TECHNICAL REPORT: CVEL-08-011.2
Survey of Current Computational Electromagnetics Techniques and
Software
T. Hubing, C. Su, H. Zeng and H. Ke
Clemson University
September 21, 2008
This work was supported by the ARMY RESEARCH OFFICE (W. Devereux Palmer)
under the auspices of the U.S. Army Research Office Scientific Services Program
administered by Battelle (Delivery Order 0271, Contract No. W911NF-07-D-0001).
Table of Contents
I.
Introduction ............................................................................................................................. 3
A.
B.
Purpose of This Report ................................................................................................................. 3
Categorizing CEM Modeling Tools .............................................................................................. 4
1.
2.
3.
4.
5.
6.
C.
II.
Time vs. Frequency Domain ........................................................................................ 4
IE vs. PDE .................................................................................................................... 4
2D vs. 3D ...................................................................................................................... 5
Static, Quasi-static, Full-wave or Asymptotic .............................................................. 5
Circuit-based Field Solvers .......................................................................................... 6
Linear vs. Higher-order Methods ................................................................................. 7
Outline of This Report .................................................................................................................. 7
The Method of Moments ........................................................................................................ 7
A.
B.
III.
IV.
V.
VI.
A.
B.
C.
D.
E.
F.
G.
H.
Equation Options .......................................................................................................................... 9
Basis and Weighting Functions..................................................................................................... 9
The Boundary Element Method ............................................................................................ 10
The Finite Element Method .................................................................................................. 13
The Finite Difference Time Domain Method ....................................................................... 15
Other Popular CEM Modeling Techniques .......................................................................... 18
The Finite Integration Technique ................................................................................................ 18
The Partial Element Equivalent Circuit Method ......................................................................... 20
The Transmission Line Matrix Method ...................................................................................... 20
The Finite Volume Time Domain Technique ............................................................................. 21
The Finite Element Time Domain Technique ............................................................................. 22
The Time Domain Method of Moments ..................................................................................... 23
The Generalized Multipole Technique........................................................................................ 24
The Finite Difference Frequency Domain Technique ................................................................ 25
VII. Asymptotic-Expansion Based Methods ................................................................................ 25
A.
B.
C.
D.
Physical optics ............................................................................................................................ 26
Geometrical Optics (Ray Optics) ................................................................................................ 27
Geometrical Theory of Diffraction ............................................................................................. 27
Uniform Theory of Diffraction ................................................................................................... 28
VIII.Hybrid Methods .................................................................................................................... 28
A.
B.
C.
Hybrid FEM/BEM ...................................................................................................................... 28
Hybrid MOM/GTD, MOM/PO ................................................................................................... 29
Hybrid FEM/PO .......................................................................................................................... 29
IX. Commercial CEM Software Survey ..................................................................................... 30
A.
B.
C.
D.
E.
F.
3D Full-Wave Codes................................................................................................................... 30
3D Quasi-Static Codes ................................................................................................................ 32
2D High-Frequency Codes.......................................................................................................... 34
2D Quasi-Static Codes ................................................................................................................ 35
Hybrid Codes .............................................................................................................................. 36
Special Purpose Codes ................................................................................................................ 36
REFERENCES ............................................................................................................................. 39
2
I.
INTRODUCTION
A. Purpose of This Report
James Clerk Maxwell’s presentation titled, “A Dynamical Theory of the Electromagnetic
Field” in 1864 established the foundation for electromagnetic field theory as we know it today
[1]. His work, which drew upon the observations of Faraday, Ampere, Oersted, Gauss and
others, introduced equations describing the relationship between electric and magnetic fields,
electric charges and electric currents. From these equations, Maxwell was able to postulate the
existence of electromagnetic waves, calculate the velocity of these waves and show that
electromagnetic waves traveled at a velocity consistent with the velocity of light; suggesting that
light could be a form of electromagnetic radiation.
The next 100 years witnessed the development of the first antennas, the wireless telegraph,
radio, radar, television and digital telemetry. Yet one of the most significant breakthroughs in
electromagnetic analysis, the ability to solve Maxwell’s equations on a computer, did not occur
until late in the 20th century.
Maxwell’s equations 1 (Table 1) govern all electric, magnetic and electromagnetic behavior.
They accurately describe electromagnetic behavior in any situation without making any
assumptions about materials, linearity or relativity. They are the basis for all electrical
interactions and they play a particularly important role in the design of antennas, EM wave
propagation analysis, microwave circuit analysis, signal integrity and electromagnetic
compatibility.
Table 1: Maxwell’s Equations
∫ E idl = − δ t ( ∫ B ids )
Integral Form
Faraday’s Law:
δ
S
Ampere’s Law:
∫ H idl = δ t ( ∫ D ids ) + ∫
δ
S
Gauss’ Law:
∫ D ids = ∫
S
Gauss’ Magnetic Law:
S
∫
S
V
ρ dV
B ids = 0
J id s
Differential Form
∇×E = −
∇×H =
δB
δt
δD
+J
δt
∇ iD = ρ
∇ iB = 0
As robust and powerful as these 4 equations are, they are virtually impossible to solve
analytically for all but a small set of canonical configurations. Virtually all of the progress in this
field in the 100 years following Maxwell’s 1864 publication was accomplished without the
1
The set of 4 equations commonly referred to as Maxwell’s equations were actually derived from Maxwell’s theory
by Oliver Heaviside and Heinrich Hertz (independently) more than 20 years later.
3
ability to find exact solutions to the problems of greatest interest. As a result most antennas
tended to resemble cones, rods, spheres and other canonical structures; and antenna design was a
mixture of science, art and trial-and-error.
With the advent of reasonably powerful computers in the mid-1960s, a new form of
electromagnetic analysis emerged; Computational ElectroMagnetic (CEM) modeling. Pioneers
in this field such as Yee [2], Harrington [3], and others demonstrated that numerical solutions of
Maxwell’s equations could be used to accurately describe the electromagnetic behavior of real
antenna configurations. For the first time, it became possible to analyze a wide range of
structures and accurately determine current and field distributions without building and
measuring these structures in a lab.
Of course, computers have come a long way since the 1960s, roughly doubling in speed and
memory capacity every couple of years. Advances in CEM modeling techniques and software
have also experienced exponential growth. A 1991 publication [4] briefly surveyed the progress
in electromagnetic modeling up to that time. This report provides an expanded summary of the
various EM modeling techniques and software that are currently available.
B. Categorizing CEM Modeling Tools
1. Time vs. Frequency Domain
All CEM modeling codes use numerical techniques to solve Maxwell’s equations in some
form. However, the specific technique employed, and the form of the equations solved, have a
tremendous impact on the suitability of a given code to analyze a given problem. For example,
some codes solve Maxwell’s equations in the frequency domain (i.e. one frequency at a time),
while others work in the time domain (usually calculating a system impulse response). This is
not unlike circuit theory, where circuits can be analyzed in the time or frequency domains.
Frequency domain techniques tend to be more efficient when modeling problems with narrow
bandwidths or high Q-factors. Time domain techniques are often more appropriate for modeling
broadband structures and sometimes larger structures with complex geometries. It is interesting
to note that the nature of the excitation (transient or swept-frequency) and the desired output
format (function of time or frequency) are relatively minor considerations when choosing the
optimum technique for a specific problem. This will become apparent as the techniques are
described in more detail later in this report.
