DEVELOPMENT OF AN ELECTROMAGNETIC

NUMERICAL SOLVER BASED ON THE FINITE

DIFFERENCE TIME DOMAIN (FDTD)

TECHNIQUE FOR RESEARCH AND TEACHING

PURPOSES

NURUL ‘ATIQAH BINTI AHMAD

UNIVERSITI TUN HUSSEIN ONN MALAYSIA

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ABSTRACT

A 1D-FDTD code was developed to support plane wave excitation in 3D-FDTD

domain and the code was developed using C++ programming language. First-order

Mur absorbing boundary condition (ABC) is applied to keep outgoing electric and

magnetic fields from being reflected into the problem space. In this thesis, the

performance of 1D-FDTD scheme is then evaluated on several medium including

free space, lossless dielectric medium, lossy dielectric medium and good conductors.

Sine-Gaussian technique is used to excite field signal in the 1D-FDTD simulation

domain and the simulation have been carried out to analyze the performance of the

scheme. From the results, the 1D-FDTD scheme shows good expected results on all

applied conditions. The integration of 1D-FDTD scheme into the 3D-FDTD solver is

realized through the implementation of Total Field Scattered Field (TFSF) technique.

The technique is used to excite plane wave into the 3D-FDTD domain for future

wave propagation studies. All simulation results presented in this work were

analyzed using OriginPro software.

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ABSTRAK

Kod Finite Difference Time Domain (FDTD) di dalam satu dimensi telah

dibangunkan untuk menghasilkan gelombang satah di dalam domain tiga dimensi

FDTD di mana gelombang satah itu bergerak dalam keadaan lurus di paksi y. Bahasa

pengaturcaraan yang digunakan dalam membangunkan kod ini ialah C++.

Seterusnya, Mur digunakan sebagai penyerap sempadan kerana ia berkebolehan

untuk memastikan medan elektrik dan medan magnet daripada terpantul semula ke

ruang simulasi. Di dalam tesis ini, prestasi skim satu dimensi FDTD telah dinilai di

dalam pelbagai ruang. Antaranya ialah ruang kosong, ruang tanpa kehilangan

dielektrik, ruang kehilangan dielektrik dan akhir sekali ialah ruang konduktor yang

baik. Teknik Sine-Gaussian telah digunakan sebagai pencetus isyarat di dalam

domain simulasi satu dimensi FDTD untuk menganalisa prestasi skim satu dimensi

FDTD. Daripada keputusan simulasi, skim satu dimensi FDTD menunjukkan

keputusan yang baik seperti yang telah dijangkakan terhadap semua ruang simulasi.

Selepas itu, skim satu dimensi FDTD digabungkan ke dalam tiga dimensi FDTD

yang sedia ada melalui pelaksanaan teknik Total Field Scattered Field (TFSF).

Teknik itu adalah untuk pengujaan gelombang satah ke dalam ruang domain tiga

dimensi FDTD untuk kajian pergerakan gelombang pada masa akan datang. Semua

keputusan simulasi dalam kajian ini dianalisis menggunakan perisian OriginPro.

CHAPTER 1

INTRODUCTION

1.1 Project Background

Over the past years, a number of numerical and analytical approaches to Maxwell's

time-dependent curl equations were broadly used with the increases in computer

memory capacity and relentless advances in computational imitating efficiency.

Consequently, the demand for efficient field modelling tools in electromagnetic

scattering problems is ceaselessly expanding. In general, computational

electromagnetic techniques have been applied to vast areas including the study of the

radiation, scattering and penetration of electromagnetic wave with 3-D objects, in

problems related to telecommunication, electromagnetic compatibility (EMC),

microwave devices, waveguide structures, medical diagnosis and many others [4].

Apparently, the finite-difference time domain (FDTD) method has become

one of the most popular methods to calculate full-wave solutions to complex

electromagnetic problems. This is because of the increase in computing power and

decreasing cost of workstations. Computer memory capacities are increasing rapidly.

While this trend positively influences all numerical techniques, it is of particular

advantage to FDTD methods, which are founded on discretizing space over a

volume, and therefore inherently require a large random access memory. The FDTD

method is also very attractive since it is accurate, robust, easily includes simple and

highly dispersive media, can easily model complex geometries, and can give

broadband solutions from a single simulation using a fast Fourier transform (FFT) on

post simulated data. The sources of error in FDTD calculations are well understood,

2

and can be bounded to permit accurate models for a very large variety of

electromagnetic wave interaction problems [3].

