The Finite Difference Time Domain Method for Electromagnetics

Karl S. Kunz, Raymond J. Luebbers

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351410472
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

The Finite-Difference Time-domain (FDTD) method allows you to compute electromagnetic interaction for complex problem geometries with ease. The simplicity of the approach coupled with its far-reaching usefulness, create the powerful, popular method presented in The Finite Difference Time Domain Method for Electromagnetics. This volume offers timeless applications and formulations you can use to treat virtually any material type and geometry.
The Finite Difference Time Domain Method for Electromagnetics explores the mathematical foundations of FDTD, including stability, outer radiation boundary conditions, and different coordinate systems. It covers derivations of FDTD for use with PEC, metal, lossy dielectrics, gyrotropic materials, and anisotropic materials. A number of applications are completely worked out with numerous figures to illustrate the results. It also includes a printed FORTRAN 77 version of the code that implements the technique in three dimensions for lossy dielectric materials.
There are many methods for analyzing electromagnetic interactions for problem geometries. With The Finite Difference Time Domain Method for Electromagnetics, you will learn the simplest, most useful of these methods, from the basics through to the practical applications.

Chapter 1

INTRODUCTION

Of the four forces in nature — strong, weak, electromagnetic, and gravitational — the electromagnetic force is the most technologically pervasive. Of the three methods of predicting electromagnetic effects — experiment, analysis, and computation — computation is the newest and fastest-growing approach. Of the many approaches to electromagnetic computation, including method of moments, finite difference time domain, finite element, geometric theory of diffraction, and physical optics, the finite difference time-domain (FDTD) technique is applicable to the widest range of problems.

This text covers the FDTD technique. Emphasis is placed on the separate field formalism, in which the incident field is specified analytically and only the scattered field is determined computationally. This approach is only slightly more complex in its basic implementation than the total field approach and more readily allows for the absorption of scattered fields at the limits of the problem space. The total field can be easily obtained from the combination of the scattered and incident fields. Though slight advantages may be found in either approach, they are very similar in concept and capabilities. The FDTD technique treats transients (e.g., pulses) in the time domain, and it is applicable over the computationally difficult-to-predict resonance region in which a wavelength is comparable to the interaction object size.

As a form of computational engineering, FDTD is part of a three-tier hierarchy consisting of:

Computer Science

• Stresses the mathematics underlying algorithms as well as the structure and development of the algorithm

Computer Engineering

• Hardware based on and concerned with hardware architecture and capabilities including parallelism and fault tolerance

Computational Engineering

• Explores various engineering problems via numerical solutions to systems of equations describing the phenomenon or process in question

Computational engineering relies on computer science and engineering, but is not hardware, language, or operating system specific. It requires a computer powerful enough to accomodate the problem in question, running within acceptable times and costs while producing the desired accuracy.

Electromagnetic computational engineering encompasses the electromagnetic modeling, simulation, and analysis of the electromagnetic responses of complex systems to various electromagnetic stimuli. It provides an understanding of the system response that allows for the better design or modification of the system.

The FDTD technique offers many advantages as an electromagnetic modeling, simulation, and analysis tool. Its capabilities include:

• Broadband response predictions centered about the system resonances

• Arbitrary three-dimensional (3-D) model geometries

• Interaction with an object of any conductivity from that of a perfect conductor, to that of a real metal, to that of low or zero conductivity

• Frequency-dependent constitutive parameters for modeling most materials

Lossy dielectrics

Magnetic materials

Unconventional materials, including anisotropic plasmas and magnetized ferrites

• Any type of response, including far fields derived from near fields, such as

Scattered fields

Antenna patterns

Radar cross-section (RCS)

Surface response

Currents, power density

Penetration/interior coupling

These capabilities are available for a variety of diverse electromagnetic stimuli covering a broad range of frequencies. Typical stimuli include:

• Lightning

• EMP (electromagnetic pulse)

• HPM (high power microwave)

• Radar

• Lasers

The systems responding to these stimuli are equally diverse. They can be small to large, inorganic or organic, in an exoatmospheric environment to a subterranean one. Samples of the diverse types of systems that can be treated are

• Aerosols

• Shelters

• Aircraft

• Humans

• Satellites

• Buried antennas

What ties the above stimuli and systems together is that typically the wavelengths of interest and the characteristic system dimensions are usually within an order of magnitude of each other. Thus, the broadband response predictions will typically encompass at least a few system resonances.

Hybrid techniques employing the geometrical theory of diffraction (GTD) or physical optics (PO) along with FDTD can in principle provide predictions from below resonance to extremely high frequencies. Numerical techniques, such as Prony’s method, allow arbitrarily long-time response prediction extensions of the FDTD generated time response predictions. Thus, resonance and below-resonance predictions can be extended to extremely low frequencies. Alternately, low frequency versions of FDTD, in which the displacement current is ignored and the equations become diffusive, can be used to extend low frequency capabilities. In short, FDTD can span the critical resonance region over more than four orders of magnitude in frequency, and with low and high frequency extensions this range can exceed six orders of magnitude.

FDTD has run on a diverse set of host computers, ranging from PCs to supercomputers, and is extremely well suited to implementation on parallel computers because only nearest-neighbor interactions are involved. The important variables are problem space size in cells required to model the system and the number of time steps needed. These determine the computer run time and computation cost. Less important are the material types modeled and the number of response locations monitored. Of little or no impact on computational capability requirements are the type of stimuli and the type of response, except in cases of far fields, which require a modest amount of postprocessing.

