Exact boundary element integrations for two-dimensional Laplace equation

This work presents exact solutions of the boundary element integral coeffic ients for the H and G influence matrices for off- and on-element boundary i ntegrations based on the direct boundary element method. The interior integ ral expressions for potential and fluxes are also presented. The exact solu tions obtained in this investigation are for the two-dimensional Laplace eq uation using the constant, linear and quadratic elements. The element geome try in all three cases is considered straight and a formulation is consider ed for the linear and quadratic elements so that boundary solutions for pot ential and flux may be discontinuous, partially discontinuous and continuou s. The exact expressions presented were verified using numerical integratio n methods of both Gauss and Romberg. Various geometric cases were also cons idered 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. M athcad and Mathematica were used to develop the analytical relationships an d verify these relationships. (C) 2000 Elsevier Science Ltd. All rights res erved.