基于平均值定理和点积分方案的自然单元法及其程序实现

自然单元法是一种基于自然邻接点插值求解偏微分方程的无网格数值方法.它使用Voronoi图或Delaunay三角形作为背景积分网格,使用几何测度构造插值点形函数并形成刚度矩阵.平均值定理定义在未知函数定义域内任何球心(或圆心)的值等于球面(或圆周)上值的平均或加权平均,对于未知函数所满足的平衡方程是充分必要的.因此用平均值定理和点积分方案将求解域内平均应变值由散度定理转化为区域周界上的环路积分,改进传统的积分格式.算例表明,这一积分方案能进一步精简计算量和提高计算效率,是一种自适应的数值计算方法.

Natural element method based on the mean value theorem and point integration and its procedures

The Natural Element Method is a mesh-free method based on the evaluation of Partial Differential Equations by the Natural Neighbor Interpolation. It uses the Voronoi chart or Delaunay triangle as the background integration mesh. The mean value theorem is defined that the value of the center of a globe (or the center of a circle) in the unknown function definition domain is equal to the average or weighted average in spherical surface (or circumference). It is fully necessary for balance equation met by the unknown function. Using the mean value theorem and the point integration, the average strain value in evaluation domain is translated through divergence theorem into perimeter integration in domain circumferences. It improves the traditional integration format. The difference between Natural Element Method and other mesh-free methods is mainly in the following two aspects: one is using geometrical measure to get the natural adjacency shape function; the other is using the dummy point integration to form the stiffness matrix.The count cases show that this integration is an adapted numerical method which can greatly simplify the calculation of the program and improve the efficiency of computation.