Previous |  Up |  Next

Article

Title: On the tangential velocity arising in a crystalline approximation of evolving plane curves (English)
Author: Yazaki, Shigetoshi
Language: English
Journal: Kybernetika
ISSN: 0023-5954
Volume: 43
Issue: 6
Year: 2007
Pages: 913-918
Summary lang: English
.
Category: math
.
Summary: In a crystalline algorithm, a tangential velocity is used implicitly. In this short note, it is specified for the case of evolving plane curves, and is characterized by using the intrinsic heat equation. (English)
Keyword: tangential velocity
Keyword: intrinsic heat equation
Keyword: crystalline algorithm
Keyword: admissible polygonal curve
MSC: 34A26
MSC: 34A34
MSC: 35K65
MSC: 53A04
MSC: 53C44
MSC: 53C80
MSC: 65L20
MSC: 65M12
MSC: 65N12
idZBL: Zbl 1139.53033
idMR: MR2388404
.
Date available: 2009-09-24T20:31:13Z
Last updated: 2013-09-21
Stable URL: http://hdl.handle.net/10338.dmlcz/135826
.
Reference: [1] Angenent S., Gurtin M. E.: Multiphase thermomechanics with interfacial structure, 2.Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989), 323–391 MR 1013461, 10.1007/BF01041068
Reference: [2] Deckelnick K.: Weak solutions of the curve shortening flow.Calc. Var. Partial Differential Equations 5 (1997), 489–510 Zbl 0990.35076, MR 1473305, 10.1007/s005260050076
Reference: [3] .Dziuk, G: Convergence of a semi discrete scheme for the curve shortening flow.Math. Models Methods Appl. Sci. 4 (1994), 589–606 Zbl 0811.65112, MR 1291140, 10.1142/S0218202594000339
Reference: [4] Giga Y.: Anisotropic curvature effects in interface dynamics.Sūgaku 52 (2000), 113–127; English transl., Sugaku Expositions 16 (2003), 135–152 MR 2019167
Reference: [5] Giga M.-H., Giga Y.: Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane.GAKUTO Internat. Ser. Math. Sci. Appl. 13 (2000), 64–79 Zbl 0957.35122, MR 1793023
Reference: [6] Gurtin M. E.: Thermomechanics of Evolving Phase Boundaries in the Plane.Clarendon Press, Oxford 1993 Zbl 0787.73004, MR 1402243
Reference: [7] Hirota C., Ishiwata, T., Yazaki S.: Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves.In: Advanced Studies in Pure Mathematics (ASPM); Proc. MSJ-IRI 2005 “Asymptotic Analysis and Singularity”, Sendai 2005 (to appear) Zbl 1143.35308, MR 2387254
Reference: [8] Hontani H., Giga M.-H., Giga, Y., Deguchi K.: Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis.Discrete Appl. Math. 147 (2005), 265–285 Zbl 1117.65036, MR 2127078, 10.1016/j.dam.2004.09.015
Reference: [9] Ishiwata T., Ushijima T. K., Yagisita, H., Yazaki S.: Two examples of nonconvex self-similar solution curves for a crystalline curvature flow.Proc. Japan Academy, Ser. A 80 (2004), 8, 151–154 Zbl 1077.53054, MR 2099341
Reference: [10] Kimura M.: Accurate numerical scheme for the flow by curvature.Appl. Math. Letters 7 (1994), 69–73 Zbl 0792.65100, MR 1349897, 10.1016/0893-9659(94)90056-6
Reference: [11] Mikula K., Ševčovič D.: Solution of nonlinearly curvature driven evolution of plane curves.Appl. Numer. Math. 31 (1999), 191–207 Zbl 0938.65145, MR 1708959, 10.1016/S0168-9274(98)00130-5
Reference: [12] Mikula K., Ševčovič D.: Evolution of plane curves driven by a nonlinear function of curvature and anisotropy.SIAM J. Appl. Math. 61 (2001), 1473–1501 Zbl 0980.35078, MR 1824511, 10.1137/S0036139999359288
Reference: [13] Taylor J. E.: Constructions and conjectures in crystalline nondifferential geometry.In: Proc. Conference on Differential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math. 52 (1991), pp. 321–336, Pitman, London 1991 Zbl 0725.53011, MR 1173051
Reference: [14] Taylor J. E.: Motion of curves by crystalline curvature, including triple junctions and boundary points.Diff. Geom.: Partial Diff. Eqs. on Manifolds (Los Angeles 1990), Proc. Sympos. Pure Math., 54 (1993), Part I, pp, 417–438, AMS, Providence MR 1216599
Reference: [15] Taylor J. E., Cahn, J., Handwerker C.: Geometric models of crystal growth.Acta Metall. 40 (1992), 1443–1474 10.1016/0956-7151(92)90090-2
Reference: [16] Ushijima T. K., Yagisita H.: Approximation of the Gauss curvature flow by a three-dimensional crystalline motion.In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005; Kuju Training Center, Oita, Japan, September 15–18, 2005, COE Lecture Note 3, Faculty of Mathematics, Kyushu University (M. Beneš, M. Kimura, and T. Nakaki, eds.), 2006, pp. 139–145 Zbl 1145.53051, MR 2279054
Reference: [17] Ushijima T. K., Yagisita H.: Convergence of a three-dimensional crystalline motion to Gauss curvature flow.Japan J. Indust. Appl. Math. 22 (2005), 443–459 Zbl 1089.53046, MR 2179771, 10.1007/BF03167494
Reference: [18] Ushijima T. K., Yazaki S.: Convergence of a crystalline approximation for an area-preserving motion.Journal of Comput. and Appl. Math. 166 (2004), 427–452 Zbl 1052.65082, MR 2041191, 10.1016/j.cam.2003.08.041
Reference: [19] Yazaki S.: Motion of nonadmissible convex polygons by crystalline curvature.Publ. Res. Inst. Math. Sci. 43 (2007), 155–170 Zbl 1132.53036, MR 2317117, 10.2977/prims/1199403812
.

Files

Files Size Format View
Kybernetika_43-2007-6_15.pdf 667.1Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo