本文基于Rods棍棒模型, 采用自洽平均场方法求解偏心圆环受限区域内液晶分子的相结构, 并进行数值模拟. 本文的主要工作有以下内容:
1.MDE方程的推导. 在偏心圆环受限区域内, 为使问题求解得到简化, 我们建立双极坐标系, 并推导了双极坐标系下的MDE方程. 双极坐标系能较好地适应偏心圆环受限区域, 有利于边界处分子排布的处理. MDE方程的推导是本文的重难点之一.
2.数值格式的稳定性分析. 对于MDE方程中的双曲部分, 我们采用迎风格式进行求解, 并证明在一定条件下格式是稳定的, 同时从方程的数值解我们也能验证该格式具有一阶时间精度. 保证数值格式的稳定是得到稳定解的重要前提.
3.相结构与相图. 通过数值计算我们得到了偏心圆环受限区域内的6种相结构, 并探究了四个参数（R/L、R/r、R/d和L²ρ₀）对相结构的影响. 近一步我们绘制了有关上述参数的相图, 并讨论各个亚稳态相出现的概率.
之前已有的工作, 如矩形、圆形与同心圆环受限区域内液晶分子的相结构的数值模拟, 对本文有较大的借鉴意义. 本文采用类似的研究思路和新的研究方法, 来求解偏心圆环受限区域内的相, 并通过相图对结果进行分析, 以寻求为理化实验提供借鉴作用.

In this paper, we establish rods model to describe the liquid crystal molecule. Then we use the self consistent mean field method to obtain the phase structure of the liquid crystal molecule confined by an eccentric ring , followed by an numerical simulation. The study can be summarized as the following three points:
1. Calculation of MDE. In the eccentric ring confinement, we establish bipolar coordinates to infer MDE in order to simplify the problem. The bipolar coordinates suit the eccentric ring confinement properly, which is useful to the permutation of rods near the boundary. The calculation of MDE is one of the emphases and difficulties of this paper.
2. Stability analysis of the numerical scheme. To solve the hyperbolic part of MDE we apply upwind scheme , the stability of which is proved under some condition. Meanwhile, we conclude that the scheme has temporal first-order accuracy according to the numerical solutions of the equation. The stability of the numerical scheme is the precondition to get reasonable solutions.
3. Phase structures and phase diagrams. We discover 6 structures through the numerical simulation, where 4 groups of parameters have great effect on them. In addition, we plot the phase diagrams of the parameters and discuss the probability of metastable phases.
The former works, such as the phase of the liquid crystal molecule confined by squared, circular and ring box, are valuable references to this paper. We apply same study frame but different study approaches to attain the structures in eccentric ring confinement, the phase diagrams of which are of significance for physical and chemical experiments.