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Mechanics of Generalized Continua pp 325–337Cite as

Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method

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Part of the book series: Advanced Structured Materials ((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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Authors and Affiliations

  1. Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstrasse 5, 91058, Erlangen, Germany

    Paul Fischer, Amirtham Rajagopal & Paul Steinmann

  2. Departments of Mechanical Engineering, Bioengineering and Cardiothoracic Surgery, Stanford University, Stanford, CA, 94305, USA

    Ellen Kuhl

Authors
  1. Paul Fischer
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  2. Amirtham Rajagopal
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  3. Ellen Kuhl
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  4. Paul Steinmann
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Corresponding author

Correspondence to Paul Fischer .

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Editors and Affiliations

  1. Halle-Wittenberg, Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität, Kurt-Mothe-Str. 1, Halle (Saale), 06099, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. , Institut Jean Le Rond d’Alembert, UPMC (Paris), Tour 55, Place Jussieu 4, Paris Cedex 05, 75252, France

    Gérard A. Maugin

  3. IMASH RAN, Belinskogo str. 85, Nizhny Novgorod, 603024, Russian Federation

    Vladimir Erofeev

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Fischer, P., Rajagopal, A., Kuhl, E., Steinmann, P. (2011). Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method. In: Altenbach, H., Maugin, G., Erofeev, V. (eds) Mechanics of Generalized Continua. Advanced Structured Materials, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19219-7_16

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