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Published in:
2011 | OriginalPaper | Chapter
Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method
Authors : Paul Fischer, Amirtham Rajagopal, Ellen Kuhl, Paul Steinmann
Published in: Mechanics of Generalized Continua
Publisher: Springer Berlin Heidelberg
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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
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continuous shape functions or a higher order
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continuous approximations to solve the fourth order equation directly. Here, the
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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
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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
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continuous version needs fewer degrees of freedom to capture the contour of the phase boundaries.