Time-domain Electric Field Integral Equation Research Articles

Finite difference delay modelling for analysis of shielding effectiveness of perforated conductive enclosure

AbstractThis paper introduces finite difference delay modelling (FDDM) for computing the shielding effectiveness (SE) of a conductive enclosure with an aperture against electromagnetic pulses. FDDM offers optimal accuracy and stability for analysing complex structures. The time domain electric field integral equation (TD‐EFIE) for the conductive enclosure is derived by imposing boundary conditions on the perfect electrical conductor (PEC) surface in the Laplace domain. Time discretization is based on finite differences, and the Laplace‐to‐Z transform mapping is utilized. Fast Fourier Transform (FFT) is employed to expedite the FDDM solution process. Both frequency domain shielding effectiveness (FD‐SE) and time domain shielding effectiveness (TD‐SE) are evaluated using this approach. Finally, to validate the accuracy of the proposed method, results are compared with simulations using CST‐MWS software and the frequency domain moment method.

Induction mechanism and propagation characteristics of current in line conductors due to an electron avalanche

Transfer of bulk electric power required in the modern world can be realized only through Extra High Voltage and Ultra High Voltage transmission lines. The scenario is dominated by overhead lines in which electromagnetic noise generated by corona is an important concern. Corona-induced currents propagate along line conductors producing electromagnetic noise, which is essential to be quantified. Towards this end, the present work aims to investigate basic mechanism of current induction using an isolated avalanche developing under space charge modulated background field. Beginning with the complete electric field produced by an avalanche, the corresponding induced currents are quantified, and the structure of electric and magnetic fields is extracted. The well-known time-domain electric field integral equation is employed for the purpose. The basic issue with long transmission line modelling is amply demonstrated. Since single avalanche is the basic mechanism behind corona, general inferences drawn are applicable for corona on transmission lines.

Modeling of Resonant Tunneling Diode Oscillators Based on the Time-Domain Boundary Element Method

We demonstrate how the coupling of a full-wave time-domain boundary element method (BEM) solver with a circuit solver can be used to model 1) the generation of high frequency oscillations in resonant tunneling diode (RTD) oscillators, and 2) the mutual coupling and synchronization of non-identical RTDs with significant differences in frequencies to achieve coherent power combination. Numerical simulations show a combined output power of up to 3.7 times a single oscillator in synchronized devices. The non-differential conductance of the RTD is modeled as a lumped component with a non-linear current-voltage relationship. The lumped element is coupled to the radiating structure using a finite-gap model in a consistent and discretisation independent manner. The resulting circuit equations are solved simultaneously and consistently with time-domain electric field integral equations that model the transient scattering of electromagnetic (EM) fields from conducting surfaces that make up the device. This paper introduces three novel elements: (i) the application of a mesh independent feed line to the modelling of feed lines of RTD devices, (ii) the coupling of the radiating system to a strongly non-linear component with negative differential resistance, and (iii) the verification of this model with circuit models where applicable and against the experimental observation of synchronisation when two RTDs are placed in close proximity. These three elements provide a methodology that create the capacity to model RTD sources and related technology.

Applying Loop-Star Functions With Multibranch Rao-Wilton-Glisson Basis Functions in the Time Domain Electric Field Integral Equation

In this paper, novel loop-star basis functions with the multibranch Rao-Wilton-Glisson (MB-RWG) basis functions are applied to obtain the low-frequency stable solution of the time domain electric field integral equation (TD-EFIE). With the help of the MB-RWG basis functions, it is possible to realize the quasi-Helmholtz decomposition in time domain between different regions that are discretized separately with nonconformal meshes, by modifying slightly the generation of the loop and star bases at the interfaces. Especially, the temporally differentiated form of the TD-EFIE is tested by the star basis, while the undifferentiated form of the TD-EFIE is tested by the loop basis. By means of the filtering technique, the components of the solution which have the potential of growing exponentially have been treated. Numerical results illustrate the accuracy and robustness of the proposed method in analyzing the transient scattering problems of perfectly electrical conducting (PEC) objects.

