finite difference time domain (fdtd) method for computational electromagnetics

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1 T.C SÜLEMAN DEMİREL UNIVERSITY FEN BİLİMLERİ ENSTİTÜSÜ Mühendislik fakültesi ELEKTRONİK VE HABERLEŞME MÜHENDİSLİĞİ COURSE SUBJECT Biological Effect of Electromagnetic Waves COURSE OFFERED By Dr. Selçuk Comlekçi Finite Difference Time Domain (FDTD) Method for Computational Electromagnetics Submitted by MSc. Student Badri - Khalid Saeed Lateef Al Student No. 1330145002

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Page 1: Finite difference time domain (fdtd) method for computational electromagnetics

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T.C

SÜLEMAN DEMİREL UNIVERSITY

FEN BİLİMLERİ ENSTİTÜSÜ

Mühendislik fakültesi

ELEKTRONİK VE HABERLEŞME MÜHENDİSLİĞİ

COURSE SUBJECT

Biological Effect of Electromagnetic Waves

COURSE OFFERED By

Dr. Selçuk Comlekçi

Finite Difference Time Domain (FDTD) Method for

Computational Electromagnetics

Submitted by MSc. Student

Badri -Khalid Saeed Lateef Al

Student No. 1330145002

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Introduction:

There are many applications for the modeling of scattering and other

radiating objects such as antennas, communications, position measurement

techniques and electromagnetic interference simulation. Clearly, Maxwell’s

equations are the starting point when simulating most electromagnetic systems,

but they quickly become impossible to use in an analytical form. Therefore,

numerical techniques have become very important in generating solutions to

Maxwell’s equations in an accurate and efficient manner.

The finite-difference time-domain method (FDTD) is one of the more popular

simulation methods in that it is simple and fairly easy to implement .

The FDTD method was first proposed by Yee in 1966 . It is based on the

substitution of each partial derivative in Maxwell’s equations in time domain

with its finite difference representation. This substitution leads to a set of

equations where each field component of a cell is evaluated at a time step as a

function of the adjacent cells’ components which were evaluated in the

preceding time steps. To this end, space and time are divided into discrete

intervals in which the electromagnetic field is supposed constant. With

reference to space, this leads to the definition of a unit cell (referred to as Yee’s

cell) in which the electromagnetic field is supposed constant.

However, as with all systems that contain a spatio-temporal (is a database that

manages both space and time information) set of equations to be solved, the

level of computation becomes excessive and onerous. For example, the

simulation that is completed using the FDTD method takes less time compering

with another method of analyses like numerical method. Considering that most

communication systems span 10s of kilometers and that large pieces of

information is being continuously encoded onto carrier signals, the FDTD

method can become impossible to compute or provide an accurate solution.

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Therefore, more efficient algorithms and simpler representations are required to

make the simulation of the electromagnetic radiation feasible to perform.

Classical methods to solve Maxwell’s equations

The classical methods to solve Maxwell’s equations are conformal

transformations and separation of variables.

a) Conformal Transformations

The limitation of conformal transformations is that, boundaries have to coincide

with equipotential lines or flux lines inclusively; i.e. boundaries have to either

coincide with equipotential lines, or coincide with flux lines, or coincide with

both equipotential and flux lines.

Furthermore, conformal transformations are limited to solve two-dimensional

electrostatic problems only.

It is also rather difficult to find the transformation equation that converts a

simple boundary into more complicated boundary containing the field which is

required to solve.

b) Separation of variables

To obtain the analytical solution of Laplace’s equation

Let u to be the product of two functions,

u(x,y) = X(x) Y(y)

Separation of variables means assuming that, the solution for u is the product of

two functions, one of which is a function of x only, and the other is a function

of y only. The separation of variables can only be applied to problems that fit

into orthonormal coordinate systems. This type of problems is limited in

number.

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Numerical Methods to solve Maxwell’s equations

In order to overcome the limitations of classical methods, numerical methods

were developed. The commonly used methods are finite element methods

(FEM), methods of moments (MOM) and finite difference time domain (FDTD)

methods.

Finite Element Method (FEM)

The origin of FEM traced back to Courant in 1943, who used triangular

elements and the principle of minimum potential energy.

The advantage of FEM is its capability to handle problems of complex

geometries and inhomogeneous materials.

