In this paper, the direct evaluation method tailored for the hyper-singular integrals arising in Galerkin surface integral equation formulations is developed.[...] The proposed method utilizes a series of coordinate transformations together with a re-ordering of the integrations in order to reduce the 4-D hyper-singular integrals to 2-D smooth integrals that can be easily computed via generalized Cartesian product rules based on standard Gauss quadratures, readily available in the literature. The direct evaluation method was originally introduced by Gray et al. for the evaluation of super hyper-singular Galerkin surface integrals arising in static problems and recently it was extended for the case of the weakly singular integrals in Galerkin MoM problems over coincident triangle elements.

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