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International Journal for Multiscale Computational Engineering
Factor de Impacto: 1.016 Factor de Impacto de 5 años: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Imprimir: 1543-1649
ISSN En Línea: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v7.i6.40
pages 523-543

Numerical Solutions of Some Diffuse Interface Problems: The Cahn-Hilliard Equation and the Model of Thomas and Windle

F. J. Vermolen
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
M. Gholami Gharasoo
Helmholz Zentrum fur Umweltforschung, Permoserstr. 15, 04318, Leipzig, Germany
Pacelli L.J. Zitha
Helmholz Zentrum fur Umweltforschung, Permoserstr. 15, 04318, Leipzig, Germany; Delft University of Technology, Department of Geotechnology, 2628 CN Delft, The Netherlands
J. Bruining
Department of Geotechnology, Delft University of Technolgy, Stevinweg 1, 2628 CN Delft, The Netherlands

SINOPSIS

We consider partial differential equations with a suddenly changing parameter. The equations that we study are the Cahn-Hilliard equation, for binary and multicomponent mixtures (i.e., vector Cahn-Hilliard equations), and a stress-enhanced diffusion equation. Numerical strategies to solve these equations are analyzed in terms of discretization and time integration. Results are presented and form the basis for further research. Next to the numerical analysis, we consider some analytic properties such as mass conservation and decrease of energy.

REFERENCIAS

  1. Thornton, K., Ågren, J., and Voorhees, P. W., Modelling the Evolution of Phase Boundaries in Solids at the Meso- and Nano-Scales. DOI: 10.1016/j.actamat.2003.08.008

  2. Ritger, P. L., and Peppas, N. A., Transport of Penetrants in the Macromolecular Structure of Coals. DOI: 10.1016/0016-2361(87)90130-X

  3. Hui, C. Y., Wu, K. C., Lasky, R. C., and Kramer, E. J., Case-II Diffusion in Polymers. I Transient Swelling. DOI: 10.1063/1.338287

  4. Hui, C. Y., Wu, K. C., Lasky, R. C., and Kramer, E. J., Case-II Diffusion in Polymers. II Steady-State Front Motion. DOI: 10.1063/1.338288

  5. Murray, W. D., and Landis, F., Numerical and Machine Solutions of Transient Heat Conduction Problems Involving Freezing and Melting.

  6. Crusius, S., Inden, G., Knoop, U., Höglund, L., and Å gren, J., On the Numerical Treatment of Moving Boundary Problems.

  7. Segal, A., Vuik, C., and Vermolen, F. J., A Conserving Discretization for the Free Boundary in a Two-Dimensional Stefan Problem. DOI: 10.1006/jcph.1998.5900

  8. Osher, S., and Sethian, J. A., Fronts Propagating with Curvature-Dependent Speed. DOI: 10.1016/0021-9991(88)90002-2

  9. Chen, S., Merriman, B., Osher, S., and Smereka, P., A Simple Level-Set Method for Solving Stefan Problems. DOI: 10.1006/jcph.1997.5721

  10. Almgren, R., Variational Algorithms and Pattern Formation in Dendritic Solidification. DOI: 10.1016/S0021-9991(83)71112-5

  11. Schmidt, A., Approximation of Crystalline Dendritic Growth in Two Space Dimensions.

  12. Javierre, E., Vuik, C., Vermolen, F. J., and van der Zwaag, S., A Comparison of Numerical Models for One-Dimensional Stefan Problems. DOI: 10.1016/j.cam.2005.04.062

  13. Van der Waals, J. D., The Thermodynamic Theory of Capillarity under the Hypothesis of a Continuous Variation of Density. DOI: 10.1007/BF01011514

  14. Cahn, J. W., and Hilliard, J. E., Free Energy of a Non-Uniform System. I. Interfacial Energy. DOI: 10.1063/1.1744102

  15. Metallurgical and Thermochemical Databank, National Physical Laboratory.

  16. Wise, S. M., Lowengrub, J. S., Friboes, H. B., and Cristini, V., Three-Dimensional Multispecies Nonlinear Tumor Growth – 1: Model and Numerical Method. DOI: 10.1016/j.jtbi.2008.03.027

  17. Macklin, P., and Lowengrub, J., Nonlinear Simulation of the Effect of Microenvironment on Tumor Growth. DOI: 10.1016/j.jtbi.2006.12.004

  18. Elliott, C. M., and Garcke, H., On the Cahn-Hilliard Equation with Degenerate Mobility. DOI: 10.1137/S0036141094267662

  19. Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics.