A novel space hybrid meshing technique combined with Laguerre polynomial to solve TD-EFIE

In this paper, we have developed a system of Time Domain Electric Field Integral Equation (TD-EFIE) for the resolution of a scattering electromagnetic problem by various shapes illuminated by an incident transient wave. The solvability of the TD-EFIE system is based on the Time Domain Method of Moments (TD-MoM) using Galerkin's method in space and time domains. In time domain, a linear combination of Laguerre functions is used as temporal basis functions. In order to approximate the unknown space variation of the transient current, we develop a novel space hybrid basis functions derived from the Rao-Wilton-Glisson function. The elaborated hybrid mesh technique using hexagonal and triangular mesh is applied for various analyzed structures: regular grid for planar structures and irregular grid for curved structures. The validity, the efficiency and the precision of developed meshing techniques are presented and discussed.

Stable and Accurate Marching-on-in-Time Solvers of Time Domain EFIE, MFIE, and CFIE Based on Quasi-Exact Integration Technique

The development of time-domain integral equation solvers with robust late-time stability properties has been a long-standing quest. Among the many methods that have been investigated, those leveraging quasi-exact integration techniques appear to be most successful. This article presents stable and accurate marching-on-in-time (MOT) solvers for time-domain electric, magnetic, and combined field integral equations (EFIE, MFIE, and CFIE) based on quasi-exact integration techniques. The novel MOT solvers exhibit excellent stability while yielding highly accurate results, as demonstrated by various numerical examples. In addition, the solvers' excellent stability and accuracy properties are used to examine spurious modes encountered when time-domain integral equations are applied to closed surfaces. It is demonstrated that MOT solutions to the time domain EFIE and MFIE often are polluted by spurious modes at the cavity's resonant frequencies whereas those of the CFIE solver are devoid of such contamination.

A comparison of three temporal basis functions for the time-domain method of moments (TD-MoM)

Three techniques are applied for the resolution of the time-domain electric field integral equation (TD-EFIE) for a thin wire antenna based on the well-known time-domain method of moments. Three temporal bases are used to approximate the unknown temporal variation. The first is derived from Dirac functions, the second from B-spline functions, and the third from Laguerre functions. Piecewise triangular functions are used for the spatial expansion and testing function for the wire structure using Galerkin’s method. The aim is to compare these three techniques in terms of their formulation, complexity, and accuracy. Numerical results are presented to identify which time technique is best suited to solve such transient electromagnetic problems.

Large Time Step and DC Stable TD-EFIE Discretized With Implicit Runge–Kutta Methods

The Time Domain-Electric Field Integral Equation (TD-EFIE) and its differentiated version are widely used to simulate the transient scattering of a time dependent electromagnetic field by a Perfect Electrical Conductor (PEC). The time discretization of the TD-EFIE can be achieved by a space-time Galerkin approach or, as it is considered in this contribution, by a convolution quadrature using Implicit Runge-Kutta methods. The solution is then computed using the Marching-On-in-Time (MOT) algorithm. The differentiated TD-EFIE has two problems: (i) the system matrix suffers from ill-conditioning when the time step increases (low frequency breakdown) and (ii) it suffers from the DC instability, i.e. the formulation allows for the existence of spurious solenoidal currents that grow slowly in the solution. In this work, we show that (i) and (ii) can be alleviated by leveraging quasi-Helmholtz projectors to separate the Helmholtz components of the induced current and rescale them independently. The efficacy of the approach is demonstrated by numerical examples including benchmarks and real life applications.

Speed-up of scattered field formulation of FDTD method combining with time-domain EFIE by using spherical harmonic expansion of Green’s function

This paper proposes a scattered field formulation of the finite-difference time domain method (S-FDTD) combining with a time-domain electric field integral equation (TD-EFIE) based on surface equivalence theorem. The scattered field formulation enables us to reduce computation cost effectively for the case that scatterers are located at a far distance from field sources of electromagnetic wave since we need to calculate the scattered field component only on the region in the vicinity of the scatterers. Then it is required for the use of the scattered field formulation that the field source for the incident field component has to be expressed by any analytical solutions such as a point dipole source, plane wave, and so on. In this work, the scattered field formulation of the FDTD method is applied to cases that there are no analytical expressions of the field sources, in which the field source contains conductors such as antennas, combining with the TD-EFIE on a virtual surface which encloses the field sources. Then, it is known that calculation of the TD-EFIE itself is time consuming. This paper considers speed-up of the S-FDTD simulation based on the TD-EFIE using a spherical harmonic expansion of Green’s function of Helmholtz equation, additionally.