Furthermore, the primary advantage of FDTD over other full-wave models is

that, FDTD being a time domain method allows broadband analysis of the antenna by

Gaussian pulse excitation. The antenna characteristics over a wide frequency range

can be obtained by taking the Fourier transform of the FDTD simulation results

obtained when a wideband Gaussian pulse is used as an excitation and most of

widely used electromagnetic simulation software's use frequency domain methods

like FEM [1].

1.2 Problem Statement

In many problems of electromagnetics the quantity of interest is the field scattered by

a localized object in an otherwise uniform background, under harmonic plane wave

illumination. Since FDTD is a near-field solver and only represents a finite portion of

the space, it is impossible to create inside it a truly plane wave of infinite extent

traveling in an arbitrary direction. Therefore, the development of the plane wave

source condition allowed the earliest engineering applications of FDTD

computational electromagnetics modeling. These applications were in the defense

and bio electromagnetics areas, and involved the interaction of complex-shaped,

inhomogeneous material structures with impinging pulsed or continuous-wave

electromagnetic fields. Initially, many problems of these types located the material

structure of interest far from the radiating antenna, where the incident illumination

could be approximated by plane wave.

The plane wave is a constant-frequency wave whose wave fronts (surfaces of

constant phase) are infinite parallel planes of constant peak-to-peak amplitude

normal to the phase velocity vector. It is not possible in practice to have a true plane

wave; only a plane wave of infinite extent will propagate as a plane wave. However,

many waves are approximately plane waves in a localized region of space. For

example, a localized source such as an antenna produces a field that is approximately

a plane wave far from the antenna in its far field region. Similarly, if the length scales

are much longer than the wave’s wavelength, as is often the case for light in the field

3

of optics, one can treat the waves as light rays which correspond locally to plane

waves.

1.3 Project Objectives

The objectives of the project are:

(i) To develop a stand-alone 1D-FDTD scheme for electromagnetic wave study.

(ii) To evaluate the performance of 1D-FDTD scheme on various materials and

medium.

(iii) To Implement TFSF decomposition into the existing 3D-FDTD scheme for

plane wave excitation in 3D space.

1.4 Scope of Project

The scopes of the project are:

(i) The 1D-FDTD scheme and the 3D-TFSF decomposition codes were

developed using C++ programming language.

(ii) Using OriginPro software for data post-processing purposes.

(iii) Mur absorbing boundary condition was applied to truncate the 1D-FDTD

simulation space to infinity.

(iv) The performance of the 1D-FDTD scheme was evaluated on several medium

and material conditions including free space, dielectric (lossy and lossless)

and PEC.

(v) Using the sine-gaussian technique to excite field signal in the 1D-FDTD

simulation domain.

(vi) The plane wave is only for normal incident propagation in 3D space.

1.5 Outlines of the Thesis

Chapter 1 covers the background of the project. The problem of the project were

stated follows by the objectives and the scopes of the project. More details overview

of FDTD is discuss in Chapter 2.

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Chapter 2 unfolds the theoretical concept of FDTD principles including the

derivation of the magnetic and electric field update equations, parameters that control

the stability and accuracy. The advantages of using FDTD, excitation source,

absorbing boundary condition (ABC) and total field scattered field (TFSF) were

discuss in general.

Chapter 3 comprises the methodology or approaches in order to achieve the

objectives of the project. It shows work flow used in completion of this project. The

derivation of electric field and magnetic field for Total/Scattered Field formulation in

3D were shown.

Chapter 4 discuss the results and analysis data which are addressed through

the simulation of the 1D-FDTD evaluation in a various medium and plane wave

implementation to the currently available FDTD code. Graphs of simulation output

and comparison of results in time steps are explained in this chapter.

Chapter 5 summarize all the simulation results and the main conclusion will

be concluded including the recommendations for future work are dealt with in this

chapter.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

Over the past few years, finite-difference time-domain (FDTD) method [1] have

become increasingly prevalent in the computational electromagnetic problems due to

its simplicity, efficiency, robustness and versatility scheme for highly complex

configuration in the computational domain. Generally, FDTD technique is the most

well-known numerical method for the solution of problems in electromagnetic

simulation ranging from RF to optical frequencies. It is considered to be one of the

most powerful numerical techniques for solving partial differential equations of any

kind. In addition, it can be utilized to solve the spatial as well as the temporal

distributions of electric and magnetic fields in various media.

