A bi-directional distance transformation for the elimination of near-singularity in boundary element method

Abstract

Purpose

In traditional methods, a one-directional distance transformation is employed to eliminate the near-singularity in the radial direction. However, when the projection point of the source point is near the boundary of the integral element, the near-singularity in the circumferential direction still exists, resulting in large errors in the numerical results. The purpose of this paper is to propose a bi-directional distance transformation for the elimination of near-singularities in two directions arising from the irregular integral element.

Design/methodology/approach

The sources of the circumferential near-singularity caused by irregular sub-triangular element are analyzed in this paper. A bi-directional distance transformation based on the (a, β) transformation is proposed to eliminate the near-singularities in the two directions. The (a, β) transformation is initially introduced to separate the integral variables and streamline the implementation of subsequent transformations. In the transformed (a, β) coordinate system, a new distance transformation applied in the circumferential direction is constructed.

Findings

With the proposed method, the radial and circumferential near-singularities are eliminated using two different distance transformations, respectively. Thus, accurate calculations of the nearly singular integrals can be achieved, irrespective of the shape of the integral element.

Originality/value

Numerical examples are presented to calculate the nearly singular integrals of different orders over both the linear integral element and the quadratic integral element. Comparative studies demonstrate that the proposed method significantly improves the accuracy of these calculations.

Citation

Dong, Y., Tan, Z., Sun, H. and Yuan, F. (2024), "A bi-directional distance transformation for the elimination of near-singularity in boundary element method", Engineering Computations, Vol. 41 No. 8/9, pp. 2169-2189. https://doi.org/10.1108/EC-06-2024-0529