Modeling and computation of two phase geometric biomembranes using surface finite elements
Modeling and computation of two phase geometric biomembranes using surface finite elements
复制标题
DOI：
10.1016/j.jcp.2010.05.014
复制
发表时间：
2010-09-01
影响因子：
4.1
通讯作者：
Stinner, Bjoern
中科院分区：
文献类型：
--
作者：
Elliott, Charles M.;Stinner, Bjoern
由多种脂质组成的生物膜可能涉及相分离现象，导致不同脂质组成的结构域共存。这种生物膜的建模涉及弹性或弯曲能量以及与相界面相关的线能量。这导致了未知平衡表面上相界面的自由边界问题，该问题使受体积和面积约束的能量泛函最小化。本文提出了一种基于L(2)松弛流计算总能量平衡的新计算工具，其中线能量由表面金兹堡-朗道相场泛函近似。弛豫动力学将膜的非线性四阶几何演化方程Willmore流型与描述横向分解的表面Allen-Cahn方程耦合在一起。导出了一种涉及二阶椭圆算子的新系统，其中场变量为曲面质点的位置、平均曲率矢量和曲面相场函数。所得的变分公式使用HI空间，我们使用三角曲面和H(1)一致性二次曲面有限元来逼近解。结合演化方程的半隐式时间离散，得到了基本上只需要线性解的迭代格式。通过计算具有横向相分离的哑铃、盘状细胞和海星等平衡态的数值实验，证明了该方法的收敛性和有效性。(C) 2010爱思唯尔公司版权所有。
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. (C) 2010 Elsevier Inc. All rights reserved.