Time-domain Surface Integral Equations Research Articles

A Wavelet-Enhanced PWTD-Accelerated Time-Domain Integral Equation Solver for Analysis of Transient Scattering From Electrically Large Conducting Objects

A wavelet-enhanced plane-wave time-domain (PWTD) algorithm for efficiently and accurately solving time-domain surface integral equations (TD-SIEs) on electrically large conducting objects is presented. The proposed scheme reduces the memory requirement and computational cost of the PWTD algorithm by representing the PWTD ray data using local cosine wavelet bases (LCBs) and performing PWTD operations in the wavelet domain. The memory requirement and computational cost of the LCB-enhanced PWTD-accelerated TD-SIE solver, when applied to the analysis of transient scattering from smooth quasi-planar objects with near-normal incident pulses, scale nearly as $O(N_{s} \log N_{s})$ and $O(N_{s}^{1.5})$ , respectively. Here, $N_{s} $ denotes the number of spatial unknowns. The efficiency and accuracy of the proposed scheme are demonstrated through its applications to the analysis of transient scattering from a 185-wavelength long NASA almond and a 123-wavelength long Airbus A-320 model.

Quantum-corrected transient analysis of plasmonic nanostructures.

A time domain surface integral equation (TD-SIE) solver is developed for quantum-corrected analysis of transient electromagnetic field interactions on plasmonic nanostructures with sub-nanometer gaps. "Quantum correction" introduces an auxiliary tunnel to support the current path that is generated by electrons tunneled between the nanostructures. The permittivity of the auxiliary tunnel and the nanostructures is obtained from density functional theory (DFT) computations. Electromagnetic field interactions on the combined structure (nanostructures plus auxiliary tunnel connecting them) are computed using a TD-SIE solver. Time domain samples of the permittivity and the Green function required by this solver are obtained from their frequency domain samples (generated from DFT computations) using a semi-analytical method. Accuracy and applicability of the resulting quantum-corrected solver scheme are demonstrated via numerical 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.

A Hybrid Scheme With Nyström Discretization for Solving Transient Electromagnetic Scattering by Conducting Objects

The interaction of transient electromagnetic (EM) waves with objects can be formulated by the integral equation approach in time domain. For conducting objects or homogeneous penetrable objects, the time-domain surface integral equations (TDSIEs) can be applied. Traditionally, the TDSIEs are solved by combining the method of moments (MoM) in spatial domain and a march-on-in-time (MoT) scheme in temporal domain. The MoM usually requires conforming meshes with a well-designed basis function and the MoT may suffer from a stability problem. In this work, we propose a hybrid scheme in which the Nystrom method is used in spatial domain while the MoM with Laguerre function as basis and testing functions is employed with a march-on-in-degree (MoD) manner in the temporal domain. The Nystrom method does not require any basis and testing functions and can use nonconforming meshes in spatial domain, whereas the MoM with Laguerre function as basis and testing functions can fully eliminate the stability problem in temporal domain. Numerical examples for EM scattering by typical conducting objects are presented to demonstrate the scheme and its merits have been verified.

A Stable Marching On-In-Time Scheme for Solving the Time-Domain Electric Field Volume Integral Equation on High-Contrast Scatterers

A time-domain electric field volume integral equation (TD-EFVIE) solver is proposed for characterizing transient electromagnetic wave interactions on high-contrast dielectric scatterers. The TD-EFVIE is discretized using the Schaubert–Wilton–Glisson (SWG) and approximate prolate spherical wave (APSW) functions in space and time, respectively. The resulting system of equations cannot be solved by a straightforward application of the marching on-in-time (MOT) scheme since the two-sided APSW interpolation functions require the knowledge of unknown “future” field samples during time marching. Causality of the MOT scheme is restored using an extrapolation technique that predicts the future samples from known “past” ones. Unlike the extrapolation techniques developed for MOT schemes that are used in solving time-domain surface integral equations, this scheme trains the extrapolation coefficients using samples of exponentials with exponents on the complex frequency plane. This increases the stability of the MOT-TD-EFVIE solver significantly, since the temporal behavior of decaying and oscillating electromagnetic modes induced inside the scatterers is very accurately taken into account by this new extrapolation scheme. Numerical results demonstrate that the proposed MOT solver maintains its stability even when applied to analyzing wave interactions on high-contrast scatterers.

Time-Domain Techniques for Transient Scattering from Dielectric Bodies and Sea Surface Governed by Jonswap's Sea Spectra

A time-domain surface integral equation (TDCFIE) approach is utilized to calculate the transient scattering from arbitrarily shaped, three-dimensional dielectric bodies. In conjunction with the marching-on-in-time (MOT) method, the TDCFIE-MOT method is used to derive explicit expressions for the present-time current as a function of the incident field. Sample results showing various geometries are presented and are compared with other numerical techniques. Finally, by an incident Gaussian plane tapered wave, transient scattering from the sea surface governed by Jonswap's sea spectra is computed.

Time domain surface impedance concept for low frequency electromagnetic problems—Part II: Application to transient skin and proximity effect problems in cylindrical conductors

A time domain boundary element formulation employing the surface impedance boundary conditions (SIBCs) is developed for the 3-dimensional transient eddy current problem of cylindrical conductors. SIBCs of different orders of approximation are implemented using the perturbation technique in the small parameter proportional to the ratio of the skin depth and characteristic size of the conductor cross-section. The formulation consists of a set of time domain surface integral equations that have identical left-hand sides and can be solved using the same program procedure. The number of equations is determined by the order of approximation of the SIBC, namely: solutions in the perfect electrical conductor (PEC) limit (lowest order) and in the so-called Rytov approximation (highest order) are given by one and four equations, respectively. It is demonstrated that each equation admits separation of variables into space and time components, a property that significantly reduces computational costs compared with traditional time domain formulations that require the integral equations to be solved at each time step. For the purpose of validation, a test problem is solved by the proposed formulation and by the ‘original’ BEM based on the time-dependent fundamental solution. Conditions of applicability are discussed and the effect of such factors as the shape of the incident current pulse and proximity effect are considered.

Analysis of transient scattering from composite arbitrarily shaped complex structures

A time-domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures. The solution method is based on the method of moments (MoM) and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions. An implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain. The usual late-time instabilities associated with the time-domain integral equations are avoided by using an implicit scheme. Detailed mathematical steps are included along with representative numerical results.