Integral and geometrical means in the analytical evaluation of the BEM integrals for a 3D Laplace equation

By using integral theorems and geometrical interpretations, the analytical formulas for the coefficients occurring in boundary element method (BEM) equations for a 3D Laplace equation were found for arbitrary planar polygonal boundary elements with constant approximation. Closed forms for the gradients of the coefficients were also obtained. In addition, an analytical formula for planar triangular boundary elements with a linear approximation of potential was given. The formulas obtained are appropriate especially in singular and nearly singular cases.