A variational method for multiphase volume-preserving interface motions. (English) Zbl 1294.53064
Summary: We develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases, whose volumes are preserved throughout time. The foundation of the method is a thresholding algorithm of the Bence-Merriman-Osher type. The original algorithm is reformulated in a vector setting, which allows for a natural inclusion of constraints, even in the multiphase case. Moreover, a new method for overcoming the inaccuracy of thresholding methods on non-adaptive grids is designed, since this inaccuracy becomes especially prominent in volume-preserving motions. Formal analysis of the method and numerical tests are presented.
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 35K55 | Nonlinear parabolic equations |
| 76T30 | Three or more component flows |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
Keywords:
mean curvature flow; volume preservation; multiphase; thresholding method; constrained variational problemReferences:
| [1] | Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20, 237-266 (1984) · Zbl 0556.53001 |
| [2] | Huisken, G., The volume preserving mean curvature flow, J. Reine. Angew. Math., 382, 35-48 (1987) · Zbl 0621.53007 |
| [3] | Bellettini, G.; Caselles, V.; Chambolle, A.; Novaga, M., The volume preserving crystalline mean curvature flow of convex sets in \(R^N\), J. Math. Pures Appl. (9), 92, 499-527 (2009) · Zbl 1178.53066 |
| [4] | Andrews, B., Volume-preserving anisotropic mean curvature flow, Indiana Univ. Math. J., 50, 783-827 (2001) · Zbl 1047.53037 |
| [5] | Mayer, U.; Simonett, G., Self-intersections for the surface diffusion and the volume preserving mean curvature flow, Differential Integral Equations, 13, 1189-1199 (2000) · Zbl 1013.53045 |
| [6] | Escher, J.; Simonett, G., The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126, 2789-2796 (1998) · Zbl 0909.53043 |
| [7] | Antonopoulou, D.; Karali, G.; Sigal, I., Stability of spheres under volume-preserving mean curvature flow, Dyn. Partial Differ. Equ., 7, 327-344 (2010) · Zbl 1219.35132 |
| [8] | Escher, J.; Feng, Z., Exponential stability of equilibria of the curve shortening flow with contact angle, Dyn. Contin. Discret Impuls. Syst. Ser. A Math. Anal., 14, 287-299 (2007) · Zbl 1127.35003 |
| [9] | Bronsard, L.; Reitich, F., On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch Ration. Mech. Anal., 124, 355-379 (1993) · Zbl 0785.76085 |
| [10] | Bronsard, L.; Wetton, B., A numerical method for tracking curve networks moving with curvature motion, J. Comput. Phys., 120, 66-87 (1995) · Zbl 0838.65087 |
| [11] | Taylor, J. E., Motion of curves by crystalline curvature, including triple junctions and boundary points, Differ Geom. Proc. Sympos. Pure Math., 51, 417-438 (1993) · Zbl 0823.49028 |
| [12] | Zhao, H.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. Comput. Phys., 127, 179-195 (1996) · Zbl 0860.65050 |
| [13] | Zhao, H.; Merriman, B.; Osher, S.; Wang, L., Capturing the behavior of bubbles and drops using the variational level set approach, J. Comput. Phys., 143, 495-518 (1998) · Zbl 0936.76065 |
| [14] | Esedoglu, S.; Smereka, P., A variational formulation for a level set representation of multiphase flow and area preserving curvature flow, Commun. Math. Sci., 6, 125-148 (2008) · Zbl 1137.76067 |
| [15] | Kublik, C.; Esedoglu, S.; Fessler, J., Algorithms for area preserving flows, SIAM J. Sci. Comput., 33, 2382-2401 (2011) · Zbl 1232.65012 |
| [16] | Smith, K.; Solis, F.; Chopp, D., A projection method for motion of triple junctions by level sets, Interfaces Free Bound, 4, 263-276 (2002) · Zbl 1112.76437 |
| [17] | Saye, R.; Sethian, J., The Voronoi implicit interface method for computing multiphase physics, Proc. Natl. Acad. Sci. USA, 108, 19498-19503 (2011) · Zbl 1256.65082 |
| [18] | Golovaty, D., The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Quart. Appl. Math., 55, 243-298 (1997) · Zbl 0878.35059 |
| [19] | Bronsard, L.; Stoth, B., Volume preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28, 769-807 (1997) · Zbl 0874.35009 |
