基于改进型双正切变换的高阶近奇异性精确积分方法研究

电磁场表面积分方程方法（SIE）中的高阶近奇异性积分是SIE精确求解的关键技术之一，但现有方法主要是处理平面单元建模中的低阶近奇异性问题，目前还没有一种可用于高阶曲面建模中3阶近奇异性的精确稳定积分方法.本文在前期提出的双正切变换方法（DAT）的基础上，针对高阶曲面建模中含有RR/R⁵、R/R⁴和1/R³等形式积分核的近奇异性问题，通过引入指数变换解决了DAT算法在近奇异点与源单元非常接近时算法不稳定的问题，并通过引入形函数变换解决了DAT近奇异点与源单元边界靠近时积分不稳定的问题，形成改进型双正切变换方法（IDAT）.相对于DAT，所提IDAT更稳定高效.所提IDAT不仅可用于曲面单元中的高阶近奇异性问题的精确积分，同时也适用于低阶近奇异积分问题.理论分析与数值算例验证了本文所提方法的精确性与稳定性.

Research on the Accurate Evaluation of Higher Order Nearly Singular Integrals Based on the Improved Double-Arctan Transformation

The accurate evaluation of higher order nearly singular integrals is one of the key technologies in accurate simulation by electromagnetic surface integral equations (SIE).However,the present methods mainly focus on low order nearly singular integrals for the planar element modeling,rather than on 3-order nearly singular integrals in higher order geometry modeling.Based on the former research of the Double Arctan-Transformation (DAT),the Improved Double Arctan-Transformation (IDAT) is proposed to improve the stability and accuracy of the nearly singular integrals with singular kernel RR/R⁵, R/R⁴ and 1/R³ for higher order geometry modeling.Specifically,the exponential transformation is utilized to stabilize the integrals when the field points are extremely close to the source surface.Furthermore,the shape-function transformation is adopted to stabilize the integrals when the projection point approaches to the border of source surface.The proposed IDAT is also effective for the lower orders of the nearly singular integral kernels.With theoretical analysis and typical testing cases,the accuracy and stability performance of IDAT is fairly evaluated.