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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. 65856612. ISSN 00219991

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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 L2 relaxation flow for the total energy in which the line energy is approximated by a surface GinzburgLandau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface AllenCahn 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 H1 conforming quadratic surface finite elements for approximating solutions. Together with a semiimplicit 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:  00219991 
Official Date:  1 September 2010 
Volume:  Vol.229 
Number:  No.18 
Number of Pages:  28 
Page Range:  pp. 65856612 
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 579/11,2 (DFG), EP/G010404 (EPSRC) 
References:  [1] D. Andelman, T. Kawakatsu, and K. Kawasaki, Equilibrium shape 
URI:  http://wrap.warwick.ac.uk/id/eprint/5509 
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