FEM analysis of problems involves the following steps:

Defining the problem’s computational domain;

Choosing the shape of discrete elements;

Generation of mesh (preprocessing);

Applying wave equation on each element;

For statics, applying laplace’s/poisson’s equations on each element;

Applying boundary conditions;

Assembling element matrices to form overall sparse matrix;

Solving matrix system;

Postprocessing field data to extract parameters, such as capacitance,

impedance, radar cross section and so on

Method of Moments (MOM)

Consider the following inhomogeneous equation

Lφ= g

Where L is an operator which can be differential or integral; g is the known

excitation or source function; and φ is the response or unknown function to be

found. It is assumed that L and g are given and the task is to determine φ the

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procedure of solving the above equation as presented by Roger F. Harrington is

called method of moments. The procedure expands φ as a series of functions

and multiples φ by a set of weighting functions wn and finally solves for an.

In other words, MOM reduces L φ = g into matrix equations by using a method

known as the method of weighted residuals.

MOM has been successfully used to model:

Scattering by wires and rods;

Scattering by two-dimensional (2d) metallic and dielectric cylinders;

Scattering by three-dimensional (3d) metallic and electric objects of

arbitrary shapes; and

Many other scattering problems.

MOM requires very huge computer memories; MOM is not suitable for

analyzing the interior of conductive enclosures.

Finite Difference Time Domain (FDTD) method

The Finite Difference Time Domain (FDTD) method is an application of the

finite difference method, commonly used in solving differential equations, to

solve Maxwell’s equations. In FDTD, space is divided into small portions called

cells. On the surfaces of each cell, there are assigned points. Each point in the

cell is required to satisfy Maxwell’s equations. In this way, electromagnetic

waves are simulated to propagate in a numerical space, almost as they do in real

physical world. FDTD is one of the commonly used methods to analyse

electromagnetic phenomena at radio and microwave frequencies. Computer

programs written in MATLAB can display electromagnetic phenomena in

movies.

The basic algorithm of FDTD was first proposed by K.S. Yee in 1966. In 1975,

Allen Taflove and M.E. Brodwin obtained the correct criterion for numerical

stability for Yee’s algorithm and, firstly solved the sinusoidal steady-state

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electromagnetic scattering problems, in two- and three-dimensions. This

solution becomes a classical computer program example in many FDTD

textbooks. Many researchers follow; and among them are G. Mur and J.P.

Berenger. Mur published the first numerically stable absorbing boundary

condition (ABC) in 1981. Berenger introduced, the best ABC for the time

being, the perfectly matched layer (PML) in 1994.

The FDTD method has gained tremendous popularity in the past decade as a

tool for solving Maxwell’s equations. It is based on simple formulations that do

not require complex asymptotic or Green’s functions. Although it solves the

problem in time, it can provide frequency-domain responses over a wide band

using the Fourier transform. It can easily handle composite geometries

consisting of different types of materials including dielectric, magnetic,

frequency-dependent, nonlinear, and anisotropic materials. The FDTD

technique is easy to implement using parallel computation algorithms. These

features of the FDTD method have made it the most attractive technique of

computational electromagnetics for many microwave devices and antenna

applications.

FDTD has been used to solve numerous types of problems arising while

studying many applications, including the following:

Biological effect calculation like SAR and medical applications

Scattering, radar cross-section

Microwave circuits, waveguides, fiber optics

Antennas (radiation, impedance)

Propagation

Shielding, coupling, electromagnetic compatibility (EMC), EM pulse

(EMP) protection

Nonlinear and other special materials

Geological applications

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Inverse scattering

Metamaterial

Plasma

Traditional Yee Method

Beginning from the continuous form of Maxwell’s equations

(1)

(2)

(3)

(4)

B = μ H (5)

D = ɛ E (6)

Where B is the magnetic flux density, E is the electric field intensity, D is the

electric flux density, H is the magnetic field intensity and ɛ is the electric

permittivity ɛo = 8.854 × 10−12

farad/meter and μ is the magnetic permeability

μo= 4π × 10−7 henry/meter. J is the electric current density vector in amperes

per square meter, M is the magnetic current density vector in volts per square

meter, ρe is the electric charge density in coulombs per cubic meter, and ρm is

the magnetic charge density in webers per cubic meter.

Yee suggest a cubic with unit cell (uniform cell size ∆ in all directions) is

shown in Fig. 1. It has the following features:

1. The electric field is defined at the edge centers of a cube;

2. The magnetic field is defined at the face centers of a cube;

3. The electric permittivity/conductivity is defined at the cube center(s);

4. The magnetic permeability/magnetic loss is defined at the cube nodes

(corners).

