Direct approach to mean-curvature flow with topological changes

Petr Pauš; Michal Beneš

Kybernetika (2009)

  • Volume: 45, Issue: 4, page 591-604
  • ISSN: 0023-5954

Abstract

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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 Γ ( t ) : S → ℝ 2 , t ≧ 0 . The curves are driven by the normal velocity v which is the function of curvature κ and the position. The evolution law reads as: v = - κ + 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.

How to cite

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Pauš, Petr, and Beneš, Michal. "Direct approach to mean-curvature flow with topological changes." Kybernetika 45.4 (2009): 591-604. <http://eudml.org/doc/37721>.

@article{Pauš2009,
abstract = {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.},
author = {Pauš, Petr, Beneš, Michal},
journal = {Kybernetika},
keywords = {mean curvature flow; dislocation dynamics; parametric approach; dislocation dynamics; mean curvature flow; parametric approach; numerical examples; backward Euler; method of lines; numerical stability; algorithm; topological changes},
language = {eng},
number = {4},
pages = {591-604},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Direct approach to mean-curvature flow with topological changes},
url = {http://eudml.org/doc/37721},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Pauš, Petr
AU - Beneš, Michal
TI - Direct approach to mean-curvature flow with topological changes
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 4
SP - 591
EP - 604
AB - 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.
LA - eng
KW - mean curvature flow; dislocation dynamics; parametric approach; dislocation dynamics; mean curvature flow; parametric approach; numerical examples; backward Euler; method of lines; numerical stability; algorithm; topological changes
UR - http://eudml.org/doc/37721
ER -

References

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  1. Shortening space curves and flow through singularities, J. Differential Geom. 35 (1992), 283–298. MR1158337
  2. On the variational approximation of combined second and fourth order geometric evolution equations, SIAM J. Sci. Comp. 29 (2007), 1006–1041. MR2318696
  3. Phase field model of microstructure growth in solidification of pure substances, Acta Math. Univ. Comenian. 70 (2001), 123–151. 
  4. Mathematical analysis of phase-field equations with numerically efficient coupling terms, Interfaces and Free Boundaries 3 (2001), 201–221. MR1825658
  5. 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. MR2520782
  6. Mean curvature flow and related topics, Frontiers in Numerical Analysis (2002), 63–108. MR2006966
  7. Course on Mean Curvature Flow, Manuscript 75 pp., Freiburg 1994. 
  8. Long-range elastic field of semi-infinite dislocation dipole and of dislocation jog, Phys. Status Solidi 9 (1965), 27–32. 
  9. Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math. 61 (2001), 5, 1473–1501. MR1824511
  10. Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Visualization Sci. 6 (2004), 4, 211–225. MR2071441
  11. Dislocation dynamics – analytical description of the interaction force between dipolar loops, Kybernetika 43 (2007), 841–854. MR2388398
  12. Numerical Simulation of dislocation dynamics by means of parametric approach, In: Proc. Czech–Japanese Seminar in Applied Mathematics (M. Beneš, J. Mikyška, and T. Oberhuber, eds.), Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague 2005, pp. 128–138. 
  13. Numerical simulation of dislocation dynamics, In: Numerical Mathematics and Advanced Applications – ENUMATH 2003 (M. Feistauer, V. Dolejší, P. Knobloch, and K. Najzar, eds.), Springer–Verlag, New York 2004, pp. 631–641. 
  14. Micromechanics of Defects in Solids, Springer–Verlag, Berlin 1987. 
  15. Finite difference scheme for the Willmore flow of graphs, Kybernetika 43 (2007), 855–867. Zbl1140.53032MR2388399
  16. Level Set Methods and Dynamic Implicit Surfaces, Springer–Verlag, New York 2003. MR1939127
  17. Numerical simulation of dislocation dynamics, In: Proceedings of Slovak–Austrian Congress, Magia (M. Vajsáblová and P. Struk, eds.), Bratislava, pp. 45–52. 
  18. Topological changes for parametric mean curvature flow, In: Proc. Algoritmy Conference (A. Handlovičová, P. Frolkovič, K. Mikula, and D. Ševčovič, eds.), Podbanské 2009, pp. 176–184. 
  19. Comparison of methods for mean curvature flow, (In preparation.) 
  20. Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge 1999. Zbl0973.76003MR1700751
  21. On a motion of plane curves with a curvature adjusted tangential velocity, In: http://www.iam.fmph.uniba.sk/institute/sevcovic/papers/ 
  22. [unknown], cl39.pdf, arXiv:0711.2568, 2007. 

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