2. IE vs. PDE
Another important way that CEM modeling techniques are categorized is based on the form
of Maxwell’s equations that they solve. As indicated in Figure 1, Maxwell’s equations can be
expressed in two fundamental forms. The integral form of Maxwell’s equations describes the
behavior of fields and currents over closed loops or closed surfaces of arbitrary size and shape.
For example, Gauss’ Law in integral form effectively states that the net electric flux passing
through any closed surface is equal to the total electric charge enclosed.
The differential form of Maxwell’s equations describes the behavior of fields and currents at
points in space. For example, Gauss’ Law in differential form simply states that the net change in
the electric flux at any point is equal to the electric charge density at that point.
4
The differential forms of each equation can be derived from the integral form by taking the
limit as the radius of a loop or surface approaches zero. Similarly, the integral forms can be
derived from the differential forms. Nevertheless, CEM modeling techniques based on solutions
of the integral form of Maxwell’s equations are fundamentally different from techniques based
on the solution of the differential form.
Techniques based on the solution of the differential form of Maxwell’s equations (Partial
Differential Equation or PDE techniques) generally analyze a problem’s geometry by dividing
the whole structure and the space around it into small pieces. The EM interactions in each region
are modeled independently. Each piece interacts only with the other pieces that are nearby.
Problems that involve radiation that propagates to infinity (open geometries) must utilize an
“absorbing boundary” that defines the edge of the problem space. The effectiveness of PDE
techniques applied to open geometries is highly dependent on the effectiveness of the absorbing
boundaries.
Integral Equation (IE) techniques generally segment only the boundaries between regions of
the problem with different electromagnetic properties. For example, in a 3D structure, the
“pieces” would be 2D surface patches or 1D wire segments.
IE techniques generally require fewer pieces than PDE techniques; however every piece in an
IE technique usually interacts with every other piece no matter where it is located, so the amount
of calculation involved to solve the problem is not necessarily lower.
3. 2D vs. 3D
CEM modeling techniques can be implemented in 1, 2, or 3 dimensions. 1-dimensional codes
are generally only of interest to students and academicians, though they can be useful in
specialized applications. 2D codes are commonly used to obtain the per-unit-length transmission
line parameters corresponding to circuit board trace geometries or cable cross-sections. They can
also be used to model certain structures with symmetry in one dimension (e.g. a biconical
antenna). Although most 2D problems could theoretically be analyzed using a 3D code (for
example by modeling a length of cable and determining the fields in a single cross-section), this
is rarely done. 2D modeling is much more efficient than 3D modeling (e.g. N2 vs. N3 unknowns
corresponding to N6 vs. N9 calculations) and 2D codes optimized for a particular application will
generally be much easier to use and more accurate than similar 3D codes.
Most problems of interest however, are 3-dimensional and require a 3-dimensional modeling
code. “Full wave” modeling codes calculate time-varying fields in 3 independent dimensions
without placing any special restrictions on the behavior of the fields in any given dimension. All
of the CEM modeling techniques described in this report can be implemented in 2 or 3
dimensions. However, 2D and 3D implementations of a given technique can be very different. It
is not necessarily true that the best technique for analyzing a particular 2D problem will be the
best technique for analyzing a “similar” 3D problem.
4. Static, Quasi-static, Full-wave or Asymptotic
Claims that a particular technique works “from DC to daylight” are not uncommon, but CEM
modeling codes are only accurate and efficient over a limited band of frequencies. Actually, it is
not the frequency itself that matters as much as the size of the structure to be modeled relative to
the wavelengths of the analysis.
5
Full wave modeling techniques, which are the main focus of this report, are most effective
when the size of the structure being analyzed is within an order of magnitude of one wavelength
(e.g. 0.1λ to 10λ). Smaller objects are often more readily modeled with static or quasi-static
codes. Larger objects are often better suited for analysis using asymptotic techniques.
There are two categories of static modeling codes: electrostatic and magnetostatic. Since the
electric and magnetic fields are uncoupled at DC, an electrostatic solver yields no information
about the magnetic fields. Similarly, a magnetostatic solver does not calculate electric field
distributions. The sources in an electrostatic analysis will be DC voltages or static charge
distributions. All materials behave like perfect conductors or perfect dielectrics. Electrostatic
solvers are used to determine induced charges or voltages, electric field distributions, or
capacitances. The sources in a magnetostatic analysis are DC current distributions. Magnetostatic
solvers are used to determine induced currents, magnetic field patterns or inductances.
Static field solvers do not employ time or frequency domain techniques, because neither time
nor frequency is variable. Static solvers can be either IE or PDE, however, the primary difference
being whether the entire volume or just the material interfaces are sectioned.
Static field analysis requires much less computation and tends to be more stable than fullwave analysis. For these reasons, static field solvers are capable of analyzing configurations with
significantly greater complexity than full-wave solvers using similar computer resources.
Quasi-static modeling codes generally employ static field solvers with some allowance made
for the solution to vary (slowly) with time. For example, an analysis of an electric motor may
employ a magnetostatic solver to recalculate the magnetic field over and over as the rotor
changes position. Similarly, an analysis of dielectric breakdown in a capacitor may rely on an
electrostatic field solver while adjusting static charge distributions in response to the calculated
field strengths and non-linear properties of the dielectric.
At high frequencies, where the structure being analyzed is many wavelengths, full-wave
solutions are less attractive because the number of electrically small “pieces” required to
represent the entire problem space can quickly grow to a number that exceeds the computer
resources available. At these frequencies, codes employing asymptotic techniques are usually
more efficient and accurate. Asymptotic techniques model radiated fields as if they were rays of
light; emanating from a source and traveling in a straight line to the point where the field is
calculated. These rays may be absorbed, reflected or diffracted by objects in the problem space.
The calculated field at a given point is the vector sum of all rays reaching that point from all
sources. There is no need to grid the entire volume as in full-wave PDE techniques; and there is
not an interaction between every pair of elements as in IE-based full-wave methods.
5. Circuit-based Field Solvers
There are at least two CEM modeling techniques that convert a field problem to an RLC
circuit analysis problem that can then be solved using SPICE-like circuit modeling tools. An
advantage of these techniques is that the resulting circuit can be analyzed in either the time or
frequency domain. Another advantage is that these techniques can make it easier to incorporate
linear or non-linear circuit elements into the field analysis.
6
6. Linear vs. Higher-order Methods
Since virtually all general purpose CEM modeling codes break the problem space into a set of
pieces (or elements), it is useful to categorize techniques depending on how the charge, current
or field quantities within these elements are defined. Most CEM modeling tools employ linear
elements, meaning that the quantities within a single element are described by a linear function
whose entire domain is confined within the boundaries of that element. Specifying the amplitude
of that function completely determines the quantity within the element independent of the
amplitudes specified outside the element.
Higher-order techniques employ elemental functions that are not confined to the boundaries
of a single element. These functions are normally nth-order polynomials where n>1. This
formulation allows complex charge, current or field distributions to be represented accurately
with fewer (i.e. larger) elements. Fewer elements generally translates to less computation time
and memory, so the primary advantage of higher-order techniques is that they solve problems
more quickly and accurately than their linear counterparts.
The primary disadvantage of higher-order techniques is that they can require significantly
more effort to segment properly and the additional “bookkeeping” required to set up a problem
for analysis can sometimes consume the resources that were saved by using the higher-order
formulation.
Virtually all of the full-wave modeling techniques discussed in this report can be formulated
with higher-order elements, but moment method and finite element codes are perhaps better
suited for this than codes based on other techniques.
