Emerging classical and quantum applications require computational electromagnetics methods that can efficiently analyze complex structures over wide bandwidths, including down to very low frequencies. This work begins to address these needs by presenting a type of charge and current integral equation that has been formulated in the time domain and is applicable to dielectric regions. This system introduces charge densities as unknowns in addition to the current densities, resulting in a system that does not exhibit a low frequency breakdown. An appropriate marching-on-in-time discretization scheme is discussed so that stable and accurate results can be achieved down to very low frequencies. Numerical results are shown to verify the accuracy and stability of this formulation.
Potential-based formulations are new approaches gaining interest for deriving computational electromagnetics methods that perform markedly better for low frequencies and complicated structures (e.g., subwavelength and multiscale geometries) compared to traditional field-based formulations. Further, these methods are also more directly applicable to coupling into quantum physics problems that are becoming more prevalent in engineering applications. These methods derive their improved performance by developing systems to be discretized directly in terms of the magnetic vector potential and electric scalar potential, which are deemed more fundamental quantities for quantum applications than the electric and magnetic fields. Performing derivations in this way has resulted in equations that can accurately capture both wave physics (where the electric and magnetic fields are tightly coupled) and quasistatic phenomena (where the electric and magnetic fields become increasingly uncoupled) at the same time. This work focuses on continuing the development of time domain integral equations (TDIEs) based on the potential-based formulation to meet the demanding bandwidth requirements needed to efficiently analyze a wide range of quantum electromagnetic physics. Past work on potential-based TDIEs were applicable to perfect electrically conducting objects, and were shown to be stable and accurate over broad frequency ranges. More recently, initial efforts at developing potential-based TDIEs for dielectric regions were introduced. However, these initial equations did not exhibit the low frequency accuracy and stability properties desired from this formulation. This work demonstrates a new set of TDIEs that overcome the limitations of the original formulation, achieving high accuracy and good stability at analyzing dielectric objects at very low frequencies. These properties of the improved formulation are demonstrated through numerical results.
The growth of applications at the intersection between electromagnetic and quantum physics is necessitating the creation of novel computational electromagnetic solvers. Work in this paper presents a new set of time domain integral equations (TDIEs) formulated directly in terms of the magnetic vector and electric scalar potentials that can be used to meet many of the requirements of this emerging area. Stability for this new set of TDIEs is achieved by leveraging an existing rigorous functional framework that can be used to determine suitable discretization approaches to yield stable results in practice. The basics of this functional framework are reviewed before it is shown in detail how it may be applied in developing the TDIEs of this work. Numerical results are presented which validate the claims of stability and accuracy of this method over a wide range of frequencies where traditional methods would fail.
Applications at the intersection of quantum and EM physics are becoming more prevalent in the engineering community. Interestingly, many of these applications require solving purely classical EM problems to characterize the most important dynamics of the system. As a result, computational electromagnetics (CEM) can play a vital role in this new area. However, the classical problems that typically need to be solved are the broadband analysis of near-field scattering problems in complicated regions with multiscale and/or subwavelength features. Recently, potential-based time domain integral equations (TDIEs) have been investigated to solve these traditionally challenging CEM problems [1], [2]. However, for these methods to be applicable, they must be robustly stable when analyzing complicated geometries over broad bandwidths.
A new potential-based TDIE formulation for dielectric regions is proposed that directly uses magnetic current densities as unknowns. This new formulation avoids the use of inconvenient integral operators that complicate the discretization, while also providing simpler computation of far-field results due to direct access to the magnetic current density. Appropriate marching-on-in-time discretization schemes are discussed so that stable results can be achieved at middle frequencies. Overall, this results in the improved performance of these new equations compared to previous formulations. The accuracy and stability of this new formulation is demonstrated through numerical results.
A new potential-based TDIE formulation for dielectric regions is proposed that directly uses magnetic current densities as unknowns. This new formulation avoids the use of inconvenient integral operators that complicate the discretization, while also providing simpler computation of far-field results due to direct access to the magnetic current density. Appropriate marching-on-in-time discretization schemes are discussed so that stable results can be achieved at middle frequencies. Overall, this results in the improved performance of these new equations compared to previous formulations. The accuracy and stability of this new formulation is demonstrated through numerical results.
As quantum theory matures, quantum applications that significantly depend on electromagnetic effects are becoming increasingly of interest to engineers. We discuss a number of diverse computations that are needed in the engineering design of various devices leveraging these quantum effects. In each case, the broadband knowledge of the classical dyadic Green's function of the problem being analyzed plays an important role. As a result, many of these applications require the extremely broadband solution of classical electromagnetic scattering problems in complex geometries to evaluate components of the dyadic Green's functions. This suggests that classical time domain computational electromagnetics methods will play an important role in the future of quantum applications.
Applications involving quantum physics are becoming an increasingly important area for electromagnetic engineering. To address practical problems in these emerging areas, appropriate numerical techniques must be utilized. However, the unique needs of many of these applications require new computational electromagnetic solvers to be developed. The A-4:1. formulation is a novel approach that can address many of these needs. This formulation utilizes equations developed in terms of the magnetic vector potential (A) and electric scalar potential (t.). The resulting equations overcome many of the limitations of traditional solvers and are ideal for coupling to quantum mechanical calculations. In this work, the A-4. formulation is extended by developing time domain integral equations suitable for multiscale perfect electric conducting objects. These integral equations can be stably discretized and constitute a robust numerical technique that is a vital step in addressing the needs of many emerging applications. To validate the proposed formulation, numerical results are presented which demonstrate the stability and accuracy of the method.
A new time domain electric field integral equation is proposed to solve low frequency problems. This new formulation uses the current and charge densities as unknowns, with a form of the continuity equation that is weighted by a Green's function as a second constraining equation. This equation can be derived from a scalar potential equivalence principle integral equation, which is in contrast to the traditional strong form of the continuity equation that has been used in an ad-hoc manner in the augmented EFIE. Numerical results demonstrate the improved stability of this approach, as well as the accuracy at low frequencies.
A new time domain electric field integral equation is proposed to solve low frequency problems. This new formulation uses the current and charge densities as unknowns, with a form of the continuity equation that is weighted by a Green's function as a second constraining equation. This equation can be derived from a scalar potential equivalence principle integral equation, which is in contrast to the traditional strong form of the continuity equation that has been used in an ad-hoc manner in the augmented EFIE. Numerical results demonstrate the improved stability of this approach, as well as the accuracy at low frequencies.