Gauss Integration Limits in Nearly Singular Integrals of BEM for Geometrically Linear Elements

The most suitable and widely used numerical integration method for boundary integrals in the BEM method is Gauss-Legendre integration. But, this integration method is not appropriate for singular and nearly singular integrations in BEM. In this study, some criteria are introduced for recognizing nearly singular integrals in the integral form of the Laplace equation. At the rst stage, a criterion is obtained for the constant element and, at the later stages, higher order elements are investigated. In the present research, the Romberg integration method is used for nearly singular integrals. The results of this numerical method have good agreement with analytical integration. The singular integrals are solved by composing the Romberg method and midpoint rule. Constant, linear and other interpolation functions of potentials over an element are a category of BEM elements. 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 the midpoint of the element.