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Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method

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Part of the Advanced Structured Materials book series (STRUCTMAT, volume 7)

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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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Chair of Applied MechanicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Departments of Mechanical Engineering, Bioengineering and Cardiothoracic SurgeryStanford UniversityStanfordUSA

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