A Semi-analytic Technique for Integration of the Newtonian Potential and Its Gradient over Triangular Surface Grid Cells

The integrals are considered that arise at solving boundary integral equations (BIEs), when the unknown Laplacian field is represented through single or double layer potential, i.e., the kernel of the BIE is the fundamental solution of the Laplace equation or its gradient. The case of piecewise-constant potential density on surface mesh panels is considered. Integrals over a panel are considered that arise in collocations method, and the calculation technique is developed for repeated integrals over two panels that arise in the Galerkin method. Exact analytical expressions suitable for practical usage are presented for single integrals; for repeated integrals a hybrid numerical-analytical scheme is proposed, that involves the analytical integration of singularities and numerical integration of smooth functions.