C. Outline of This Report
The next five sections of this report describe various CEM modeling techniques commonly
employed by commercial software or used to model specific problems of interest. The focus of
these sections is on full-wave EM modeling, though several of these techniques can also be
formulated to solve quasi-static problems. Section VII reviews high-frequency (asymptotic)
techniques. Section VIII discusses hybrid methods, which combine two or more techniques to
create codes that can analyze a wider variety of problem geometries. Finally, Section IX provides
a brief summary of commercially available CEM modeling software available as of the date of
this report.
II.
THE METHOD OF MOMENTS
In the 1960s, R.F. Harrington [5] and others applied a technique called the Method of
Moments to the solution of electromagnetic field problems. The Method of Moments (also called
the Method of Weighted Residuals) is a technique for solving linear equations of the form,
L (φ ) = f
(II-1)
where L ( •) is a linear operator, f is a known excitation or forcing function, and φ is an unknown
quantity. To solve this problem on a digital computer, we start by expressing the unknown
solution as a series of basis or expansion functions, v n ,
7
φ = ∑ an vn
N
(II-2)
n =1
where a n are unknown coefficients describing the amplitude of each term in the series.
Now instead of one equation with a continuous unknown quantity, f , we have an equation
with N scalar unknowns,
L ( a1v1 + a 2 v 2 + … + a N v N ) = f
(II-3)
To solve for the values of a n , we need N linearly independent equations; so N different
weighting or testing functions, w n , are applied. This yields the following system of N equations
in N unknowns:
< w 1 , L (φ ) > = < w 1 , f >
< w 2 , L (φ ) > = < w 2 , f >
(II-4)
< w N , L (φ ) > = < w N , f >
Or, expressed in matrix form,
⎡ < w 1 , L ( v1 )) > < w 1 , L ( v 2 )) >
⎢ < w , L ( v )) > < w , L ( v )) >
2
1
2
2
⎢
⎢
⎢
⎣< w N , L ( v1 )) > < w N , L ( v 2 )) >
[ Z ][ A ] = [ B ]
< w 1 , L ( v N )) > ⎤ ⎡ a1 ⎤ ⎡ < w 1 , f1 > ⎤
< w 2 , L ( v N )) > ⎥⎥ ⎢⎢ a 2 ⎥⎥ ⎢⎢ < w 2 , f 2 > ⎥⎥
=
⎥
⎥⎢ ⎥ ⎢
⎥
⎥⎢ ⎥ ⎢
< w N , L ( v N )) > ⎦ ⎣a N ⎦ ⎣< w N , f N > ⎦
(II-5)
This linear system of equations has the form,
(II-6)
where the elements of [ Z ] are known quantities that can be calculated from the linear
operator, L ( •) , and the chosen basis and weighting functions. The elements of [ B ] are
determined by applying the weighting functions to the known forcing function. The unknown
elements of [ A ] can be found by solving the matrix equation. After solving for [ A ] (i.e. the
unknown coefficients, a n ), the value of φ is determined using Equation (II-2).
The Method of Moments (MoM) can be used to solve a wide range of equations involving
linear operations including integral and differential equations. This numerical technique has
many applications other than electromagnetic modeling; however the MoM is widely used to
solve equations derived from Maxwell’s equations. In general, moment method codes generate
and solve large, dense matrix equations and most of the computational resources required are
devoted to filling and solving this matrix equation. The particular form of the equations that is
solved and the choice of basis and weighting functions have a great impact on the size of this
matrix and ultimately the suitability of a given moment method code to model a given geometry.
8
A. Equation Options
The most common equation form solved by CEM modeling codes based on the Method of
Moments is the Electric Field Integral Equation (EFIE). This is an equation of the form,
E = fe (J ,M)
(II-7)
where E is the impressed (i.e. source) electric field and J and M are the induced electric and
magnetic current densities, respectively. The EFIE will be discussed further in the section
describing the Boundary Element Method. Generally, codes that solve a form of the EFIE excel
at modeling open (unbounded) geometries in which the electric field dominates in the near-field
region of the source.
Another equation solved by Moment Method codes is the Magnetic Field Integral Equation
(MFIE), which has the general form,
H = fm (J , M)
(II-8)
where H is the impressed (i.e. source) magnetic field intensity. Codes that solve a form of the
MFIE are best suited for modeling geometries with circulating currents, where the magnetic near
field is dominant.
Moment Method codes based on the EFIE or MFIE alone, may exhibit unstable behavior
when the modeling surfaces form a resonant cavity at a particular frequency. To avoid this, many
moment method codes solve a linear combination of the EFIE and MFIE known as a Combined
Field Integral Equation (CFIE). This requires more calculations to fill the matrix, but results in a
more stable solution when the modeling surface is large enough to support an interior resonance.
Some CEM modeling codes employ the Method of Moments to solve other equations. For
example, static modeling codes often solve a form of Laplace’s equation relating electric field
strengths to charge densities or magnetic field strengths to current densities. The Generalized
Multiple Technique (GMT), which is described in another section of this report, employs a
moment method to solve equations for the electric field generated by multipole sources.
B. Basis and Weighting Functions
An appropriate choice of basis and weighting functions can make a tremendous difference in
the number of elements, N, required to obtain an accurate solution. Since the solution is
represented as a summation of basis functions [see Eq. (II-2)], it is important to choose basis
functions that accurately represent the solution with a small number of terms. For example, when
solving for the current distribution on a surface, the basis functions should be current distribution
elements that can be summed together in a way that is able to efficiently approximate any overall
current distribution that might result from the analysis.
Weighting functions should be chosen that maximize the linear independence of the various
weighted forms of the equation. Often, the best choices of weighting functions are functions that
are identical to the basis functions. Moment method techniques that employ identical basis and
weighting functions are called Galerkin techniques.
9
III.
THE BOUNDARY ELEMENT METHOD
Boundary Element Method (BEM) codes use the method of moments to solve an EFIE, MFIE
or CFIE for electric and/or magnetic currents on the surfaces forming the interfaces between any
two dissimilar materials. Most CEM modeling codes that bill themselves as simply “moment
method” codes employ a boundary element method.
i
E,H
s
E,H
i
i
S
μ, ε , σ
s
E,H
i
Js
EQUIVALENCE THEOREM
s
E,H
Object with properties different from free space
s
Ms
Object replaced by free space and surface currents
Figure 1: Principle of Equivalence.
The first step in a boundary element analysis is to represent the problem geometry as a
distribution of equivalent surface currents in a homogeneous medium (usually free space). As
illustrated in Figure 1, the fields exterior to an object consist of fields incident on the object,
fields reflected from the object and fields emanating from the object. The Equivalence Theorem
[6] states that any field distribution exterior to an object can be exactly duplicated by removing
the object and replacing it with a set of equivalent electric and magnetic currents on the boundary
surface.
Since the forms of EFIE and MFIE used by boundary element methods are only valid for
current distributions in a uniform homogeneous medium, all objects in the problem space must
be removed and replaced with (initially unknown) surface currents conforming to their
boundaries.
Equations (III-1) and (III-2) below show a common form of the EFIE and MFIE employed by
boundary element method codes that model metallic objects only (i.e. there are no equivalent
magnetic surface currents);
E(r ) =
H (r ) =
− jη
J S ( r ')iG e ( r , r ') dS '
4π k ∫S
1
4π
∫
S
J S ( r ') × ∇ ' G m ( r , r ') dS ' .