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Equation (1) and (2) are composed of two vector equations, and each vector

equation can be decomposed into three scalar equations for three-dimensional

space. Therefore, Maxwell’s curl equations can be represented with the

following six scalar equations in a Cartesian coordinate system (x, y, z):

Figure 1

𝜕𝑡 𝑀

𝜕𝑡 𝐽

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The material parameters εx , εy , and εz are associated with electric field

components Ex , Ey , and Ez through constitutive relations Dx = εxEx , Dy = εy Ey ,

and Dz = εzEz, respectively. Similarly, the material parameters μx , μy , and μz are

associated with magnetic field components Hx , Hy , and Hz through constitutive

relations Bx = μxHx , By = μyHy , and Bz = μzHz, respectively.

Similar decompositions for other orthogonal coordinate systems are possible,

but they are less attractive from the applications point of view.

Electric and Magnetic Field Interactions with Living Tissues

One of the most important aspects of bio-electromagnetics is how

electromagnetic fields interact with materials, for example, how the electric (E)

and magnetic (H) fields affect the human body. Because E and H were defined

to account for forces among charges, the fundamental interaction of E and H

with materials is that they exert forces on the charges in the materials.

Biological materials are lossy, and this loss changes the way the wave interacts

with the material and its propagation behavior. A material is lossy if the

conductivity σ ≠ 0. Power will be dissipated in the lossy material as a wave

passes through it, thus causing loss to the propagating wave. If power is

dissipated in the material, the material will heat up, and this is what raises the

concerns of RF waves’ effect on human tissues.

In many electromagnetic field interactions, energy transfer is of prime

consideration and concern. For example, in hyperthermia for cancer therapy, the

electric field is transformed in the body into heat, which is the desired outcome

of the therapy. For cell phones, the energy transfer must be below some

predefined regulations. The E field can transfer energy to electric charges

through the forces it exerts on them, but the H field does not transmit energy to

charges. H effect is not prominent in EM biological interactions. For steady-

state EM fields, the electric power density is given by

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Specific Absorption Rate (SAR) The specific absorption rate (SAR) is defined as the dissipated power

divided by the mass of the object. SAR is the basic parameter that institutions

take into consideration for the evaluation of the exposure hazards in the RF and

microwave range. “Specific” refers to the normalization to mass, and

“absorption rate” refers to the rate of energy absorbed by the object.

The maximum temperature generally occurs in the tissue region with high heat

deposition. However, one should note that SAR and temperature distribution

may not have the same profile, since temperature distribution can also be

affected by the environment or imposed boundary conditions. Several methods

can be used to determine the SAR distribution induced by various heating

applicators.

One method is the experimental determination of the SAR distribution based on

the heat conduction equation. The experiment is generally performed on a tissue

equivalent phantom gel. The applicability of the SAR and temperature elevation

distributions measured in the phantom gel (to that in the living tissue) depends

on the electrical properties of the phantom gel. The electrical properties depend

on the electromagnetic wave frequency and water content of the tissue. The

ingredients of the gel can be selected to achieve the same electrical

characteristics of the tissue for a specific electromagnetic wavelength.

The simplest experimental approach to determine the SAR distribution is from

the temperature change when power source is ON. In this approach, temperature

sensors are placed at different spatial locations within the gel. Before the

experiment, the gel is allowed to establish a uniform temperature distribution

within itself. As soon as the initial heating power level is applied, the gel

temperature is elevated and the temperatures at all sensor locations are

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measured and recorded by a computer. The transient temperature field in the gel

can be described by the heat conduction equation:

where T is the temperature of the tissue, K is the thermal conductivity of the

tissue (W/(K.m)), Cp is the specific heat of the tissue (J/(K.Kg)) and ρ is the

tissues density (Kg/m3). Within a very short period after the heating power is

ON, heat conduction can be negligible if the phantom gel is allowed to reach

equilibrium with the environment before the heating. Thus, the SAR can be

determined by the slope of the initial temperature rise, i.e.,

Neglecting the heat conduction and assuming that SAR at each spatial location

is constant during the heating, the temperature rise at each location is expected

to increase linearly. It should be noted that this method is an approximate

method that assumes a linear relationship between SAR and temperature

elevation. Some other assumptions are also made that makes this method

inaccurate.

Another method to find SAR distribution is by deriving it from Maxwell’s

equations. E and H are first determined analytically or numerically from

Maxwell’s equations. The SAR (W/kg) is then calculated by the following

equation:

Where ρi is the ith tissue density (Kg/m

3).Since the electric fields are now

available, the dissipated power density in each layer can also be calculated using

the following equation.

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The values of SAR vary directly with the conductivity. Generally speaking, the

tissues with higher water content, such as skin, are more lossy for a given

electric field magnitude than drier tissues, such as bone and fat.

