A mesh-adapted closed-form regular kernel for 3D singular integral equations

The Green's functions employed in the method of moments (MoM) diverge when observation and source points coincide; this is at the origin of the difficulties in computing the MoM matrix entries, and in handling the near-field interactions in fast Fourier transform (FFT)-based fast methods and other sampling-based methods. In this paper, we show that this singularity can be avoided, and a modified regular Green's function can be used instead. This latter is obtained from the spectral representation of the usual Green's function via windowing of its spectrum; the width of the spectral window depends on the size of the mesh employed for discretizing the problem, so that the proposed regular Green's function is a mesh-adapted regular kernel. We address a general 3D problem; we relate the MoM reaction integrals to the 2D Fourier spectrum of the Green's function, that allows to discuss the necessary spectral bandwidth for the windowed Green's function. We employ a tapered window, and present a closed-form expression for the spatial Green's function. Numerical results are presented for 3D antenna and scattering problems discretized with Rao-Wilton-Glisson (RWG) functions, and for uniform and nonuniform meshing. They show that the proposed method yields accurate solutions also for the antenna input impedance. The meaning of the regularized Green's function is also discussed and put in perspective.