Analytical Integration for Logarithmic Spatial Singularities in the Time Domain Boundary Element Method

The treatment of logarithmic spatial singular integrals is a key challenge affecting the reliability of results when the time domain boundary element method (TD-BEM) is used to solve elastodynamic problems. To address this problem, this paper derives and establishes a set of analytically rigorous integration formulas for logarithmic spatial singularities based on the fundamental properties of the Heaviside function, which enables the direct spatiotemporal analytical solution of such singular integrals in TD-BEM. The formulas fill the research gap of the absence of direct analytical solutions for logarithmic spatial singular integrals in elastodynamic problems of TD-BEM, and enrich the theoretical system of the treatment of singular integrals for TD-BEM. Three typical elastodynamic engineering problems, including a fixed–fixed beam under a uniform sudden load, an infinite domain with a single cavity under a boundary blasting load, and a double tunnel beneath valley topography subjected to metro vibration load, are selected for numerical verification. The calculation results of the proposed method are compared with the reference solutions. It is shown that the calculation results of the proposed method are in good agreement with the reference solutions, which effectively verifies the correctness and engineering applicability of the analytical integration formulas.