Computationally efficient solution to the Cahn-Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

Journal of Computational Physics

Volumn 230, Issue 15, 2011, Pages 6037-6060

  Wodo, Olga ^a   Ganapathysubramanian, Baskar ^a  

^a Iowa State University of Science and Technology   (United States)

Author keywords

Adaptive time stepping; Cahn Hilliard equation; Isoperimetric problem; Numerical analysis; Spinodal decomposition

Indexed keywords

DOMAIN DECOMPOSITION METHODS; RUNGE KUTTA METHODS; SENSITIVITY ANALYSIS; SPINODAL DECOMPOSITION;

ADAPTIVE TIME STEPPING; ANALYSIS OF VARIOUS; CAHN-HILLIARD EQUATION; COMPUTATIONALLY EFFICIENT; ISOPERIMETRIC PROBLEM; MESH-SENSITIVITY; SUB PROCESS; TEST PROBLEM; THREE-DIMENSIONAL PROBLEMS; TWO-DIMENSIONAL PROBLEM;

FINITE ELEMENT METHOD;

EID: 79956091698     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.04.012     Document Type: Article

References (52)

1. Cahn J.W., Hilliard J.E. Free energy of a nonuniform system. I: Interfacial energy. J. Chem. Phys. 1958, 28:258.
2. Cahn J.W., Hilliard J.E. Free energy of a nonuniform system. II: Thermodynamic basis. J. Chem. Phys. 1959, 30:1121-1135.
3. Saxena R., Caneba G.T. Studies of spinodal decomposition in a ternary polymer-solvent-nonsolvent systems. Polym. Eng. Sci. 2002, 42:1019-1031.
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. 2008, 253:524-543.
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. 2006, 268:150-164.
7. Tremaine S. On the origin of irregular structure in Saturn's rings. Astron. J. 2003, 125:894.
8. Bertozzi A., Esedoglu S., Gillette A. Inpainting of binary images using the Cahn-Hilliard equation. IEEE Trans. Image Process. 2007, 16:285-291.
9. Moelans N., Blanpain B., Wollants P. An introduction to phase-field modeling of microstructure evolution. Comput. Coupling Phase Diagrams Thermochem. 2008, 32:268-294.
10. Langer J.S., Bar-on M., Miller H.D. New computational method in the theory of spinodal decomposition. Phys. Rev. A 1975, 11:1417-1429.
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. 2007, 3:851-864.
12. Cueto-Felgueroso L., Peraire J. A time-adaptive finite volume method for the Cahn-Hilliard and Kuramoto-Sivashinsky equations. J. Comput. Phys. 2008, 227:9985-10017.
13. Kim J., Kang K. A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility. Appl. Numer. Math. 2009, 59:1029-1042.
14. Elliot C., French D., Milner F. A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 1989, 54:575-590.
15. Elliott C.M., Garcke H. On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 1996, 27:404-423.
16. Elliott C.M., French D.A. A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 1989, 26:884-903.
17. Zhang S., Wang M. A nonconforming finite element method for the Cahn-Hilliard equation. J. Comput. Phys. 2010, 229:7361-7372.
18. He Y., Liu Y., Tang T. On large time-stepping methods for the Cahn-Hilliard equation. Appl. Numer. Math. 2007, 57:616-628. (special issue for the International Conference on Scientific Computing).
19. Gomez H., Calo V., Bazilevs Y., Hughes T. Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 2008, 197:4333-4352.
20. Deibel C., Dyakonov V. Polymer-fullerene bulk heterojunction solar cells. Rep. Progr. Phys. 2010, 73:096401.
21. Chen L.-Q. Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 2002, 32:113-140.
22. Rajagopal A., Fischer P., Kuhl E., Steinmann P. Natural element analysis of the Cahn-Hilliard phase-field model. Comput. Mech. 2010, 46:471-493.
23. Wells G.N., Kuhl E., Garikipati K. A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys. 2006, 218:860-877.
24. Banas L., Nurnberg R. A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy. ESAIM 2009, 43:1003-1026.
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.
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. 2009, 47:193-205.
27. Khiari N., Achouri T., Ben Mohamed M., Omrani K. Finite difference approximate solutions for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Eqs. 2007, 23:437-455.
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. 2007, 225:1849-1862.
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. 2002, 179:313-329.
31. C.A. Kennedy, M.H. Carpenter, Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations, Technical Report TM-2001-211038, NASA, 2001.
32. Soderlind G., Wang L. Adaptive time-stepping and computational stability. J. Comput. Appl. Math. 2006, 185:225-243. (International Workshop on the Technological Aspects of Mathematics).
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. 2002, 14:219-240.
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. Modern Software Tools in Scientific Computing 1997, 163-202. Birkhäuser Press. E. Arge, A.M. Bruaset, H.P. Langtangen (Eds.).
37. Amestoy P.R., Duff I.S., L'Excellent J.Y. Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 2000, 184:501-520.
38. Badalassi V.E., Ceniceros H.D., Banerjee S. Computation of multiphase systems with phase field models. J. Comput. Phys. 2003, 190:371-397.
39. Choksi R., Sternberg P. Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems. Interfaces Free Bound. 2006, 8:371-392.
40. Ros A. Stable periodic constant mean curvature surfaces and mesoscopic phase separation. Interfaces Free Bound. 2007, 9:355-365.
41. Modica L. The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 1987, 98:123-142.
42. Jeong J.-H., Goldenfeld N., Dantzig J.A. Phase field model for three-dimensional dendritic growth with fluid flow. Phys. Rev. E 2001, 64:041602.
43. Wang J., Yang G. Phase-field modeling of isothermal dendritic coarsening in ternary alloys. Acta Mater. 2008, 56:4585-4592.
44. Nauman E.B., He D.Q. Nonlinear diffusion and phase separation. Chem. Eng. Sci. 2001, 56:1999-2018.
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. 1979, 20:197-200.
46. Chella R., Vi nals V. Mixing of a two-phase fluid by a cavity flow. Phys. Rev. E 1996, 53:3832.
47. Ceniceros H.D., Nós R.L., Roma A.M. Three-dimensional, fully adaptive simulations of phase-field fluid models. J. Comput. Phys. 2010, 229:6135-6155.
48. Nauman E., He D.Q. Morphology predictions for ternary polymer blends undergoing spinodal decomposition. Polymer 1994, 35:2243-2255.
49. Thomas E.L., Anderson D.M., Henkee C., Hoffman D. Periodic area-minimizing surfaces in block copolymers. Nature 1988, 334:598-601.
50. Lenz P., Bechinger C., Schafle C., Leiderer P., Lipowsky R. Perforated wetting layers from periodic patterns of lyophobic surface domains. Langmuir 2001, 17:7814-7822.
51. Brakke K. The surface evolver. Exp. Math. 1992, 1:141-165.
52. A. Ros, The isoperimetric problem, in: Proceedings from the Clay Mathematics Institude Summer School, MSRI, Berkley, CA, 2001. http://www.ugr.es/~ros.