In principle, FDTD is a method that divides the solution domain into finite

discrete points and then replaces the partial differential equation with a set of

difference equations. It has successfully been applied to many problems of

propagation, radiation and scattering of electromagnetic waves such as antenna,

radar, wireless communication system, high speed electronic, photonic, radiography,

x-ray crystallography, bio-electromagnetic and geophysical imaging. A good

measure of its success lies in the fact that many of papers on the subject have been

published in journals and international symposium, apart from the books and

tutorials devoted to it. Moreover, much specific and general purpose commercial

software is available on the market which further extends its appeal globally. The

books used as the main references in this project is written by Taflove and Hagness

[1].

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2.2 Rise of FDTD Methods

The FDTD method, introduced by Yee in 1966 [1], was the first technique in this

class, and has remained the subject of continuous development. Since 1990, when

engineers in the general electromagnetics community became aware of the modeling

capabilities afforded by FDTD and related techniques, the interest in this area has

expanded well beyond defense technology.

There are seven primary reasons for the expansions of interest in FDTD and

related computational solution approaches for Maxwell’s equations:

(i) FDTD uses no linear algebra:

Being a fully explicit computation, FDTD avoids the difficulties with linear

algebra that limit the size of frequency-domain integral-equation and finite-

element electromagnetics models to generally fewer than

electromagnetic field unknowns. FDTD models with as many as field

unknowns have been run; there is no intrinsic upper bound to this number [4].

(ii) FDTD is accurate and robust:

The sources of error in FDTD calculations are well understood, and can be

bounded to permit accurate models for a very large variety of electromagnetic

wave interaction problems [4].

(iii) FDTD treats impulsive behavior naturally:

Being a time-domain technique, FDTD directly calculates the impulse

response of an electromagnetic system. Therefore, a single FDTD simulation

can provide either ultra wideband temporal waveforms or the sinusoidal

steady-state response at any frequency within the excitation spectrum [4].

(iv) FDTD treats nonlinear behavior naturally:

Being a time-domain technique, FDTD directly calculates the nonlinear

response of an electromagnetic system.

(v) FDTD is a systematic approach:

With FDTD, specifying a new structure to be modeled is reduced to a

problem of mesh generation rather than the potentially complex reformulation

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of an integral equation. For example, FDTD requires no calculation of

structure-dependent Green functions [4].

(vi) Computer memory capacities are increasing rapidly:

While this trend positively influences all numerical techniques, it is of

particular advantage to FDTD methods, which are founded on discretizing

space over a volume, and therefore inherently require a large random access

memory.

(vii) Computer visualization capabilities are increasing rapidly:

While this trend positively influences all numerical techniques, it is of

particular advantage to FDTD methods, which generate time-marched arrays

of field quantities suitable for use in color videos to illustrate the field

dynamics.

Generally, the algorithm used by Yee was described by the electric field

component which was spatially and temporally offset from the magnetic field

component to acquire the update equations. These equations were used in a leapfrog

manner to propagate the electric and magnetic fields ahead in time. The equations

provide the present fields in terms of the past fields all over the computational

domain. After Yee’s publication, the approach was widely used with different

endeavor [2-6].

The boundaries of the computational domain in FDTD need to be carefully treated

when simulating problems in open regions. Spurious reflections will generally occur

from the termination of the grid.

The problem can be solved by means of the well-known method called the

absorbing boundary condition (ABC). It is generally meant to absorb any outgoing

propagating waves without ideally producing spurious reflections. The ABCs was

first proposed in 1971 by Merewether [7] to solve the open region difficulties. The

development chronicle to magnify the practicability study of the technique was

continued in the literature by [8-12] which were based of nonmaterial type. In

contrast, Berenger presented a new idea in 1994 called the perfectly matched layer

(PML) ABC which was based on material category [13]. The state of the art of

Berenger’s PML contributes to notably better precision when compared to the other

ABCs in the written works [14-15] for broad assortment of applications.

