A generalized Calderón preconditioner for the Electric Field Integral Equation

Abstract

Among all integral equations pertinent to the analysis of scattering from three-dimensional perfect electrically conducting surfaces, the Electric Field Integral Equation (EFIE) remains the most widely used. Unfortunately, the EFIE operator is ill-posed as its eigenvalues accumulate at zero and infinity. For this reason, the linear system matrix that results upon discretization of the EFIE using boundary elements is highly ill-conditioned whenever the surface mesh is dense. The need for dense surface discretizations commonly arises when analyzing electromagnetic phenomena on structures with sub-wavelength geometric features, e.g. millimeter and microwave integrated circuits, antenna feeds and beam-forming networks, aircraft fuselage details, etc. The literature abounds with techniques to precondition EFIEs. One such technique, the Calderon Multiplicative Preconditioner (CMP) proposed in [1] can be trivially integrated into existing boundary element codes as the discretized operator comprises the multiplication of two standard EFIE matrices (produced with a Rao-Wilton-Glisson code) weighted by sparse Gram and projection matrices, containing O(N) nonzero elements. Unfortunately, the CMPs proposed to date only apply to triangular meshes and their barycentric refinement.