Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with <i>C</i><sup>1</sup> elements

Stogner, Roy H.; Carey, G. F.; Murray, Bruce T.

doi:10.1002/nme.2337

Cited by 50 publications

(34 citation statements)

References 37 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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“…In such an implementation, the estimates are derived for the fully discrete system for advancing a time step, but they estimate errors only due to the spatial part. Any a posteriori error estimator that is suitable for the spatial operator can be employed, e.g., the Zienkiewicz-Zhu gradient-recovery estimator (Provatas et al, 1999), or an explicit residual-based estimator (Stogner et al, 2008).…”

Section: Adaptive Mesh and Time-step Refinementmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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“…Cahn-Hilliard Equation 567 q [126], [134], [135], [136], [138], [139], [141], [142], [143], [144], [145], [146], [147], [152], [158], [191], [200], [201], [203], [208], [209], [211] and [217] for the numerical analysis and simulations of the Cahn-Hilliard equation (and several of its generalizations).…”

Section: Vol79 (2011)mentioning

confidence: 99%

The Cahn-Hilliard Equation with Logarithmic Potentials

2011

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Our aim in this article is to discuss recent issues related with the Cahn- Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials; in particular, we are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Neumann boundary conditions and then dynamic boundary conditions which account for the interactions with the walls in confined systems and have recently been proposed by physicists. We also present, in the case of dynamic boundary conditions, some numerical results

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Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C¹ elements

Cited by 50 publications

References 37 publications

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

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

Computational Phase‐Field Modeling

The Cahn-Hilliard Equation with Logarithmic Potentials

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Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements

Cited by 50 publications

References 37 publications

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

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

Computational Phase‐Field Modeling

The Cahn-Hilliard Equation with Logarithmic Potentials

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Product

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Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C¹ elements