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The main handicap of FDTD lies in the truth that only consistent grids can be

used. Accordingly, the geometry resemblance in FDTD is restricted to staircase-

shaped boundaries which lead to a large number of computer memory requirements

and the CPU time particularly when dealing with curvature geometries with fine

features [16]. The total number of cells in the computational domain grows

significantly due to a global fine mesh. Another FDTD weakness is the presence of

error due to numerical dispersion [17-18].

2.3 Finite Difference Time Domain Method in 1D

The finite-difference time-domain (FDTD) method is arguably the simplest, both

conceptually and in terms of implementation, of the full-wave techniques used to

solve problems in electromagnetics. It can accurately tackle a wide range of

problems. However, as with all numerical methods, it does have its share of artifacts

and the accuracy is contingent upon the implementation. The FDTD method can

solve complicated problems, but it is generally computationally expensive. Solutions

may require a large amount of memory and computation time. The FDTD method

loosely fits into the category of “resonance region” techniques, in example, ones in

which the characteristic dimensions of the domain of interest are somewhere on the

order of a wavelength in size. If an object is very small compared to a wavelength,

quasi-static approximations generally provide more efficient solutions. Alternatively,

if the wavelength is exceedingly small compared to the physical features of interest,

ray-based methods or other techniques may provide a much more efficient way to

solve the problem.

The FDTD method employs finite differences as approximations to both the

spatial and temporal derivatives that appear in Maxwell’s equations (specifically

Ampere’s and Faraday’s laws). Consider the Taylor series expansions of the function

expanded about the point with an offset of

(2.1)

(2.2)

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where the primes indicate differentiation. Subtracting the second equation from the

first yields:

(2.3)

Dividing by produces:

(2.4)

Thus the term on the left is equal to the derivative of the function at the point plus

a term which depends on plus an infinite number of other terms which are not

shown. For the terms which are not shown, the next would depend on and all

subsequent terms would depend on even higher powers of . Rearranging slightly,

this relationship is often stated as:

(2.5)

The “O” term represents all the terms that are not explicitly shown and the value in

parentheses, in example., , indicates the lowest order of in these hidden terms. If

is sufficiently small, a reasonable approximation to the derivative may be obtained

by simply neglecting all the terms represented by the “O” term. Thus, the central-

difference approximation is given by:

(2.6)

Note that the central difference provides an approximation of the derivative of the

function at , but the function is not actually sampled there. Instead, the function is

sampled at the neighboring points and . Since the lowest power of

being ignored is second order, the central difference is said to have second-order

accuracy or second-order behavior. This implies that if is reduced by a factor of 10,

the error in the approximation should be reduced by a factor of 100 (at least

approximately). In the limit as goes to zero, the approximation becomes exact.

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One can construct higher-order central differences. In order to get higher-

order behavior, more terms, in example, more sample points, must be used. The use

of higher-order central differences in FDTD schemes is certainly possible, but there

are some complications which arise because of the increased “stencil” of the

difference operator. For example, when a PEC is present, it is possible that the

difference operator will extend into the PEC prematurely or it may extend to the

other side of a PEC sheet. Because of these types of issues, it will only consider the

use of second-order central difference.

2.3.1 Update Equations in 1D

Consider a one-dimensional space where there are only variations in the x direction.

Assume that the electric field only has a z component. In this case Faraday’s law can

be written:

(2.7)

Thus must be the only non-zero component of the magnetic field which is time

varying. (Since the right-hand side of this equation has only a component, the

magnetic field may have non-zero components in the and directions, but they

must be static. It will not be concerned with static fields here.) Knowing this,

Ampere’s law can be written as:

(2.8)

The two scalar equations obtained from (2.7) and (2.8) are:

(2.9)

(2.10)

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The first equation gives the temporal derivative of the magnetic field in terms of the

spatial derivative of the electric field. Conversely, the second equation gives the

temporal derivative of the electric field in terms of the spatial derivative of the

magnetic field. As will be shown, the first equation will be used to advance the

magnetic field in time while the second will be used to advance the electric field. A

method in which one field is advanced and then the other, and then the process is

repeated, is known as a leapfrog method.

The next step is to replace the derivatives in (2.9) and (2.10) with finite

differences. To do this, space and time need to be discretized. The following notation

will be used to indicate the location where the fields are sampled in space and time:

(2.11)

(2.12)

where is the spatial offset between sample points and is the temporal offset.

