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Modeling and computation of two phase geometric biomembranes using surface finite elements

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Elliott, Charles M. and Stinner, Björn. (2010) Modeling and computation of two phase geometric biomembranes using surface finite elements. Journal of Computational Physics, Vol.229 (No.18). pp. 6585-6612. ISSN 0021-9991

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Official URL: http://dx.doi.org/10.1016/j.jcp.2010.05.014

Abstract

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L-2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen-Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses HI spaces, and we employ triangulated surfaces and H-1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics
Q Science > QP Physiology

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Bilayer lipid membranes -- Mathematical models, Finite element method

Journal or Publication Title:

Journal of Computational Physics

Publisher:

Academic Press Inc. Elsevier Science

ISSN:

0021-9991

Official Date:

1 September 2010

Dates:

Date	Event
1 September 2010	Published

Volume:

Vol.229

Number:

No.18

Number of Pages:

Page Range:

pp. 6585-6612

Identification Number:

10.1016/j.jcp.2010.05.014

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Restricted or Subscription Access

Funder:

Deutsche Forschungsgemeinschaft (DFG), Engineering and Physical Sciences Research Council (EPSRC)

Grant number:

STI 579/1-1,2 (DFG), EP/G010404 (EPSRC)

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