  20. Barrett, J. W., Blowey, J. F., and Garcke, H., Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility. DOI: 10.1137/S0036142997331669

  21. Chan, P. K., and Rey, A. D., A Numerical Method for the Nonlinear Cahn-Hilliard Equation with Nonperiodic Boundary Conditions. DOI: 10.1016/0927-0256(94)00076-O

  22. Wells, G. N., Kuhl, E., and Garakipati, K., A Discontinuous Galerkin Method for the Cahn- Hilliard Equation. DOI: 10.1016/j.jcp.2006.03.010

  23. Ceniceros, H. D., and Roma, A. M., A Nonstiff, Adaptive Mesh Refinement-Based Method for the Cahn-Hilliard Equation. DOI: 10.1016/j.jcp.2007.02.019

  24. Garcke, H., Rumpf, M., and Weikard, U., The Cahn-Hilliard Equation with Elasticity: Finite Element Approximation and Qualitative Studies.

  25. Garcke, H., Maier-Paape, S., and Weikard, U., Spinodal Decomposition in the Presence of Elastic Interactions. DOI: 10.1007/978-3-642-55627-2_32

  26. Cueto-Felgueroso, L., and Peraire, J., A Time- Adaptive Finite Volume Method for the Cahn-Hilliard and Kuramoto-Sivashinsky Equations. DOI: 10.1016/j.jcp.2008.07.024

  27. Elliott, C. M., and Garcke, H., Diffusional Phase Transitions in Multicomponent Systems with a Concentration Dependent Mobility Matrix. DOI: 10.1016/S0167-2789(97)00066-3

  28. Eyre, D. J., Systems of Cahn-Hilliard Equations. DOI: 10.1137/0153078

  29. Maier-Paape, S., Stoth, B., and Wanner, T., Spinodal Decomposition for Multicomponent Cahn-Hilliard Systems. DOI: 10.1023/A:1018687811688

  30. Mikelic, A., and Bruining, J., Analysis of Model Equations for Stress-Enhanced Diffusion in Coal Layers. Part I: Existence of a Weak Solution.

  31. Pego, R. L., Front Migration in the Nonlinear Cahn-Hilliard Equation. DOI: 10.1098/rspa.1989.0027

  32. Mauri, R., Shinnar, R., and Triantafyllou, G., Spinodal Decomposition in Binary Mixtures. DOI: 10.1103/PhysRevE.53.2613

  33. Lowengrub, J., and Truskinovski, L., Quasi- Incompressible Cahn-Hilliard Fluids and Topological Transitions. DOI: 10.1098/rspa.1998.0273

  34. Verschueren, M., A Diffuse-Interface Model for Structure Development in Flow.

  35. Ubachs, R. L. J. M., Schreurs, P. J. G., and Geers, M. G. D., A Nonlocal Diffuse Interface Model for Microstructural Evolution of Tin- Lead Solder. DOI: 10.1016/j.jmps.2004.02.002

  36. Kim, J., A Diffuse-Interface Model for Axi-Symmetric Immiscible Two-Phase Flow. DOI: 10.1016/j.amc.2003.11.020

  37. Andersson, D. M., McFadden, G. B., and Wheeler, A. A., Diffuse-Interface Methods in Fluid Mechanics. DOI: 10.1146/annurev.fluid.30.1.139

  38. Mackenzie, J. A., and Robertson, M. L., A Moving Mesh Method for the Solution of the One-Dimensional Phase-Field Equations. DOI: 10.1006/jcph.2002.7140

  39. Burman, E., Picasso, M., and Rappaz, J., Analysis and Computation of Dendritic Growth in Binary Alloys Using a Phase-Field Model. DOI: 10.1007/978-3-642-18775-9_18

  40. Cheng, X. L., Han,W., and Huang, H. C., Some Mixed Finite Element Methods for Biharmonic Equation. DOI: 10.1016/S0377-0427(99)00342-8

  41. Gholami-Ghorashoo, M., Finite Element Analysis of Cahn-Hilliard Equations.

  42. de Mello, E. V. L., and Teixeira da Silveira Filho, O., Numerical Study of the Cahn-Hilliard Equation in One, Two and Three Dimensions. DOI: 10.1016/j.physa.2004.08.076

  43. Eyre, D. J., An Unconditionally Stable One Step Scheme for Gradient Systems.

  44. He, Y., Liu, Y., and Tang, T., On Large Time-Stepping Methods for the Cahn-Hilliard Equation. DOI: 10.1016/j.apnum.2006.07.026

  45. Elliott, C. M., French, D. A., and Milner, F. A., A Second Order Splitting Method for the Cahn-Hilliard Equation. DOI: 10.1007/BF01396363

  46. Stoth, B. E. E., Convergence of the Cahn-Hilliard Equation to the Mullins-Sekerka Problem in Spherical Geometry.


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