带强奇异边界积分方程的迦辽金边界元解法

采用双层位势来表示二维Laplace方程Neumann问题的解,导致求解含超强奇异性的边界积分方程,将其转换为边界上的Galerkin变分方程求解.针对超强奇异积分的计算,运用分步积分,详细地推导了基于边界旋度的变分公式及边界旋度的表达式,最终把超强奇异的积分计算转化为弱奇异积分的数值计算.当采用线性边界单元来离散Galerkin变分公式时,在每个离散的单元上边界旋度成为常向量,因此,数值积分变得很简单.数值算例验证了方法的有效性和实用性.

Galerkin boundary element method for solving the boundary integral equation with hypersingularity

A Galerkin boundary elements method is applied to solve the integral equation with hypersingularity,which can be deduced from the double layer solution for the Neumann problem of Laplace equation.The scheme of integration by parts in the sense of distributions is performed to reduce the hypersingularity integral into a weak one,which shifts the partial derivatives of hypersingular kernel to the unknown function in the variational formulation.Thus,the boundary rotation of an unknown function is used to substitute for the original unknown function in the variational equation.When linear boundary elements are used in two-dimensional cases,the boundary rotation can be discretized into a constant vector on each element,so that the integrations can be performed in a simple way.The numerical tests illustrate the effectiveness and practicality of the scheme presented.