A Selective Survey of the Finite-Difference Time-Domain Literature
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paper_title: The inclusion of wall loss in finite-difference time-domain thin-slot algorithms

paper_content:

The authors present two methods for incorporating slot wall loss into the finite-difference time-domain (FDTD) calculations. The walls are assumed to be good conductors. Loss is only applied to the current component that flows axially along the slot walls, which is generally the dominant component for slots that are long and narrow. The first method modifies an FDTD equation internal to the slot to include a surface-impedance contribution. This method is appropriate for the usual FDTD thin-slot formalisms. The second method includes the losses into a half-space integral equation that can be used by the recently introduced hybrid thin-slot algorithm. Results based on the two methods are compared for a variety of slot parameters and wall conductivities. >
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paper_title: Enhanced thin-slot formalism for the FDTD analysis of thin-slot penetration

paper_content:

The inclusion of the singularities of the field variation near a thin-slot is validated to be a useful enhancement of the standard thin-slot formalism for the finite-difference time-domain analysis of thin-slot penetration. >
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paper_title: Small-hole formalism for the FDTD simulation of small-hole coupling

paper_content:

Based on Bethe's (1944) small hole coupling theory, a small-hole formalism (SHF) for the finite-difference time-domain simulation of small hole coupling is presented and validated. Much of the computing time and computer resources are saved by the SHF presented here. >
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paper_title: Small aperture modeling for EMI applications using the finite-difference time-domain technique

paper_content:

A method for modeling small aperture scattering in the Finite-Difference Time-Domain (FDTD) method is described and evaluated. The method replaces a small aperture in a perfectly conducting screen with pairs of oppositely directed electric and magnetic dipoles on either side of the short-circuited screen. Equations are derived which determine the electric and magnetic currents induced by arbitrary fields within the FDTD computational domain. These currents are induced by the short-circuit fields at the aperture. To determine the short-circuit fields it is necessary to subtract out the fields of the electric and magnetic dipoles. An analytical solution for dipole radiation is used to determine the dipole fields, and it is shown that the FDTD dipole fields deviate from the continuum dipole fields at distances close to the dipole. The method is evaluated for an isolated circular aperture and the results are compared with other methods. An analysis of the various errors inherent in the method is given. The an...
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paper_title: Hybrid thin-slot algorithm for the analysis of narrow apertures in finite-difference time-domain calculations

paper_content:

A technique to incorporate a half-space aperture integral equation into a finite-difference time-domain (FDTD) code based on the offset Yee mesh (see K.S. Yee, ibid., vol. AP-14, p.302-7, 1966) is presented. To introduce the technique, linear apertures that are electrically narrow in both width and depth are discussed. The method incorporates an independent time-marching solution for the aperture problem into the FDTD code so that the aperture formally does not exist within the main FDTD mesh. A feedback scheme is introduced so that full exterior and interior coupling is included in the aperture solution. The technique is particularly useful for the analysis of apertures that are narrow both in width and depth with regard to the FDTD spatial cell. Previous thin-slot methods are shown to significantly underestimate the transverse gap electric field for this case, and an explanation for this is provided with the aid of the hybrid algorithm. >
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paper_title: Finite-Difference Solution of Maxwell's Equations in Generalized Nonorthogonal Coordinates

paper_content:

This article describes a time-domain finite-difference algorithm for solving Maxwell's equations in generalized nonorthogonal coordinates. We believe this approach would be most useful for applications where a uniform, uncurved, but oblique, meshing scheme could be applied in lieu of staircasing. This algorithm is similar to conventional leapfrog-differencing schemes, but now an additional leapfrogging between covariant and contravariant field representations becomes necessary.
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paper_title: Propagation in linear dispersive media: finite difference time-domain methodologies

paper_content:

