Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method

Fischer, Paul and Rajagopal, Amirtham and Kuhl, Ellen and et al, . (2011) Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method. ‘Mechanics of Generalized Continua, Advanced Structural Materials, 7. pp. 325-337. ISSN 1869-8433

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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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