Exact boundary element integrations for two-dimension Laplace equation

摘要

This work presents exact solutions of the boundary element integral coefficients for the H and G influence matrices for off- and on-element boundary integration's bassoon 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 Lapace 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 method of both Gauss and Romberg. Various geometric cases were also considered and included positing 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. Mathacad and Mathematical were used to develop the analytical relationships and verity these relationships.