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首页> 外文期刊>Journal of Computational Physics >Modeling and computation of two phase geometric biomembranes using surface finite elements
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Modeling and computation of two phase geometric biomembranes using surface finite elements

机译：使用表面有限元对两相几何生物膜进行建模和计算

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摘要

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 H~1 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.
机译：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域共存。这种生物膜的建模涉及弹性或弯曲能以及与相界面有关的线能。这导致未知平衡表面上相界面的自由边界问题，从而使受体积和面积约束的能量功能最小化。在本文中，我们提出了一种基于L〜2弛豫流计算总能量的新计算工具，其中线能量由表面Ginzburg-Landau相场函数近似。弛豫动力学将膜的Willmore流动类型的非线性四阶几何演化方程与描述横向分解的表面Allen-Cahn方程耦合在一起。推导了涉及二阶椭圆算子的新颖系统，其中场变量是表面的实体点的位置，平均曲率矢量和表面相场函数。所得的变分公式使用H〜1空间，我们采用三角曲面和H〜1符合二次曲面的有限元来近似求解。与演化方程的半隐式时间离散化一起，获得了基本上仅需要线性求解器的迭代方案。数值实验通过计算平衡度，例如哑铃，盘状细胞和具有横向相分离的海星，展示了这种新方法对两种组分几何生物膜的收敛性和功效。

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