The index corresponds to the spatial step, effectively the spatial location, while the

index corresponds to the temporal step. When written as a superscript still

represents the temporal step and it is not an exponent. When implementing FDTD

algorithms, it will see that the spatial indices are used as array indices while the

temporal index, which is essentially a global parameter, is not explicitly specified for

each field location. Hence, it is reasonable to keep the spatial indices as an explicit

argument while indicating the temporal index separately.

Figure 2.1 shows the arrangement of electric and magnetic field nodes in

space and time. The electric field nodes are shown as circles and the magnetic-field

nodes as triangles. The indicated point is where the difference equation is expanded

to obtain an update equation for [3]. It is the answer to the question of how

should the electric and magnetic field sample points, also known as nodes, be

arranged in space and time? Although it only have one spatial dimension, time can be

thought of as another dimension. Thus this is effectively a form of two-dimensional

problem. Assume that all the fields below the dashed line are known and they are

considered to be in the past while the fields above the dashed line are future fields

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and hence unknown. The FDTD algorithm provides a way to obtain the future fields

from the past fields.

Figure 2.1: Arrangement of electric and magnetic field nodes in space and time [3].

As indicated in Figure 2.1, consider Faraday’s law at the space-time point

:

(2.13)

The temporal derivative is replaced by a finite difference involving

and

(in example, the magnetic field at a fixed location but two

different times) while the spatial derivative is replaced by a finite difference

involving and

(in example, the electric field at two different

locations but one time). This yields:

(2.14)

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Solving this for

yields:

(2.15)

This is known as an update equation, specifically the update equation for the

field. It is a generic equation which can be applied to any magnetic-field node. It

shows that the future value of depends on only its previous value and the

neighboring electric fields. Figure 2.2 shows the space-time after updating the

magnetic field. The dividing line between future and past values has moved forward

a half temporal step. The indicated point is where the difference equation is written

to obtain an update equation for [3]. After applying (2.15) to all the magnetic-

field nodes, the dividing line between future and past values has advanced a half

time-step. The space-time grid thus appears as shown in Figure 2.2 which is identical

to Figure 2.1 except for the advancement of the past or future dividing line.

Figure 2.2: Space-time after updating the magnetic field [3].

Now consider Ampere’s law (2.10) applied at the space-time point

(which is indicated in Figure 2.2:

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(2.16)

Replacing the temporal derivative on the left with a finite difference involving

and z

and replacing the spatial derivative on the right with a finite

difference involving

and

yields:

(2.17)

Solving for yields:

(2.18)

Equation (2.18) is the update equation for the field. The indices in this equation

are generic so that the same equation holds for every node. Similar to the update

equation for the magnetic field, here it will see that the future value of depends on

only its past value and the value of the neighboring magnetic fields.

After applying (2.18) to every electric-field node in the grid, the dividing line

between what is known and what is unknown moves forward another one-half

temporal step. One is essentially back to the situation depicted in Figure 2.1 where

the future fields closest to the dividing line between the future and past are magnetics

fields. They would be updated again, then the electric fields would be updated, and

so on.

It is often convenient to represent the update coefficients

and

in terms

of the ratio of how far energy can propagate in a single temporal step to the spatial

step. The maximum speed electromagnetic energy can travel is the speed of light in

free space and hence the maximum distance energy can travel in one

time step is (in all the remaining discussions the symbol will be reserved for

the speed of light in free space). The ratio is often called the Courant number

which is label . It plays an important role in determining the stability of a

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simulation (hence the use of ) and will be considered further later. Letting

and the coefficients in (2.18) and (2.15) can be written as:

(2.19)

(2.20)

where

is the characteristic impedance of free space.

In FDTD simulations there are restrictions on how large a temporal step can

be. If it is too large, the algorithm produces unstable results (in example, the numbers

obtained are completely meaningless and generally tend quickly to infinity). At this

stage it is not consider a rigorous analysis of stability. However, thinking about the

way fields propagate in an FDTD grid, it seems logical that energy should not be

able to propagate any further than one spatial step for each temporal step, .

This is because in the FDTD algorithm each node only affects its nearest neighbors.

