Approximation of Cahn-Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 5, 2008, Pages 636-661

  Stogner, Roy H ^a   Carey, Graham F ^a   Murray, Bruce T ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b State University of New York at Binghamton   (United States)

Author keywords

Adaptive; Cahn Hilliard; Diffuse interface; Finite elements; Multiscale

Indexed keywords

FINITE ELEMENT METHOD; PHASE SEPARATION; SPINODAL DECOMPOSITION; THREE DIMENSIONAL; THREE DIMENSIONAL COMPUTER GRAPHICS;

ADAPTIVE; CAHN-HILLIARD; DIFFUSE INTERFACE; FINITE ELEMENTS; MULTISCALE;

PHASE INTERFACES;

EID: 60949111199     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2337     Document Type: Article

References (37)

1. Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics 1958; 28(2):258-267.
2. Thorton K, Agren J, Voorhees PW. Modelling the evolution of phase boundaries in solids at the meso- and nano-scales. Acta Materialia 2003; 51:5675-5710.
3. Ubachs RLJM, Schreurs PJG, Geers MGD. A nonlocal diffuse interface model for microstructure evolution of tin-lead solder. Journal of the Mechanics and Physics of Solids 2004; 52:1763-1792.
4. Yu HC, Lu W. Dynamics of the self-assembly of nanovoids and nanobubbles in solids. Acta Materialia 2005; 53:1799-1807.
5. Eyre DJ. Systems of Cahn-Hilliard equations. SIAM Journal on Applied Mathematics 1993; 53(6):1686-1712.
6. Kessler D, Scheid JF, Schimperna G, Stefanelli U. Study of a system for the isothermal separation of components in a binary alloy with change of phase. IMA Journal of Applied Mathematics 2004; 69:233-257.
7. Lowengrub J, Truskinovsky L. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proceedings of the Royal Society of London, Series A 1998; 454:2617-2654.
8. Anderson DM, McFadden GB, Wheeler AA. Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics 1998; 30:139-165.
9. Zhao P, Heinrich JC, Poirier DR. Fixed mesh front-tracking methodology for finite element simulations. International Journal for Numerical Methods in Engineering 2004; 61:928-948.
10. Elliot CM, French DA. Numerical studies of the Cahn-Hilliard equation for phase separation. IMA Journal of Applied Mathematics 1987; 38:97-128.
11. Wells GN, Kuhl E, Garikipati K. A discontinuous Galerkin method for the Cahn-Hilliard equation. Journal of Computational Physics 2006; 218:860-877.
12. Barrett JW, Garcke H, Nurnberg R. Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid. Mathematics of Computation 2006; 75:7-41.
13. Kirk BS, Peterson JW, Stogner RH, Carey GF. libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Engineering with Computers 2006; 22(3):237-254.
14. Cahn JW. On spinodal decomposition. Acta Materialia 1961; 9:795-801.
15. Huggins ML. Solutions of long chain compounds. Journal of Chemical Physics 1941; 9(5):440.
16. Flory PJ. Thermodynamics of high polymer solutions. Journal of Chemical Physics 1941; 9(8):660.
17. Carey GF, Oden JT. Finite Elements. Prentice-Hall: Englewood Cliffs, NJ, 1981-1986.
18. Ralston A. A First Course in Numerical Analysis. McGraw-Hill: New York, 1965.
19. Valli AMP, Carey GF, Coutinho ALGA. Control strategies for timestep selection in simulation of coupled viscous flow and heat transfer. Communications in Numerical Methods in Engineering 2002; 18(2):131-139.
20. Kay D, Welford R. A multigrid finite element solver for the Cahn-Hilliard equation. Journal of Computational Physics 2006; 212:288-304.
21. Elliott CM, Larsson S. Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation. Mathematics of Computation 1992; 58(198):603-630.
22. Petera J, Pittman JFT. Isoparametric Hermite elements. International Journal for Numerical Methods in Engineering 1994; 37(20):3489-3519.
23. Jia RQ. Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh. Transactions of the American Mathematical Society 1986; 295(1):199-212.
24. Ženíšek A. Polynomial approximation on tetrahedrons in the finite element method. Journal of Approximation Theory 1973; 7:334-351.
25. Powell MJD, Sabin MA. Piecewise quadratic approximations on triangles. ACM Transactions on Mathematical Software 1977; 3:316-325.
26. Clough R, Tocher J. Finite element stiffness matrices for analysis of plates in blending. Proceedings of Conference on Matrix Methods in Structural Analysis, Wright-Patterson Air Force Base, Ohio, AFFDL-TR-66-80, 26-28 October 1965.
27. 1 macroelements in adaptive finite element methods. International Journal for Numerical Methods in Engineering 2007; 70(9):1076-1095.
28. Carey GF. A mesh refinement scheme for finite element computations. Computer Methods in Applied Mechanics and Engineering 1976; 7(1):73-105.
29. Carey GF. Computational Grids. Taylor & Francis: New York, 1997.
30. Ainsworth M, Oden JT. A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience: New York, 2000.
31. Estep DJ, Larson MG, Williams RD. Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations. vol. 146. Memoirs of the American Mathematical Society: Providence, RI, July 2000; 1-109.
32. Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. International Journal for Numerical Methods in Engineering 1992; 33:1331-1364.
33. Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. International Journal for Numerical Methods in Engineering 1992; 33:1365-1382.
34. Carey V. A posteriori error estimation for the finite element method via local averaging. Ph.D. Dissertation, Cornell University, 2005.
35. Castellano C, Glotzer SC. On the mechanism of pinning in phase-separating polymer blends. Journal of Chemical Physics 1995; 103(21):9363-9369.
36. Cahn JW, Elliott CM, Novick-Cohen A. The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. European Journal of Applied Mathematics 1996; 7:287-301.
37. Pego RL. Front migration in the nonlinear Cahn-Hilliard equation. Proceedings of the Royal Society of London, Series A 1989; 422:261-278.