9 - Computational Electromagnetics: The Finite-Difference Time-Domain Method

The finite-difference time domain (FDTD) method is a numerical technique based on the finite difference concept. It is employed to solve Maxwell's equations for the electric and magnetic field distributions in both the time and spatial domains. The FDTD method utilizes the central difference approximation to discretize two of Maxwell's curl equations, namely, Faraday's and Ampere's laws, in both the time and spatial domains, and then solve the resulting equations numerically to derive the electric and magnetic field distributions at each time step and spatial point using the explicit leap-frog scheme. The FDTD solution, thus derived, is second-order accurate, although the difference formulation is first order, and is stable if the time step size is chosen to satisfy the special criterion.

This contribution studies a particular solution of Maxwell wave equations known as Hopfions by making use of the well-known Finite Difference Time Domain (FDTD) method. Hopfions are a family of solutions, in which the field lines are closed, forming different knot topologies that are preserved during its time evolution. These solutions have been studied in different analytical ways (e.g. [1]), but which to the best of our knowledge they have never been simulated before. Their simulation can be of interest because they are demonstrated to exist, not from purely analytical arguments, but from the direct numerical resolutions of the Maxwell's curl equations. This numerical solution would ease the study of how these fields could interact with other structures made of different materials, with other Hopfions, and on how they could be possibly be generated with antenna arrays.

Summary form only given. We present our recent results on the calculation of the distribution of the light scattered by red blood cells in the full range of scattering angle between 0 to 180 degrees using an accurate quantitative finite-difference-time-domain (FDTD) method. The FDTD algorithm allows computation of scattering from cells of any shape and with any index of refraction profile. The algorithm discretizes Maxwell's curl equations in space and time, resulting in a set of explicit finite-difference equations.

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described here. The method employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice. Small air-dielectric loss factors are introduced to improve the lattice truncation conditions and to accelerate convergence of cavity interior fields to the sinusoidal steady state. This method is evaluated with comparison to classical theory, method-of-moment frequency-domain numerical theory, and experimental results via application to a dielectric sphere and acylindrical metal cavity with an aperture. Results are also given for a missile-like cavity with two different types of apertures illuminated by an axial-incidence plane wave.

For over 100 years after the publication of Maxwell's equations in 1865, essentially all solution techniques for electromagnetic fields and waves were based on Fourier‐domain concepts, assuminga prioria time‐harmonic (sinusoidal steady‐state) field variation and possibly the existence of a particular Green's function or a set of spatial modes. In 1966, Kane Yee's seminal paper introduced a complete paradigm shift in how to solve Maxwell's equations, reporting a field evolution‐in‐time technique that subsequently evolved into the finite‐difference time‐domain (FDTD) method. In the decades since the publication of Yee's paper, there has been an explosion of interest in FDTD and related grid‐based time‐marched solutions of Maxwell's equations among scientists and engineers. During this period, FDTD modeling has evolved to an advanced stage enabling large‐scale simulations of full‐wave time‐domain electromagnetic wave interactions with volumetrically complex structures over large frequency ranges, spatial scales, and timescales. Currently, FDTD modeling spans the electromagnetic spectrum from ultralow frequencies to visible light. FDTD modeling is routinely conducted as an invaluable virtual laboratory bench in scientific inquiry and exploration in electrodynamics; as an integral part of the electromagnetic engineering design and optimization process; and as a powerful forward solver in imaging and sensing inverse problems. This article reviews the technical basis of the key features of FDTD solution techniques for Maxwell's equations and provides 18 modeling examples spanning the electromagnetic spectrum to illustrate the power, flexibility, and robust nature of FDTD computational electrodynamics simulations.

Partial discharge (PD) in transformer can be detected on-line by UHF technique. However, the relationship between the UHF signal received by UHF sensors and the PD sources is not very clear. It is very important studying the propagation properties of the PD signal. In this paper, the electromagnetic waves emitted by PD sources are investigated. The finite difference time domain (FDTD) technique is used to model wave transients. The model consists of a tank and a PD source. The wave transients at different time and position are simulated. The results show the propagation transients of the PD signal in the tank and wave propagation attenuates with distance between the source and the sensors.

In this paper, a difierential wavelet-based operator deflned on an interval is presented and used in evaluating the electromagnetic fleld described by Maxwell's curl equations, in time domain. The wavelet operator has been generated by using Daubechies wavelets with boundary functions. A spatial difierential scheme has been performed and it has been applied in studying electromagnetic phenomena in a lossless medium. The proposed approach has been successfully tested on a bounded axial-symmetric cylindrical domain.

