Computing minimal surfaces by mean curvature flow with area-oriented tangential redistribution
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Abstract
In this paper we use a surface evolution model for construction of minimal surfaces with given boundary curves. The initial surface topologically equivalent to a desired minimal surface is evolved by mean curvature ow. To improve the quality of the mesh we propose an area-oriented tangential redistribution of the grid points. We derive the numerical scheme and present several numerical experiments.
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Tomek, L., Remešíková, M., & Mikula, K.
(2018).
Computing minimal surfaces by mean curvature flow with area-oriented tangential redistribution.
Acta Mathematica Universitatis Comenianae, 87(1), 55-72.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/591/535
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References
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9. Dziuk G., Algorithm for evolutionary surfaces Numer. Math. 58 (1991), pp. 603-611.
10. Dziuk G. and Elliot C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), pp. 262-292.
11. Elliott C. M. and Fritz H., On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick, SMAI J.
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12. Evans L. C. and Spruck J., Motion of level sets by mean curvature I, J. Diff. Geom. 33 (1991), 635-681.
13. Hou T. Y., Klapper I. and Si H., Removing the stiffness of curvature in computing 3-d fllaments, J. Comput. Phys., 143 (1998), pp. 628-664.
14. Hou T. Y., Lowengrub J., and Shelley M., Removing the stiffness from interfacial flows and surface tension, J. Comput. Phys., 114 (1994), pp. 312-338.
15. Huska M., Medla M., Mikula K., Novysedlak P. and Remesikova M., A new form-finding method based on mean curvature ow of surfaces, in Proceedings of ALGORITMY 2012, 19th Conference on Scientific Computing, Podbansk, Slovakia, (2012), pp. 120-131.
16. Jiao X., Colombi A., Ni X. and Hart J. C., Anisotropic mesh adaptation for evolving triangulated surfaces, Engineering with Computers, Vol 26(4): 363-376, 2009
17. Kimura M., Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., 14 (1997), pp. 373-398.
18. Mantegazza C., Lecture Notes on Mean Curvature Flow, Addison-Wesley publishing company, Birkhauser Verlag, (2011), 166 s. ISBN 978-3-0348-0144-7
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22. Mikula K. and Sevcovic D., Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), pp. 1473-1501.
23. Mikula K. and Sevcovic D., A direct method for solving an anisotropic mean curvature flow of planar curve with an external force, Math. Methods Appl. Sci., 27 (2004), pp. 1545-1565.
24. Mikula K. and Sevcovic D., Evolution of curves on surface driven by the geodesic curvature and external force, Appl. Anal., 85 (2006), pp. 345362.
25. Mikula K. and Urban J., 3D curve evolution algorithm with tangential redistribution for a fully automatic finding of an ideal camera path in virtual colonoscopy, in Proceedings
of the Third International Conference on Scale Space Methods and Variational Methods in Computer Vision 2011, Lecture Notes in Comput. Sci. 6667, Springer-Verlag, Berlin, 2011.
26. Morigi S., Geometric surface evolution with tangential contribution, J. Comput. Appl. Math., 233 (2010), pp. 12771287.
27. Mullins W.M., Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900-904.
28. Osher S. and Fedkiw R., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, Berlin, (2003).
29. Plateau J. A. F., Statique experimentale et teoretique des liquides soumis aux seule forces molceculaires, volume I, II. Gauthier-Villars, Paris, 1873.
30. Sethian J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science, Cambridge University Press, New York, (1999).
31. Sevcovic D. and Yazaki S., Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), pp. 1784-1798.
32. Van der Vorst H. A., Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of fonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp.
631644.
2. Barrett J. W., Garcke H. and Nurnberg R., On the parametric finite element approximation of evolving hypersurfaces in R3, J. Comput. Phys., 227 (2008), pp. 42814307.
3. Barrett J. W., Garcke H. and Nurnberg R., The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute, Numer. Methods Partial Differential Equations, 27 (2011), pp. 130.
4. Bauer M., Harms P., and Michor P. W., Almost local metrics on shape space of hypersurfaces in n-space, SIAM J. Imaging Sci., 5(1):244310, 2012.
5. Benninghoff H. and Garcke H., Efficient image segmentation and restoration using parametric curve evolution with junctions and topology changes, SIAM Journal on Imaging Sciences, 7(3), (2014), 1451-1483.
