基于网格等分布与对齐的曲线、曲面和区域统一移动网格方法
A unifying moving mesh method for curves, surfaces, and domains based on mesh equidistribution and alignment
AI总结：
提出一种基于等分布与对齐的统一移动网格方法，可处理无需参数表示的曲线、曲面和区域，并证明非奇异性、实现网格点集中控制。
摘要：
本文针对d维空间（d≥1且1≤m≤d）中的一般m维几何对象（包括曲线、曲面和区域），发展了一种统一的移动网格方法。该方法基于网格等分布与对齐，不需要底层几何对象具有可用的解析参数表示。文中推导了m-单纯形形状与尺寸的数学刻画，以及相应边矩阵和仿射映射的性质。针对m-单纯形网格，以统一形式给出了等分布条件和对齐条件。网格移动方程基于移动网格PDE方法定义，并采用对节点网格速度的适当投影，以确保网格点保持在底层几何对象上。网格速度的解析表达式以紧凑的矩阵形式得到。文中还证明了移动网格的非奇异性。给出了二维和三维中曲线（m=1）与曲面（m=2）的数值结果，以展示所发展方法能够在不引起奇异性的情况下移动网格点，并控制其集中程度。
英文摘要：
A unifying moving mesh method is developed for general $m$-dimensional geometric objects in $d$-dimensions ($d \ge 1$ and $1\le m \le d$) including curves, surfaces, and domains. The method is based on mesh equidistribution and alignment and does not require the availability of an analytical parametric representation of the underlying geometric object. Mathematical characterizations of shape and size of $m$-simplexes and properties of corresponding edge matrices and affine mappings are derived. The equidistribution and alignment conditions are presented in a unifying form for $m$-simplicial meshes. The equation for mesh movement is defined based on the moving mesh PDE approach, and suitable projection of the nodal mesh velocities is employed to ensure the mesh points stay on the underlying geometric object. The analytical expression for the mesh velocities is obtained in a compact matrix form. The nonsingularity of moving meshes is proved. Numerical results for curves ($m=1$) and surfaces ($m=2$) in two and three dimensions are presented to demonstrate the ability of the developed method to move mesh points without causing singularity and control their concentration.
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