Efficient evaluation of near-singular integrals of the eXtended Isogeometric Boundary Element Method for three-dimensional fracture mechanics

The Boundary Element Method (BEM) has proven to be a suitable method for the numerical assessment of numerous engineering problems in both two-dimensional and three-dimensional forms. In particular, BEM robustly handles discontinuities in the mechanical response present in fracture mechanics applications. Additionally, its boundary-only nature simplifies the re-meshing process during crack growth modelling. Furthermore, coupling BEM with isogeometric analysis (IgA) is straightforward, as both strategies rely on boundary representation, enabling the direct use of Computer-Aided Design (CAD) geometry as the mesh for the method. IgA basis functions can accurately represent complex surfaces such as spheres and toroids. In this context, the Isogeometric Boundary Element Method (IGABEM) emerges as a reliable tool for solving engineering problems, combining the advantages of BEM with the improved geometric accuracy provided by IgA. In addition, a relevant approach in fracture mechanics is using enrichment functions that incorporate known solution fields, addressing response aspects that standard isogeometric basis functions cannot capture. Combining the enrichment strategy with IGABEM leads to the eXtended IGABEM (XIGABEM) method. Specifically, enriching functions that account for the square-root inverse behaviour on crack surfaces promotes several improvements, such as improved convergence rate and removal of a non-physical jump displacement in the crack front. Another advantage is the direct extraction of Stress Intensity Factors (SIF) as system variables, eliminating the need for costly extraction techniques in three-dimensional problems. However, near-singular integrals appear after the Singularity Subtraction Technique (SST) of the enriching terms in 3D XIGABEM. This study proposes using the hyperbolic sine transformation for the near singular surface integrals that arise from the SST to evaluate them efficiently. Numerical applications demonstrate a reduction in the required integration points for these integrals compared to traditional methods. Therefore, this advancement in the XIGABEM framework reduces the computational cost of this strategy while maintaining accuracy.