(III-1)
(III-2)
10
In these equations,
∫ dS ' represents integration over all boundary surfaces. Note that since
S
the boundary only exists at places where an interface between two different materials occurred,
the size of the boundary is limited (i.e. there is no integration to infinity).
The functions Ge and Gm in these equations relate source currents to the electric and
magnetic field generated by those currents, respectively. Ge and Gm are called Green’s functions
[7] and they play a central role in boundary element analysis. Free-space Green’s functions
express the field emanating from the surface current represented by an individual basis function
(e.g. the current on an individual surface patch). However, other Green’s functions can be
employed to express the field emanating from more complex structures that are common to a
particular problem geometry. For example, a geometry consisting of metal surfaces coated with a
thin dielectric may employ a special Green’s function that expresses the fields emanating from
the combined surfaces of the metal-dielectric and dielectric-air interfaces. This can substantially
reduce the number of surface elements required to model the problem.
All Green’s functions are approximate expressions that are accurate for a limited range of
frequencies, distances and source geometries. Many boundary element methods employ several
Green’s functions to model different regions of a problem or different problem environments.
General purpose 3D BEM codes usually employ basis and weighting functions that are linear
current distributions on a rectangular or triangular surface patch. There are generally two
unknowns per patch corresponding to two orthogonal current vectors. Thin wires can be
represented very efficiently with a single unknown representing the amplitude of the current
distribution on each wire segment.
Point matching techniques employ basis and weighting functions that are simple impulse
functions often at the center of each patch or segment. Pulse matching techniques employ basis
and weighting functions that have a constant value everywhere on the patch or segment. More
accurate implementations employ basis and weighting functions that transition smoothly from
one patch to the next. Rao-Wilton-Glisson (RWG) [8] basis functions are a popular choice for
codes that employ triangular surface patch elements. Roof-top basis functions [9] are often
employed by codes that use rectangular elements.
Generally, CEM software employing a boundary element method excels at modeling
unbounded problems, particularly when it is not necessary to model regions of great complexity
in detail. Structures that can be adequately represented with a wire grid can be analyzed very
effectively using boundary element methods, because these methods model wires very
efficiently.
Table 2 lists various strengths and weakness of BEM modeling techniques. Note that the
capabilities of any particular modeling software depend strongly on the form of the integral
equation solved, the choice of basis and weighting functions, the Green’s function(s) employed,
and the matrix solver and any optimization techniques employed.
11
Table 2: Strengths and Weaknesses of the Boundary Element Method
BEM Modeling Strengths
BEM Modeling Weaknesses
− Excellent for modeling
unbounded (radiation) problems.
− Does not model inhomogeneous or
complex materials well.
− Good for modeling structures
with lumped circuit elements
included.
− CFIE formulation required to model
enclosed structures of resonant size.
− Excellent for modeling metal
plates and thin wires.
− Not good for modeling problems that
combine small detailed geometries
with larger objects.
Table 3 lists various CEM modeling codes that are based on a boundary element method. The
codes listed and the comments in Table 3 are based on the information available to the authors as
of the publication date of this report.
Table 3: CEM Modeling Codes that use the Boundary Element Method
Software Title
Description
Source
Amperes/Coulomb/Faraday
3D quasi-static
Integrated Eng. Software
Antenna Model
wire structures
Teri Software
CableMod
2D cable modeling
CST/Simlab
CONCEPT-II
3D full-wave
TU Hamburg-Harburg
Efield FD start-pack
3D full-wave
efield
Electro/Oersted
2D quasi-static
Integrated Eng. Software
EM3DS
MMIC Modeling
MEM Research
EZNEC Pro
Wire structures
EZNEC
FEKO
3D full-wave
EM Software and Systems
GEMACS
3D full-wave (wires and plates) Advanced Electromagnetics
NEC2
3D full-wave (wires and plates) Open Source
OPERA 2D
2D quasi-static
Vector Fields
OPERA 3D
3D quasi-static
Vector Fields
PCBMod
2D circuit board structures
CST/Simlab
SuperNEC
Wire structures
Poynting Antennas
WIPL-D Pro
3D full-wave
WIPL-D
12
IV.
THE FINITE ELEMENT METHOD
Scalar finite element methods have been used by civil and mechanical engineers to analyze
material and structural problems since the 1940s. However it wasn’t until the 1960s that FEM
codes were developed to solve problems in electromagnetics. Some of the pioneers in this field
were Silvester [10], Zienkiewicz [11], and Wexler [12]. Initial FEM-based CEM modeling codes
were applied to problems in electrostatics and magnetostatics. Later they were used to solve
high-frequency problems in 2 dimensions. Practical 3-dimensional codes did not appear until the
1980s due largely to problems with vector parasites [13, 14] and unreliable absorbing boundary
conditions [15]. Unwanted reflections from absorbing boundaries continue to be a problem with
full-wave 3D FEM codes even today.
Like BEM techniques, finite element methods can be based on different formulations (even
the method of moments). However BEM techniques always solve an integral equation and FEM
techniques always solve a differential equation. Every FEM code divides the entire problem
domain into small elements. For 2D problems the elements are usually triangles or rectangles.
For 3D problems, the elements are usually tetrahedra (4 faces) or bricks (6 faces). The domain
must be finite and bounded. Modeling an unbounded (e.g. radiation) problem requires that the
problem domain be bounded with special elements that absorb all incident energy. These
elements are called ABC (Absorbing Boundary Condition) elements.
The unknowns in scalar FEM codes are the three orthogonal components of the field at the
“nodes” (vertices) of each element. The unknowns in vector FEM codes are the field components
along the edges of each element. Scalar codes are conceptually simpler, but they are unsuitable
for full-wave modeling, because they are susceptible to spurious solutions that can cause
significant and unpredictable errors in the solution. Vector FEM codes are much less likely to
exhibit these parasitics.
To form a linear system of equations, the governing differential equation and associated
boundary conditions are converted to an integro-differential form using either a variational
method or a weighted-residual (moment) method. Variational methods solve for the unknown
quantity by minimizing an energy functional. Weighted-residual methods multiply a weak form
of Maxwell’s equations by a weighting function and integrate over each element. Ultimately, a
matrix equation is generated in the form,
[ A ][ X ] = [ B ]
(IV-1)
where [ X ] is a vector of the unknown field quantities, [B ] is a vector of source terms, and [A ] is a
sparse matrix whose only non-zero values correspond to positions in the matrix corresponding to
edges that share an element.
Generally, the matrices generated by FEM codes are must larger than the matrices generated
by BEM codes applied to similar geometries. This is because gridding an entire problem volume
requires many more elements than gridding just the material interfaces. However, because FEM
matrices are very sparse, they do not necessarily require more storage or computing resources to
solve than the small, but dense, matrices generated by BEM codes.
As indicated previously, modeling unbounded problems requires special absorbing elements
(ABCs). Many formulations of these elements have been proposed [15-22]. The ABCs that have
13
been developed for 2D FEM codes work very well; however 3D FEM ABCs work well only at
prescribed angles of incidence resulting in the need to locate the boundaries sufficiently far from
other structures. Hybrid FEM/BEM codes terminate open surfaces of the FEM volume with a
BEM surface negating the need for ABCs [23]. Unfortunately, the BEM portion of the resulting
matrix is dense, which can significantly increase the amount of computational resources
required.
