Space-time energetic Galerkin BEM for the numerical solution of 3D elastodynamic problems: overcoming challenges of the strongly singular integral operator

To advance the space-time Energetic Boundary Element Method (EBEM) in 3D elastodynamics, in this paper we study the characteristic singularity of the double-layer operator, which is involved in the direct boundary integral formulation. On the basis of a decomposition of the time-dependent point load traction Green’s function, we employ a regularized Boundary Integral Equation (BIE) and we discretize it by means of a Galerkin-type EBEM with double analytical integration in the time variable. However, one of the main difficulties of this approach is the efficient approximation of remaining weakly singular double space integrals, whose accurate computation is a key issue for the stability of the method. In particular, the integration domains are generally delimited by the wave fronts of the primary and the secondary waves. By analyzing the geometric characteristic of these domains, we develop an ad-hoc quadrature strategy, where the outer integrals are computed efficiently by standard quadrature (with a small number of points), while the inner integrals are evaluated with respect to polar coordinates and expressed by analytical formulations. The effectiveness of the proposed approach is illustrated via two benchmark finite-domain and one infinite-domain problems.