Rank Deficiency of EFIE Matrix Blocks for Objects Embedded in Lossy Layered Medium

Fast direct solvers for integral equations usually rely on a hierarchical data-sparse representation of impedance matrix and the rank-deficient property of matrix blocks is the key to their success. In this paper, we investigate the rank deficiency of impedance matrix blocks in the discretization of electric field integral equation (EFIE) for modeling objects in a lossy planarly layered medium (PLM). A bunch of basis functions defined on a rectangular plane is utilized as a source/observer cluster and four source-observer configurations are designed to represent matrix blocks resultant from four typical coupling scenarios in a general PLM. A comprehensive analysis of singular value distributions of these matrix blocks is conducted, highlighting the influence of medium parameters, layered medium Green’s functions (LMGFs) and other factors that are closely related to PLM. It is noted that the rank-deficient property of matrix blocks resultant from coupling of source-observer clusters in PLM is much more complicated than that in free space. In particular, a worse rank deficiency for basis function clusters straddling different layers of PLM is observed, and corresponding reason for this deterioration is verified and analyzed. This study may offer guidelines for designing more efficient fast direct solvers for scattering problems in a PLM background.