Design of a new scheme to indicate the domain of applicability of near singular approach in 2D BEM

The Gauss–Legendre integration is not appropriate for singular and nearly singular integrations in BEM. In this study, some criteria are introduced for recognizing the nearly singular integrals in integral form of Laplace equation. At first, a criterion is obtained for constant element and consequently higher order elements are investigated. To indicate this near singular approach, there are different formulations amongst which the Romberg method was selected due to its compatibility with analytical integration. The singular integrals were carried out by composing the Romberg method and midpoint rule. The potential functions over geometrically linear BEM elements can be defined in the form of constant, linear or other types of interpolation functions. In those elements, the Gauss–Legendre integration will be accurate, if the source point is placed out of the circle with a diameter equal to element length and its center matched to midpoint of the element. Also, some criteria are obtained for parabolic function of geometry over an element.