Analytical integrations in 3D BEM for elliptic problems: evaluationand implementation

The present publication deals with 3D elliptic boundary value problems (potential, Stokes, elasticity) in the framework of linear, isotropic, and homogeneous materials. Numerical approximation of the unique solution is achieved by 3D boundary element methods (BEMs). Adopting polynomial test and shape functions of arbitrary degree on flat triangular discretizations, the closed form of integrals that are involved in the 3D BEMs is proposed and discussed. Analyses are performed for all operators (single layer, double layer, hypersingular). The Lebesgue integrals are solved working in a local coordinate system. For singular integrals, both a limit to the boundary as well as the finite part of Hadamard (Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press: New Haven, CT, U.S.A., 1923) approach have been considered.