Finite difference time-domain (FDTD) methodologies are presented for electromagnetic wave propagation in two different kinds of linear dispersive media: an Nth order Lorentz and an Mth order Debye medium. The temporal discretization is accomplished by invoking the central difference approximation for the temporal derivatives that appear in the first-order differential equations. From this, the final equations are temporally advanced using the classical leapfrog method. One-dimensional scattering from a dielectric slab is chosen for a test case. Provided that the maximum operating frequency times the time step is small and that the wave is adequately resolved in space, as shown in the error analysis, the agreement between the computed and exact solutions will be excellent. The attached data, which are associated with the four pole Lorentz dielectric and the five pole Debye medium, verify this assertion. >
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paper_title: Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target

paper_content:

Several improvements to the finite-difference time-domain (FDTD) method for calculating the radar cross section (RCS) of a perfectly conducting target are presented. Sinusoidal and pulsed FDTD excitations are compared to determine an efficient method of finding the frequency response of targets. The maximum cell size, the minimum number of external cells, and a method to eliminate field storage in the shielded internal volume of perfect conductors to reduce the computer storage requirements of FDTD are discussed. The magnetic-field DC offset induced by surface currents on perfectly conducting objects is observed, and its effects are removed by postprocessing to achieve convergence of the RCS calculations. RCS calculations using the FDTD method in two dimensions are presented for both square and circular infinite cylinders illuminated by both transverse electric and transverse magnetic polarized plane waves. The RCS of a metal cube in three dimensions is also presented. Good agreement between FDTD calculations and theoretical values was achieved for all cases, and parameters necessary to achieve this agreement are examined. >
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paper_title: A Novel Method to Analyze Electromagnetic Scattering of Complex Objects

paper_content:

The finite-difference time-domain (FD-TD) method is proposed as a means of accurately computing electromagnetic scattering by arbitrary-shaped extremely complex metal or dielectric objects excited by an external plane wave. In the proposed method, one first uses the FD-TD method to compute the near total fields within a rectangular volume which fully encloses the object. Then, an electromagnetic-field equivalence principle is invoked at a virtual surface of this rectangular volume to transform the tangential near scattered fields to the far field. To verify the feasibility of this method, the surface currents, near scattered fields, far scattered fields, and radar cross section of two canonical two-dimensional objects are presented. For these cases, it is shown that the FD-TD method provides magnitude of current and field predictions which are within ± 2.5 percent and further phase values within ± 30 of values predicted by the method of moments ( MOM) at virtually every point including in shadow regions.
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paper_title: Validation of FD-TD modeling of the radar cross section of three-dimensional structures spanning up to nine wavelengths

paper_content:

The first experimental validation is reported of the finite-difference time-domain (FD-TD) method for modeling the monostatic radar cross section (RCS) of three-dimensional conducting structures. The structures modeled and tested span up to nine free-space wavelengths ( k_{0}s =57 ). This represents a thirty-fold increase in electrical size over the previous analytically validated case of FD-TD modeling of radar cross section. It appears that the cases studied represent the largest detailed three-dimensional numerical scattering models of any type ever verified wherein a uniformly fine spatial resolution and the ability to treat nonmetallic composition is incorporated in the model. It was found that FD-TD provided a high modeling accuracy of 1 dB (with respect to the measurements) over at least a 40-dB dynamic range of radar cross section values for the nine-wavelength size objects, which exhibited such scattering physics as edge and corner diffraction, corner reflection (double bounce), and cavity penetration.
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paper_title: Finite-difference time-domain method for antenna radiation

paper_content:

The finite-difference time-domain (FDTD) method is used to model and predict the radiation patterns of wire and aperture antennas of three basic configurations. A critical step in each is the modeling of the feed. Alternate suggestions are made and some are implemented. The first antenna is a quarter-wavelength monopole and the second is a waveguide aperture antenna. In both bases the antenna is mounted on ground planes, either perfectly conducting or of composite material. The results obtained using the FDTD technique are compared with results obtained using the geometrical theory of diffraction (GTD) and measurements. The third configuration of interest is a pyramidal horn antenna. To model the flared parts of the horn, a staircase approximation was applied to the antenna surface. The computed radiation patterns compared well with measurements. >
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paper_title: Accurate computation of the radiation from simple antennas using the finite-difference time-domain method

paper_content:

The analysis of a group of three simple antennas is used to illustrate the accuracy of the FDTD (finite-difference time-domain) method and to show that various geometrical features are handled correctly by the method. Each antenna is a well-posed electromagnetic boundary value problem that corresponds to a realizable experimental model. The three antennas considered are of increasing complexity: an open-ended parallel plate waveguide, a cylindrical monopole, and a conical monopole. The FDTD calculations for these antennas were compared with analytical results (open-ended parallel plate waveguide) and accurate measurements in the time and frequency domains (cylindrical and conical monopoles). In all cases the agreement was excellent. >
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paper_title: Time-domain analysis of yagi-uda antennas using the FD-TD method

paper_content:

The fundamental importance of the Yagi-Uda antenna is its structural simplicity and its director and reflector actions. To fully comprehend the working principle of this antenna, it is important to be able to visualize the radiation mechanism directly in the time domain. In this paper, transient analysis of the Yagi-Uda antenna is performed using the FD-TD method. First, the input impedances and directivities obtained by this method are compared with the moment method, good agreement was found between the two methods, and the validity of this method was demonstrated. Next, the quasi-impulse responses of the antenna are calculated. The field pattern obtained by this method was visualized in the time domain. The radiation from the antenna edge is shown in the early time responses, and dispersion of the pulse shape with the passage of time is shown in late time responses. Moreover, the difference of characteristics due to the length of parasitic elements is shown by time-domain observation.
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paper_title: EM interaction of handset antennas and a human in personal communications

paper_content:

In personal communications, the electromagnetic interaction between handset-mounted antennas and the nearby biological tissue is a key consideration. This paper presents a thorough investigation of this antenna-tissue interaction using the finite-difference time-domain (FDTD) electromagnetic simulation approach with detailed models of real-life antennas on a transceiver handset. The monopole, side-mounted planar inverted F, top-mounted bent inverted F, and back-mounted planar inverted F antennas are selected as representative examples of external and internal configurations. Detailed models of the human head and hand are implemented to investigate the effects of the tissue location and physical model on the antenna performance. Experimental results are provided which support the computationally obtained conclusions. The specific absorption rate (SAR) in the tissue is examined for several different antenna/handset configurations. It is found that for a head-handset separation of 2 cm, the SAR in the head has a peak value between 0.9 and 3.8 mW/g and an average value between 0.06 and 0.10 mW/g for 1 W of power delivered to the antenna. Additionally, the head and hand absorb between 48 and 68% of the power delivered to the antenna. >
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paper_title: Performance analysis of antennas for hand-held transceivers using FDTD

paper_content:

The design of antennas for hand-held communications devices depends on the implementation of simulation tools that can accurately model general topologies. The paper presents the analysis of small antennas mounted on hand-held transceivers using the finite-difference time-domain (FDTD) method. The key features of the FDTD implementation are discussed, with particular emphasis placed upon modeling of the source region. The technique is used to predict the gain patterns and broadband input impedance behavior of monopole, planar inverted F, and loop antenna elements mounted on the handset. Effects of the conducting handset chassis, the plastic casing around the device, and lumped elements integrated into the antenna design are illustrated. Experimental results are provided to verify the accuracy of the computational methodology. The concept of antenna diversity is discussed, and key assumptions and expressions are provided that characterize the multipath fading fields. Several computational examples demonstrate the diversity performance of two receiving antennas on a single handset. >
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paper_title: Effects on portable antennas of the presence of a person

paper_content:

