IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 2, FEBRUARY 2004 373 Improved Quadrature Formulas for Boundary Integral Equations With Conducting or Dielectric Edge Singularities Paolo Burghignoli, Member, IEEE, Lara Pajewski, Fabrizio Frezza, Senior Member, IEEE, Alessandro Galli, Member, IEEE, and Giuseppe Schettini, Member, IEEE Abstract—In this paper we derive new two-dimensional (2–D) quadrature formulas for the discretization of boundary integral equations in the presence of conducting or dielectric edges. The proposed formulas allow us to exactly integrate polynomials of de- gree less than or equal to five, multiplied by an algebraic singular factor that diverges along one side of the triangular integration domain. This is the kind of singularity that occurs when physical edges are present in both conducting and dielectric bodies. Nu- merical tests are performed on the presented formulas, in order to validate the achieved improvement in accuracy, and examples are given of their application to the determination of radar cross-sec- tion of 3-D metallic objects. Index Terms—Boundary integral equations (BIEs), edge singu- larities, numerical integration, quadrature formulas, triangular meshes. I. INTRODUCTION N UMERICAL integration of functions defined over two-di- mensional (2-D) domains is usually required in the dis- cretization of boundary integral equations (BIEs) for scattering and radiation problems involving 3-D conducting and dielectric bodies [1], [2]. The unknowns of such BIEs are typically the equivalent electric and magnetic surface currents defined on the interfaces between different media; these are linked to the tan- gential 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 [3], [4]. In this case, it is therefore necessary to integrate functions which are singular along one edge of their integration domain. In this paper we derive 2-D quadrature formulas of the Radon type [5] for planar triangles, which usually occur in the mesh used to discretize the boundaries of the involved 3-D objects. The new formulas exactly integrate polynomials of degree less than or equal to five, multiplied by a weight function that di- verges 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 Manuscript received June 18, 2001; revised January 13, 2003. P. Burghignoli, L. Pajewski, F. Frezza, and A. Galli are with the Department of Electronic Engineering, University of Rome “La Sapienza,” 00184 Rome, Italy (e-mail: burghignoli@die.uniroma1.it). G. Schettini is with the Department of Applied Electronics, University of Rome “Roma Tre,” 00146 Rome, Italy (e-mail: g.schettini@uniroma3.it). Digital Object Identifier 10.1109/TAP.2004.824001 of the wedge and on the constitutive parameters of the involved media [3]. The proposed quadrature formulas allow us to take into ac- count 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 [6], or by di- rectly discretizing the boundary integral operators in the Nys- tröm method [7]. In this paper 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 bound- aries of the involved objects, and therefore the total number of unknowns, with a consequent significant advantage in terms of memory and computing-time requirements. This paper is organized as follows. In Section II the analyt- ical details of the derivation of the novel quadrature formulas are described, and examples of quadrature points and weights are given for some significant cases. In Section III numerical tests are performed to assess the accuracy of the proposed for- mulas, and an example is given of their application to the so- lution of the magnetic field integral equation (MFIE) by means of the Nyström method. Finally, in Section IV conclusions are presented on the results of this paper. II. DERIVATION OF THE QUADRATURE FORMULAS A. Background As is known, numerical integration (quadrature) formulas en- able the evaluation of multiple integrals by means of approxima- tions of the form [8] (1) where is a given region in an -dimensional Euclidean space, is a given weight function, are the integration points of the formula, and are the weight factors of the formula: both the integration points and weight factors do not depend on . To construct a formula of degree , it is necessary to calculate coordinates and weight factors for a suitable number of integration points, so that polynomial functions of degree up to are exactly integrated. 0018-926X/04$20.00 © 2004 IEEE