6. Benninghoff H. and Garcke H., Segmentation and Restoration of Images on Surfaces by Parametric Active Contours with Topology Changes, Journal of Mathematical Imaging and Vision, 55(1), (2016), pp. 105-124.
7. Caselles V., Kimmel R. and Sapiro G., Geodesic active contours, Int. J. Comput. Vis., 22 (1997), pp. 61-79.
8. Costa A., Examples of a Complete Minimal Immersion in R3 of Genus One and Three Embedded Ends. Bil. Soc. Bras. Mat. 15, 47-54, 1984.
9. Dziuk G., Algorithm for evolutionary surfaces Numer. Math. 58 (1991), pp. 603-611.
10. Dziuk G. and Elliot C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), pp. 262-292.
11. Elliott C. M. and Fritz H., On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick, SMAI J.
of Comput. Math., Volume 2 (2016), pp. 141-176.
12. Evans L. C. and Spruck J., Motion of level sets by mean curvature I, J. Diff. Geom. 33 (1991), 635-681.
13. Hou T. Y., Klapper I. and Si H., Removing the stiffness of curvature in computing 3-d fllaments, J. Comput. Phys., 143 (1998), pp. 628-664.
14. Hou T. Y., Lowengrub J., and Shelley M., Removing the stiffness from interfacial flows and surface tension, J. Comput. Phys., 114 (1994), pp. 312-338.
15. Huska M., Medla M., Mikula K., Novysedlak P. and Remesikova M., A new form-finding method based on mean curvature ow of surfaces, in Proceedings of ALGORITMY 2012, 19th Conference on Scientific Computing, Podbansk, Slovakia, (2012), pp. 120-131.
16. Jiao X., Colombi A., Ni X. and Hart J. C., Anisotropic mesh adaptation for evolving triangulated surfaces, Engineering with Computers, Vol 26(4): 363-376, 2009
17. Kimura M., Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., 14 (1997), pp. 373-398.
18. Mantegazza C., Lecture Notes on Mean Curvature Flow, Addison-Wesley publishing company, Birkhauser Verlag, (2011), 166 s. ISBN 978-3-0348-0144-7
19. Meyer M., Desbrun M., Schroeder P. and Barr A.H., Discrete differential geometry operators for triangulated 2-manifolds, Visualization and Mathematics III (H.-C. Hege and K. Polthier, eds.) (2003), pp. 35-57
20. Mikula K., Peyrieras N., Remeskova M. and Smisek M., 4D numerical schemes for cell image segmentation and tracking, in Proceedings of Finite Volumes in Complex Applications VI, Problems & Perspectives, Springer-Verlag, Berlin, 2011, pp. 693702.
21. Mikula K., Remesikova M., Sarkoci P. and Sevcovic D., Manifold evolution with tangential redistribution of points, SIAM J. Scientific Computing, Vol. 36, No.4 (2014), pp. A1384-A1414
22. Mikula K. and Sevcovic D., Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), pp. 1473-1501.
23. Mikula K. and Sevcovic D., A direct method for solving an anisotropic mean curvature flow of planar curve with an external force, Math. Methods Appl. Sci., 27 (2004), pp. 1545-1565.
24. Mikula K. and Sevcovic D., Evolution of curves on surface driven by the geodesic curvature and external force, Appl. Anal., 85 (2006), pp. 345362.
25. Mikula K. and Urban J., 3D curve evolution algorithm with tangential redistribution for a fully automatic finding of an ideal camera path in virtual colonoscopy, in Proceedings
of the Third International Conference on Scale Space Methods and Variational Methods in Computer Vision 2011, Lecture Notes in Comput. Sci. 6667, Springer-Verlag, Berlin, 2011.
26. Morigi S., Geometric surface evolution with tangential contribution, J. Comput. Appl. Math., 233 (2010), pp. 12771287.
27. Mullins W.M., Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900-904.
28. Osher S. and Fedkiw R., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, Berlin, (2003).
29. Plateau J. A. F., Statique experimentale et teoretique des liquides soumis aux seule forces molceculaires, volume I, II. Gauthier-Villars, Paris, 1873.
30. Sethian J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science, Cambridge University Press, New York, (1999).
31. Sevcovic D. and Yazaki S., Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), pp. 1784-1798.
32. Van der Vorst H. A., Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of fonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp.
631644.