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.


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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[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.
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