Abstract：The boundary element method (BEM) is a numerical computation method developed from the ideas of element division and discretization of the finite element method (FEM). The BEM is challenging to handle alongside other numerical methods, so it is important to leverage its strengths in high precision and computational efficiency. The singular/nearly singular integration problem, unique to BEM, has significantly affected its accuracy. In particular, the nearly singular integrals are often more difficult to handle than singular integrals due to their mathematical properties and the complexity of numerical computation. This paper proposes ageneralized numerical computation method for nearly singular integrals to improve the accuracy of BEM, based on the curved boundary element method (CBEM) and coordinate transformation. The method subdivides the element in a scaled form, using the field-point projections onto the integration element as the basis for subdivision. For slender elements that often appear in surface meshing, the scaling is optimized by considering the longer side's dimensions to balance sub-element dimensions in all directions, thereby improving computational accuracy. Since the subdivision process is carried out in the parameter domain, the method can be applied to all planar and curved elements. In addition, adaptive methods are used to determine the number of subdivisions, ensuring high accuracy at any field point location. In the single-element calculation, the planar triangular element and the spherical quadrilateral element are considered. The results show that, compared with the ordinary 4-point Gaussian integration and Gaussian point subdivision methods, the proposed method achieves higher accuracy and better adaptability to different field point locations, especially on curved elements. Through iterative convergence, the new method can effectively ensure a sufficient number of subdivisions. A model of an indoor ring conductor is constructed. In the comparative analysis, the proposed method effectively improves overall field accuracy and the maximum value. Subsequently, this paper takes the electric-field analysis problem of the actual converter valve tower as an example. The calculations are accelerated using the fast multipole algorithm, and the results are obtained on anordinary PC. The results show that the method achieves higher accuracy when using a similar number of Gaussian points, especially for the maximum field strength, which is of greater concern in engineering. In addition, the advantages of BEM in terms of computational time and cost are demonstrated in this calculation example. The element subdivision method based on projection points effectively improves the accuracy of the nearly singular integrals, thereby reducing the overall computational error of BEM. This improvement highlights the advantage of BEM, demonstrating great potential across a wide range of applications. Therefore, this method is expected to provide a more accurate and efficient solution for the computation of complex engineering problems.
段一伟, 王泽忠. 曲面边界元中基于投影点单元细分奇异积分计算[J]. 电工技术学报, 2026, 41(10): 3221-3229.
Duan Yiwei, Wang Zezhong. Calculation of Nearly Singular Integrals in Curved Boundary Element Method Based on Element Subdivision at Projection Points. Transactions of China Electrotechnical Society, 2026, 41(10): 3221-3229.
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2026, Vol. 41 