Over 1 million cells can be accommodated on personal workstations for a 3-D problem space 100 × 100 × 100 cells large. At a typical 10 FDTD cells per wavelength, this space is a 10-wavelength cube. The limits of today’s supercomputers are reached at roughly 100 million cells with computation times on the order of hours.

The advantages of FDTD can be summarized as its ability to work with a wide range of frequencies, stimuli, objects, environments, response locations, and computers. To this list can be added the advantage of computational efficiency for large problems in comparison with other techniques such as the method of moments, especially when broadband results are required. Further, the FDTD code, while inherently volumetric, has successfully treated thin plates and thin wire antennas. Its accuracy, using a sufficiency of cells, can be made as high as desired. Conversely, engineering estimates of a few decibels’ accuracy can be made with surprisingly few cells. Finally, powerful visualization tools are being developed to enhance the user’s understanding of the essential physics underlying the various processes that FDTD can model, simulate, and analyze.

The basis of the FDTD code is the two Maxwell curl equations in derivative form in the time domain. These equations are expressed in a linearized form by means of central finite differencing. Only nearest-neighbor interactions need be considered as the fields are advanced temporally in discrete time steps over spatial cells of rectangular shape (as emphasized here, other cell shapes are possible, as are reduced 2- and 1-D treatments).

It should also be noted that at least six kinds of electromagnetic computational problems exist:

• Generation (power, devices, klystrons, etc.)

• Transmission (transmission lines, waveguides, etc.)

• Reception/detection/radiation (antennas)

• Coupling/shielding/penetration

• Scattering

• Switching/nonlinearities

All but the first can be treated using FDTD. The first area either requires, as for the klystron, the addition of charged particles or a 60-Hz calculation for power frequencies which require inordinately many time steps (on the order of 1 billion) for analysis.

While FDTD is most suited to computing transient responses, FDTD may be the computational approach of choice even when a single frequency or continuous wave (CW) response is sought. This is especially the case when complex geometries or difficult environments, such as an antenna that is buried or dielectrically clad, are considered. Interestingly, interior coupling into metallic enclosures is also a situation wherein FDTD is the method of choice. A CW analysis, using the method of moments, for example, will most likely fail to capture the highly resonant behavior of a metallic enclosure, even when made at many frequency points. The highly resonant nature of interior coupling was verified first experimentally and then computationally with FDTD. Indeed, low frequency resonances may be poorly characterized experimentally because of their extremely resonant behavior, but are revealed by FDTD runs of 1 million time steps.

This book is organized into an introduction followed by five sections:

1. Fundamental concepts

2. Basic applications

3. Special capabilities

4. Advanced applications

5. Mathematical basis of FDTD and alternate methods

The first section treats the most basic aspects of FDTD, providing the reader with the formalism and the basic procedures for FDTD operation. Along with the introduction describing FDTD’s utility and areas of application, the two chapters of the first section allow the reader to apply FDTD to a host of problems using the nondispersive lossy dielectric FDTD code listed in Appendix B:

• Chapter 2: Scattered Field FDTD Formulation. Discusses the discretized central differenced or “leap frog” Maxwell curl equations upon which the separate field formalism is based. The equations are formulated for lossy dielectrics, which in the limit of infinite conductivity become perfect conductors. The rudimentary computer code requirements and architecture needed to support the formalism in an operational code are presented as well. Much more detail about these issues is given in later chapters.

• Chapter 3: FDTD Basics. Provides guidance for applying the FDTD formulations of Chapter 2. This includes limitations on cell and time step sizes, specifying the incident field and the object to be analyzed, estimating the computer resources required, and applying an outer boundary condition at the extremities of the FDTD computation space.

The second section treats the basic applications of the basic formulation of FDTD given in the first section:

• Chapter 4: Coupling Effects. The scattered field formulation of FDTD was first applied to an F-111 aircraft to calculate the induced surface currents and charges from a simulated EMP field. This brought together all the elements representative of FDTD modeling, and for this reason and for some sense of history the modeling effort is discussed in detail. Only exterior coupling is treated with this example and to complete the discussion of coupling, interior coupling modeling of a simple shield, penetrated by an aperture and containing an interior wire is presented. The response is examined above and below aperture cutoff with resonant and highly resonant behavior noted in the two regimes. A strongly stressed point is the large number of time steps needed to accurately characterize the highly resonant behavior of the currents induced on the interior wire.

• Chapter 5: Waveguide Aperture Coupling. A transient wave propagating in a waveguide and coupled to a second waveguide via circular aperture(s) is examined in this section. At issue is whether FDTD can be employed successfully to model waveguide behavior. It is shown to work well and FDTD modeling is being rapidly extended into this modeling regime.

• Chapter 6: Lossy Dielectric Scattering. Chapter 2 develops the scattered field FDTD formulation for perfect electric conductors and lossy dielectrics without frequency-dependent constitutive parameters. Here, the capability for modeling lossy dielectrics is applied to a simple sphere geometry where there are known analytic surface responses and to a complex human body geometry where there are experimental results available for comparison. Excellent to good agreement is obtained in these early modeling efforts. As will be seen in later chapters, the agreement has only gotten better. Where FDTD once had to “prove itself” it has become something of a standard. Results for the models of complex geometries are treated as nearly exact, with the limits established mainly by the skill of the practitioner in defining the geometry and in ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Chapter 1. Introduction
• Part 1: Fundamental Concepts
• Part 2: Basic Applications
• Part 3: Special Capabilities
• Part 4: Advanced Applications
• Part 5: Mathematical Basis of FDTD and Alternate Methods
• Appendix A. Other Coordinate Systems and Reduced Dimensions
• Appendix B. FORTRAN Listings
• Index