Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method
- 1.4k Downloads
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.
Unable to display preview. Download preview PDF.
- 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
- 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
- 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