American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[1]

Mohammad Safdari. The regularity of some vector-valued variational inequalities with gradient constraints. Communications on Pure and Applied Analysis, 2018, 17 (2) : 413-428. doi: 10.3934/cpaa.2018023
[2]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial and Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174
[3]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[4]

Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011
[5]

Y. Goto, K. Ishii, T. Ogawa. Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature. Communications on Pure and Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311
[6]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[7]

Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487
[8]

Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schrödinger operators in Lp-spaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934/dcdss.2020086
[9]

Christiane Pöschl, Jan Modersitzki, Otmar Scherzer. A variational setting for volume constrained image registration. Inverse Problems and Imaging, 2010, 4 (3) : 505-522. doi: 10.3934/ipi.2010.4.505
[10]

Michele Campiti. Korovkin-type approximation of set-valued and vector-valued functions. Mathematical Foundations of Computing, 2022, 5 (3) : 231-239. doi: 10.3934/mfc.2021032
[11]

Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112
[12]

Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313
[13]

Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098
[14]

Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591
[15]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108
[16]

Emmanuel Hebey. The Lin-Ni's conjecture for vector-valued Schrödinger equations in the closed case. Communications on Pure and Applied Analysis, 2010, 9 (4) : 955-962. doi: 10.3934/cpaa.2010.9.955
[17]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[18]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153
[19]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016
[20]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (175)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy