Introduction to the finite-difference time-domain (FDTD) method for electromagnetics (Q2904685)

The book consists of eight chapters on 236 pages and two appendices, a glossary of relevant terms, and a biography of the author. The chapters are completed by references for additional reading. Problems to improve the skills of the reader are formulated at the end of all chapters, except the introduction. Solutions to the problems are not given.NEWLINENEWLINE The well-written book is intended for students, researchers, and professional engineers. It can be used to teach a course at the undergraduate or early graduate level. The described implementation of the finite-difference time-domain (FDTD) in \texttt{Matlab} allows the solution of complex three-dimensional electrodynamic problems.NEWLINENEWLINEThe author starts the Introduction with a brief, but accurate history of the FDTD method, primarily proposed by \textit{K. S. Yee} [IEEE Trans. Antennas Propag., 14, No. 3, 302--307 (1966; Zbl 1155.78304)] using staggered grids for the electric and magnetic fields in space and time and the central difference approximation of the spatial and temporal derivatives in Maxwell's equations. The method has been improved parallel to the advances in computer technology until this day including the use of parallel computers and the integration of new features such as perfect matched layer absorbing boundary conditions and subcell techniques. Lists of publications with the most important innovations in advancing the FDTD algorithm are given.NEWLINENEWLINEAdvantages and limitations of the FDTD method are outlined. A number of available FDTD based software packages are mentioned. Two alternative solution methods, the MoM (method of moments) and the FEM (finite element method), are shortly compared with the FDTD algorithm.NEWLINENEWLINEThe contents of the following 7 chapters and 2 appendices are briefly characterized.NEWLINENEWLINEChapter 2 is devoted to the propagation of electromagnetic waves in transmission lines. The corresponding time-dependent one-dimensional transmission line equations, a set of two first-order partial differential equations, describe the line currents and voltages at a certain time at a position along the transmission line axis. Using a Taylor series approximation, the central difference approximation of the first-order derivatives of partial differential equations is introduced in detail including the proof of the second-order accuracy resulting in a recursive formulation for the determination of the line currents and voltages in time, also known as ``leap-frog time update strategy''. The staggered grid and the stencil of points are illustrated by figures.NEWLINENEWLINEIn order to characterize the error of the discrete solution, the numerical dispersion is treated considering a known monochromatic plane wave propagating on the transmission line. Inserting the corresponding discrete plane wave into the discretized wave equation results in a dispersion relation. Studying this relation useful results for the choice of the time and spatial steps are derived. Especially, the so-called ``magic time step'', for that the errors due the discretization in time and space are exactly cancelled, independent of frequency and step size, is depicted, but it is a result which is only valid for a single lossless transmission line.NEWLINENEWLINEUsing an eigenvalue analysis of the second-order difference operator the stability of the recursive procedure is studied resulting in the Courant-Friedrichs-Lewy (CFL) stability criterion for the explicit update scheme.NEWLINENEWLINEThe voltage and the current are not known at the same point in time and space. Thus, to retain the accuracy and stability special load and source conditions are derived in the last section of this chapter.NEWLINENEWLINEChapter 3 starts with Maxwell's equations, consisting of Faraday's law, Ampere's law, Gauss's laws, and the continuity equations for the electric and magnetic current densities, in both differential and integral form. MKS units are used. The tangential and normal boundary conditions for the electromagnetic fields and the flux densities are derived directly from Maxwell's equations in their integral form differentiating between surfaces that support surface current densities and/or surface charge densities and surfaces that are free from them, including the special boundary conditions for perfectly conducting media.NEWLINENEWLINEAfter formulating the foundational concepts of the FDTD method in Chapter 2, the Yee algorithm for the three-dimensional Maxwell's equations based on their differential form is derived using uniform staggered grids and the central difference approximation for both the space and time. The six first-order difference equations for the electric and magnetic field intensities, the discrete forms of Faraday's and Ampere's laws, have similar to the one-dimensional transmission line equations second-order accuracy. It is shown that the discrete approximations of Maxwell's equations satisfy the discrete form of Gauss' laws independent of the step size, an advantage of the Yee algorithm, otherwise spurious modes could occur.NEWLINENEWLINEAlternatively, the derivation of the Yee algorithm from the integral form, the so-called finite integration technique, pioneered by \textit{T. Weiland} [``Discretization method for the solution of Maxwell's equations for six-component fields'' (German), AEU International Journal of Electronics and Communications, 31, 116--120 (1977)], is given.NEWLINENEWLINESimilar to Chapter 2, a stability analysis leads to the well-known CFL condition, and a corresponding analysis of the numerical dispersion using a uniform monochromatic plane wave characterize the behavior of the errors resulting in recommendations for the space discretization. The phase and group velocities are shown to be frequency dependent and anisotropic.NEWLINENEWLINEThe question of how the boundary conditions for Maxwell's equations are satisfied in the discrete FDTD space is treated in detail including the special case of a perfectly conducting media.NEWLINENEWLINEBringing forward the argument that, in the case of lossy media the losses of most physical problems can be treated as conductor losses, Ampere's law is written taking into account a loss term, and it is shown that in this case via a linear time averaging the second-order accuracy can be maintained. Hints are given how the CFL stability criterion for this lossy case can be satisfied and this can be verified. Corresponding tips are outlined for inhomogeneous lossy media.NEWLINENEWLINEA special FDTD algorithm is derived for dispersive media, i.e., for media with frequency dependent permeabilities or permittivities. Strongly dispersive media are for instance meta-materials. The corresponding FDTD method is based on the so-called auxiliary differential equation and a changed Ampere's law taking into account an auxiliary electric polarization vector caused by a linear isotropic Debye media model.NEWLINENEWLINEThere are problems that need very fine grids only in some regions. The use of a fine grid everywhere leads to large computational costs. Thus, nonuniform staggered grids are of interest. The corresponding FDTD algorithm is formulated and the appropriate CFL limit is presented. The full second-order accuracy can not be maintained in this case.NEWLINENEWLINEThe solution of real world problems with the FDTD method requires additionally the treatment of source excitations. Thus, Chapter 4 is devoted to a number of source excitations which approximate the real engineering sources within the discrete FDTD space. For example, there are simulations that require a signal with a finite bandwidth about a centered frequency. Presented are source signatures, such as the Gaussian pulse, the Blackman-Harris window and corresponding differentiated pulses, discrete voltage and current source excitations, lumped circuit source excitations, plane wave excitations, and so on.NEWLINENEWLINEChapter 5 is focused on absorbing boundary conditions (ABCs) to simulate electromagnetic fields by the FDTD method. Using a finite dimensional grid, unbounded domains have to be truncated with a special boundary condition which avoids ideally reflection errors. There are a number of absorbing boundary conditions that fulfil sufficiently this requirement. The ABCs are classified into three categories, i.e., local ABCs, global ABCs, and absorbing media. The global ABCs are the most computationally expensive methods. Thus, the author focuses on the two other techniques. The idea of absorbing media consists in the use of a physical layer as boundary with a material property that absorbs the outgoing waves like an ``anechoic chamber''. This powerful method is treated in an own chapter.NEWLINENEWLINEThe local ABCs apply local derivatives of the fields on the truncated boundary for the calculation of reduced reflections. Three methods are described. To simplify the understanding, the principle of ABCs the author starts with the derivation of the first-order Sommerfeld ABC followed by two improved second-order formulations of so-called Mur-ABCs, namely the Higdon ABC taking into account additionally layered media, and the Betz-Mittra ABC that improves the last annihilating also evanescent modes.NEWLINENEWLINEThe relative high refection errors of the local ABCs cause an outer truncation boundary of the unbounded domain that have to be sufficiently far of the device with the consequence that the number of grids to be taken into account increases. This disadvantage has been overcome by the introduction of the perfectly matched layer absorbing media (PML). Chapter 6 is devoted to PMLs. The constitutive parameters of the PMLs are selected such that waves are impinging into these absorbing media without reflection. Thus, the outer truncation boundaries can be selected closer to the device. The PML was pioneered by \textit{J.-P. Berenger}, [J. Comput. Phys. 114, No. 2, 185--200 (1994; Zbl 0814.65129)] using a split-field formulation of Maxwell's equations. Berenger's PML can also represented in a stretched coordinate frame with complex metric-tensor coefficients and as an anisotropic medium. The properties of the PMLs are deduced, and it is outlined that these are equivalent in practice for both anisotropic media and the stretched coordinate form. The reflection errors of the PMLs are much smaller than one of the local ABCs. Hints are given of how to select the layers of PML and how to scale the constitutive parameters of the layers in order to reduce the reflection errors which are caused by the piecewise constant approximation of the discrete fields and the staggered grids.NEWLINENEWLINEIntroducing new auxiliary variables consistently with the Yee-algorithm, the author presents a more robust and computationally efficient method of the PML for the FDTD method, the so-called convolutional-PML (C-PML) [\textit{J. A. Roden} and \textit{S. D. Gedney}, ``Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media'', Microwave and Optical Technology Letters 27, No. 5, 334--339 (2000; \url{doi:10.1002/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A})] that absorbs both propagating and evanescent waves at high and low frequencies. An example is presented to verify the performance of the C-PML.NEWLINENEWLINESo-called subcell models are treated in Chapter 7. Practical problems contain often parts with local geometric features that are finer than the grid cells required for the large complex structure to be simulated. The uniformity of the gridding would require a very fine grid spacing for the whole structure, i.e., a rigorous increase of the number of grid cells. To avoid a drastically increase of computational costs a number of methods are presented that locally approximate the fields near the geometric finer parts and/or use deformations of the rectangular grid. Presented are the thin-wire subcell model and conformal FDTD methods.NEWLINENEWLINEA thin wire is a wire with a radius much smaller than its length and the wavelength, and in the model also smaller than the grid cell. After the description of the basic model special cases are considered, such as the curvature correction for cylindrical faces, the treatment of the wire ends, and wire antennas exited by transmission line feeds.NEWLINENEWLINETo model more general geometries conformal FDTD methods are treated, which achieve for instance a more computational efficiency compared to the stair case approximation of a sphere. The aim consists in an improved approximation of arbitrary surface geometries. First approaches suffer from late-time instabilities. The author presents three modern methods that overcome this disadvantage, i.e., the Dey-Mittra conformal FDTD method with an concentration on the accuracy, the Yu-Mittra algorithm with a reduced stability, and the algorithm of Benkler-Chavannes-Kuster, that is oriented towards a balance between accuracy and stability.NEWLINENEWLINEChapter 8 is devoted to the selection of some post-processing methods which provide special quantities of the device under investigation based on the time-dependent electromagnetic fields calculated by the FDTD algorithm.NEWLINENEWLINE The discrete port parametrization extracts the port voltage, current and impedance, which characterize a discrete network, from a FDTD simulation via line integrals. Other topics are the calculation of admittance (Y) parameters, of scattering (S) parameters, and near-field to far-field transformations.NEWLINENEWLINEThe two appendices contain \texttt{Matlab}-codes of the FDTD algorithm.NEWLINENEWLINEAn implementation of the one-dimensional transmission line equation for lossless media described in Chapter 2 is given in form of a set of routines in Appendix A including explanations of how to manage the indexing or the boundary conditions. This appendix has the aim to develop the understanding of the reader of how a numerical algorithm can be translated into a computer program.NEWLINENEWLINEAn efficient \texttt{Matlab}-code of the full three-dimensional FDTD algorithm is presented in Appendix B taking into account adsorbing boundaries, lossy and inhomogeneous media, conformal boundaries, and subcell models. The implementation of the algorithm is elucidated.