ševčovič, Podbanské 2009, pp. 176-184. · Zbl 1171.65346 [19] P. Pauš and M. Beneš: Comparison of methods for mean curvature flow. [20] J. A. Sethian: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge 1999. · Zbl 0973.76003 [21] D. Ševčovič and S. Yazaki: On a motion of plane curves with a curvature adjusted tangential velocity. · Zbl 1291.35109 [22] cl39.pdf, arXiv:0711.2568, 2007.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.