Evolution of Space Curves by Parametric Method with Natural and Uniform Redistribution
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Published
Sep 21, 2024
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Abstract
Space curve evolution occurs frequently in various domains of science and engineering such as computer graphics, navigation, or vortex motion. This paper focuses on the parametric method for evolving space curves by normal curvature and force. We first introduce the concept of curve evolution and its parametrization. Subsequently, we present a numerical scheme based on method of lines and show several computational studies of the forced curvature flow in space.
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Narayanan, M., & Beneš, M.
(2024).
Evolution of Space Curves by Parametric Method with Natural and Uniform Redistribution.
Proceedings Of The Conference Algoritmy, , 109 - 118.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2160/1032
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References
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[2] S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, J. Differential Geometry, 35 (1992), pp. 283–298.
[3] M. Beneš, M. Kolář, and D. Ševčovič, Qualitative and numerical aspects of a motion of a family of interacting curves in space, SIAM Journal on Applied Mathematics, 82 (2022), pp. 549–575.
[4] M. Beneš, M. Kolář, and D. Ševčovič, Curvature driven flow of a family of interacting curves with applications, Mathematical Methods in the Applied Sciences, 43 (2020), pp. 4177–4190.
[5] K. Deckelnick, Weak solutions of the curve shortening flow, Calculus of Variations and Partial Differential Equations, 5 (1997), pp. 489–510.
[6] B. Devincre, Three dimensional stress field expression for straight dislocation segments, Solid State Communications, 93 (1995), pp. 875–878.
[7] R. T. Farouki and C. Giannelli, Spatial camera orientation control by rotation-minimizing directed frames, Computer Animation and Virtual Worlds, 20 (2009), pp. 457–472.
[8] J. Fierling, A. Johner, I. M. Kulic, H. Mohrbach, and M. M. Mueller, How bio-filaments twist membranes, Soft Matter, 12 (2016), pp. 5747–5757.
[9] J.-H. He, Y. Liu, L.-F. Mo, Y.-Q. Wan, and L. Xu, Electrospun Nanofibres and Their Applications, iSmithers, Shawbury, 2008.
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[12] R. L. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, J. Eur. Math. Soc., 017 (2015), pp. 1487–1515.
[13] M. Kolář, M. Beneš, J. Kratochvı́l, and P. Pauš, Modeling of double cross-slip by means of geodesic curvature driven flow, Acta Physica Polonica A, 134 (2018), pp. 667–670.
[14] M. Kolář, M. Beneš, and D. Ševčovič, Computational studies of conserved mean-curvature flow, Mathematica Bohemica, 139 (2014), pp. 677–684.
[15] M. Kolář, M. Beneš, and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Mathematics and Computers in Simulation, 126 (2016), pp. 1–13.
[16] M. Kolář, P. Pauš, J. Kratochvı́l, and M. Beneš, Improving method for deterministic treatment of double cross-slip in fcc metals under low homologous temperatures, Computational Materials Science, 189 (2021), p. 110251.
[17] M. Kolář and D. Ševčovič, Evolution of multiple closed knotted curves in space, in ALGORITMY 2024, Proceedings of contributed papers and posters, Bratislava, 2024, p. submitted. (peer-reviewed).
[18] C. M. Elliott and H. Fritz, On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick, IMA Journal of Numerical Analysis, 37 (2016), pp. 543–603.
[19] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Computing and Visualization in Science, 6 (2004),
[20] K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Mathematical Methods in the Applied Sciences, 27 (2004), pp. 1545–1565.
[21] J. Minarčı́k and M. Beneš, Long-term behavior of curve shortening flow in R3, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 1221–1231.
[22] P. Pauš and M. Beneš, Direct approach to mean-curvature flow with topological changes, Kybernetika, 45 (2009), pp. 591–604.
[23] P. Pauš, M. Beneš, M. Kolář, and J. Kratochvı́l, Dynamics of dislocations described as evolving curves interacting with obstacles, Modelling and Simulation in Materials Science and Engineering, 24 (2016), p. 035003.
[24] D. H. Reneker and A. L. Yarin, Electrospinning jets and polymer nanofibers, Polymer, 49 (2008), pp. 2387–2425.
[25] D. Ševčovič and K. Mikula, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM Journal on Applied Mathematics, 61 (2001), pp. 1473–1501.
[26] P. Strachota, Implementation of the MR tractography visualization kit based on the anisotropic Allen-Cahn equation, Kybernetika, 45 (2009), pp. 657–669.
[27] P. Strachota and A. Wodecki, High resolution 3D phase field simulations of single crystal and polycrystalline solidification, Acta Phys. Pol. A, 134 (2018), pp. 653–657.
[28] A. Yarin, B. Pourdeyhimi, and S. Ramakrishna, Fundamentals and applications of micro and nanofibers, Cambridge University Press, Cambridge, 2014.
[29] S. Yazaki, M. Kolář, and K. Sakakibara, Image segmentation of flame front of smoldering experiment by gradient flow of curves, in ALGORITMY 2024, Proceedings of contributed papers and posters, Bratislava, 2024, p. submitted. (peer-reviewed).
