Efficiency improvement of the polar coordinate transformation for evaluating BEM singular integrals on curved elements

摘要

The polar coordinate transformation (PCT) method has been extensively used to treat various singular integrals in the boundary element method (BEM). However, the resultant integrands tend to become nearly singular when (1) the aspect ratio of the element is large or (2) the field point is closed to the element boundary. In this paper, the first problem is circumvented by using a conformal transformation so that the geometry of the curved physical element is preserved in the transformed domain. The second problem is alleviated by using a sigmoidal transformation, which makes the quadrature points more concentrated around the near singularity. By combining the proposed two transformations with the Guiggiani method in Guiggiani et at. (1992) [8], one obtains an efficient and robust numerical method for computing the weakly, strongly and hyper-singular integrals in high-order BEM. Numerical integration results show that, compared with the original PCT, the present method can reduce the number of quadrature points considerably, for given accuracy. For further verification, the method is incorporated into a 2-order Nystrom BEM code for solving acoustic Burton-Miller boundary integral equation. It is shown that the method can retain the convergence rate of the BEM with much less quadrature points than the existing PCT.