| [20] | Brassel, M.; Bretin, E., A modified phase field approximation for mean curvature flow with conservation of the volume, Math. Methods Appl. Sci., 34, 1157-1180 (2011) · Zbl 1235.49082 |
| [21] | Garcke, H.; Nestler, B.; Stoth, B., A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions, SIAM J. Appl. Math., 60, 295-315 (1999) · Zbl 0942.35095 |
| [22] | Merriman, B.; Bence, J.; Osher, S., Motion of multiple junctions: a level set approach, J. Comput. Phys., 112, 334-363 (1994) · Zbl 0805.65090 |
| [23] | Ruuth, S.; Wetton, B., A simple scheme for volume-preserving motion by mean curvature, J. Sci. Comput., 19, 373-384 (2003) · Zbl 1035.65097 |
| [24] | Svadlenka, K.; Omata, S., Construction of solutions to heat-type problems with volume constraint via the discrete Morse flow, Funkcial. Ekvac., 50, 261-285 (2007) · Zbl 1156.35394 |
| [25] | Barles, G.; Georgelin, C., A simple proof of convergence of an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32, 484-500 (1995) · Zbl 0831.65138 |
| [26] | Evans, L., Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42, 533-557 (1993) · Zbl 0802.65098 |
| [27] | Goto, Y.; Ishii, K.; Ogawa, T., Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature, Comm. Pure Appl. Anal., 4, 311-339 (2005) · Zbl 1077.35070 |
| [28] | Slepčev, D., Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Anal., 52, 79-115 (2003) · Zbl 1028.35068 |
| [29] | Vivier, L., Convergence of an approximation scheme for computing motions with curvature dependent velocities, Differential Integral Equations, 13, 1263-1288 (2000) · Zbl 0983.65112 |
| [30] | Heida, M.; Málek, J.; Rajagopal, K., On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Z. Angew. Math. Phys., 1-25 (2011) |
| [31] | Rothe, E., Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102, 650-670 (1930) · JFM 56.1076.02 |
| [32] | Alt, H.; Caffarelli, L.; Friedman, A., Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282, 431-461 (1984) · Zbl 0844.35137 |
| [33] | Caffarelli, L.; Lederman, C.; Wolanski, N., Uniform estimates and limits for a two phase parabolic singular perturbation problem, Indiana Univ. Math. J., 46, 453-489 (1997) · Zbl 0909.35012 |
| [34] | Caffarelli, L.; Lederman, C.; Wolanski, N., Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem, Indiana Univ. Math. J., 46, 719-739 (1997) · Zbl 0909.35013 |
| [35] | Weiss, G., A parabolic free boundary problem with double pinning, Nonlinear Anal., 57, 153-172 (2004) · Zbl 1072.35210 |
| [36] | Aguilera, N.; Alt, H.; Caffarelli, L., An optimization problem with volume constraint, SIAM J. Control Optim., 24, 191-198 (1986) · Zbl 0588.49005 |
| [37] | Tilli, P., On a constrained variational problem with an arbitrary number of free boundaries, Interfaces Free Bound, 2, 201-212 (2000) · Zbl 0995.49002 |
| [38] | Leonardi, G.; Tilli, P., On a constrained variational problem in the vector-valued case, J. Math. Pures Appl. (9), 85, 251-268 (2006) · Zbl 1086.49011 |
| [39] | Weiss, G., A gradient flow approach to a free boundary problem with volume constraint, RIMS Kôkyûroku Bessatsu, 1198, 117-121 (2001) · Zbl 0985.65544 |
| [40] | Kikuchi, N., An approach to the construction of Morse flows for variational functionals, (Coron, J.-M. G.J.-M.; Hélein, F., NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. (1991), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht, Boston, London), 195-198 · Zbl 0850.76043 |
| [41] | Esedoglu, S.; Ruuth, S.; Tsai, R., Diffusion generated motion using signed distance functions, J. Comput. Phys., 229, 1017-1042 (2010) · Zbl 1181.65031 |
| [42] | Ruuth, S., Efficient algorithms for diffusion-generated motion by mean curvature, J. Comput. Phys., 144, 603-625 (1998) · Zbl 0946.65093 |
| [43] | Ginder, E.; Omata, S.; Svadlenka, K., A variational method for diffusion-generated area-preserving interface motion, Theor. Appl. Mech. Japan, 60, 265-270 (2011) |
| [44] | Ishii, K., Mathematical analysis to an approximation scheme for mean curvature flow, (Omata, S.; Svadlenka, K., International Symposium on Computational Science 2011. International Symposium on Computational Science 2011, Mathematical Sciences and Applications, vol. 34 (2011), GAKUTO International Series), 67-85 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