In one complete cycle of updating the fields, the furthest a disturbance could

propagate is one spatial step. It turns out that the optimum ratio for the Courant

number (in terms of minimizing numeric errors) is also the maximum ratio. Hence,

for the one-dimensional simulations considered initially, it will use:

(2.21)

When first obtaining the update equations for the FDTD algorithm, it is

helpful to think in terms of space-time. However, treating time as an additional

dimension can be awkward. Thus, in most situations it is more convenient to think in

terms of a single spatial dimension where the electric and magnetic fields are offset a

half spatial step from each other. This is depicted in Figure 2.3. The temporal offset

between the electric and magnetic field is always understood whether explicitly

shown or not.

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Figure 2.3: A one-dimensional FDTD space showing the spatial offset between the

magnetic and electric fields [3].

2.4 Maxwell’s Equations in 3D

A pioneering way of describing the electromagnetic phenomena was introduced by

James Clerk Maxwell in 1865 [4]. Later in 1873, he published an article called

“Treatise on Electricity and Magnetism” in which the discoveries of Coulomb,

Oersted, Ampere and Faraday were united into four refined constructed mathematical

equations known as Maxwell’s equations. The differential time domain Maxwell’s

equations in a linear medium are given by:

(2.22)

(2.23)

In (2.22) and (2.23), the following symbols are defined:

(i) is the magnetic flux density in webers/meter2.

(ii) is the electric flux density in coulombs/meter2.

(iii) is the electric field in volts/meter.

(iv) H is the magnetic field in amperes/meter.

(v) is the magnetic current density in volts/meter2 and is defined to relate any

magnetic loss to the field.

(vi) is the electric current density in amperes/meter2 and is defined to relate any

electric loss to the field.

and are respectively given by:

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(2.24)

(2.25)

Where * is magnetic resistivity in Ω/m and σ is the electric conductivity in S/m. In

materials with field-independent, direction-independent and frequency-independent

electric and magnetic properties, the following proportions apply:

(2.26)

(2.27)

μ is the magnetic permeability in H/m and ε is the electric permittivity in F/m.

Inserting

(2.24) and (2.26) to (2.22) and dividing by μ gives:

(2.28)

is the magnetic losses which may exist inside the medium. Inserting (2.25) and

(2.27) to (2.23) and dividing by ε gives:

(2.29)

Where is the electric losses which may exist inside the medium. In Cartesian

coordinates, equations (2.28) and (2.29) yield the following six scalar equations:

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

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The system of six coupled partial differential equations of (2.30-2.32) and

(2.33-2.35) forms the basis of the FDTD numerical algorithm for electromagnetic

wave interaction with general 3D objects. The FDTD algorithm need not explicitly

enforce the Gauss law relations indicating zero free electric and magnetic charge.

This is because these relations are theoretically a direct consequence of the curl

equations. However, the FDTD space lattice must be structured so that the Gauss law

relations are implicit in the positions of the electric and magnetic components, and in

the numerical space derivative operations upon these components that model the

action of the curl operator.

2.5 The Yee Algorithm

The FDTD algorithm as first proposed by Kane Yee in 1966 employs second-order

central differences. The algorithm can be summarized as follows:

(i) Replace all the derivatives in Ampere’s and Faraday’s laws with finite

differences. Discretize space and time so that the electric and magnetic fields

are staggered in both space and time.

(ii) Solve the resulting difference equations to obtain “update equations” that

express the (unknown) future fields in terms of (known) past fields.

(iii) Evaluate the magnetic fields one time-step into the future so they are now

known (effectively they become past fields).

(iv) Evaluate the electric fields one time-step into the future so they are now

known (effectively they become past fields).

(v) Repeat the previous two steps until the fields have been obtained over the

desired duration.

At this stage, the summary is probably a bit too abstract. One really needs an

example to demonstrate the simplicity of the method. However, developing the full

set of three-dimensional equations would be overkill and thus the algorithm will first

be presented in one-dimension. As it can be seen, the extension to higher dimensions

is quite simple.

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A space point in a rectangular grid is defined from Yee's original notation [5] as:

(2.36)

Let F denote any function of discrete space and time given by:

(2.37)

Δx, Δy and Δz are the grid space increments in x, y and z directions respectively, and

Δt is the time increment. Using a central finite-difference approximation, space and

time derivatives of F can be written as:

(2.38)

(2.39)

In equation (2.38), is the error term that represents all the remaining terms in

a Taylor series expansion. It is known as a central finite difference scheme in space

with second-order accuracy. Similarly, (2.39) is second-order accurate in time.

