Time domain integral equations (TDIEs) are an attractive framework from which to analyze electromagnetic scattering problems. Casting problems in the time domain enables study of systems with nonlinearities, characterization of transient behavior both at the early and late time, and broadband analysis within a single simulation. Integral equation frameworks have the advantages of restricting the computational domain to the scatterer surface (boundary integral equations) or volume (volume integral equations), implicitly satisfying the radiation boundary condition, and being free of numerical dispersion error. Despite these advantages, TDIE solvers are not widely used by computational practitioners; principally because TDIE solutions are susceptible to late-time instability. While a plethora of stabilization schemes have been developed, particularly since the early 1980s, most of these schemes either do not guarantee stability, are difficult to implement, or are impractical for certain problems. The most promising methods seem to be the {\em space-time Galerkin} schemes. These are very challenging to implement as they require the accurate evaluation of 4-dimensional spatial integrals. The most successful recent approach to implementing these schemes has been to approximate a subset of these integrals, and evaluate the remaining integrals analytically. This approach describes the {\em quasi-exact integration} methods [Shanker et al. IEEE TAP 2009, Shi et al. IEEE TAP 2011]. The method of [Shanker et al. IEEE TAP 2009] approximates 2 of the 4 dimensions using numerical quadrature. The remaining integrals are evaluated analytically by determining shadow boundaries on the domain of integration. In [Shi et al. IEEE TAP 2011], only 1 dimension is approximated, but the procedure also relies on analytical integration between shadow boundaries. These two characteristics-the need to find shadow boundaries and develop analytical integration rules-prevent these methods from being extended to higher order tessellations of scattering surfaces. This is an important restriction as the use of curvilinear elements can greatly improve the accuracy of the geometric representation. The need for a method to accurately evaluate the spatial integrals involved in these formulations on higher order surface tessellations motivates this thesis.The major novelty of this thesis is a space-time separated expansion of the convolution with the retarded potential Green's function. This separation leads to integrands that are smooth over the entire domain of integration. This implies that integration can be accurately carried out via numerical quadrature (not analytically) and shadow boundaries do not need to be found, unlike in the quasi-exact integration methods. The numerical nature of the method allows it to trivially be implemented on higher order surface descriptions. In this thesis, we will detail the procedure of the separable expansion and investigate, both numerically and analytically, the error incurred in truncating the expansion to a given upper limit. We will validate the stability of the resulting scheme by (1) observing the late time behavior of solutions to scattering from a variety of objects and (2) deriving and implementing an eigenvalue analysis to demonstrate the absence of growing terms. Additionally, this thesis will detail the use of the separable expansion in tandem with the plane wave time domain algorithm to accelerate the solution. Also, we will present extension of the space-time separation to analysis of penetrable materials using the PMCHWT formulation. A prescription for integrating singular and near-singular kernels over curved elements will also be given. The final contribution of this thesis is the application of the space-time separation to the generalized method of moments (GMM) with smooth surface parameterization.
Read
