Particles and biomembranes : a variational PDE approach
Particles and biomembranes : a variational PDE approach
复制标题
颗粒和生物膜：变分偏微分方程方法
DOI：
--
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
G. Hobbs
中科院分区：
文献类型：
--
作者：
G. Hobbs
我们研究膜的小变形的数学模型。首先，我们从数学的角度回顾了物理上建立良好的模型，提出了在蒙赫规范。我们产生了一个变分框架，在其中适定性可以研究和有限元方法。该方法被用来研究点力，点位移约束和点曲率约束的影响。这样的模型是适合于研究所包含的细胞骨架中的细丝和嵌入的蛋白质夹杂物引起的变形。特别是，我们研究了膜介导的长丝之间的相互作用，也夹杂物之间。
然后，我们介绍了一个新的线性模型，它描述了小变形的封闭表面是最小的Helfrich型能量。变形表面被描述为Helfrich最小化未变形表面上的图形。这是蒙日规范对初始曲面的自然推广。我们专注于一个Willmore能源，从而产生领域和一个家庭的环面作为未变形的表面，也介绍了表面张力的领域。再次，我们研究变形引起的细丝。一个变分制定这是类似的蒙氏规范的情况下，我们制定了一个数值方法来研究膜介导的相互作用。
最后，我们介绍了一个抽象的分裂方法，它允许一个高阶偏微分方程求解的等价系统的低阶方程。我们给出的条件，确保适定性的系统，并产生一个有限元方法，其解决方案收敛到整个系统的解决方案。该理论被应用于显示收敛的数值方法用于表面变形模型。我们提供的例子表明，实现了理论误差估计。
We examine mathematical models for small deformations of membranes. First we review physically well-established models, posed in the Monge gauge, from a mathematical perspective. We produce a variational framework in which well posedness can be studied and finite element methods applied. The methods are used to investigate the effects of point forces, point displacement constraints and point curvature constraints. Such models are suitable for the study of deformations induced by filaments contained in the cell cytoskeleton and by embedded protein inclusions. In particular we study the membrane mediated interactions between filaments and also between inclusions.
We then introduce a new linearised model which describes small deformations of closed surfaces that are minimisers of Helfrich-type energies. The deformed surface is described as a graph over the Helfrich minimising undeformed surface. This is the natural generalisation of the Monge gauge to initially curved surfaces. We focus on a Willmore energy which gives rise to spheres and a family of tori as undeformed surfaces and also introduce surface tension on a sphere. Again we study deformations induced by filaments. A variational formulation is produced which is similar to the Monge gauge case and we formulate a numerical method to study membrane mediated interactions.
Finally we introduce an abstract splitting method which allows a high order PDE to be solved by an equivalent system of lower order equations. We give conditions which ensure well posedness of the system and produce a finite element method whose solution converges to the solution of the full system. The theory is applied to show convergence for the numerical methods used for the surface deformations model. We provide examples which show the theoretical error estimates are achieved.