奇异边界法:一个新的、简单、无网格、边界配点数值方法

物理力学控制方程的基本解有源点奇异性.因而,传统的观点认为奇异基本解一般不能用做控制方程数值解的基函数;除非源点布置在物理域以外的虚假边界上,与物理边界上的配点分离.后者就是近年来受到广泛关注的基本解方法的基本思路.与这些传统方法不同,文中直接使用基本解做为插值基函数,且源点和配点是同一组物理边界上的离散点.本项工作的一个基本假设是源点奇异时的源点强度因子的存在性.利用待求问题控制方程的已知简单解,提出了一个计算源点强度因子的数值方法,并发现源点强度因子确实存在,且是一个有限值,其大小依赖于边界离散点的分布和边界条件类型.由此,文中提出了一个计算微分方程问题的新数值方法,称为奇异边界方法.该方法数学简单,编程容易,是一个真正的无网格方法.初步数值试验显示该方法精度高,收敛速度快.但有关该方法的数学物理基础还不是十分清楚.

SINGULAR BOUNDARY METHOD; A NOVEL,SIMPLE, MESHFREE, BOUNDARY COLLOCATION NUMERICAL METHOD

The fundamental solution of governing equations in physics and mechanics encounters so-called singularity at origin. Not surprisingly, the traditional view is that the fundamental solution can not be used as the basis function in the numerical solution of a partial differential equation, except that the source nodes are placed on the fictitious boundary outside the physical domain and are separated from the collocation nodes on the physical boundary, which underlies the method of fundamental solution, a popular method in recent years. In this paper, a breakthrough on this traditional view is made in that we simply u-ses the fundamental solution as the interpolation basis function while keeping the same source and collocation nodes on the physical boundary. The fundamental assumption of this research is the existence of the origin intensity factor (OIF) upon the singularity of the coincident source-collocation nodes. By employing a simple known solution of the governing equation of interest, we present a numerical approach to evaluate OIF. Our findings are that OIF does exist and its value is of a finite value, depending on the distribution of discrete boundary knots and respective boundary conditions. Based on the above findings, this paper proposes a novel numerical method for partial differential equations, called the singular boundary method. The method is mathematically simple, easy-to-program, and truly meshfree. Our preliminary numerical experiments show that the method is highly accurate and fast convergent. But mathematical physics underlying this method remains an open issue.