Partial integration based regularization in BEM for 3D elastostatic problems: The role of line integrals

The Boundary Element Method (BEM) is a powerful numerical approach for solving 3D elastostatic problems, particularly advantageous for crack propagation in fracture mechanics and half-space problems. Despite its benefits, BEM faces significant challenges related to dense system matrices and singular integral kernels. The computational expense can be mitigated using various fast methods; this study employs the Chebyshev interpolation-based Fast Multipole Method (FMM). To handle singular kernels, several analytical and numerical integration or regularization techniques exist. One such technique combines partial integration with Stokes' theorem to transform hyper-singular and strong singular kernels into weakly singular ones. However, applying Stokes' theorem introduces line integrals in half-space problems and with FMM, where the geometry is partitioned into near-field and far-field regions and must be treated as an open surface. In this paper, the necessary line integrals for strongly singular and hyper-singular kernels are presented and their significance in the aforementioned problems is demonstrated.