Perhaps the most attractive feature of the finite element method is its ability to model
configurations that have complicated geometries and incorporate various materials. The
electrical properties of each element are defined independently and elements can be as small or
as large as needed to facilitate the analysis.
Table 4 lists various strengths and weakness of FEM modeling techniques. Note that the
capabilities of any particular modeling software depend on the specific formulation, the matrix
solver and any optimization techniques employed.
Table 4: Strengths and Weaknesses of the Finite Element Method
FEM Modeling Strengths
− Excels at modeling
inhomogeneous or complex
materials
− Excels at modeling problems that
combine small detailed geometries
with larger objects
FEM Modeling Weaknesses
− Absorbing boundary required for
modeling unbounded (radiation)
problems
− Difficult to model thin wires accurately
− Excels at modeling structures in
resonant cavities or waveguides
Table 5 lists various CEM modeling codes that are based on the finite element method. The
codes listed and the comments in Table 5 are based on the information available to the authors as
of the publication date of this report.
14
Table 5: CEM Modeling Codes that use the Finite Element Method
Software Title
Description
Source
Amaze (HiPhi/Magnum)
3D quasi-static
Field Precision
Analyst
2D and 3D full-wave
STAAR
Microwave Studio –FDS
3D full-wave
CST
EMDS
3D full-wave
Agilent Technologies
EMS
3D quasi-static
ElectromagneticWorks, Inc.
FEMM
2D quasi-static
Dr. David Meeker
Flux2D/3D
2D and 3D quasi-static
Magsoft
HFSS
3D full-wave
Ansys / Ansoft
JCMsuite
optics
JCMwave
MagNet / ElecNet
3D quasi-static
Infolytica
Magneto / Oersted
2D quasi-static
Integrated Eng. Software
Maxwell
2D/3D quasi-static
Ansys / Ansoft
Opera 2D/3D
2D/3D quasi-static
Vector Fields
Q3D Extractor
2D/3D quasi-static
Ansys / Ansoft
QuickField
2D quasi-static
Tera Analysis
Tricomp EStat/PerMag
2D quasi-static
Field Precision
Tricomp WaveSim
2D high-frequency
Field Precision
V.
THE FINITE DIFFERENCE TIME DOMAIN METHOD
The Finite Difference Time Domain (FDTD) method, as first proposed by Yee [2], is a direct
solution of Maxwell’s time dependent curl equations. It uses simple central-difference
approximations to evaluate the space and time derivatives. A basic element of the FDTD space
lattice is illustrated in Figure 2. An electric-field grid is offset from a magnetic-field grid in both
space and time. A first-order central-difference approximation can be expressed as,
μ
1
⎡ Ez1 ( t ) + E y 2 ( t ) − Ez 3 ( t ) − E y 4 ( t ) ⎦⎤ = − 0 ⎡⎣ H x 0 ( t + Δt ) − H x 0 ( t − Δt ) ⎤⎦
⎣
2Δt
Δl
(V-1)
where Δl is the length of one side of the cubical cell in Figure 2. Hxo(t+Δt) is the only unknown
in this equation, since all other quantities were found in a previous time step. In this way, the
electric field values at time t are used to find the magnetic field values at time t+Δt. A similar
central-difference approximation of Equation (V-1) can then be applied to find the electric field
15
values at time t+2Δt from the magnetic field values at time t+Δt. By alternately calculating the
electric and magnetic fields at each time step, fields are propagated throughout the grid.
Time stepping is continued until a steady state solution or the desired response is obtained.
The required computer storage and running time is proportional to the electrical size of the
volume being modeled and the grid resolution.
Figure 2: Basic Element of the FDTD Space Lattice.
For an open region problem, an absorbing boundary condition (ABC) is used to truncate the
computational domain. One technique, which is obtained by factoring the wave equation to
permit only outgoing waves, is differential based ABCs, such as those proposed by Engquist
[24], Lindman [25], Mur [26], Liao [27], Keys [28], and Higdon [29]. Another is material based
ABCs that are constructed so that the fields are dampened as they propagate into an absorbing
medium. Rappaport [30] proposed an ABC employing pyramid-shaped absorber material. In
1994, Berenger [31] introduced the perfectly matched layer (PML) absorbing boundary
condition. This ABC outperforms any that had been proposed previously and is widely used
today. Andrew [32] compared the accuracy of the Berenger perfectly matched layer and the
Lindman higher-order ABCs for the FDTD method. Accuracy studies of ABCs have also been
conducted for dispersive media [33, 34]. In 2003, Diaz [35] introduced a new radiation boundary
condition for FDTD based on self-teleportation of fields.
Because the basic elements are cubes, curved surfaces on a scatterer must be staircased. For
many configurations this does not present a problem. However for configurations with sharp,
acute edges, this approximation may lead to significant errors [36-37], and an adequately
staircased approximation may require a very small grid size. Surface-conforming FDTD
techniques with non-rectangular elements have been introduced to combat this problem [38-45].
Since all of the elements in an FDTD analysis must generally be the same size, the size of the
elements is determined by the smallest structural details that need to be modeled. If an object
under consideration contains small-scale geometries, such as a narrow slot or a very thin wire, an
excessively fine grid would have to be used to accurately model the associated fields. To
overcome these shortcomings, sub-cellular structures [46-55] have been introduced. Sub-cellular
structures are essentially special FDTD cells whose boundary conditions have been altered to
model small structures contained within the cells.
16
One major advantage of the FDTD method is the ability to obtain wideband results using a
transient excitation in one simulation. Frequency domain results can be obtained by applying a
discrete Fourier transform to the time domain results. Since many materials have frequency
dependent properties, it is necessary to take special precautions to model these materials
correctly with a time-domain technique. In 1990, Luebbers [56] used a recursive convolution
scheme to model a Debye media; this was the first frequency dependent FDTD formation.
Kashiwa and co-workers [57-59] published the first papers utilizing the auxiliary differential
equation (ADE) method to model Debye media, Lorentz media, and media obeying the ColeCole Circular Arc law. In 1992, Sullivan [60] proposed a dispersive formulation based on Z
transforms. Petropoulos [61] provided a comparison of the stability and phase error among
frequency dispersive FDTD methods.
Lossy surfaces can be modeled in FDTD codes by utilizing a surface impedance boundary
condition (SIBC) [62-64]. A thin material sheet model has also been developed for the FDTD
method [65-67].
A primary advantage of FDTD methods is their great flexibility. Arbitrary signal waveforms
can be modeled as they propagate through complex configurations of conductors, dielectrics, and
lossy non-linear non-isotropic materials. Another advantage of FDTD techniques is that they are
readily implemented on massively parallel computers, particularly vector processors and SIMD
(single-instruction-multiple-data) machines.
Time stepping techniques like FDTD are subject to dispersion errors when the time step is too
large for the given problem size. Many researchers have studied the numerical dispersion error
inherent in the FDTD method [68-76], but it is easily controlled by using appropriately small
time steps.
Table 6 lists various strengths and weakness of FDTD modeling techniques. Table 7 lists
various CEM modeling codes that employ an FDTD solver.
Table 6: Strengths and Weaknesses of the Finite Difference Time Domain Method
FDTD Modeling Strengths
FDTD Modeling Weaknesses
− Excels at modeling
inhomogeneous or complex
materials
− Absorbing boundary required for
modeling unbounded problems, but
PML boundaries work very well.
− Runs efficiently on highly parallel
computers
− Uniform cells must be small enough
to model necessary detail, but still fill
the entire volume.