This work deals with the influence the presence of a person has on the performance of antennas for hand-held portable telephones. It is shown how input impedance, far-field radiation pattern, radiation efficiency, and the magnitude of the near field are affected by the presence of the human body. The investigations have been carried out by applying the finite-difference time-domain (FDTD) technique to a model of the telephone with a lambda /4 monopole antenna and a simple hand-head model of the operator. Results from this configuration are compared with results for the telephone box and antenna alone. Measurements have been performed to verify the results obtained by simulations. The results show that a shadow effect in the far-field pattern occurs in the direction of the person and that the radiation pattern, including polarization, changes considerably from the telephone with the antenna alone. The impedance results show a change in resonance frequency when the telephone is placed next to a person. Results for radiation efficiency show that close to half of the energy is absorbed by hand and head. On average a system loss of 3-4 dB should be included in a link budget, and there is considerable fading, even in a radio-anechoic chamber, when persons move around in a natural manner. >
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paper_title: FDTD analysis of power deposition patterns of an array of interstitial antennas for use in microwave hyperthermia

paper_content:

The finite-difference-time-domain (FDTD) method is used to calculate EM power deposition patterns in inhomogeneous tissue models of tumors. The radiated near-field patterns from an array of uniformly and step-insulated interstitial antennas were used as incident fields on the excitation planes to calculate the scattered fields and SAR patterns in tumors. A comparison of data shows that the FDTD solution, in this particular application, overcomes some of the modeling difficulties encountered in the method of moments. Results for some specific tumor geometries are also presented to show the effectiveness of the microwave interstitial heating techniques in treating large tumors. Other advantages of the FDTD method, such as improved accuracy in modeling dielectric interfaces and the ability to model large tumors, are illustrated. >
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paper_title: Calculations of heating patterns of an array of microwave interstitial antennas

paper_content:

The heating (temperature) distribution patterns of an array of uniformly and step-insulated interstitial antennas located in inhomogeneous tissue and cancerous regions of the human body are calculated. Specifically, the bioheat equation, which takes into account various heat exchange mechanisms such as blood flow rate, heat conduction, and metabolic heat generation, is solved using the finite difference method, and the specific absorption rate in tissue heated using an array of interstitial antennas is determined using the finite-difference-time-domain method. Numerical results that validate the computer program are presented, and the effect of varying parameters such as the blood-flow rate on the resulting heating rate and patterns is examined. Possible clinical implementation of the temperature-distribution-EM-power-deposition-pattern computer code in treatment planning is described. >
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paper_title: Rigorous analysis of microstrip phased array antennas using a new FDTD method

paper_content:

A finite-difference time-domain (FDTD) algorithm for radiating antennas that involve multiple, phased excitation sources is presented. The algorithm has the potential to analyse phased array antennas in the time domain. The authors apply it to a two-element microstrip antenna (MSA) phased array on a finite substrate, and compare the results as gained with an existing method and experiments.
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paper_title: FD-TD analysis of Vivaldi flared horn antennas and arrays

paper_content:

This paper summarizes a detailed computational study of Vivaldi flared horn antenna designs including single element, double element crossed-pair subarray elements, and linear arrays using the finite-difference time-domain (FD-TD) method. FD-TD, which numerically solves Maxwell's time dependent curl equations, accounts for the complex geometrical and electrical characteristics associated with this antenna design and array implementation. Validations have been made relative to a moment method (MM) model of an electrically large linearly tapered slot antenna. Also, good correlation is shown to exist in the primary features of the antenna patterns between computed and measured data for all cases. This work has achieved a number firsts for the FD-TD methodology. It represents the first "exact" computational model of a single quad Vivaldi slot antenna; the first "exact" computational model of a phased array of Vivaldi quad elements; and the first FD-TD model to demonstrate grating lobes for a phased array antenna of any sort. Lastly, this research represents an extensive study of the largest grid-based antenna models conducted to date. >
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paper_title: Contour path FDTD method for analysis of pyramidal horns with composite inner E-plane walls

paper_content:

The contour path finite-difference time-domain (FDTD) method is used for modeling pyramidal horn antennas with or without composite E-plane inner walls. To model the pyramidal horn surface, a locally distorted grid is used. Modified equations are obtained based on the locally distorted grid and the assumptions of the contour path method. The developed algorithm is validated by comparing computed antenna gain patterns, with and without the presence of composite material, with available measurements. >
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paper_title: FDTD analysis of electromagnetic wave radiation from systems containing horn antennas

paper_content:

