On the regularization of single source combined 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 are by far the most popular, solve a coupled pair of electric, magnetic, or mixed/combined field integral equations for electric and magnetic surface currents. Single source techniques on the other hand, solve one electric, magnetic, or combined field integral equation for an electric or magnetic surface current density. Equations of the first kind involve hypersingular operators which lead to ill-conditioned matrices when discretized, therefore are susceptible to dense mesh and low frequency breakdown. Moreover, they exhibit resonances; that is, their solution is not unique at a set of discrete frequencies that grows increasingly dense as the electrical size of the scatterer increases. Second kind equations on the other hand, do not suffer from dense mesh nor low frequency breakdown, but they are still susceptible to resonances and hence problematic when applied to the analysis of electrically large scatterers. A linear combination of first and second kind single source equations has been proposed, yielding a resonant free formulation. Unfortunately, this equation still contains a hypersingular electric field integral operator rendering the entire equation hypersingular and susceptible to dense mesh breakdown. In this paper, a second kind single source equation for analyzing scattering from homogeneous penetrable objects is discretized, while a first kind single source equation is regularized, which combined with the former yields a resonance free formulation that is not susceptible to dense mesh breakdown.