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. doi:10.1016/j.jcp.2010.05.014

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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:	28
Page Range:	pp. 6585-6612
DOI:	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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