Analysis of natural transformations for regularizing nearly singular and hypersingular boundary integrals

A long-standing issue in computational mechanics is the accurate quadrature of nearly singular and hypersingular integrals. This issue arises, for example, in the stress boundary integral equation ( σ -BIE) when the collocation point lies close to the integration element, so that the kernels become nearly singular ( 1 / r ) or nearly hypersingular ( 1 / r 2 ). An analysis of the natural transformations for regularizing such integrals is presented. Here, natural transformations are changes of variables whose Jacobian coincides with the near-singularity denominator, thereby canceling the singular factor. The order of the singularity is readily analyzed using complex variables, avoiding classical normalization along the tangent line and cumbersome closest-point computations. Accordingly, the tangent and sinh variable changes are identified as the natural transformations for 1 / r 2 near-hypersingular and 1 / r near-singular kernels, respectively. The regularization power is further enhanced by studying compositions of these transformations, such as tan–p3c and the well-known iterated sinh–sinh. After identifying the most suitable transformation for each kernel moment, robust schemes for practical numerical integration are established based on per-block error tolerances and nondimensional proximity thresholds. These schemes enable efficient and reliable evaluation of internal stresses in 2D elastostatics, as demonstrated for models with both straight and curved elements. • Tangent and sinh compared for 1 / r 2 , 1 / r near-singularities down to 1 0 − 6 and NPG. • Composite tan–p3c and sinh–sinh transformations are analyzed for improvements. • Complex-plane pole analysis avoids tangent-line and closest-point computations. • Tangent best for S1 on straight (15 NPG); tan–p3c for curved; sinh–sinh else. • Robust stress evaluation for 2D elastostatics on straight and curved elements.