A numerical method for the nonlinear Cahn-Hilliard equation with nonperiodic boundary conditions

Chan, Philip K.; Rey, Alejandro D.

doi:10.1016/0927-0256(94)00076-o

Cited by 55 publications

(112 citation statements)

References 21 publications

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“…Moreover, the higher-order continuity that is provided by these approximations allows for a direct discretisation without additional degrees of freedom. Different from earlier approaches based on Hermite elements [24,25] it can straightforwardly be applied to two-and three-dimensional problems.…”

Section: Introductionmentioning

confidence: 99%

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Kästner

Metsch

Borst

2016

Journal of Computational Physics

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Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. AbstractHerein, we present a numerical convergence study of the Cahn-Hilliard phase-field model within an isogeometric finite element analysis framework. Using a manufactured solution, a mixed formulation of the Cahn-Hilliard equation and the direct discretisation of the weak form, which requires a C 1 -continuous approximation, are compared in terms of convergence rates. For approximations that are higher than second-order in space, the direct discretisation is found to be superior. Suboptimal convergence rates occur when splines of order p = 2 are used. This is validated with a priori error estimates for linear problems. The convergence analysis is completed with an investigation of the temporal discretisation. Second-order accuracy is found for the generalised-α method. This ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the Cahn-Hilliard equation. The isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition.

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Section: Introductionmentioning

confidence: 99%

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Kästner

Metsch

Borst

2016

Journal of Computational Physics

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“…Similarly, the set of natural boundary conditions is expressed as [40] @c Ã @x Ã ¼ 0 at t Ã > 0; and x Ã ¼ 0 and…”

Section: Model Formulation and Methods Of Solutionmentioning

confidence: 99%

“…The boundary conditions used in this study are zero mass flux and the natural boundary condition. The set of zero mass flux boundary conditions are [40] …”

Section: Model Formulation and Methods Of Solutionmentioning

confidence: 99%

“…[40] They used the Monte Carlo simulation technique, incorporating the linearization approximation and the equipartition theorem to mimic the existing infinitesimally small concentration fluctuations (also called the homophase thermal fluctuations), that are present in the polymer solution initially. The dimensionless form of the initial condition is expressed as…”

Section: Model Formulation and Methods Of Solutionmentioning

confidence: 99%

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The Ludwig‐Soret Effect on the Thermally Induced Phase Separation Process in Polymer Solutions: A Computational Study

Kukadiya

Chan

Mehrvar

2009

Macro Theory & Simulations

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The Ludwig‐Soret effect was investigated in the thermally induced phase separation process via SD in polymer solutions under an externally imposed spatial linear temperature gradient using mathematical modeling and computer simulation. The mathematical model incorporated non‐linear Cahn‐Hilliard theory for SD, Flory‐Huggins theory for thermodynamics, and the Ludwig‐Soret effect for thermal diffusion. 2D simulation results revealed that the Ludwig‐Soret effect had negligible impact on the phase separation mechanism in binary polymer solutions under a non‐uniform temperature field, as reflected by the time evolution of the dimensionless structure factor and the transition time from the early to the intermediate stages of SD.magnified image

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“…Des travaux précédents ont appliqué la méthode des éléments finis pour l'étude de transformations de phases telles que la résolution de l'équation de Cahn-Hilliard [4], la simulation de la solidification [5] et l'étude des matériaux ferroélectriques [6].…”

Section: Introductionunclassified

Formulation éléments finis des modèles de champ de phases basée sur la théorie de l'équilibre des microforces

Ammar

Appolaire

Cailletaud

et al. 2009

PlastOx 2007 - Mécanismes Et Mécanique Des Interactions Plasticité - Environnement

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Résumé. Nous présentons une formulation éléments finie des modèles de champ de phases pour les alliages binaires, développée dans le cadre général de la thermodynamique. Cette formulation est basée sur la théorie de l'équilibre des microforces, proposée par Gurtin [1]. En utilisant les principes de la thermodynamique des processus irréversibles, les lois de comportement et les équations d'évolution sont clairement exposées et séparées dans la formulation. En outre, des conditions initiales et des conditions aux limites en paramètre d'ordre et en concentration ainsi que leurs quantités duales sont énoncées. L'ensemble des équations d'évolution couplées, à savoir l'équation de champ de phases et l'équation de conservation de la masse, est résolu en utilisant la méthode des éléments finis pour la discrétisation spatiale et un schéma implicite des différences finies pour la discrétisation temporelle.Le modèle est appliqué à l'étude de la croissance des couches d'oxyde dans un système binaire Zr-O à 350C. Des calculs 1D montrent une bonne concordance de la cinétique d'oxydation avec des solutions analytiques. De plus, des calculs bidimensionnels, du même processus, ont été faits pour l'étude de la stabilité morphologique d'une couche d'oxyde ondulée afin d'illustrer les avantages de la méthode des éléments finis avec des frontières complexes et des conditions aux limites variées.

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A numerical method for the nonlinear Cahn-Hilliard equation with nonperiodic boundary conditions

Cited by 55 publications

References 21 publications

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

The Ludwig‐Soret Effect on the Thermally Induced Phase Separation Process in Polymer Solutions: A Computational Study

Formulation éléments finis des modèles de champ de phases basée sur la théorie de l'équilibre des microforces

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