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March  2021, 14(3): 893-907. doi: 10.3934/dcdss.2020390

## Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

 1 Department of Mathematical Sciences, Shibaura Institute of Technology, Fukasaku 309, Minuma-ku, Saitama 337-8570, Japan 2 Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, Aramaki-machi 4-2, Maebashi, 371-8510 Gunma, Japan

* Corresponding author: Takeshi Ohtsuka

Received  January 2019 Revised  February 2020 Published  June 2020

In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized $L^1$ norm of the difference of step-like functions which are branches of $\arg (x)$ whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.

Citation: Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
##### References:
  F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42 (1995), 1-22.  doi: 10.4310/jdg/1214457030.  Google Scholar  F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.  Google Scholar  S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. Ⅱ. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391.  doi: 10.1007/BF01041068.  Google Scholar  G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.  doi: 10.14492/hokmj/1351516749.  Google Scholar  A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218.  doi: 10.4171/IFB/97.  Google Scholar  A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc., 32 (2019), 779-824.  doi: 10.1090/jams/919.  Google Scholar  A. Chambolle, M. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114.  doi: 10.1002/cpa.21668.  Google Scholar  B. Engquist, A.-K. Tornberg and R. Tsai, Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), 28-51.  doi: 10.1016/j.jcp.2004.09.018.  Google Scholar  M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333.  doi: 10.1007/s002050100154.  Google Scholar  Y. Giga, Surface Evolution Equations: A Level Set Approach, Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. Google Scholar  Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631–698, http://projecteuclid.org/euclid.ade/1462298654. Google Scholar  S. Goto, M. Nakagawa and T. Ohtsuka, Uniqueness and existence of generalized motion for spiral crystal growth, Indiana University Mathematics Journal, 57 (2008), 2571-2599.  doi: 10.1512/iumj.2008.57.3350.  Google Scholar  M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Google Scholar  T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62.  doi: 10.3934/dcdss.2014.7.53.  Google Scholar  T. Ishiwata and T. Ohtsuka, Evolution of spiral-shaped polygonal curve by crystalline curvature flow with a pinned tip, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5261-5295.  doi: 10.3934/dcdsb.2019058.  Google Scholar  A. Oberman, S. Osher, R. Takei and R. Tsai, Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation, Commun. Math. Sci., 9 (2011), 637-662.  doi: 10.4310/CMS.2011.v9.n3.a1.  Google Scholar  T. Ohtsuka, Y.-H. R. Tsai and Y. Giga, A level set approach reflecting sheet structure with single auxiliary function for evolving spirals on crystal surfaces, Journal of Scientific Computing, 62 (2015), 831-874.  doi: 10.1007/s10915-014-9877-2.  Google Scholar  T. Ohtsuka, A level set method for spiral crystal growth, Advances in Mathematical Sciences and Applications, 13 (2003), 225-248. Google Scholar  T. Ohtsuka, Minimizing movement approach for spirals evolving by crystalline curvature using level set functions, Oberwolfach Reports, 14 (2017), 314-317.   Google Scholar  T. Ohtsuka, Minimizing movement approach without using distance function for evolving spirals by the crystalline curvature with driving force, RIMS Kôkyûroku No.2121, 74–87, http://www.kurims.kyoto-u.ac.jp/ kyodo/kokyuroku/contents/pdf/2121-06.pdf. Google Scholar  T. Ohtsuka, Y.-H. R. Tsai and Y. Giga, Growth rate of crystal surfaces with several dislocation centers, Crystal Growth & Design, 18 (2018), 1917-1929.  doi: 10.1021/acs.cgd.7b00833. Google Scholar  R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. Google Scholar  J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, 52 (1991), 321-336. Google Scholar

