On the discretization of single source integral equations for analyzing scattering from homogeneous penetrable objects

The literature abounds with integral equation techniques for analyzing scattering from homogeneous penetrable objects. Dual source techniques, which solve a coupled pair electric, magnetic, or mixed/combined field integral equations for electric and magnetic surface currents, are by far the most popular. Single source techniques, which solve one integral equation for an electric or magnetic surface current density, never gained much traction even though they are conceptually appealing. The first kind single source equations proposed by Marx [1] and Glisson [2] involve hypersingular operators; hence they are susceptible to dense mesh breakdown and of limited use when solving low-frequency problems. The second kind single source equations anticipated by these authors and formally proposed by Yeung [3] involve operator products that cannot be discretized using established procedures for discretizing single electric, magnetic, and combined field operators. This problem is not unlike that arising in the discretization of the Calderon identities [6], and its solution requires a well-conditioned mapping from div- to curl-conforming function spaces. In this paper, a second kind single source equation for analyzing scattering from homogeneous penetrable objects is discretized via a recently developed technique that achieves such mapping [6]. The proposed discretization procedure fully respects the space mapping properties of the operators involved, thereby guaranteeing accuracy and stability; in addition, it permits the development of numerically viable techniques for solving regularized first kind and combined field single source equations.