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Description
To further advance the regularisation and numerical treatment of the double-layer operator in three-dimensional elastodynamics within the space-time Energetic Boundary Element Method (EBEM) framework [1], this work extends the analysis to the singular behaviour of the adjoint double-layer and hypersingular operators arising in mixed boundary value problems. The mixed formulation is characterised by the simultaneous prescription of displacement and traction conditions on complementary parts of the domain boundary, resulting in a coupled system of time-dependent Boundary Integral Equations (BIEs) involving all four elastodynamic boundary integral operators. By exploiting a suitable decomposition of the traction-displacement and traction-traction Green’s functions, a fully regularised system of BIEs is derived. The resulting formulation is discretised using a Galerkin-type EBEM, originally introduced for 3D elastodynamics in [2], combined with exact analytical integrations in the time variable. A central challenge of the proposed approach lies in the efficient and stable evaluation of the remaining weakly singular double integrals in space, whose accurate approximation is essential to ensure the robustness of the mixed formulation. In this context, the space-time integration domains are generally delimited by the wave fronts of primary and secondary elastic waves, resulting in complex geometric configurations. By analyzing the geometric characteristics of these domains, we develop an ad-hoc quadrature strategy, where the outer integrals are computed efficiently by Gaussian quadrature, 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 problems.
[1] L. Coppolino, L. Desiderio. Space-time energetic Galerkin BEM for the numerical solution of 3D elastodynamic problems: overcoming challenges of the strongly singular integral operator. Comput Mech 76, 1689–1714 (2025).
[2] A. Aimi, S. Dallospedale, L. Desiderio, C. Guardasoni. A space-time Energetic BIE method for 3D Elastodynamics. The Dirichlet case. Computational Mechanics, 72(5), 2023, pp. 885–905.