Abstract
Anisotropic mean curvature motion and in particular anisotropic surface diffusion play a crucial role in the evolution of material interfaces. This evolution interacts with conservations laws in the adjacent phases on both sides of the interface and are frequently expected to undergo topological chances. Thus, a level set formulation is an appropriate way to describe the propagation. Here we recall a general approach for the integration of geometric gradient flows over level set ensembles and apply it to derive a variational formulation for the level set representation of anisotropic mean curvature motion and anisotropic surface flow. The variational formulation leads to a semi-implicit discretization and enables the use of linear finite elements.
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References
M. Rost, (this volume).
U. Clarenz, The Wulff-shape minimizes an anisotropic Willmore functional. DFG-Forschergruppe Freiburg, Preprint 15 (2003).
M. Droske, M. Rumpf, A level set formulation for Willmore flow. Interfaces and Free Boundaries 6 (2004), 361–378.
D.L. Chopp, J.A. Sethian, Motion by intrinsic Laplacian of curvature. Interfaces Free Bound. 1 (1999), 107–123.
M. Khenner, A. Averbuch, M. Israeli, M. Nathan, Numerical solution of grain boundary grooving by level set method. J. of Comput. Phys. 170 (2001), 764–784.
P. Smereka, Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion J. of Sci. Comput. 19 (2003), 439–456.
S. Osher, J.A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. of Comput. Phys. 79 (1988), 12–784.
R. Rusu, An algorithm for the elastic flow of surfaces. Mathematische Fakultät Freiburg, Preprint 35 (2001).
L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, 2000.
L.C. Evans, J. Spruck, Motion of Level Sets by Mean Curvature I. J. Diff. Geom. 33 (1991), 635–681.
U. Clarenz, G. Dziuk, M. Droske, M. Rumpf, Level set formulation for anisotropic geometric evolutions problems manuscript, in preparation
S. Osher, N. Paragios, Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, 2003.
G. Wulff, Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Kristallflächen. Zeitschrift der Kristallographie 34 (1901), 449–530.
I. Fonseca, S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinb. A 119 (1991), 125–136.
S. Yoshizawa, A.G. Belyaev, Fair Triangle Mesh Generation with Discrete Elastica. in Geometric Modeling and Processing, RIKEN, Saitama, 2001, 119–123
T.F. Chan, S.H. Kang, J. Shen, Euler’s Elastica and curvature-based inpainting. SIAM Appl. Math. 63 (2002), 564–592.
K. Deckelnick, G. Dziuk, A fully discrete numerical scheme for weighted mean curvature flow. Numer. Math. 91 (2002), 423–452.
L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, 1992.
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© 2005 Birkhäuser Verlag Basel/Switzerland
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Clarenz, U., Haußer, F., Rumpf, M., Voigt, A., Weikard, U. (2005). On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion. In: Voigt, A. (eds) Multiscale Modeling in Epitaxial Growth. ISNM International Series of Numerical Mathematics, vol 149. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7343-1_14
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DOI: https://doi.org/10.1007/3-7643-7343-1_14
Publisher Name: Birkhäuser Basel
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