Causality condition relevant functions-orientated analytical treatment on singularities in 3D TD-BEM

A direct analytical treatment on singularities in the 3D TD-BEM formulation is proposed, where the wavefront singularity and the dual singularity are analytically expressed. In the process of the solution of the spatial singularity, the integration domain is divided into the regular part and the singular part. The singularities in the singular part are analytically eliminated by the direct method of the concept of the finite part of an integral (Hadamard principal integral), while the singular integrals in the regular part are solved by the convenient Gaussian integration. Due to the increase of the dimension and the additional causality relevant function, the 3D TD-BEM formulation is further more intricate than the 2D one. In order to reduce the complexity, in the process of the solution of the boundary integral equation in the 3D TD-BEM formulation, a new coordinate transformation method is proposed to analytically transform the coefficient integrals on the spatial surface element into the 2D plane element. The 3D TD-BEM formulation based on the proposed analytical treatment on singularities is verified to be correct by three examples.