The analytically time integrable time-space domain (ATI-TSD) is discovered based on which the minimum time-space domain is identified for treatment on singularities in the three-dimensional time-domain boundary element method (3D TD-BEM) formulation. A direct method to solve singular integrals in the 3D TD-BEM formulation for elastodynamic problems is proposed. The wavefront singularity can be analytically eliminated in ATI-TSD, while the dual singularity can be treated by the direct method using Kutt’s quadrature in the identified minimum time-space domain. Three benchmark examples are presented to verify the correctness and the applicability of the direct method for solving the singular integrals in 3D TD-BEM.

This study extends the calculation of unknown quantities at boundary points to the computation of internal displacements and stresses. The methodological approach is broadly consistent with boundary point calculations. However, the computation of internal unknowns does not involve spatial singularities; the internal stress boundary integral equation is first derived. Subsequently, the obtained boundary integral equation is numerically processed, requiring discretization in both time and space, followed by assembly and solution. When solving the elements in the influence coefficient matrices, the displacement influence coefficient matrix S and the traction influence coefficient matrix D exhibit only wavefront singularities. These wavefront singularities are treated analytically in the time domain, and the spatial integrals are handled using Gaussian numerical integration. The correctness of the algorithm and theory is verified through two classical numerical examples. A theoretically sound and accurate three-dimensional elastodynamic time domain boundary element method and its corresponding computational program are established, providing a reliable tool for guiding engineering design.