[30] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Mathematical Methods in the Applied Sciences, 35 (2012), pp. 1784–1798.
[2] S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, J. Differential Geometry, 35 (1992), pp. 283–298.
[3] M. Beneš, M. Kolář, and D. Ševčovič, Qualitative and numerical aspects of a motion of a family of interacting curves in space, SIAM Journal on Applied Mathematics, 82 (2022), pp. 549–575.
[4] M. Beneš, M. Kolář, and D. Ševčovič, Curvature driven flow of a family of interacting curves with applications, Mathematical Methods in the Applied Sciences, 43 (2020), pp. 4177–4190.
[5] K. Deckelnick, Weak solutions of the curve shortening flow, Calculus of Variations and Partial Differential Equations, 5 (1997), pp. 489–510.
[6] B. Devincre, Three dimensional stress field expression for straight dislocation segments, Solid State Communications, 93 (1995), pp. 875–878.
[7] R. T. Farouki and C. Giannelli, Spatial camera orientation control by rotation-minimizing directed frames, Computer Animation and Virtual Worlds, 20 (2009), pp. 457–472.
[8] J. Fierling, A. Johner, I. M. Kulic, H. Mohrbach, and M. M. Mueller, How bio-filaments twist membranes, Soft Matter, 12 (2016), pp. 5747–5757.
[9] J.-H. He, Y. Liu, L.-F. Mo, Y.-Q. Wan, and L. Xu, Electrospun Nanofibres and Their Applications, iSmithers, Shawbury, 2008.
[10] J. P. Hirth and J. Lothe, Theory of Dislocations, Wiley, 1982.
[11] T. Ishiwata and K. Kumazaki, Structure-preserving finite difference scheme for vortex filament motion, in Proceedings of Algoritmy 2012, 19th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, September 9-14, 2012, K. Mikula, ed., Slovak University of Technology in Bratislava, 2012, pp. 230–238.
[12] R. L. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, J. Eur. Math. Soc., 017 (2015), pp. 1487–1515.
[13] M. Kolář, M. Beneš, J. Kratochvı́l, and P. Pauš, Modeling of double cross-slip by means of geodesic curvature driven flow, Acta Physica Polonica A, 134 (2018), pp. 667–670.
[14] M. Kolář, M. Beneš, and D. Ševčovič, Computational studies of conserved mean-curvature flow, Mathematica Bohemica, 139 (2014), pp. 677–684.
[15] M. Kolář, M. Beneš, and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Mathematics and Computers in Simulation, 126 (2016), pp. 1–13.
[16] M. Kolář, P. Pauš, J. Kratochvı́l, and M. Beneš, Improving method for deterministic treatment of double cross-slip in fcc metals under low homologous temperatures, Computational Materials Science, 189 (2021), p. 110251.
[17] M. Kolář and D. Ševčovič, Evolution of multiple closed knotted curves in space, in ALGORITMY 2024, Proceedings of contributed papers and posters, Bratislava, 2024, p. submitted. (peer-reviewed).
[18] C. M. Elliott and H. Fritz, On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick, IMA Journal of Numerical Analysis, 37 (2016), pp. 543–603.
[19] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Computing and Visualization in Science, 6 (2004),
[20] K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Mathematical Methods in the Applied Sciences, 27 (2004), pp. 1545–1565.
[21] J. Minarčı́k and M. Beneš, Long-term behavior of curve shortening flow in R3, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 1221–1231.
[22] P. Pauš and M. Beneš, Direct approach to mean-curvature flow with topological changes, Kybernetika, 45 (2009), pp. 591–604.
[23] P. Pauš, M. Beneš, M. Kolář, and J. Kratochvı́l, Dynamics of dislocations described as evolving curves interacting with obstacles, Modelling and Simulation in Materials Science and Engineering, 24 (2016), p. 035003.
[24] D. H. Reneker and A. L. Yarin, Electrospinning jets and polymer nanofibers, Polymer, 49 (2008), pp. 2387–2425.
[25] D. Ševčovič and K. Mikula, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM Journal on Applied Mathematics, 61 (2001), pp. 1473–1501.
[26] P. Strachota, Implementation of the MR tractography visualization kit based on the anisotropic Allen-Cahn equation, Kybernetika, 45 (2009), pp. 657–669.
[27] P. Strachota and A. Wodecki, High resolution 3D phase field simulations of single crystal and polycrystalline solidification, Acta Phys. Pol. A, 134 (2018), pp. 653–657.
[28] A. Yarin, B. Pourdeyhimi, and S. Ramakrishna, Fundamentals and applications of micro and nanofibers, Cambridge University Press, Cambridge, 2014.
[29] S. Yazaki, M. Kolář, and K. Sakakibara, Image segmentation of flame front of smoldering experiment by gradient flow of curves, in ALGORITMY 2024, Proceedings of contributed papers and posters, Bratislava, 2024, p. submitted. (peer-reviewed).
[30] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Mathematical Methods in the Applied Sciences, 35 (2012), pp. 1784–1798.