An adaptive meshfree method for phase-field models of biomembranes. Part II: a Lagrangian approach for membranes in viscous fluids

Author (s): C. Peco, A. Rosolen and M. Arroyo
Journal: Journal of Computational Physics

Volume: 249
Pages: 320 – 336
Date: 2013

Abstract:
We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phase-field tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms base on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phase-field as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D.

Keywords: phase field models, biomembranes, vesicles, meshfree methods, variational methods, adaptivity

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Bibtex:

@article{2013-JCP-PRA,
author = "C. Peco and A. Rosolen and M. Arroyo",
title = "An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids ",
journal = "Journal of Computational Physics ",
volume = "249",
number = "",
pages = "320 - 336",
year = "2013",
note = "",
issn = "0021-9991",
doi = "http://dx.doi.org/10.1016/j.jcp.2013.04.038",
url = "http://www.sciencedirect.com/science/article/pii/S0021999113003239",
}