Exact boundary element integrations for two-dimensional Laplace equation

This work presents exact solutions of the boundary element integral coefficients for the H and G influence matrices for off- and on-element boundary integrations based on the direct boundary element method. The interior integral expressions for potential and fluxes are also presented. The exact solutions obtained in this investigation are for the two-dimensional Laplace equation using the constant, linear and quadratic elements. The element geometry in all three cases is considered straight and a formulation is considered for the linear and quadratic elements so that boundary solutions for potential and flux may be discontinuous, partially discontinuous and continuous. The exact expressions presented were verified using numerical integration methods of both Gauss and Romberg. Various geometric cases were also considered and included positioning the source point on and off the element and repositioning the Cartesian coordinate origin. Furthermore, two benchmark problems were also considered to verify the exact integrations presented. Mathcad and Mathematica were used to develop the analytical relationships and verify these relationships.