Abstract-The marching-on-in-time solution of the time domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of issues, including the emergence of spurious static currents (DC instability) and ill-conditioning at large time steps (low frequencies). In this contribution, a spacetime Galerkin discretization of the TD-EFIE is proposed, which separates the loop and star components of both the equation and the unknown. Judiciously integrating or differentiating these components with respect to time leads to an equation which is free from DC instability. By choosing the correct temporal basis and testing functions for each of the components, a stable marchingon-in-time system is obtained. Furthermore, the scaling of these basis and testing functions ensure that the system remains wellconditioned for large time steps. The loop-star decomposition is performed using quasi-Helmholtz projectors in order to avoid the explicit transformation to the unstable bases of loops and stars (or trees), and to avoid the search for global loops, which is a computationally expensive operation.Index Terms-time domain, electric field integral equation, DC instability, low frequency breakdown.

Abstract-The time domain combined field integral equation (TD-CFIE), which is constructed from a weighted sum of the time domain electric and magnetic field integral equations (TD-EFIE and TD-MFIE) for analyzing transient scattering from closed perfect electrically conducting bodies, is free from spurious resonances. The standard marching-on-in-time technique for discretizing the TD-CFIE uses Galerkin and collocation schemes in space and time, respectively. Unfortunately, the standard scheme is theoretically not well understood: stability and convergence have been proven for only one class of spacetime Galerkin discretizations. Moreover, existing discretization schemes are non-conforming, i.e., the TD-MFIE contribution is tested with divergence conforming functions instead of curl conforming functions. We therefore introduce a novel spacetime mixed Galerkin discretization for the TD-CFIE. A family of temporal basis and testing functions with arbitrary order is introduced. It is explained how the corresponding interactions can be computed efficiently by existing collocation-in-time codes. The spatial mixed discretization is made fully conforming and consistent by leveraging both Rao-Wilton-Glisson and BuffaChristiansen basis functions and by applying the appropriate bi-orthogonalization procedures. The combination of both techniques is essential when high accuracy over a broad frequency band is required.

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following problems: (i) ill-conditioning when the frequency is low; (ii) ill-conditioning when the discretization density is high; (iii) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); (iv) incorrect solution at low frequencies due to current cancellations; (v) presence of spurious resonances. In this paper, quasi-Helmholtz projectors are leveraged to obtain a magnetic field integral equation (MFIE) formulation that is immune to drawbacks (i)-(iv). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector combined field integral equation (CFIE) is obtained that also is immune to (v). Numerical results corroborate the theory and show the practical impact of the newly proposed formulations.

Classical Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) formulations for modeling radiation and scattering from penetrable objects suffer from ill-conditioning when the frequency is low or when the mesh density is high. The most effective techniques to solve these problems, unfortunately, either require the explicit detection of the so-called global loops of the structure, or suffer from numerical cancellation at extremely lowfrequency. In this contribution, a novel regularization method for the PMCHWT equation is proposed, which is based on the quasi-Helmholtz projectors. This method not only solves both the low frequency and the dense mesh ill-conditioning problems of the PMCHWT, but it is immune from low-frequency numerical cancellations and it does not require the detection of global loops. This is obtained by projecting the range space of the PMCHWT operator onto a dual basis, by rescaling the resulting quasi-Helmholtz components, by replicating the strategy in the dual space, and finally by combining the primal and the dual equations in a Calderón like fashion. Implementation-related treatments and details alternate the theoretical developments in order to maximize impact and practical applicability of the approach. Finally, numerical results corroborate the theory and show the effectiveness of the new schemes in real case scenarios.

Stability of time domain integral equation (TDIE) solvers has remained an elusive goal for many years. Advancement of this research has largely progressed on four fronts: (1) Exact integration, (2) Lubich quadrature, (3) smooth temporal basis functions, and (4) Space-time separation of convolutions with the retarded potential. The latter method was explored in [1]. This method's efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was demonstrated on first order surface descriptions (flat elements) in tandem with 0th order functions as the temporal basis. In this work, we develop the methodology necessary to extend to higher order surface descriptions as well as to enable its use with higher order temporal basis functions. These higher order temporal basis functions are used in a Galerkin framework. A number of results that demonstrate convergence, stability, and applicability are presented.