An Improved MOD Discontinuous Galerkin Method for Solving Time-Domain Electric Field Integral Equation

An improved marching-on-in-degree discontinuous Galerkin method for solving time-domain electric field integral equation is presented in this letter. By adopting half-RWGs as spatial basis functions, both conformal and nonconformal discretization scattering problems can be simulated accurately. A combination of three weighted Laguerre polynomials is utilized as new temporal basis functions to derive final equations, where some accumulation terms are eliminated. Therefore, the time spent on computing accumulation term during the iterative procedure is reduced significantly, with the total solution process accelerated. Some numerical examples are presented to demonstrate both accuracy and efficiency of our method.

Analyses of Initial Condition Problem and Linear Loop Modes in the Solution of the Derivative Form of Time-Domain Electric Field Integral Equation

The marching-on-in-time solution of the derivative form of time-domain electric field integral equation has long been suffering from three typical problems: internal resonant modes, initial condition problem, and linear loop modes. This letter focuses on the last two problems. The possible appearance of initial condition problem is attributed to the inappropriate choice of temporal basis function; to avoid it, a constraint condition for temporal basis functions is presented in this letter. Additionally, linear loop modes defined as the constant and linearly increasing components of current density in time are studied in theory. Theoretical analysis indicates that limitation on machine precision makes linear loop modes approximately belong to the null space of discretized derivative form of time-domain electric field integral equation. It is also shown that errors due to space–time discretization (except time step size) and double surface integration have no visible influence on linear loop modes. As numerical experiments will demonstrate, the magnitudes of linear loop modes are approximately proportional to the incident field at the initial time and the reciprocal of time step size.

Accurate Simulation of Shielding Effectiveness of Metallic Cabins Using an Improved Calderón Preconditioner-Based Time-Domain Integral Equation Method

An efficient simulation of shielding effectiveness of some metallic cabins or enclosures is performed using an improved Calderon preconditioner-based time-domain integral equation method. During its implementation, Rao–Wilton–Glisson and Buffa–Christiansen functions are used as the spatial basis functions for solving the Calderon preconditioned time-domain electric field integral equation, which generate a set of well-conditioned system matrices irrespective of the mesh density. Moreover, a set of approximate prolate spheroidal wave functions is chosen as the temporal basis functions; therefore, both late-time stability and accuracy are improved significantly for simulating electromagnetic pulse responses in some typical cabins. Numerical results are validated in comparison with those obtained by some commercial software. These results are useful for further evaluating electromagnetic environment effects in some complex systems.

Discontinuous Galerkin Method Using Laguerre Polynomials for Solving a Time-Domain Electric Field Integral Equation

A discontinuous Galerkin time-domain electric field integral equation method based on a marching-on-in-degree (MOD) scheme for simulating transient responses of perfect electric conductor objects is proposed in this letter. With the help of half-Rao–Wilton–Glisson spatial basis functions, we can simulate transient scattering responses of three-dimensional structures with either conformal or nonconformal meshes. The weighted Laguerre polynomials are chosen as temporal basis functions and implemented for the MOD scheme. Under such circumstances, the developed algorithm eliminates late-time instabilities and the limitation of Courant–Friedrichs–Lewy criteria. In addition, we adopt a fast-Fourier-transform-based blocking scheme to accelerate the temporal convolution of the MOD procedure. Some numerical examples are given to validate its stability and accuracy.

A Fast Marching-on-in-Degree Solution for Analysis of Conductors Coated With Thin Dispersive Dielectric

In this paper, a novel marching-on-in-degree-based time-domain electric field integral equation (TD-EFIE) is proposed for the analysis of conductors coated with thin dispersive dielectric. The algorithm begins with representing the polarization electric current/charge of the dielectric part in terms of the surface electric current of the conductor. Then, the TD-EFIE can only be formulated with the surface electric current by using the thin dielectric sheet approach. Since the spatial unknowns are only assigned to the surface of the conductor, the computational resources can be saved significantly. Besides, the analytical solution can be obtained for the convolution of the medium susceptibility and the divergence of surface electric current. Moreover, the time variables can be eliminated by using the Galerkin testing with the weighted Laguerre polynomials. As a result, the late time stability can be guaranteed. Numerical examples of Debye, Lorentz, and Drude media are given to demonstrate the validity of the proposed algorithm.

