Analysis of transient scattering from PEC objects using the Generalized Method of Moments

摘要

Surface integral formulations for the anlysis of transient scattering have become very prevalent over the recent years. These formulations are typically discretized by employing basis functions that are products of spatial and temporal functions. While the design and construction of optimal, stable and accurate temporal basis functions have been the object of considerable study, the de-facto standard for spatial discretization is the traditional set of Rao-Wilton-Glission (RWG) functions. These functions have been carried over to time domain integral equations (TDIE) from their frequency domain counterparts and have been shown to be highly effective in a variety of problems. These functions are defined on a tessellation of the geometry and have spawned a set of higher order and singular variants, though these variations have seen limited use in time domain. The close marriage between the basis function construction and the tessellation stems from the need to satisfy continuity constraints across the interior edges of the tessellation. While this has several advantages, this nature of construction is inherently restrictive. Indeed, even the use of mixed orders of polynomial basis functions is significantly involved and usually requires the careful design of patches and "transition" areas. In the frequency domain, a technique called the Generalized Method of Moments (GMM) was recently introduced to resolve some of these problems [1]. The GMM was designed as an umbrella framework to include arbitrary functions in the basis space. In addition, [ 1] demonstrates that the basis functions proposed result in a matrix system with stable condition numbers to very low frequencies. In this work, we develop the GMM scheme for the discretization of the TDIE. We will describe a scheme for the extension of the GMM to time domain using products of GMM functions (for spatial basis functions) and interpolatory polynomials (for temporal basis functions). First, we will show that these basis functions provide accurate scattering results over a wide variety of geometries. Further, we will demonstrate the ability of the GMM scheme to arbitrarily mix basis functions across the geometry. We will show that using different basis functions in different areas of the tessellation can be trivially achieved by simply changing an input flag. While the results obtained in this paper are using the magnetic field equation, it will also be shown at the conference that this scheme leads to a stable discretization scheme for the electric field integral equation (EFIE) as well.