Transformation techniques for weakly and nearly singular integrals: Comparative performance and numerical implementation
Description
Accurate evaluation of singular and nearly singular integrals is crucial for the successful application of numerical techniques such as the Boundary Element Method (BEM) and the Generalized/eXtended Finite Element Method (G/XFEM). Employing a unified framework based on Duffy-type mappings and an asymptotic analysis, we identify distinct radial and angular near-singular behaviors. Several established variable transformation techniques are analyzed and compared for evaluating weakly singular and nearly singular integrals over spatial surface elements. Comprehensive numerical experiments are conducted to assess the accuracy and robustness of various radial (Polynomial, Sinh, Exponential, L_1^{-1/5} , and Xiao’s [1]) and angular (Xiao’s [1], Sigmoidal and PART’s) transformations for weakly singular integrals and for weakly/strongly nearly singular integrals over planar and representative curved surfaces. The results demonstrate that Sinh and Xiao’s radial transformations offer superior performance for (nearly) singular integrals across diverse scenarios. In contrast, Polynomial and L_1^{-1/5} transformations exhibit limitations, particularly for stronger near-singularities or challenging geometries. Furthermore, Xiao’s angular transformation is shown to effectively mitigate angular near-singularities arising from edge/corner proximity within the employed coordinate system. These comparative insights provide practical guidance for selecting optimal integration strategies. Based on the analysis, we present a computational algorithm and its implementation in the MATLAB/Octave package, offering a versatile framework for accurately evaluating these challenging integrals in computational mechanics.