Shape minimisation problems in liquid crystals

We consider a class of liquid crystal free-boundary problems for which both the equilibrium shape and internal configuration of a system must simultaneously be determined, for example in films with air–liquid or fluid–liquid crystal interfaces and elastomers. We develop a finite element algorithm to solve such problems with dynamic mesh control, achieved by supplementing the free energy with an auxiliary functional that promotes mesh quality and is minimised in the null space of the energy. We apply this algorithm to a flexible capacitor, as well as to determine the shape of liquid crystal tactoids as a function of the surface tension and elastic constants. These are compared with theoretical predictions and experimental observations of tactoids from the literature.我们研究一类液晶自由边界问题（liquid crystal free-boundary problems），需同时确定系统的平衡形貌（equilibrium shape）与内部构型（internal configuration），例如包含气-液界面、液-液晶界面及弹性体（elastomers）的薄膜体系。我们开发了一种集成动态网格控制的有限元算法（finite element algorithm）以求解此类问题：通过向自由能（free energy）中引入一项可提升网格质量的辅助泛函（auxiliary functional），并在原能量的零空间（null space）内对该泛函进行极小化，以此实现动态网格控制。我们将该算法应用于柔性电容器（flexible capacitor）场景，同时用于求解液晶 tactoids（liquid crystal tactoids）的形貌随表面张力（surface tension）与弹性常数（elastic constants）的变化关系。将上述计算结果与文献中关于液晶 tactoids的理论预测及实验观测结果进行对比。