Two-dimensional quadrature formulas for the discretization of boundary integral equations in the presence of conducting and dielectric edges

Numerical integration of functions defined over two-dimensional domains is usually required in the discretization of Boundary Integral Equations (BIEs) for scattering and radiation problems involving three-dimensional conducting and dielectric bodies. The unknowns of such BIEs are typically the equivalent electric and magnetic surface currents defined on the interfaces between different media; these currents are linked to the tangential values of the magnetic and electric fields, respectively, and, as is known, some components of these fields may diverge in the neighborhood of an edge in a conducting or a dielectric body. In this case, it is therefore necessary to integrate functions which are singular along one edge of their integration domain. In this work we derive two-dimensional quadrature formulas of the Radon type for planar triangles, which usually occur in the mesh used to discretize the boundaries of the involved three-dimensional objects. The new formulas integrate exactly polynomials of degree less than or equal to five, multiplied by a weight function which diverges algebraically along one side of the triangle. This is the kind of singularity to be expected for the transverse components of the electromagnetic field in the vicinity of a physical edge, with the singularity exponent depending on the internal angle of the wedge and on the constitutive parameters of the involved media. The proposed quadrature formulas allow us to take into account the field singularity due to the presence of edges in the adopted numerical discretization method, e.g., by using suitable vector basis functions in the Method of Moments or by directly discretizing the boundary integral operators in the Nystr¨om method. In this work we show how the use of the new formulas improves the accuracy of the calculated solutions in radiation and scattering problems. It is thus possible to reduce the number of triangles of the mesh used to represent the boundaries of the involved objects, and therefore the total number of unknowns, with a consequent significant advantage in terms of memory and computing-time requirements.