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.