The application of the finite-difference time-domain (FDTD) method to various radiating structures is considered. These structures include two- and three-dimensional waveguides, flared horns, a two-dimensional parabolic reflector, and a two-dimensional hyperthermia application. Numerical results for the horns, waveguides, and parabolic reflectors are compared with results from using the method of moments (MM). The results for the hyperthermia application are shown as extensions of the previously validated models. This new application of the FDTD method is shown to be useful when other numerical or analytic methods cannot be applied. >
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paper_title: Optimization of a conical antenna for pulse radiation: an efficient design using resistive loading

paper_content:

The conical monopole antenna with a section of continuous resistive loading is considered as a radiator for temporally short, broad-bandwidth pulses. The geometrical details of the coaxial feed and the resistive loading are varied to optimize this structure for pulse radiation. Compared with the perfectly conducting cone, the optimized resistive cone radiates a better reproduction of the pulse excitation with no loss in amplitude, and has internal reflections that are much smaller in amplitude. Graphical displays of the field surrounding the antenna are used to give insight into the physical processes for transient radiation from this antenna. Experimental models were constructed to verify the optimization and demonstrate the practicality of the design. Measurements of both the reflected voltage in the feed line and the time-varying radiated field are in excellent agreement with the theoretical calculations. >
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paper_title: A study of transient radiation from the Wu-King resistive monopole-FDTD analysis and experimental measurements

paper_content:

The cylindrical monopole antenna with a continuous resistive loading is considered as a radiator for temporally short, broad-bandwidth pulses. Specifically, the variation of the resistance used along the monopole is one proposed by Wu and King (1965). This antenna is analyzed by the finite-difference-time-domain (FDTD) method utilizing a new, efficient technique for handling the thin-walled, conducting tube that forms the resistance. The electromagnetic field in the space surrounding the antenna is determined as a function of time, and quantities useful for describing the performance of the antenna are then calculated from these results. Graphical displays of the results are used to give new insight into the physical processes for transient radiation form this antenna. An experimental model is constructed using a discretized version of the Wu-King profile formed from a set of precision, high-frequency resistors. Measurements of both the reflected voltage in the feed line and the time-varying radiated field are in excellent agreement with the theoretical calculations. >
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paper_title: Accurate characterization of planar printed antennas using finite-difference time-domain method

paper_content:

The finite-difference-time-domain method (FD-TD) is used to characterize complex planar printed antennas with various feed structures, which include coaxial probe feed, microstrip line feed, and aperture coupled feed structures. A coaxial probe model is developed by using a three-dimensional FD-TD technique. This model is shown to be an efficient and accurate tool for modeling coaxial line fed structures. A novel use of a dispersive absorbing boundary condition is presented for a printed antenna with a high dielectric constant. All the numerical results obtained by the FD-TD method are compared with experimental results, and the comparison shows excellent agreement over a wide frequency band. >
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paper_title: Input impedance, radiation pattern, and radar cross section of spiral antennas using FDTD

paper_content:

A finite-difference time-domain (FDTD) analysis of spiral antennas is performed to calculate input impedance, antenna gain, and scattering. A semicircular spiral mounted on a dielectric substrate was simulated for computing the input impedance versus frequency. The gain and scattering computations were performed on a square Archimedean spiral mounted in a ground plane with a cavity backing. Total-field FDTD calculations are used to compute the impedance and gain patterns, while a specially modified scattered-field approach for aperture antennas in infinite ground planes is used for the scattering results. Comparisons are made with published impedance measurements and gain and scattering calculations done with a finite element method. Good results were obtained for impedance, radiation, and scattering. >
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paper_title: Radiation and Scattering of a Square Archimedean Spiral Antenna Using FDTD

paper_content:

