finite difference time domain (fdtd) method for computational electromagnetics
TRANSCRIPT
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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