− Excels at modeling very large
problems
− Difficult to model thin wires
− High Q structures are not modeled
efficiently
17
Table 7: CEM Modeling Codes that use the Finite Difference Time Domain Method
Software Title
Description
Source
ApsimFDTD
3D full-wave
Applied Simulation Tech.
CST M. Studio - TS
3D full-wave
CST
EMA3D
3D full-wave
Electromagnetic Appl.
EMPIRE XCcel
3D full-wave
Empire
EZ-FDTD
3D full-wave
EMS-Plus
Fidelity
3D full-wave
Zeland Software
GEMS
3D full-wave
2COMU
LC
3D full-wave
Cray Research
PAM-CEM
3D full-wave
ESI Group
SEMCAD X
3D full-wave
Schmid & Partner Eng.
Toy
3D full-wave
The CEMTACH Group
XFDTD
3D full-wave
Remcom
VI.
OTHER POPULAR CEM MODELING TECHNIQUES
A. The Finite Integration Technique
The Finite Integration Technique (FIT) is a consistent formulation for the discrete
representation of Maxwell's equations on spatial grids. First proposed by Weiland [77] in 1977,
the finite integration technique can be viewed as a generalization of the FDTD method. It is also
similar to the finite element method.
Weiland [77, 78] proposed exact algebraic analogues to Maxwell’s equations that guarantee
physical properties of computed fields and lead to a unique solution. By discretizing the integral
form of Maxwell’s equations on a pair of dual interlaced discretization grids, the finite
integration technique generates so-called Maxwell’s Grid Equations (MGEs) that guarantee the
physical properties of computed fields and lead to a unique solution.
Ce = −
Ch =
d
b
dt
(VI-1)
d
d+ j
dt
(VI-2)
Sb = 0
Sd = q
(VI-3)
(VI-4)
18
where e is the electric voltage between the grid points and h is the magnetic voltage between
dual grid points. d , b and j are fluxes over grid or dual grid faces. The allocation of the
voltage and flux components on the dual grids is shown in Figure 3.
dual grid element
h, b
e, d
grid element
Figure 3: Allocation of the voltage and flux components in the mesh.
Due to the consistent transformation, the analytical properties of the fields are maintained
resulting in corresponding discrete topological operators on the staggered grid duplet. The
topology matrices C , C , S and S correspond to the curl- and the div- operators. The tilde
means that the operator is performed on the dual grid.
After discretization, the material property relations become
d = Mε e
(VI-5)
b = Mμh
(VI-6)
j = M κ e + jA
(VI-7)
where Mε, Mμ and Mk are matrices describing the material properties. The relations in (VI-1) (VI-4) are exact on a given mesh, however, the material matrices contain the unavoidable
approximations of any numerical procedure. In addition, these matrices have diagonal form [78].
Employing a so-called leap-frog scheme which samples values of e and h at times separated
by a half time step, the MGEs can be rewritten as a set of two recursion formulas:
hi +1 = hi − ΔtM μ−1Cei +1/ 2
(
ei +3/ 2 = ei +1/ 2 + ΔtM ε−1 Chi +1 − j i +1
)
(VI-8)
(VI-9)
The recursion is stable if the time step inside an equidistant grid is restricted by the Courant
criterion to
Δt ≤
1
1
1
1
c
+ 2+ 2
2
Δx Δy
Δz
(VI-10)
19
The calculation of each further time step only requires one matrix-vector multiplication. Thus
it has the advantage being an explicit algorithm.
The FIT can be applied to different mesh types [79, 80]. On Cartesian grids, the time-domain
FIT is equivalent to FDTD.
B. The Partial Element Equivalent Circuit Method
The Partial Element Equivalent Circuit (PEEC) Method was developed by Ruehli [81, 82] in
the 1970s and 1980s. This electromagnetic modeling technique does not solve Maxwell’s
equations directly. Instead, it models electric field interactions as capacitances and magnetic field
interactions as inductances. Metallic geometries are segmented on their surfaces, much as they
would be using other integral equation techniques. Dielectrics (other than free space) are
segmented throughout their volume. Each segment is then represented as a node in a large circuit
model. The currents flowing in these segments are determined by analyzing the circuit using a
SPICE-like circuit solver.
This technique has the advantage that it is reasonably intuitive to many electrical engineers
and it is easy to integrate the field solver with real circuit elements. Problems can be solved in
the time domain or in the frequency domain,
Static modeling with PEEC codes is particularly powerful and intuitive [83-86]. Full-wave
modeling requires a circuit solver that can handle time-retarded values of mutual capacitance and
inductance. Since this is not a normal feature of SPICE solvers, the circuit solver for full-wave
PEEC codes must generally be distributed with the PEEC software.
C. The Transmission Line Matrix Method
The Transmission Line Matrix (TLM) method, introduced by Johns [87], is similar to the
FDTD method in terms of its capabilities, but its approach is unique. Like FDTD, analysis is
performed in the time domain and the entire region of the analysis is gridded. Instead of
interleaving E-field and H-field grids however, a single grid is established and the nodes of this
grid are interconnected by virtual transmission lines. Excitations at the source nodes propagate to
adjacent nodes through these transmission lines at each time step.
The symmetrical condensed node formulation introduced by Johns [88] has become the
standard for three-dimensional TLM analysis. The basic structure of the symmetrical condensed
node is illustrated in Figure 4. Each node is connected to its neighboring nodes by a pair of
orthogonally polarized transmission lines.
Figure 4: The Symmetrical Condensed Node.
20
Absorbing boundaries are easily constructed in TLM meshes by terminating each boundary
node transmission line with its characteristic impedance. Simons and Bridges [89, 90] derived
2D TLM absorbing boundaries, Saguet [91] proposed matched load simulation based on a Taylor
series expansion technique for 2D TLM waveguide problems. Morente et al. [92] investigated
TLM absorbing boundaries based on a one-way equation technique for three-dimensional
problems. Chen [93] implemented absorbing and connecting boundary conditions into a 3D
TLM simulation based on Higdon’s absorbing conditions, a Taylor expansion algorithm, and
connecting boundary conditions. Kukutsu [94] presented a super absorption boundary condition
for guided waves. Eswarappa [95] and Pena [96] describe an algorithm that interfaces the threedimensional (3D) transmission-line matrix (TLM) with an absorbing-boundary condition (ABC)
based on the perfectly matched-layer (PML) approach. Shao [97] implemented a Z-transform
based absorbing boundary condition.
Generally, dielectric loading is accomplished by loading the nodes with reactive stubs. These
stubs are usually half the length of the mesh spacing and have a characteristic impedance
appropriate for the amount of loading desired. Lossy media can be modeled by introducing loss
into the transmission line equations or by loading the nodes with lossy stubs. De Menenes [98100] presented the modeling of a nonlinear Lorentz dielectric and a frequency independent
dielectric with a Kerr nonlinearity. Hein [101, 102] developed the TLM model for propagation in
both magnetized plasma and ferrite. Paul [103-105] designed TLM algorithms for onedimensional (1D) and three-dimensional (3D) models involving linear frequency-dependent
isotropic dielectric media, anisotropic materials, and frequency dependent nonlinear dielectric
materials.
The strengths of the TLM method are similar to those of the FDTD method. Complex,
nonlinear materials are readily modeled. Impulse responses and the time-domain behavior of
systems are determined explicitly. And, like FDTD, this technique is suitable for implementation
on massively parallel machines.