SUMMARYIn this paper, a numerical meshless particle method is presented in order to solve the magnetoencephalography forward problem for analyzing the complex activation patterns in the human brain. The forward problem is devoted to compute the scalp potential and magnetic field distribution generated by a set of current sources representing the neural activity, and in this paper, it has been approached by means of the smoothed particle hydrodynamics method suitably handled. The Poisson equation generated by the quasi‐stationary Maxwell's curl equations, by assuming Neumann boundary conditions has been considered, and the current sources have been simulated by current dipoles. The adopted meshless particle model has provided good results in agreement with the analytical ones and by overcoming the drawback of the mesh generation. The numerical model has been validated, at first, in computing the electric potential and the external magnetic field for a dipole plunged near the upper surface of a homogeneous sphere simulating the human brain. Simulation results obtained by simulating two concentric spheres with different conductivities are also reported. Moreover, in order to better assess the validity of the proposed approach, a realistic human brain cortex model has been also simulated and compared with boundary element method results. A satisfactory agreement has been reached. Copyright © 2012 John Wiley &amp; Sons, Ltd.

In this paper an efficient algorithm is presented for the development of compact and passive macro‐models of electromagnetic devices through the systematic reduction of the order of discrete models for these devices obtained through the use of finite elements. The proposed methodology is founded on a new finite element formulation that casts Maxwell's curl equations in a state‐space form. Such state‐space representations are very compatible with a class of robust model order reduction techniques based on Krylov subspaces. However, the advantage of this compatibility appears to be hindered by the fact that the state matrix of the discretized Maxwellian system is of dimension almost twice that obtained from the finite element approximation of the electromagnetic vector wave equation. It is shown in this paper that the apparent penalty in both memory and computation efficiency can be avoided by a proper selection of the finite element expansion functions used for the discretization of the electromagnetic fields. More specifically, it is shown that the proper selection of expansion functions renders the state‐space form of the Maxwellian system equivalent to the discrete problem obtained from the approximation of the vector wave equation using tangentially continuous vector finite elements. This equivalence is then used to effect Krylov‐based model order reduction directly from the finite element approximation of the vector wave equation. In particular, a passive model order reduction algorithm is used for this purpose. The proposed reduced order macro‐modelling algorithm is demonstrated through its application to a variety of microwave passive components. Copyright © 2002 John Wiley &amp; Sons, Ltd.

The time-harmonic Discontinuous Galerkin Method (DGM) is a flexible forward solver for inverse scattering problems in electromagnetics. Standard DGM discretizations of Maxwell's curl equations, simultaneously solving for both electric and magnetic fields, are generally more accurate but less efficient than the DGM applied to the electric vector wave equation. A natural extension is to mix DGM formulations such that either the curl equation formulation or the wave equation formulation is locally adopted to preserve accuracy. Herein we present such a mixed DGM formulation for time-harmonic scattering problems, where the formulation choice is locally based on the presence of electric or magnetic constitutive parameters that differ from the background medium.

The frequency response of thin metallic wires that are found in networking cables is an important parameter required for the EMC/EMI evaluation of associated devices. One of the popular numerical methods used for such studies is the finite difference time domain (FDTD) method. In the FDTD technique, the contour-path thin wire model, based on Faraday's law, is widely used to simulate the effect of thin wires. This technique and its variants use perfect electric conductor (PEC) approximation for the wires and are not suitable to calculate the frequency response of thin power supply wires inside cables. This is due to the fact that the PEC approximation would result in an inaccurate estimation of the frequency response of such wires in the calculation of EMI. In this paper, we demonstrate the application of the surface impedance boundary condition (SIBC) along with the classical contour-path thin wire approximation in the FDTD method in order to provide improved reflection and transmission coefficients of thin wires that could be used for enhanced EMI prediction.

The finite-difference time domain (FDTD) technique and the Pade approximation with Baker's algorithm are used to calculate the mode frequencies and quality factors of cavities. Comparing with the fast Fourier transformation/Pade method, we find that the Pade approximation and the Baker's algorithm can obtain exact resonant frequencies and quality factors based on a much shorter time record of the FDTD output.

We present a 3‐D finite‐difference time domain (FDTD) solution of Maxwell's equations for modeling the near‐field radiation of dipole antennas in ground‐penetrating radar surveys. The antenna is represented as a conducting wire with a narrow gap in the middle. The electromagnetic field in the immediate vicinity of the wire is evaluated by numerical solution of an integral representation of Maxwell's equations. Integration contours and surfaces are defined by the finite‐difference cells surrounding the wire. Everywhere else the electromagnetic wavefield is calculated using conventional FDTD approximations of Maxwell's equations. The accuracy of the algorithm has been tested by comparing the simulated near‐field radiation of a dipole antenna between two dielectric media with corresponding experimental observations.

We present first-time calculations from the time-domain vector Maxwell's equations of spatial optical soliton propagation and mutual deflection, including carrier waves, in a 2-D homogeneous Kerr-type nonlinear dielectric. The nonlinear Schrodinger equation predicts that two co-propagating, in-phase spatial solitons remain bound to each other, executing a periodic separation. This disagrees with our new extensively tested finite-difference time-domain (FD-TD) solution of Maxwell's equations. FD-TD shows that co-propagating in-phase spatial solitons become unbound, i.e. diverge to arbitrarily large separations, if the ratio of soliton beamwidth to wavelength is order 1 or less. Not relying upon paraxial approximations or analogies to temporal soliton interactions, FD-TD appears to be a robust means of obtaining detailed models of the interaction of sub-picosecond pulsed light beams in nonlinear media directly in the space-time domain. >