无网格广义有限差分法模拟三维位势问题

Generalized Finite Difference Method for Three-dimensional Potential Problems

广义有限差分法是一种新型的无网格数值离散方法.该方法基于多元函数泰勒级数展开和加权最小二乘拟合,将控制方程中未知参量的各阶偏导数表示为相邻节点函数值的线性组合,克服了传统有限元等基于网格的方法对网格的依赖性.本文以三维位势问题为例,引入一种新的优化选点技术,克服了传统广义有限差分法在模拟三维复杂几何域问题时遇到的"病态选点问题",极大地提高了该方法的计算精度与数值稳定性.

The generalized finite difference method（GFDM）is a relatively new meshless method for the numerical solution of certain boundary value problems.The main idea of the GFDM is to combine the Taylor series expansions and the moving-least squares（MLS）approximation to derive explicit formulate for the required partial derivatives of unknown variables.The derivatives of unknown variables at a node,and then,can be approximated by a linear combination of function values with respect to its neighboring nodes.This paper introduced a new distance criterion for adaptive selection of nodes in the GFDM simulations.Preliminary numerical experiments show that the proposed method is very promising for accurate and efficient simulations for problems with complex geometry.