Both the TLM and FDTD techniques are powerful and widely used. For many types of EM
problems they represent the only practical methods of analysis. Deciding whether to utilize a
TLM or FDTD technique is often based on personal preference. Many engineers find the
transmission line analogies of the TLM method to be more intuitive and easier to work with. On
the other hand, others prefer the FDTD method due to its simple, direct approach to the solution
of Maxwell’s field equations. For modeling propagation in complex materials, TLM may offer a
more straightforward solution than FDTD. Also, the TLM method generally does a better job of
modeling complex boundary geometries, because both E and H are calculated at every boundary
node. However, TLM methods require more computer memory per node than FDTD.
D. The Finite Volume Time Domain Technique
The Finite Volume Time Domain (FVTD) method was first applied to electromagnetic
problems in the early 1990’s [106, 107]. This technique is based on Maxwell’s curl equations in
their conservative form [108],
−
∂
Bdυ =
∂t ∫∫∫
V
∫∫ n × Eda
∂V
(VI-11)
21
∂
Ddυ =
∂t ∫∫∫
V
∫∫ n × Hda
∂V
(VI-12)
where ∂V represents the boundary enclosing V. The FVTD method solves the above form of
Maxwell’s equations numerically by integration over small elementary volumes. Because there
are no limitations for selecting the shape of the elementary volumes, the FVTD is well suited for
implementation with unstructured meshes. It has become a powerful alternative to the finite
difference time domain (FDTD) method for electromagnetic problems where conformal meshing
is advantageous.
Like FDTD, FVTD methods can take advantage of PML absorbing boundaries. Bonnet [109]
presented a vertex-centered FVTD model of the PML for scattering problems. Sankaran [110]
extended the PML concept to the cell-centered FVTD approach and systematically characterized
its performance using both structured and unstructured finite volume meshes. He introduced a
uniaxial Maxwellian absorber using PML to solve the waveguide truncation problem [111] and
extended the absorber to incorporate radial anisotropy for modeling cylindrical geometries [112].
Fumeaux [113] presented a spherical perfectly matched absorber for finite-volume 3D domain
truncation. Pinto [114] incorporated the uniaxial perfectly matched layer in the analysis of light
propagation in photonic bandgap devices.
The FVTD method is a promising numerical technique with good potential for the simulation
of a variety of complex electromagnetic problems [115-118]. Bonnet [119] presented a method
for the resolution of electromagnetic diffraction by complex structures; results obtained for an
aircraft were compared with results from a classical FDTD code. Lacour [120] described a multidomain decomposition method using an FVTD technique for the resolution of an
electromagnetic problem on vehicles and evaluated the current on a cable inside the volume of
an airplane. Applications of the technique in microwave engineering require both the
implementation of electromagnetic sources and the characterization of ports. Baumann [121,
122] introduced new schemes for full-wave field excitation and full-wave S-parameter extraction
that make the FVTD method especially well suited for microwave device simulations. To
improve the computational efficiency of the FVTD method, Fumeaux [123] introduced a new
generalized local time-stepping scheme, which is based on an automatic partition of the
computational domain into subdomains where local time steps of the type 2l-1∆t ( l = 1 2 3 . . .)
can be applied without violating the stability condition.
E. The Finite Element Time Domain Technique
The finite-element time-domain (FETD or TDFEM) method combines the advantages of a
time-domain technique with the versatile spatial discretization options of the finite element
method. A variety of FETD methods have been proposed. These schemes generally fall into two
categories. Methods in the first category directly discretize the time-dependent Maxwell’s
equations, yielding an explicit, conditionally stable, time-marching algorithm that can be viewed
as a generalization of the finite-difference time-domain (FDTD) method for unstructured grids
[124-133]. Methods in the second category discretize the second-order vector wave (curl-curl)
equation, obtained by eliminating one of the field variables from Maxwell’s equations. The
22
solution of a linear system of equations is required at each time step, but this implicit method can
be formulated to be unconditionally stable [134-146].
The explicit FETD has less computational complexity, however the maximum time-step must
be constrained to insure stability and it can be relatively difficult to achieve convergence. In
implicit FETD methods, the time step is not constrained by a stability criterion and these
methods can be extended to higher-orders in a relatively straightforward manner. However,
implicit schemes have greater computational complexity because they require a global linear
system of equations to be solved at each time step. This can make the simulation of large-scale
electromagnetic problems relatively inefficient.
Lou [147] presented a dual-field FETD formulation that computes both the electric and
magnetic fields in a leapfrog fashion. This formulation has the advantages of implicit FETD
schemes while reducing the computational complexity significantly when the computational
domain is split into non-overlapping smaller subdomains. He extended the domain
decomposition to the element level in [148].
For unbounded problems, the truncated boundaries of the FETD computational domain need
to be properly treated. This is usually done by using conventional absorbing boundary operators
[149, 150], boundary integral (BI) methods [151], or PML boundaries [152-160]. As discussed in
Section IV, absorbing boundary operators are easy to implement but can exhibit large reflection
errors [150]. Boundary integral methods are theoretically exact but computationally expensive.
PML implementations yield very small reflection errors in other time-domain methods, such as
the FDTD method. However, the FETD formulation of the PML has not been thoroughly
investigated.
F. The Time Domain Method of Moments
Most moment method codes solve integral equations in the frequency domain, but it is also
possible to use the method of moments to solve time-domain integral equations [161-190].
Consider a perfect electric conductor in free space excited by an incident field E i ( r , t ) . This
incident field induces a current J (r , t ) on the surface S of the conductor that in turn radiates a
scattered field. Enforcing the boundary condition on the total magnetic field or electric field on S
gives rise to a time domain magnetic field integral equation (TDMFIE) or a time domain electric
field integral equation (TDEFIE), respectively.
J ( r , t ) = 2nˆ × H i ( r , t ) +
⎡ J ( r ′, t ′ ) × R 1 ∂J ( r ′, t ′ )
⎤
1
+
× Rˆ ⎥ds′
nˆ × ∫ ⎢
2
s
2π
R
Rc
∂t ′
⎢⎣
⎥⎦
4π ∂ i
1 ∂2
E (r ,t )
=
c ∂t 2
η ∂t
Tangent to S
J ( r ′, t ′ )
⎛1⎞
∫s R ds′ − c ∫s ∇′ ⋅ J ( r ′, t ′) ∇ ⎜⎝ R ⎟⎠ds′
(
)
(VI-13)
(VI-14)
Note that in the principal value, we essentially exclude the part where the source and
observation points are the same (i.e. R=0). Since t ′ = t − R / c , and R ≠ 0 , it is always true that
t ′ < t . The main difficulty in extending the approach used to solve the frequency domain integral
equations comes from the retarded time variable. However, the TDIEs can be solved numerically
by means of a marching-on-in-time (MOT) procedure.
23
Like the method of moments in the frequency domain, the MoM-TD method discretizes the
scatterers or targets into segments or patches. The time axis is then divided into equal increments
or time steps. The triangular patches and vector basis functions proposed by Rao–Wilton–
Glisson (RWG) [8] are commonly used to discretize the current in space and time by expanding
the current J (r , t ) as a finite linear combination of products of spatial basis functions S n ( r ) and
temporal basis functions Tk ( t )
J ( r , t ) = ∑∑ I j ,n Sn ( r )Tk ( t )
Nt
Ns
j = k n =1
(VI-15)
where the temporal basis functions are generally versions of the same function shifted by a
certain number of time steps, Tk ( t ) = T ( t − k Δt ) with T ( t ) = 0 , ∀ t < Δt .
