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

TL;DR

This paper explores the role of boundary line integrals in BEM for 3D elastostatic problems, demonstrating their importance in certain cases and redundancy in FMM-based approaches, enhancing regularization techniques.

Contribution

It introduces boundary line integrals into partial integration regularization in BEM and analyzes their significance in different problem settings.

Findings

Boundary line integrals are significant in pure half-space problems.

In FMM-based BEM, boundary line integrals are redundant due to geometry partitioning.

The approach improves handling of singular kernels in elastostatic BEM.

Abstract

The Boundary Element Method (BEM) is a powerful numerical approach for solving 3D elastostatic problems, particularly useful for crack propagation in fracture mechanics and half-space problems. A key challenge in BEM lies in handling singular integral kernels. Various analytical and numerical integration or regularization techniques address this, including one that combines partial integration with Stokes' theorem to reduce hyper-singular and strong singular kernels to weakly singular ones. This approach typically assumes a closed surface, omitting the boundary integrals from Stokes' theorem. In this paper, these usually neglected boundary line integrals are introduced and their significance is demonstrated, first in a pure half-space problem, and then shown to be redundant in fast multipole method (FMM) based BEM, where geometry partitioning produces pseudo open surfaces.