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

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

Rhudaina Z. Mohammad 1, and  Karel Švadlenka 2,

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.
Keywords: Multiphase mean curvature flow, vector-valued signed distance, constrained variational method, volume preservation, BMO thresholding..
Mathematics Subject Classification: Primary: 53C44, 76T30; Secondary: 35K5.
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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1512/iumj.1993.42.42024.  Google Scholar

[9]

M. Gage, Curve shortening makes convex curves circular,, Invent. Math., 76 (1984), 357.  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.   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.  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), 34 (2011), 67.   Google Scholar

[14]

B. Merriman, J. Bence and S. Osher, Motion of multiple junctions: A level set approach,, J. Comp. Physics, 112 (1994), 334.  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.   Google Scholar

[16]

E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben,, Math. Ann., 102 (1930), 650.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1512/iumj.1993.42.42024.  Google Scholar

[9]

M. Gage, Curve shortening makes convex curves circular,, Invent. Math., 76 (1984), 357.  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.   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.  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), 34 (2011), 67.   Google Scholar

[14]

B. Merriman, J. Bence and S. Osher, Motion of multiple junctions: A level set approach,, J. Comp. Physics, 112 (1994), 334.  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.   Google Scholar

[16]

E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben,, Math. Ann., 102 (1930), 650.  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.  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.  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.  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.  doi: 10.1006/jcph.1997.5810.  Google Scholar

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