Efficient parallel analysis of transient plane wave scattering from nonplanar doubly periodic structures

This paper presents a fast scheme for solving a time-domain electric field integral equation pertinent to the analysis of transient plane wave scattering from nonplanar doubly periodic structures. The proposed scheme efficiently evaluates scattered fields from periodic structures by decomposing the sources that produce them into present and past components. The fields radiated by present sources are evaluated classically. In contrast, fields generated by past sources are expanded into time domain Floquet waves with modal amplitudes that are evolved in both space and time using a spectral time-stepping scheme complemented by Huygens’ principle based boundary conditions. The kernel equations and efficient parallel evaluation associated with the scattered fields generated by past sources are validated and effectiveness of the resulting algorithm are demonstrated by a number of examples.

Time-Domain Analysis of Transient Electromagnetic Scattering From Dielectric Objects Based on Electric Field Integral Equations

Time-domain analysis for transient electromagnetic scattering from dielectric objects is formulated through time-domain electric field integral equations (TDEFIEs). The TDEFIEs are usually solved by combining the method of moments (MoM) in space domain and march-on-in-time (MoT) scheme in time domain. In the space-domain MoM, the electric current density is represented with the Rao-WiltonGlisson (RWG) basis function, while the magnetic current density is expanded by the n×RWG basis function, which may not be good, since it is not divergenceor curl-conforming. Also, the MoT scheme has a well-known late-time instability problem, which will aggravate in timedomain surface integral equations with penetrable media. We present a different strategy to solve the TDEFIEs by using the Nystrom method to discretize the space domain and using the Galerkin method with Laguerre basis and testing functions to discretize the time domain. The proposed method can fully overcome the problems of conventional methods as the presented numerical examples have demonstrated.

EFIE Solution of Transmitted Electromagnetic Field Through a Perforated Metallic Shield by a Marching-on-in-Degree Approach

An accurate modeling technique is proposed to solve the problem of the penetrated transient field through a perforated metallic shield. In this technique, the method of moments (MoMs) is adopted to treat the governing time-domain electric field integral equation for the unknown electric current density on the metallic surface of the shield. The weighted Laguerre polynomials are used as entire-domain temporal basis functions in the MoM-based technique to set the solution approach by the marching-on-in-degree (MOD) method. To examine the accuracy of the proposed modeling technique, simulated results are presented for a finite and an infinite perfect electric conductor (PEC) plates as well as for a metallic box illuminated by a modulated/unmodulated incident wave. The accuracy of the proposed technique is demonstrated by comparing the simulated results with those obtained using the well-known Computer Simulation Technology (CST) finite integration technique and FEKO frequency-domain MoM.

A DC-Stable, Well-Balanced, Calderón Preconditioned Time Domain Electric Field Integral Equation

The marching-on-in-time (MOT) solution of the time domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of problems, including: 1) instability; 2) spurious static contributions plaguing the solution; 3) low-frequency breakdown; and 4) dense discretization breakdown. The first issue can be resolved by employing proper space-time Galerkin discretization schemes and accurate quadrature methods. The second and the third issue have been resolved by the quasi-Helmholtz Projected TD-EFIE (qHP-TDEFIE). This contribution introduces a multiplicative preconditioner which can be applied to the qHP-TDEFIE, without further modifying the original scheme. This preconditioner is based on Calderon techniques and guarantees that the MOT system can be solved efficiently using iterative methods, not only for large time step sizes but also for dense spatial discretizations, and for both simply and multiply connected geometries.

A DC Stable and Large-Time Step Well-Balanced TD-EFIE Based on Quasi-Helmholtz Projectors

The marching-on-in-time (MOT) 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 space-time 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 MOT system is obtained. Furthermore, the scaling of these basis and testing functions ensure that the system remains well conditioned for large-time steps. The loop–star decomposition is performed using quasi-Helmholtz projectors 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.

Marching-on in degree method for electromagnetic coupling analysis of carbon nanotubes (CNT) dipoles array

Transient electromagnetic analysis of carbon nanotubes (CNT) dipoles is presented. The thin wire approximation and the scattering theory are employed to formulate the problem. Therefore, an accurate system of coupled time domain electric field integral equations (TD-EFIE) is developed in order to predict the responses of the CNT dipoles excited by transient electromagnetic waves. To obtain stable and accurate solutions based on proposed system, the marching-on in degree method is applied with piecewise functions as space basis and weighted Laguerre Polynomials as temporal basis. Numerical results involving transient currents on each CNT dipole are presented. The electromagnetic coupling analysis between the CNT dipoles is conducted depending on the separate distances and the number of the CNT dipoles. Hence, the physical coupling effects are proved.