show all references

##### References:
  F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42 (1995), 1-22.  doi: 10.4310/jdg/1214457030.  Google Scholar  F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.  Google Scholar  S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. Ⅱ. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391.  doi: 10.1007/BF01041068.  Google Scholar  G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.  doi: 10.14492/hokmj/1351516749.  Google Scholar  A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218.  doi: 10.4171/IFB/97.  Google Scholar  A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc., 32 (2019), 779-824.  doi: 10.1090/jams/919.  Google Scholar  A. Chambolle, M. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114.  doi: 10.1002/cpa.21668.  Google Scholar  B. Engquist, A.-K. Tornberg and R. Tsai, Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), 28-51.  doi: 10.1016/j.jcp.2004.09.018.  Google Scholar  M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333.  doi: 10.1007/s002050100154.  Google Scholar  Y. Giga, Surface Evolution Equations: A Level Set Approach, Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. Google Scholar  Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631–698, http://projecteuclid.org/euclid.ade/1462298654. Google Scholar  S. Goto, M. Nakagawa and T. Ohtsuka, Uniqueness and existence of generalized motion for spiral crystal growth, Indiana University Mathematics Journal, 57 (2008), 2571-2599.  doi: 10.1512/iumj.2008.57.3350.  Google Scholar  M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Google Scholar  T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62.  doi: 10.3934/dcdss.2014.7.53.  Google Scholar  T. Ishiwata and T. Ohtsuka, Evolution of spiral-shaped polygonal curve by crystalline curvature flow with a pinned tip, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5261-5295.  doi: 10.3934/dcdsb.2019058.  Google Scholar  A. Oberman, S. Osher, R. Takei and R. Tsai, Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation, Commun. Math. Sci., 9 (2011), 637-662.  doi: 10.4310/CMS.2011.v9.n3.a1.  Google Scholar  T. Ohtsuka, Y.-H. R. Tsai and Y. Giga, A level set approach reflecting sheet structure with single auxiliary function for evolving spirals on crystal surfaces, Journal of Scientific Computing, 62 (2015), 831-874.  doi: 10.1007/s10915-014-9877-2.  Google Scholar  T. Ohtsuka, A level set method for spiral crystal growth, Advances in Mathematical Sciences and Applications, 13 (2003), 225-248. Google Scholar  T. Ohtsuka, Minimizing movement approach for spirals evolving by crystalline curvature using level set functions, Oberwolfach Reports, 14 (2017), 314-317.   Google Scholar  T. Ohtsuka, Minimizing movement approach without using distance function for evolving spirals by the crystalline curvature with driving force, RIMS Kôkyûroku No.2121, 74–87, http://www.kurims.kyoto-u.ac.jp/ kyodo/kokyuroku/contents/pdf/2121-06.pdf. Google Scholar  T. Ohtsuka, Y.-H. R. Tsai and Y. Giga, Growth rate of crystal surfaces with several dislocation centers, Crystal Growth & Design, 18 (2018), 1917-1929.  doi: 10.1021/acs.cgd.7b00833. Google Scholar  R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. Google Scholar  J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, 52 (1991), 321-336. Google Scholar A figure of two spirals (the solid and dashed lines) and the interposed region by them. The function $\mathcal{D} (t)$ indicates the area of the gray regions Description of $\Gamma_D = \bigcup_{j = 0}^k L_j (t)$. Note that, for the simplicity, the variable $t$ of $L_j$ and $y_j$ is omitted in the above figure Construction of $\theta_D (t, x)$; we construct a branch of $\arg (x)$ whose discontinuities are only on $\Gamma (t)$(the dashed line in (1)). For this purpose we first construct $\vartheta (x) = \arg (x)$ whose discontinuities are only on $\mathcal{L}_k (t)$ (the solid line in (2)). Then, we make go down the height of $\vartheta (x)$ on $R_{j} (t)$ (the gray region in (3) or (4)) with the jump-height $2 \pi$ from $j = k-1$ to $j = 0$ inductively to remove illegal discontinuities. The solid line in figure (3) or (4) denotes the discontinuity of $\Theta_{k,k-1}$ or $\Theta_{k,k-2}$, respectively Profiles of the square spiral at $t = 1$. The level set method is calculated using $\rho = 0.02$ and $\Delta x = 0.0050$ Graphs of functions $\mathcal{D} (t)$ for the square spiral with a fixed center radius $\rho = 0.02$(left), and with a reduced center radius $\rho = 2 \Delta x$(right) Profiles of the diagonal spiral at $t = 1$. The level set method is calculated using $\rho = 0.02$ and $\Delta x = 0.0050$ Graphs of $\mathcal{D}(t)$ for the diagonal spiral with a fixed center radius $\rho = 0.02$(left), and with a reduced center radius $\rho = 4 \Delta x$(right) Profiles of the triangle spiral at $t = 0.8$. The level set method is calculated using $\rho = 0.02$ and $\Delta x = 0.0050$ Graphs of $\mathcal{D} (t)$ for the triangle spiral with a fixed center radius $\rho = 0.02$(left), and with a reduced center radius $\rho = 4\Delta x$(right)
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