October  2015, 8(5): 969-988. doi: 10.3934/dcdss.2015.8.969

Multiphase volume-preserving interface motions via localized signed distance vector scheme

1. 

Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan

2. 

Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan

Received  December 2013 Revised  August 2014 Published  July 2015

We develop a signed distance vector approach for approximating volume-preserving mean curvature motions of interfaces separating multiple phases -- a variant of the BMO (Bence-Merriman-Osher) thresholding dynamics. We adopt a variational method employing the idea of a vector-type discrete Morse flow, which allows us to easily treat volume constraint via penalization without having to change the threshold value. Moreover, employing a vector-valued analogue of the signed distance function, the scheme is designed to allow subgrid accuracy on uniform grids without adaptive refinement; thereby, alleviating the well-known BMO time and grid restrictions. Finally, we present numerical tests and examples.
Citation: Rhudaina Z. Mohammad, Karel Švadlenka. Multiphase volume-preserving interface motions via localized signed distance vector scheme. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 969-988. doi: 10.3934/dcdss.2015.8.969
References:
[1]

N. Aguilera, H. W. Alt and L. A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control and Optimization, 24 (1986), 191-198. doi: 10.1137/0324011.  Google Scholar

[2]

H. W. Alt, L. A. Caffarelli and A. Friedman, Variational prrblems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461. doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar

[3]

J. W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of willmore flow and related geometric evolution equations, SIAM J. Sci. Comput., 31 (2008), 225-253. doi: 10.1137/070700231.  Google Scholar

[4]

G. Barles and C. Georgelin, A simple proof of convergence of an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020.  Google Scholar

[5]

I. C. Dolcetta, S. F. Vita and R. March, Area preserving curve shortening flows: From phase separation to image processing, Interfaces and Free Boundaries, 4 (2002), 325-343. doi: 10.4171/IFB/64.  Google Scholar

[6]

M. Elsey, S. Esedoglu and P. Smereka, Diffusion generated motion for grain growth in two and three dimensions, J. Comp. Physics, 228 (2009), 8015-8033. doi: 10.1016/j.jcp.2009.07.020.  Google Scholar

[7]

S. Esedoglu, S. Ruuth and R. Tsai, Diffusion-generated motion using signed distance functions, J. Comp. Physics, 229 (2010), 1017-1042. doi: 10.1016/j.jcp.2009.10.002.  Google Scholar

[8]

L. Evans, Convergence of an algorithm for mean curvature motion, Indiana University Mathematics Journal, 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024.  Google Scholar

[9]

M. Gage, Curve shortening makes convex curves circular, Invent. Math., 76 (1984), 357-364. doi: 10.1007/BF01388602.  Google Scholar

[10]

E. Ginder, S. Omata and K. Švadlenka, A variational method for diffusion-generated area-preserving interface motion, Theoretical and Applied Mechanics Japan, 60 (2011), 265-270. Google Scholar

[11]

E. Ginder, A Variational Approach to Volume-Controlled Evolutionary Equations, PhD Thesis, 2013. Google Scholar

[12]

J. Hass, M. Hutchings and R. Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc., 1 (1995), 98-102. doi: 10.1090/S1079-6762-95-03001-0.  Google Scholar

[13]

K. Ishii, Mathematical analysis to an approximation scheme for mean curvature flow, in International Symposium on Computational Science 2011 (eds. S. Omata and K. Svadlenka), Mathematical Sciences and Applications, GAKUTO International Series, 34 (2011), 67-85.  Google Scholar

[14]

B. Merriman, J. Bence and S. Osher, Motion of multiple junctions: A level set approach, J. Comp. Physics, 112 (1994), 334-363. doi: 10.1006/jcph.1994.1105.  Google Scholar

[15]

R. Z. Mohammad and K. Švadlenka, On a penalization method for an evolutionary free boundary problem with volume constraint, Adv. Math. Sci. Appl., 24 (2014), 85-101. Google Scholar

[16]

E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102 (1930), 650-670. doi: 10.1007/BF01782368.  Google Scholar

[17]

S. Ruuth and B. Wetton, A simple scheme for volume-preserving motion by mean curvature, J. Scientific Computing, 19 (2003), 373-384. doi: 10.1023/A:1025368328471.  Google Scholar

[18]

K. Švadlenka, E. Ginder and S. Omata, A variational method for multiphase area-preserving interface motions, J. Comp. Appl. Math, 257 (2014), 157-179. doi: 10.1016/j.cam.2013.08.027.  Google Scholar