Applying Yee's finite-difference scheme to (2.30) gives:

(2.40)

The

field component in (2.40) is evaluated at time step n. However, the value

of

at time step n is not available and hence the following interpolated

approximation is used:

(2.41)

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By substituting equation (2.41) in (2.40), leaving

on the left hand side and

passing the all remaining terms to the right, assuming cubical FDTD cells are used,

the finite difference updating equation for the magnetic and electric field components

can be derived as:

(2.42)

(2.43)

(2.44)

(2.45)

(2.46)

(2.47)

It can be seen that the coefficients on the left hand side are referred to as Yee's

updating coefficients. The electric field coefficients are given by:

(2.48)

(2.49)

The magnetic updating coefficients can be written as:

(2.50)

(2.51)

The subscript p can be x, y or z and is the cell size in the p-direction. Assuming

the structure under investigation contains different types of material such as

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dielectric or magnetic, electric and magnetic field updating coefficients can be easily

calculated from equations (2.48) to (2.51) before the FDTD time stepping algorithm

starts. The orientation of the fields in Figure 2.4 is known as the FDTD lattice or Yee

cell. The magnetic and electric fields are located on the faces and the edges of the

cube respectively. Each electric field vector component is surrounded by four

circulating magnetic field vector components and vice versa. Both the electric and

magnetic field vector components are located half a cell from each other. In addition,

this arrangement permits easy implementations of the central finite difference

approximations and the integral form of the Faraday's law and the Ampere's law. The

system of difference equations is solved at the nodes. Figure 2.5 illustrates the typical

relationship between field components within a quarter of a cell and on a plane

distinctly helpful when handling boundary conditions of a closed region.

Figure 2.4: The electric and magnetic field components distribution on the FDTD

lattice [6].

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Figure 2.5: Relationship between field components: (a) within a quarter of a unit cell,

(b) on a plane [6].

Figure 2.6: Space-time chart of the Yee algorithm in a leapfrog arrangement [1].

The field quantities are solved with a leapfrog arrangement where a half time

step separates the solutions of the electric and magnetic fields as shown in Figure

2.6. The time-stepping of the FDTD algorithm is continued until the desired late time

pulse response or steady state behavior is reached. Figure 2.7 depicts the time-

stepping FDTD algorithms flow chart. Apparently, a suitable size of the time step

should be chosen properly to avoid the late time instability of the algorithms, after

determining the spatial resolution based on the geometrical features and the operating

23

frequency. It can be seen that a leapfrog arrangement between the electric and

magnetic field vector components is used to implement the time step of the FDTD

algorithm. The grids of the electric and magnetic fields are displaced half a cell

between them indicating that the computer must work through them in turn.

Figure 2.7: The flow chart of the time stepping FDTD algorithm [3].

2.6 Numerical Dispersion and Stability

The accuracy and stability of FDTD method is controlled by two significant

parameters namely the space resolution and the time resolution . Waves

of different frequencies will propagate at slightly different speeds through the grid

due to the approximations inherent in FDTD. As a result, the grid dispersion error

will occur in the FDTD. The difference in the propagation speed also depends on the

direction of propagation relative to the grid. Hence, the grid dispersion error must be

reduced to an acceptable level to increase the accuracy and stability of the results. To

ensure the FDTD numerical accuracy, the spatial increment h used in the simulation

is required to be small enough compared to the wavelength. Usually, h is less than or

24

equal to λmin/10, where λmin is the shortest wavelength at the frequency of

computation. The numerical grid produces a certain amount of numerical dispersion

error, when the grid size and the angle of propagating wave are increased as

illustrated in Figure 2.8. Numerical dispersion is observed to be approximately 1%

when the cell size is exactly λ/10.

Figure 2.8: Variation of the numerical phase velocity with wave propagation angle in

two-dimensional FDTD grid [1].

The time increment must satisfy the CFL criterion [1, 3] to ensure the stability of

the computation process, which sets the relation between the time and space

resolution for three-dimensional FDTD given by:

(2.52)

is the maximum phase velocity of the propagating wave in any material in the

model. For the three-dimensional cubic cell space lattice with

in a d-dimensional spatially homogeneous FDTD grid, the time step is bounded by:

(2.53)

92

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