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An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions

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10.1016/j.jcp.2013.04.046
 
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hdl:2117/20105

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Peco Regales, Christian
Arroyo Balaguer, MarinoMés informacióMés informacióMés informació
Rosolen, Adrian Martin
Document typeArticle
Defense date2013-09
Rights accessRestricted access - publisher's policy
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
ProjectPREDMODSIM - Predictive models and simulations in nano- and biomolecular mechanics: a multiscale approach (EC-FP7-240487)
Abstract
We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the enclosed volume of a vesicle are written as functionals of a continuous phase-field, which describes the interface in a smeared manner. Such functionals involve up to second-order spatial derivatives of the phase-field, leading to fourth-order Euler–Lagrange partial differential equations (PDE). The solutions develop sharp internal layers in the vicinity of the putative interface, and are nearly constant elsewhere. Thanks to the smoothness of the local maximum-entropy (max-ent) meshfree basis functions, we approximate numerically this high-order phase-field model with a direct Ritz–Galerkin method. The flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions. Thus, the proposed approach is more efficient than common tensor product methods (e.g. finite differences or spectral methods), and simpler than unstructured Cº finite element methods, applicable by reformulating the model as a system of second-order PDE. The proposed method, implemented here under the assumption of axisymmetry, allows us to show numerical evidence of convergence of the phase-field solutions to the sharp interface limit as the regularization parameter approaches zero. In a companion paper, we present a Lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid.
CitationPeco, C.; Arroyo, M.; Rosolen, A. An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions. "Journal of computational physics", Setembre 2013, vol. 249, núm. 15, p. 303-319. 
URIhttp://hdl.handle.net/2117/20105
DOI10.1016/j.jcp.2013.04.046
ISSN0021-9991
Publisher versionhttp://www.sciencedirect.com/science/article/pii/S0021999113003483
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