High-Order Calderon Preconditioning of Integral Equations for the Analysis of Scattering from PEC and Homogeneous Penetrable Objects.

Abstract

Boundary integral equations often are used to analyze transient and time-harmonic scattering from perfect electrically conducting (PEC) as well as homogeneous penetrable objects. When discretized, these equations give rise to linear systems of equations that can be solved for the induced currents on the surface of the scatterer. Unfortunately, standard formulations have the property that whenever the mesh discretization increases, the frequency approaches zero, or it is close to a resonance, the condition number of these systems grows rapidly. From a numerical perspective these properties are problematic as they slow down or even preclude the use of iterative solvers. Previous work in this area has provided with new (regularized) equations that overcome these problems, but their applicability has been limited exclusively to zeroth-order representations of the current expansions. The objective of this research is to extend the applicability of existing regularized equations applicable to PEC structures, and to provide with new equations suitable for homogeneous penetrable objects. First, a new set of high-order div- and quasi curl-conforming basis functions is presented. With these functions, several existing equations for PEC structures are implemented using high-order representations for the surface of the scatterer and the current expansions. Time-domain and frequency-domain regularized equations are explored. Numerical results demonstrate that high-order implementation of these equations preserves the desired properties of their predecessors, yet they exhibit the advantage of using high-order representations. Second, a new regularized single source equation for analyzing scattering from homogeneous penetrable objects in frequency-domain is presented. The proposed equation is immune to low-frequency and densemesh breakdown, and free from spurious resonances. This single source equation is discretized using a combination of basis functions (including those presented here), thereby fully respecting the space mapping properties of the operators involved, and guaranteeing accuracy and stability. Numerical results show that the proposed equation and discretization technique give rise to rapidly convergent solutions. They also validate the equation’s resonant free character. With this same discretization scheme, (non-regularized) time-domain single source equations are also explored.