Mechanics of Generalized Continua pp 325-337 | Cite as
Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method
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Abstract
In this work, we present an application of two versions of the natural element method (NEM) to the Cahn-Hilliard equation. The Cahn-Hilliard equation is a nonlinear fourth order partial differential equation, describing phase separation of binary mixtures. Numerical solutions requires either a two field formulation with C 0 continuous shape functions or a higher order C 1 continuous approximations to solve the fourth order equation directly. Here, the C 1 NEM, based on Farin’s interpolant is used for the direct treatment of the second order derivatives, occurring in the weak form of the partial differential equation. Additionally, the classical C 0 continuous Sibson interpolant is applied to a reformulation of the equation in terms of two coupled second order equations. It is demonstrated that both methods provide similar results, however the C 1 continuous version needs fewer degrees of freedom to capture the contour of the phase boundaries.
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References
- 1.Cahn, JW.: Free energy of a non uniform system ii: thermodynamic basis. J. Chem. Phys. 30, 1121–1124 (1959)CrossRefGoogle Scholar
- 2.Falk, F.: Cahn-Hilliard theory and irreversible thermodynamics. J. Non-Equilib. Thermodyn. 17(1), 53–65 (1992)CrossRefGoogle Scholar
- 3.Dolcetta, IC., Vita, SF., March, R.: Area preserving curve shortening flows: from phase transitions to image processing. Interfaces Free Boundaries 4(4), 325–343 (2002)CrossRefGoogle Scholar
- 4.Kuhl, E., Schmid, DW.: Computational modeling of mineral unmixing and growth- an application of the Cahn-Hilliard equation. Comput. Mech. 39, 439–451 (2007)CrossRefGoogle Scholar
- 5.Khain, E., Sander, LW.: Generalized Cahn-Hilliard equation for biological applications. Phys. Rev. Lett. E 77(5), 1–7 (2008)Google Scholar
- 6.Wu, XF., Dzenis, WA.: Phase-field modeling of the formation of lamellar nanostructures in diblock copolymer thin film under inplanar electric field. Phys. Rev. Lett. E 77(3,4), 1–10 (2008)Google Scholar
- 7.Ostwald, W.: Ăśber die vermeintliche Isometrie des roten und gelben Quecksilberoxyds und die Oberflächenspannung fester Körper. Z. Phys. Chem. 6, 495–503 (1900)Google Scholar
- 8.Rajagopal, A., Fischer, P., Kuhl, E., P. Steinmann, P.: Natural element analysis of the Cahn-Hilliard phase-field model ÂComput. Mech. 46, 471–493 (2010)Google Scholar
- 9.Stogner, RH., Carey, GF., Murray, BT.: Approximation of Cahn-Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with \(C^1\) elements. Int. J. Numer. Methods Eng. 76, 636–661 (2008)Google Scholar
- 10.Gomez, H., Calo, VM., Bazilevs, Y., Hughes, TJR.: Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Meth. Appl. Mech. Eng. 197, 4333–4352 (2008)CrossRefGoogle Scholar
- 11.Wells, G.N., Kuhl, E., Garikipati, K.: A discontinuous galerkin method for Cahn-Hilliard equation. J. Comput. Phys 218, 860–877 (2006)CrossRefGoogle Scholar
- 12.Elliott, CM.: French DA. A non-conforming finite element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26(4), 884–903 (1989)Google Scholar
- 13.Ubachs, RLJM., Schreurs, RJG., Geers, MGD.: A nonlocal diffuse interface model for microstructure evolution in tin-lead solder. J. Mech. Phys. Solids 52(8), 1763–1792 (2004)CrossRefGoogle Scholar
- 14.Farin, G.: Surfaces over Dirichlet tessellations. Comp. Aided Geom. D. 7, 281–292 (1990)CrossRefGoogle Scholar
- 15.Fischer, P., Mergheim0, J., Steinmann, P.: On the \(C^1\) continuous discretization of nonlinear gradient elasticity: a comparison of NEM and FEM based on Bernstein-Bezier patches. Int. J. Numer. Methods Eng. 82(10), 1282–1307 (2010)Google Scholar
- 16.Sukumar, N., Moran, B.: \(C^1\) natural neighbor interpolant for partial differential equations. Numer. Meth. Part. D. E. 15, 417–447 (1999)Google Scholar
- 17.Sibson, R.: A vector identity for the Dirichlet tessellation. Math. Proc. Cambridge 87, 151–153 (1980)CrossRefGoogle Scholar