ABSTRACT A Finite-Difference, Time-Domain (FDTD) analysis of a square Archimedean spiral antenna is performed to calculate antenna gain and scattering patterns. The spiral is mounted in a large, conducting ground plane and is backed by an air-filled or absorber-filled cavity. The effects of resistive sheets placed over the spiral are also evaluated for both radiation and scattering. Total-field FDTD calculations are used to compute the gain patterns, while a specially-modified scattered-field approach for aperture antennas in infinite ground planes is used for the scattering results. Comparisons are made with calculations done with a Finite Element method and excellent results are obtained.
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paper_title: Analysis of microstrip patch antennas using finite difference time domain method

paper_content:

The study of microstrip patch antennas is directly treated in the time domain, using a modified finite-difference time-domain (FDTD) method. Assuming an appropriate choice of excitation, the frequency dependence of the relevant parameters can readily be found using the Fourier transform of the transient current. The FDTD method allows a rigorous treatment of one or several dielectric interfaces. Different types of excitation can be taken into consideration (coaxial, microstrip lines, etc.). Plotting the spatial distribution of the current density gives information about the resonance modes. The usual frequency-dependent parameters (input impedance, radiation pattern) are given for several examples. >
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paper_title: FDTD analysis of an active antenna

paper_content:

Coupled FDTD-SPICE simulations are performed for an active antenna problem. The results are comparable to previously published results using FDTD in conjunction with special integration techniques for the nonlinear elements. Some differences occur, and better agreement with experiment is observed for our newer approach. The main advantages are that all of the SPICE device models are directly available for FDTD modeling and the efficient SPICE integration schemes can be used directly. No user intervention is required for either the device models or the integration schemes. >
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paper_title: Combining modal analysis and the finite-difference time-domain method in the study of dielectric waveguide problems

paper_content:

A numerical approach that combines the finite-difference time-domain (FDTD) method with a compatible method of extracting the modal information from the FDTD results in the analysis of dielectric waveguide problems is presented. The junction is analyzed by the FDTD method; the result obtained from the FDTD method is in the form of total field distributions at the discretized locations and time intervals. The modal amplitudes at the various waveguides are extracted from the total field distribution by a least-squares approach. While this approach exploits the flexibilities of FDTD method, which can analyze arbitrarily shaped geometries that may contain multiple inputs and outputs and/or anisotropies, the results can still be expressed in the simple form of the modal solution. The methodology for model extraction is presented. >
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paper_title: A method of moments analysis and a finite-difference time-domain analysis of a probe-sleeve fed rectangular waveguide cavity

paper_content:

A multifilament method of moments (MOM) analysis and a finite-difference time-domain (FE-TD) analysis are used to numerically calculate the input impedance of a probe-sleeve fed rectangular waveguide which has been short-circuited on one side. The input impedance of the system is determined by using the above methods for several probe-sleeve configurations, and reasonable agreement between the two methods for the cases studied is found. An MOM Green's function formulation which is based on scattering superposition and which allows the input impedance of a probe-sleeve feed to be calculated when the waveguide is terminated in a given load is derived. The MOM results and FD-TD numerical results are compared for this loaded waveguide input impedance case, and reasonable agreement between the methods is found. A comparison of theory and experiment is given for when the waveguide is terminated in a ground plane aperture. >
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paper_title: Field and power-density calculations in closed microwave systems by three-dimensional finite differences

paper_content:

A three-dimensional finite-difference scheme is used to solve Maxwell's equations in closed microwave systems. The calculation of the sinusoidal steady-state amplitude and phase of the EM-field, and the derivation of the power density are discussed. The method is tested on some canonical examples which can be solved analytically. Finally, some nontrivial three-dimensional examples are given.
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paper_title: Application of the FD-TD method to the analysis of H-plane waveguide discontinuities

paper_content:

A direct finite-difference time-domain (FD-TD) method is applied to the analysis of waveguide discontinuities. Both the validity and efficiency of the method ate demonstrated in the case of an H-plane right-angle corner bend with and without dielectric loading. Comparisons of the present results with those obtained using the spectral-domain methods are made and they are shown to be in good agreement.
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paper_title: Analysis of H-plane waveguide discontinuities with an improved finite-difference time domain algorithm

paper_content:

A finite-difference time domain scheme for the analysis of H-plane waveguide discontinuities is described. The two-dimensional treatment of this kind of discontinuities and the use of a non-monochromatic excitation allow us to develop an efficient algorithm with a low demand of computer resources. The method is tested by comparing the numerical results for a right corner bend and a T-junction, the choice of the absorbing boundary conditions is discussed for each case.
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paper_title: The finite-difference−time-domain method and its application to eigenvalue problems

paper_content:

This paper describes the application of the finite-difference method in the time domain to the solution of three-dimensional (3-D) eigenvalue problems. Maxwell's equations are discretized in space and time, and steady-state solutions are then obtained via Fourier transform. While achieving the same accuracy and versatility as the TLM method, the finite-difference-time-domain (FD-TD) method requires less than half the CPU time and memory under identical simulation conditions. Other advantages over the TLM method include the absence of dielectric boundary errors in the treatment of 3-D inhomogeneous planar structures, such as microstrip. Some numerical results, including dispersion curves of a microstrip on anisotropic substrate, are presented.
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Title: A Selective Survey of the Finite-Difference Time-Domain Literature

Section 1: Fundamental issues
Description 1: Discuss the basic principles and foundational aspects of the Yee FDTD algorithm, including accuracy, stability, and field calculations.

Section 2: Absorbing boundary conditions
Description 2: Examine the use of absorbing boundary conditions to model open-region problems and reduce boundary reflections in FDTD simulations.

Section 3: Material ABCs
Description 3: Explore material-based absorbing boundary conditions, including the development and advancements of these techniques for FDTD methods.

Section 4: Orthogonal grids
Description 4: Address the use of Cartesian and orthogonal grids in the FDTD method to handle smoothly varying surfaces and small-scale structures.

Section 5: Other grids
Description 5: Discuss the extension of FDTD algorithms to non-orthogonal coordinates and alternative grid systems for more complex simulations.

Section 6: Material modeling
Description 6: Review the modeling of complicated materials in FDTD, including frequency-dependent, anisotropic, and nonlinear materials.

Section 7: Frequency-dispersive material
Description 7: Cover techniques for modeling frequency-dependent materials in FDTD simulations, such as recursive convolution and other methods.

Section 8: Impedance-boundary conditions
Description 8: Present formulations for surface-impedance-boundary conditions and their applications in FDTD.

Section 9: Active and passive device modeling
Description 9: Highlight recent advancements in modeling active and passive devices using the FDTD method.

Section 10: Transformations
Description 10: Describe various transformations used in FDTD modeling to handle radar cross-sections, frequency-domain scattering parameters, and efficiency improvements.

Section 11: Error analysis and algorithm comparisons
Description 11: Investigate numerical-dispersion errors and compare different FDTD algorithms and their accuracy.

Section 12: Higher-order techniques
Description 12: Introduce higher-order FDTD techniques to reduce numerical phase errors and improve accuracy.

Section 13: Radiating structures
Description 13: Analyze the application of FDTD methods to the design and analysis of antennas, including simple antennas, hand-held antennas, arrays, and other structures.

Section 14: Microwave devices and guiding structures
Description 14: Examine the use of FDTD in modeling microwave devices, waveguides, resonators, and other guiding structures.

Section 15: Miscellaneous applications
Description 15: Discuss various other applications of the FDTD method, such as scattering from random surfaces and non-destructive evaluation.

Section 16: Hybrid techniques
Description 16: Explore the integration of FDTD with other numerical techniques to enhance simulation capabilities.

Section 17: On-line FDTD bibliographic database
Description 17: Provide details on the on-line database of FDTD-related publications, its contents, and how to access and utilize this resource.

Section 18: Conclusions
Description 18: Summarize the major contributions and robustness of the FDTD algorithm over the past few decades, and its broad range of applications.