Evolution of curves on a surface driven by the geodesic curvature and external force

References

• Angenent , SB . 1990 . Nonlinear analytic semiflows . Proceedings of the Royal Society of Edinburgh. Section A , 115 : pp. 91 – 107 .
   Google Scholar
• Beneš , M . 2001 . Mathematical and computational aspects of solidification of pure crystallic materials . Acta Mathematica Universitatis , 70 : 123 – 151 .
   Google Scholar
• Caselles , V , Catté , F , Coll , T and Dibos , F . 1993 . A geometric model for active contours in image processing . Numerische Matematik , 66 : 1 – 31 .
   Web of Science ®Google Scholar
• Caselles , V , Kimmel , R , Sapiro , G and Sbert , C . 1997 . Minimal surfaces: a geometric three dimensional segmentation approach . Numerische Matematik , 77 : 423 – 451 .
   Web of Science ®Google Scholar
• Deckelnick , K . 1997 . Weak solutions of the curve shortening flow . Calculus of Variations and Partial Differential Equations , 5 : 489 – 510 .
   Web of Science ®Google Scholar
• Dziuk , G . 1994 . Convergence of a semi discrete scheme for the curve shortening flow . Mathematical Models and Methods in Applied Sciences , 4 : 589 – 606 .
   Web of Science ®Google Scholar
• Dziuk , G . 1999 . Discrete anisotropic curve shortening flow . SIAM Journal on Numerical Analysis , 36 : 1808 – 1830 .
   Web of Science ®Google Scholar
• De Turck , D . 1983 . Deforming metrics in the direction of their Ricci tensors . Journal of Differential Geometry , 18 : 157 – 162 .
   Web of Science ®Google Scholar
• Epstein , CL and Gage , M . 1987 . “ The curve shortening flow ” . In Wave Motion: Theory, Modelling and Computation , Mathematical Sciences Research Institute Publications Edited by: Chorin , A and Majda , AJ . Vol. 7 , 15 – 59 . New York : Springer .
   Google Scholar
• Gage , M and Hamilton , RS . 1986 . The heat equation shrinking convex plane curves . Journal of Differential Geometry , 23 : 69 – 96 .
   Web of Science ®Google Scholar
• Grayson , M . 1987 . The heat equation shrinks embedded plane curves to round points . Journal of Differential Geometry , 26 : 285 – 314 .
   Web of Science ®Google Scholar
• Hou , TY , Lowengrub , J and Shelley , M . 1994 . Removing the stiffness from interfacial flows and surface tension . Journal of Computational Physics , 114 : 312 – 338 .
   Web of Science ®Google Scholar
• Hou , TY , Klapper , I and Si , H . 1998 . Removing the stiffness of curvature in computing 3-d filaments . Journal of Computational Physics , 143 : 628 – 664 .
   Web of Science ®Google Scholar
• Kass , M , Witkin , A and Terzopulos , D . 1987 . Snakes: active contour models . International Journal of Computer Vision , 1 : 321 – 331 .
   Web of Science ®Google Scholar
• Kichenassamy , S , Kumar , A , Olver , P , Tannenbaum , A and Yezzi , A . 1996 . Conformal curvature flows: from phase transitions to active vision . Archive for Rational Mechanics and Analysis , 134 : 275 – 301 .
   Web of Science ®Google Scholar
• Kimura , M . 1997 . Numerical analysis for moving boundary problems using the boundary tracking method . Japan Journal of Industrial and Applied Mathematics , 14 : 373 – 398 .
   Google Scholar
• Malladi , R , Sethian , J and Vemuri , B . 1995 . Shape modeling with front propagation: a level set approach . IEEE Transactions on Pattern Analysis and Machine Intelligence , 17 : 158 – 174 .
   Web of Science ®Google Scholar
• Mikula , K and Kačur , J . 1996 . Evolution of convex plane curves describing anisotropic motions of phase interfaces . SIAM Journal on Scientific Computing , 17 : 1302 – 1327 .
   Web of Science ®Google Scholar
• Mikula , K . 1997 . Solution of nonlinear curvature driven evolution of plane convex curves . Applied Numerical Mathematics , 21 : 1 – 14 .
   Google Scholar
• Mikula , K and Ševčovič , D . 1999 . Solution of nonlinearly curvature driven evolution of plane curves . Applied Numerical Mathematics , 31 : 191 – 207 .
   Web of Science ®Google Scholar
• Mikula , K and Ševčovič , D . 2001 . Evolution of plane curves driven by a nonlinear function of curvature and anisotropy . SIAM Journal on Applied Mathematics , 61 : 1473 – 1501 .
   Web of Science ®Google Scholar
• Mikula , K and Ševčovič , D . 2004 . A direct method for solving an anisotropic mean curvature flow of plane curves with an external force . Mathematical Methods in the Applied Sciences , 27 ( 13 ) : 1545 – 1565 .
   Web of Science ®Google Scholar
• Mikula , K and Ševčovič , D . 2004 . Computational and qualitative aspects of evolution of curves driven by curvature and external force . Computing and Visualization in Science , 6 ( 4 ) : 211 – 225 .
   Google Scholar
• Osher , S and Fedkiw , R . 2003 . Level Set Methods and Dynamic Implicit Surfaces , New York : Springer-Verlag .
   Google Scholar
• Sapiro , G . 2001 . Geometric Partial Differential Equations and Image Analysis , New York : Cambridge University Press .
   Google Scholar
• Sethian , JA . 1999 . Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science , New York : Cambridge University Press .
   Google Scholar