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Computationally efficient solution to the Cahn-Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem. (English) Zbl 1416.65364
Summary: We present an efficient numerical framework for analyzing spinodal decomposition described by the Cahn-Hilliard equation. We focus on the analysis of various implicit time schemes for two and three dimensional problems. We demonstrate that significant computational gains can be obtained by applying embedded, higher order Runge-Kutta methods in a time adaptive setting. This allows accessing time-scales that vary by five orders of magnitude. In addition, we also formulate a set of test problems that isolate each of the sub-processes involved in spinodal decomposition: interface creation and bulky phase coarsening. We analyze the error fluctuations using these test problems on the split form of the Cahn-Hilliard equation solved using the finite element method with basis functions of different orders. Any scheme that ensures at least four elements per interface satisfactorily captures both sub-processes. Our findings show that linear basis functions have superior error-to-cost properties.
This strategy – coupled with a domain decomposition based parallel implementation – let us notably augment the efficiency of a numerical Cahn-Hillard solver, and open new venues for its practical applications, especially when three dimensional problems are considered. We use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions. The framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3D-sphere, cylinder, lamella, doubly periodic surface with genus two (Lawson surface) and triply periodic minimal surface (P Schwarz surface).

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I: interfacial energy, J. chem. phys., 28, 258, (1958)
[2] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. II: thermodynamic basis, J. chem. phys., 30, 1121-1135, (1959)
[3] Saxena, R.; Caneba, G.T., Studies of spinodal decomposition in a ternary polymer – solvent – nonsolvent systems, Polym. eng. sci., 42, 1019-1031, (2002)
[4] D. Cogswell, A Phase-Field Study of Ternary Multiphase Microstructures, Ph.D. Thesis, MIT, 2010.
[5] Wise, S.; Lowengrub, J.; Frieboes, H.; Cristini, V., Three-dimensional multispecies nonlinear tumor growth. I: model and numerical method, J. theor. biol., 253, 524-543, (2008) · Zbl 1398.92135
[6] Zhou, B.; Powell, A., Phase field simulation of early stage structure formation during immersion precipitation of polymeric membranes in 2D and 3D, J. membr. sci., 268, 150-164, (2006)
[7] Tremaine, S., On the origin of irregular structure in saturn’s rings, Astron. J., 125, 894, (2003)
[8] Bertozzi, A.; Esedoglu, S.; Gillette, A., Inpainting of binary images using the cahn – hilliard equation, IEEE trans. image process., 16, 285-291, (2007) · Zbl 1279.94008
[9] Moelans, N.; Blanpain, B.; Wollants, P., An introduction to phase-field modeling of microstructure evolution, Comput. coupling phase diagrams thermochem., 32, 268-294, (2008)
[10] Langer, J.S.; Bar-on, M.; Miller, H.D., New computational method in the theory of spinodal decomposition, Phys. rev. A, 11, 1417-1429, (1975)
[11] Saylor, D.; Kim, C.-S.; Patwardhan, D.; Warren, J., Diffuse-interface theory for structure formation and release behavior in controlled drug release systems, Acta biomater., 3, 851-864, (2007)
[12] Cueto-Felgueroso, L.; Peraire, J., A time-adaptive finite volume method for the cahn – hilliard and kuramoto – sivashinsky equations, J. comput. phys., 227, 9985-10017, (2008) · Zbl 1153.76043
[13] Kim, J.; Kang, K., A numerical method for the ternary cahn – hilliard system with a degenerate mobility, Appl. numer. math., 59, 1029-1042, (2009) · Zbl 1171.65068
[14] Elliot, C.; French, D.; Milner, F., A second order splitting method for the cahn – hilliard equation, Numer. math., 54, 575-590, (1989) · Zbl 0668.65097
[15] Elliott, C.M.; Garcke, H., On the cahn – hilliard equation with degenerate mobility, SIAM J. math. anal., 27, 404-423, (1996) · Zbl 0856.35071
[16] Elliott, C.M.; French, D.A., A nonconforming finite-element method for the two-dimensional cahn – hilliard equation, SIAM J. numer. anal., 26, 884-903, (1989) · Zbl 0686.65086
[17] Zhang, S.; Wang, M., A nonconforming finite element method for the cahn – hilliard equation, J. comput. phys., 229, 7361-7372, (2010) · Zbl 1197.65139
[18] He, Y.; Liu, Y.; Tang, T., On large time-stepping methods for the cahn – hilliard equation, Appl. numer. math., 57, 616-628, (2007), (special issue for the International Conference on Scientific Computing) · Zbl 1118.65109
[19] Gomez, H.; Calo, V.; Bazilevs, Y.; Hughes, T., Isogeometric analysis of the cahn – hilliard phase-field model, Comput. methods appl. mech. eng., 197, 4333-4352, (2008) · Zbl 1194.74524
[20] Deibel, C.; Dyakonov, V., Polymer – fullerene bulk heterojunction solar cells, Rep. progr. phys., 73, 096401, (2010)
[21] Chen, L.-Q., Phase-field models for microstructure evolution, Ann. rev. mater. res., 32, 113-140, (2002)
[22] Rajagopal, A.; Fischer, P.; Kuhl, E.; Steinmann, P., Natural element analysis of the cahn – hilliard phase-field model, Comput. mech., 46, 471-493, (2010) · Zbl 1398.74059
[23] Wells, G.N.; Kuhl, E.; Garikipati, K., A discontinuous Galerkin method for the cahn – hilliard equation, J. comput. phys., 218, 860-877, (2006) · Zbl 1106.65086
[24] Banas, L.; Nurnberg, R., A posteriori estimates for the cahn – hilliard equation with obstacle free energy, Esaim, 43, 1003-1026, (2009) · Zbl 1190.65137
[25] van der Zee, K.G.; Tinsley Oden, J.; Prudhomme, S.; Hawkins-Daarud, A., Goal-oriented error estimation for cahn – hilliard models of binary phase transition, Numer. methods partial diff. eqs., (2010) · Zbl 1428.35398
[26] Asai, S.; Majumdar, S.; Gupta, A.; Kargupta, K.; Ganguly, S., Dynamics and pattern formation in thermally induced phase separation of polymer – solvent system, Comput. mater. sci., 47, 193-205, (2009)
[27] Khiari, N.; Achouri, T.; Ben Mohamed, M.; Omrani, K., Finite difference approximate solutions for the cahn – hilliard equation, Numer. methods partial differ. eqs., 23, 437-455, (2007) · Zbl 1119.65080
[28] D. Eyre, An unconditionally stable one-step scheme for gradient systems, preprint, 1997.
