改进的奇异边界法模拟三维位势问题

奇异边界法是与基本解法相对应的一种边界型无网格数值离散方法. 该方法提出了源点强度因子的概念, 克服了传统基本解方法中最复杂最头疼的虚拟边界问题.基于边界元法中处理奇异积分的数值处理技术, 导出了源点强度因子的解析表达式, 提出了改进的无网格奇异边界法, 并进一步将该方法应用于三维位势问题. 该方法消除了传统方法中样本点的选取, 在不增加计算量的前提下, 极大地提高了奇异边界法的计算精度与稳定性. 

IMPROVED SINGULAR BOUNDARY METHOD FOR THREE DIMENSIONAL POTENTIAL PROBLEMS

The singular boundary method(SBM) is a relatively new meshless boundary collocation method for the numerical solution of certain boundary value problems.The key idea is to introduce the concept of the origin intensity factor to isolate the singularity of the fundamental solutions,so that the source points directly coincide with the collocation points on the realistic boundary.This overcomes a perplexing fictitious boundary outside physical domain in the recently popular method of fundamental solutions(MFS).However,the inverse interpolation technique requires the placement of a cluster of sample nodes inside or outside the physical domain for either interior or exterior problems.Our recent numerical experiments indicate that the overall accuracy of this SBM formulation is,to a certain degree,sensitive to the location of such sample nodes.To remedy the above-mentioned drawbacks,this paper proposes an improved SBM formulation for three-dimensional potential problems to circumvent sample nodes in the inverse interpolation technique with the traditional SBM.Numerical experiments demonstrate its significantly improved accuracy and stability in comparison with the traditional SBM formulation.