Matlab somlation example:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %1D electromagnetic finite-difference time-domain (FDTD) program. %Assumes Ey and Hz field components propagating in the x direction. %Fields, permittivity, permeability, and conductivity %are functions of x. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% close all; clear all; Lingth_of_domain = 5; Number_of_sample = 505; Number_of_iteration = 800; source_frequency = 300e6; spatial_step = Lingth_of_domain/Number_of_sample; time_step = spatial_step/300e6; permittivity0 = 8.854e-12; %permittivity of free space permeability0 = pi*4e-7; %permeability of free space x_coordinate = linspace(0,Lingth_of_domain,Number_of_sample); %scale factors for E and H a_electric =

ones(Number_of_sample,1)*time_step/(spatial_step*permittivity0); a_magnatic =

ones(Number_of_sample,1)*time_step/(spatial_step*permeability0); a_scattering = ones(Number_of_sample,1); e_permittivity = ones(Number_of_sample,1); m_permeability= ones(Number_of_sample,1); condactvity = zeros(Number_of_sample,1); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Here we specify the epsilon, sigma, and mu profiles. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i=1:Number_of_sample e_permittivity(i) = 1; m_permeability(i) = 1; w1 = 0.5; w2 = 1.5; if (abs(x_coordinate(i)-Lingth_of_domain/2)<1.5) e_permittivity(i)=1+3*(1+cos(pi*(abs(x_coordinate(i)... -Lingth_of_domain/2)-w1)/(w2-w1)))/2; end if (abs(x_coordinate(i)-Lingth_of_domain/2)<0.5) e_permittivity(i)=4; end if (x_coordinate(i)>Lingth_of_domain/2) condactvity(i) = 0.005; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a_electric = a_electric./e_permittivity; a_magnatic = a_magnatic./m_permeability; a_electric =

a_electric./(1+time_step*(condactvity./e_permittivity)/(2*permittivity0));

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a_scattering = (1-

time_step*(condactvity./e_permittivity)/(2*permittivity0))... ./(1+time_step*(condactvity./e_permittivity)/(2*permittivity0)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %plot the permittivity, permeability, and conductivity profiles figure(1) subplot(3,1,1); plot(x_coordinate,e_permittivity); grid on; axis([3*spatial_step Lingth_of_domain min(e_permittivity)*0.9

max(e_permittivity)*1.1]); title('relative permittivity'); subplot(3,1,2); plot(x_coordinate,m_permeability); grid on; axis([3*spatial_step Lingth_of_domain min(m_permeability)*0.9

max(m_permeability)*1.1]); title('relative permeabiliity'); subplot(3,1,3); plot(x_coordinate,condactvity); grid on; axis([3*spatial_step Lingth_of_domain min(condactvity)*0.9-0.001

max(condactvity)*1.1+0.001]); title('conductivity');

%initialize fields to zero Hz = zeros(Number_of_sample,1); Ey = zeros(Number_of_sample,1); figure(2); set(gcf,'doublebuffer','on'); %set double buffering on for smoother

graphics plot(Ey); grid on;

for iter=1:Number_of_iteration Ey(3) = Ey(3)+2*(1-exp(-((iter-

1)/50)^2))*sin(2*pi*source_frequency*time_step*iter); %absorbing boundary conditions for left-propagating waves Hz(1) = Hz(2); for i=2:Number_of_sample-1 %update H field Hz(i) = Hz(i)-a_magnatic(i)*(Ey(i+1)-Ey(i)); end %absorbing boundary conditions for right propagating waves Ey(Number_of_sample) = Ey(Number_of_sample-1); for i=2:Number_of_sample-1 %update E field Ey(i) = a_scattering(i)*Ey(i)-a_electric(i)*(Hz(i)-Hz(i-1)); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% figure(2) hold off plot(x_coordinate,Ey,'b'); axis([3*spatial_step Lingth_of_domain -2 2]); grid on; title('E (blue) and 377*H (red)'); hold on plot(x_coordinate,377*Hz,'r'); xlabel('x (m)'); pause(0); end

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Theatrical plot %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% phase = cumsum((e_permittivity).^0.5)*spatial_step; beta0 = 2*pi*source_frequency/(300e6); theory = sin(2*pi*source_frequency*(Number_of_iteration+4)*time_step-

beta0*phase)... ./(e_permittivity.^0.25); figure(3) plot(x_coordinate,theory,'b.'); theory = sin(2*pi*source_frequency*(Number_of_iteration+4)*time_step-

beta0*phase)... .*(e_permittivity.^0.25); figure(4) plot(x_coordinate,theory,'r.'); title('E (blue), 377*H (red), WKB theory (points)');

Theoretical resulte