Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
anisotropic curve shortening flow; finite element method; stability
anisotropic curve shortening flow; finite element method; stability
Summary:
Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.
References:
[1] Alfaro, M., Garcke, H., Hilhorst, D., Matano, H., Schätzle, R.: Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation. Proc. Roy. Soc. Edinburgh Sect. A 140 (4) (2010), 673–706. MR 2672065
[2] Barrett, J.W., Blowey, J.F.: Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77 (1) (1997), 1–34. DOI 10.1007/s002110050276
[3] Barrett, J.W., Garcke, H., Nürnberg, R.: Numerical approximation of anisotropic geometric evolution equations in the plane. IMA J. Numer. Anal. 28 (2) (2008), 292–330. DOI 10.1093/imanum/drm013 | MR 2401200
[4] Barrett, J.W., Garcke, H., Nürnberg, R.: Numerical approximation of gradient flows for closed curves in ${\mathbb{R}}^d$. IMA J. Numer. Anal. 30 (1) (2010), 4–60. DOI 10.1093/imanum/drp005 | MR 2580546
[5] Barrett, J.W., Garcke, H., Nürnberg, R.: The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute. Numer. Methods Partial Differential Equations 27 (1) (2011), 1–30. DOI 10.1002/num.20637 | MR 2743598
[6] Barrett, J.W., Garcke, H., Nürnberg, R.: Stable phase field approximations of anisotropic solidification. IMA J. Numer. Anal. 34 (4) (2014), 1289–1327. DOI 10.1093/imanum/drt044 | MR 3269427
[7] Barrett, J.W., Garcke, H., Nürnberg, R.: Numerical approximation of curve evolutions in Riemannian manifolds. IMA J. Numer. Anal. 40 (3) (2020), 1601–1651. DOI 10.1093/imanum/drz012 | MR 4122486
[8] Barrett, J.W., Garcke, H., Nürnberg, R.: Parametric finite element approximations of curvature driven interface evolutions. Handb. Numer. Anal. (Bonito, A., Nochetto, R.H., eds.), vol. 21, Elsevier, Amsterdam, 2020, pp. 275–423. MR 4378429
[9] Bellettini, G.: Anisotropic and crystalline mean curvature flow. A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, 2004, pp. 49–82. MR 2132657
[10] Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (3) (1996), 537–566. DOI 10.14492/hokmj/1351516749 | Zbl 0873.53011
[11] Beneš, M., Mikula, K.: Simulation of anisotropic motion by mean curvature-comparison of phase field and sharp interface approaches. Acta Math. Univ. Comenian. (N.S.) 67 (1) (1998), 17–42.
[12] Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22 (1) (1997), 61–79. DOI 10.1023/A:1007979827043
[13] Clarenz, U., Dziuk, G., Rumpf, M.: On generalized mean curvature flow in surface processing. Geometric Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, 2003, pp. 217–248. MR 2008341
[14] Deckelnick, K., Dziuk, G.: On the approximation of the curve shortening flow. Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994) (Bandle, C., Bemelmans, J., Chipot, M., Paulin, J.S.J., Shafrir, I., eds.), Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1995, pp. 100–108.
[15] Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005), 139–232. DOI 10.1017/S0962492904000224 | MR 2168343 | Zbl 1113.65097
[16] Deckelnick, K., Nürnberg, R.: A novel finite element approximation of anisotropic curve shortening flow. arXiv:2110.04605 (2021).
[17] Deimling, K.: Nonlinear functional analysis. Springer-Verlag, Berlin, 1985.
[18] Dziuk, G.: Convergence of a semi-discrete scheme for the curve shortening flow. Math. Models Methods Appl. Sci. 4 (4) (1994), 589–606. DOI 10.1142/S0218202594000339
[19] Dziuk, G.: Discrete anisotropic curve shortening flow. SIAM J. Numer. Anal. 36 (6) (1999), 1808–1830. DOI 10.1137/S0036142998337533 | Zbl 0942.65112
[20] Ecker, K.: Regularity Theory for Mean Curvature Flow. Birkhäuser, Boston, 2004. MR 2024995 | Zbl 1058.53054
[21] Elliott, C.M., Fritz, H.: On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick. IMA J. Numer. Anal. 37 (2) (2017), 543–603. MR 3649420
[22] Elliott, C.M., Stuart, A.M.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (6) (1993), 1622–1663. DOI 10.1137/0730084
[23] Garcke, H., Lam, K.F., Nürnberg, R., Signori, A.: Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies. arXiv:2111.14070 (2021).
[24] Giga, Y.: Surface evolution equations. vol. 99, Birkhäuser, Basel, 2006. MR 2238463
[25] Gurtin, M.E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993. Zbl 0787.73004
[26] Haußer, F., Voigt, A.: A numerical scheme for regularized anisotropic curve shortening flow. Appl. Math. Lett. 19 (8) (2006), 691–698. DOI 10.1016/j.aml.2005.05.011 | MR 2232241
[27] Mantegazza, C.: Lecture notes on mean curvature flow. Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2815949
[28] Mikula, K., Ševčovič, D.: Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61 (5) (2001), 1473–1501. DOI 10.1137/S0036139999359288 | MR 1824511
[29] Mikula, K., Ševčovič, D.: A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl. Sci. 27 (13) (2004), 1545–1565. DOI 10.1002/mma.514 | MR 2077443
[30] Mikula, K., Ševčovič, D.: Computational and qualitative aspects of evolution of curves driven by curvature and external force. Computing Vis. Sci. 6 (4) (2004), 21–225. DOI 10.1007/s00791-004-0131-6 | MR 2071441
[31] Pozzi, P.: Anisotropic curve shortening flow in higher codimension. Math. Methods Appl. Sci. 30 (11) (2007), 1243–1281. DOI 10.1002/mma.836 | MR 2334978
[32] Taylor, J.E., Cahn, J.W., Handwerker, C.A.: Geometric models of crystal growth. Acta Metall. Mater. 40 (7) (1992), 1443–1474. DOI 10.1016/0956-7151(92)90090-2
[33] Wu, C., Tai, X.: A level set formulation of geodesic curvature flow on simplicial surfaces. IEEE Trans. Vis. Comput. Graph. 16 (4) (2010), 647–662. DOI 10.1109/TVCG.2009.103
Search
Partner of