[29] Ceniceros, H.D.; Roma, A.M., A nonstiff, adaptive mesh refinement-based method for the cahn – hilliard equation, J. comput. phys., 225, 1849-1862, (2007) · Zbl 1343.65109
[30] Bijl, H.; Carpenter, M.; Vatsa, V.N.; Kennedy, C., Implicit time integration schemes for the unsteady compressible navier – stokes equations: laminar flow, J. comput. phys., 179, 313-329, (2002) · Zbl 1060.76079
[31] C.A. Kennedy, M.H. Carpenter, Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations, Technical Report TM-2001-211038, NASA, 2001. · Zbl 1013.65103
[32] Soderlind, G.; Wang, L., Adaptive time-stepping and computational stability, J. comput. appl. math., 185, 225-243, (2006), (International Workshop on the Technological Aspects of Mathematics) · Zbl 1077.65086
[33] K. Schloegel, G. Karypis, V. Kumar, Parallel multilevel algorithms for multi-constraint graph partitioning, in: Euro-Par, pp. 296-310.
[34] Schloegel, K.; Karypis, G.; Kumar, V., Parallel static and dynamic multi-constraint graph partitioning, Concurr. comput.: pract. exp., 14, 219-240, (2002) · Zbl 1012.68146
[35] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, 2009. <http://www.mcs.anl.gov/petsc>.
[36] Balay, S.; Gropp, W.D.; McInnes, L.C.; Smith, B.F., Efficient management of parallelism in object oriented numerical software libraries, (), 163-202 · Zbl 0882.65154
[37] Amestoy, P.R.; Duff, I.S.; L’Excellent, J.Y., Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. methods appl. mech. eng., 184, 501-520, (2000) · Zbl 0956.65017
[38] Badalassi, V.E.; Ceniceros, H.D.; Banerjee, S., Computation of multiphase systems with phase field models, J. comput. phys., 190, 371-397, (2003) · Zbl 1076.76517
[39] Choksi, R.; Sternberg, P., Periodic phase separation: the periodic cahn – hilliard and isoperimetric problems, Interfaces free bound., 8, 371-392, (2006) · Zbl 1109.35092
[40] Ros, A., Stable periodic constant Mean curvature surfaces and mesoscopic phase separation, Interfaces free bound., 9, 355-365, (2007) · Zbl 1142.53013
[41] Modica, L., The gradient theory of phase transitions and the minimal interface criterion, Arch. ration. mech. anal., 98, 123-142, (1987) · Zbl 0616.76004
[42] Jeong, J.-H.; Goldenfeld, N.; Dantzig, J.A., Phase field model for three-dimensional dendritic growth with fluid flow, Phys. rev. E, 64, 041602, (2001)
[43] Wang, J.; Yang, G., Phase-field modeling of isothermal dendritic coarsening in ternary alloys, Acta mater., 56, 4585-4592, (2008)
[44] Nauman, E.B.; He, D.Q., Nonlinear diffusion and phase separation, Chem. eng. sci., 56, 1999-2018, (2001)
[45] Rowlinson, J.S., Translation of J.D. van der Waals “the thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”, J. stat. phys., 20, 197-200, (1979) · Zbl 1245.82006
[46] Chella, R.; Vi nals, V., Mixing of a two-phase fluid by a cavity flow, Phys. rev. E, 53, 3832, (1996)
[47] Ceniceros, H.D.; Nós, R.L.; Roma, A.M., Three-dimensional, fully adaptive simulations of phase-field fluid models, J. comput. phys., 229, 6135-6155, (2010) · Zbl 1425.76179
[48] Nauman, E.; He, D.Q., Morphology predictions for ternary polymer blends undergoing spinodal decomposition, Polymer, 35, 2243-2255, (1994)
[49] Thomas, E.L.; Anderson, D.M.; Henkee, C.; Hoffman, D., Periodic area-minimizing surfaces in block copolymers, Nature, 334, 598-601, (1988)
[50] Lenz, P.; Bechinger, C.; Schafle, C.; Leiderer, P.; Lipowsky, R., Perforated wetting layers from periodic patterns of lyophobic surface domains, Langmuir, 17, 7814-7822, (2001)
[51] Brakke, K., The surface evolver, Exp. math., 1, 141-165, (1992) · Zbl 0769.49033
[52] A. Ros, The isoperimetric problem, in: Proceedings from the Clay Mathematics Institude Summer School, MSRI, Berkley, CA, 2001. <www.ugr.es/∼ros>.
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