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phase transitions; area-preserving mean-curvature flow; parametric method
The paper presents the results of numerical solution of the evolution law for the constrained mean-curvature flow. This law originates in the theory of phase transitions for crystalline materials and describes the evolution of closed embedded curves with constant enclosed area. It is reformulated by means of the direct method into the system of degenerate parabolic partial differential equations for the curve parametrization. This system is solved numerically and several computational studies are presented as well.
[1] Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 1084-1095 (1979). DOI 10.1016/0001-6160(79)90196-2
[2] Beneš, M.: Diffuse-interface treatment of the anisotropic mean-curvature flow. Mathematical and computer modeling in science and engineering. Appl. Math., Praha 48 (2003), 437-453. DOI 10.1023/B:APOM.0000024485.24886.b9 | MR 2025297
[3] Beneš, M., Kimura, M., Pauš, P., Ševčovič, D., Tsujikawa, T., Yazaki, S.: Application of a curvature adjusted method in image segmentation. Bull. Inst. Math., Acad. Sin. (N.S.) 3 (2008), 509-523. MR 2502611 | Zbl 1170.53040
[4] Beneš, M., Kratochvíl, J., Křišťan, J., Minárik, V., Pauš, P.: A parametric simulation method for discrete dislocation dynamics. European Phys. J. ST 177 177-192 (2009). DOI 10.1140/epjst/e2009-01174-7
[5] Beneš, M., Yazaki, S., Kimura, M.: Computational studies of non-local anisotropic Allen-Cahn equation. Math. Bohem. 136 (2011), 429-437. MR 2985552 | Zbl 1249.35153
[6] Cahn, J. W., Hilliard, J. E.: Free energy of a nonuniform system. III. Nucleation of a two-component incompressible fluid. J. Chem. Phys. 31 688-699 (1959). DOI 10.1063/1.1730447
[7] Dolcetta, I. Capuzzo, Vita, S. Finzi, March, R.: Area-preserving curve-shortening flows: From phase separation to image processing. Interfaces Free Bound. 4 (2002), 325-343. MR 1935642
[8] Deckelnick, K., Dziuk, G., Elliott, C. M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numerica 14 (2005), 139-232. DOI 10.1017/S0962492904000224 | MR 2168343 | Zbl 1113.65097
[9] Esedo\={g}lu, S., Ruuth, S. J., Tsai, R.: Threshold dynamics for high order geometric motions. Interfaces Free Bound. 10 (2008), 263-282. MR 2453132 | Zbl 1157.65330
[10] Gage, M.: On an area-preserving evolution equation for plane curves. Nonlinear Problems in Geometry, Proc. AMS Spec. Sess., Mobile/Ala. 1985 Contemp. Math. 51 American Mathematical Society, Providence (1986), 51-62 D. M. DeTurck. DOI 10.1090/conm/051/848933 | MR 0848933 | Zbl 0608.53002
[11] Grayson, M. A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26 (1987), 285-314. MR 0906392 | Zbl 0667.53001
[12] Henry, M., Hilhorst, D., Mimura, M.: A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation. Discrete Contin. Dyn. Syst., Ser. S 4 (2011), 125-154. MR 2746398 | Zbl 1207.35189
[13] McCoy, J.: The surface area preserving mean curvature flow. Asian J. Math. 7 (2003), 7-30. MR 2015239 | Zbl 1078.53067
[14] Minárik, V., Beneš, M., Kratochvíl, J.: Simulation of dynamical interaction between dislocations and dipolar loops. J. Appl. Phys. 107 Article No. 061802, 13 pages (2010). DOI 10.1063/1.3340518
[15] Osher, S., Sethian, J. A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988), 12-49. DOI 10.1016/0021-9991(88)90002-2 | MR 0965860 | Zbl 0659.65132
[16] Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48 (1992), 249-264. DOI 10.1093/imamat/48.3.249 | MR 1167735 | Zbl 0763.35051
[17] Ševčovič, D.: Qualitative and quantitative aspects of curvature driven flows of planar curves. Topics on Partial Differential Equations Jindřich Nečas Center for Mathematical Modeling Lecture Notes 2 Matfyzpress, Praha 55-119 (2007), P. Kaplický et al. MR 2856665
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