[19]

P. Tilli, On a constrained variational problem with an arbitrary number of free boundaries, Interfaces Free Bound, 2 (2000), 201-212. doi: 10.4171/IFB/18.  Google Scholar

[20]

H.-K. Zhao, B. Merriman, S. Osher and L. Wang, Capturing the behavior of bubbles and drops using the variational level set approach, J. Comp. Phys., 143 (1998), 495-518. doi: 10.1006/jcph.1997.5810.  Google Scholar

show all references

References:
[1]

N. Aguilera, H. W. Alt and L. A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control and Optimization, 24 (1986), 191-198. doi: 10.1137/0324011.  Google Scholar

[2]

H. W. Alt, L. A. Caffarelli and A. Friedman, Variational prrblems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461. doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar

[3]

J. W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of willmore flow and related geometric evolution equations, SIAM J. Sci. Comput., 31 (2008), 225-253. doi: 10.1137/070700231.  Google Scholar

[4]

G. Barles and C. Georgelin, A simple proof of convergence of an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020.  Google Scholar

[5]

I. C. Dolcetta, S. F. Vita and R. March, Area preserving curve shortening flows: From phase separation to image processing, Interfaces and Free Boundaries, 4 (2002), 325-343. doi: 10.4171/IFB/64.  Google Scholar

[6]

M. Elsey, S. Esedoglu and P. Smereka, Diffusion generated motion for grain growth in two and three dimensions, J. Comp. Physics, 228 (2009), 8015-8033. doi: 10.1016/j.jcp.2009.07.020.  Google Scholar

[7]

S. Esedoglu, S. Ruuth and R. Tsai, Diffusion-generated motion using signed distance functions, J. Comp. Physics, 229 (2010), 1017-1042. doi: 10.1016/j.jcp.2009.10.002.  Google Scholar

[8]

L. Evans, Convergence of an algorithm for mean curvature motion, Indiana University Mathematics Journal, 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024.  Google Scholar

[9]

M. Gage, Curve shortening makes convex curves circular, Invent. Math., 76 (1984), 357-364. doi: 10.1007/BF01388602.  Google Scholar

[10]

E. Ginder, S. Omata and K. Švadlenka, A variational method for diffusion-generated area-preserving interface motion, Theoretical and Applied Mechanics Japan, 60 (2011), 265-270. Google Scholar

[11]

E. Ginder, A Variational Approach to Volume-Controlled Evolutionary Equations, PhD Thesis, 2013. Google Scholar

[12]

J. Hass, M. Hutchings and R. Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc., 1 (1995), 98-102. doi: 10.1090/S1079-6762-95-03001-0.  Google Scholar

[13]

K. Ishii, Mathematical analysis to an approximation scheme for mean curvature flow, in International Symposium on Computational Science 2011 (eds. S. Omata and K. Svadlenka), Mathematical Sciences and Applications, GAKUTO International Series, 34 (2011), 67-85.  Google Scholar

[14]

B. Merriman, J. Bence and S. Osher, Motion of multiple junctions: A level set approach, J. Comp. Physics, 112 (1994), 334-363. doi: 10.1006/jcph.1994.1105.  Google Scholar

[15]

R. Z. Mohammad and K. Švadlenka, On a penalization method for an evolutionary free boundary problem with volume constraint, Adv. Math. Sci. Appl., 24 (2014), 85-101. Google Scholar

[16]

E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102 (1930), 650-670. doi: 10.1007/BF01782368.  Google Scholar

[17]

S. Ruuth and B. Wetton, A simple scheme for volume-preserving motion by mean curvature, J. Scientific Computing, 19 (2003), 373-384. doi: 10.1023/A:1025368328471.  Google Scholar

[18]

K. Švadlenka, E. Ginder and S. Omata, A variational method for multiphase area-preserving interface motions, J. Comp. Appl. Math, 257 (2014), 157-179. doi: 10.1016/j.cam.2013.08.027.  Google Scholar

[19]

P. Tilli, On a constrained variational problem with an arbitrary number of free boundaries, Interfaces Free Bound, 2 (2000), 201-212. doi: 10.4171/IFB/18.  Google Scholar

[20]

H.-K. Zhao, B. Merriman, S. Osher and L. Wang, Capturing the behavior of bubbles and drops using the variational level set approach, J. Comp. Phys., 143 (1998), 495-518. doi: 10.1006/jcph.1997.5810.  Google Scholar

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