To determine the expansion coefficients I j ,n , Galerkin testing functions are applied in space
and point matching is applied at times t j = j Δt , j = 1, 2,3... , leading to a set of matrix equations
that can be written as,
[V ] = [ Z ][ I ] .
(VI-16)
The vector [V] contains the known incident field quantities and the terms of the Z-matrix are
functions of the geometry. The unknown coefficients of the induced current are the terms of the
[I] vector. These values are obtained by solving the system of equations iteratively. For example,
Andriulli [188] proposed an explicit iterative scheme,
Z 0 I j = V j − ∑ Z k I j −k
j
k =1
(VI-17)
where I j is a N s ×1 vector, Z k is a N s × N s matrix relating the currents on the body at time
tk = k Δt . The current coefficient vector I j can be obtained once the current coefficient vectors
I k , k = 1,..., j − 1 are known.
The TDIEs have applications and limitations similar to their frequency domain counterparts.
The EFIE is suitable for closed and open bodies, while the MFIE is only suitable for smooth,
closed bodies. MoM-TD techniques are not very effective when applied to arbitrary
configurations with complex geometries or inhomogeneous dielectrics. They also are not wellsuited for analyzing the interior of conductive enclosures or thin plates with wire attachments on
both sides. However, the time domain MoM is especially well suited for dealing with fast
transient electromagnetic fields incident on or radiated from structures in free space.
G. The Generalized Multipole Technique
The Generalized Multipole Technique (GMT) was developed in the 1980s by C. Hafner [191]
and implemented in software known as the Multiple Multipole (MMP) programs. It is essentially
a frequency-domain moment-method technique where the basis functions are analytic solutions
of the fields generated by sources located some distance away from the surface where the
boundary conditions is being enforced. These basis functions are spherical wave field solutions
24
corresponding to multipole sources. By locating these sources away from the boundary, the field
solutions form a smooth set of basis functions on the boundary and singularities on the boundary
are avoided.
Like the method of moments, a system of linear equations is developed and then solved to
determine the coefficients of the basis functions that yield the best solution. Since the basis
functions are already field solutions, it is not necessary to do any further computation to
determine the fields. Conventional moment methods determine the currents and/or charges on
the surface first and then must integrate these quantities over the entire surface to determine the
fields. This integration is not necessary at any stage of the GMT solution.
There is little difference in the way dielectric and conducting boundaries are treated by the
GMT. The same multipole expansion functions are used. For this reason, a general purpose
implementation of the GMT models configurations with multiple dielectrics and conductors
much more readily than a general purpose moment method technique.
Despite the advantages of this technique for certain types of modeling, it is not widely used.
This may be partly because this technique is a little less intuitive to use and it can be difficult to
learn to locate the multipole sources optimally.
Note that the Generalized Multipole Technique should not be confused with the Fast
Multipole Method (FMM), which is a technique for exploiting symmetry or periodicity in
structures to accelerate some types of electromagnetic modeling codes.
H. The Finite Difference Frequency Domain Technique
Like the finite difference time domain method, the finite difference frequency domain
(FDFD) method is based on a finite differential approximation of the derivative operators in the
Maxwell curl equations. However, in FDFD, the time-harmonic version of these equations is
employed. While time-domain finite difference schemes are very popular, the finite difference
frequency domain method has received little attention in the literature. It is essentially similar to
the finite element method, but it requires a uniform grid.
VII.
ASYMPTOTIC-EXPANSION BASED METHODS
The techniques described in the previous sections are exact methods in that the error in the
numerical solution only comes from the discretization. The numerical solution approaches the
exact solution as the discretization is refined. However, as the number of unknowns grows, the
demand for computer memory and calculation time also grows. This prohibits these methods
from being applied to high frequency problems where the size of the object is much larger than
the wavelength. The methods described in this section are based on asymptotic high-frequency
expansions of Maxwell’s equations. They are high frequency methods that are only accurate
when the dimensions of the objects being analyzing are large compared to the wavelength of the
field. The asymptotic techniques introduced in the following sections include physical optics,
geometrical optics, geometrical theory of diffraction, and uniform theory of diffraction.
25
A. Physical Optics
The Physical Optics (PO) approximation is a well known and efficient method for analyzing
large scatters [192]. PO reduces the cost of memory and CPU-time by performing a high
frequency approximation. It is a current-based method in which the physical optics
approximation is used to obtain the current density induced on a surface. The surface current
density, Js, can be determined by,
(
J S = n̂ × H i + H r
)
(VII-1)
where Hi and Hr represent the incident and reflected magnetic field components evaluated on the
surface. n̂ is the unit vector normal to the surface. If the surface can be approximated as an
infinite plane surface, then by image theory,
nˆ × H i = nˆ × H r
(
(VII-2)
)
and Equation (VII-1) reduces to
J S = nˆ × H i + H r = 2nˆ × H i = 2n̂ × H r
(VII-3)
The electric and magnetic field radiated by the surface current on the illuminated side of the
reflector can be determined by [194],
E(r ) =
w ε0
j
∇ × ∇ × ∫ J S ( r , r ' ) g ( r , r ') dS '
Sr
H ( r ) = ∇ × ∫ J S ( r , r ' ) g ( r , r ') dS '
Sr
where g ( r , r ') =
(VII-4)
(VII-5)
jk r −r '
e
.
r −r '
Equation (VII-3) is exact only when the surface is infinitely large. The accuracy of the
approximation depends on the transverse dimensions of the reflecting surface, the radius of
curvature, location of edges, and the angle of the incident field. Generally, PO works well for
large, smooth surfaces with low curvature. The implicit assumption for the physical optics
approximation is that the incident field is treated as a locally planar wave. Also, it assumes that
the reflector surface is perfectly conducting.
It has been found that PO provides an accurate prediction of far-field patterns of reflected
antennas in the main beam region and out to several side lobes [196]. The major disadvantage of
PO is that the integration over the surface of the reflector may be quite complicated and time
consuming when the feed is placed off-axis or the feed pattern is asymmetric [197]. Moreover,
the radiation integral has to be evaluated each time the observation point is changed.
Fast and efficient evaluation of the radiation integral was proposed using a fast series
approach [196], incorporating a multilevel fast multipole method [197], or decomposing the
scatterer into subdomains [198]. Initially applied in the frequency domain, PO has also been
extended into the time domain [199].
26
B. Geometrical Optics (Ray Optics)
Geometrical Optics (GO) [202] or geometrical optics with aperture integration (GO/AI) is a
ray-based method intended for the consideration of electrically large dielectric structures in
applications like the analysis of reflector antennas. In GO analysis, geometrical optics techniques
(ray tracing) are used to set up equivalent currents on an aperture plane which is normal to the
axis of the reflector. Then, the tangential aperture fields are constructed and used to determine
the radiated fields utilizing the Fourier transform. Different formulations are obtained based on
the use of aperture electric fields, magnetic fields or their combinations [201]. The advantage of
the GO/AI method is that the integration over the aperture plane can be performed with equal
ease for any feed pattern or feed position [195].
The relationship between GO and PO was demonstrated in [201]. It was shown that the PO
integral can be represented as a summation of many Fourier transforms, such that the first few
terms resemble the GO representation. Using the “extinction theorem” [194], the fields predicted
by the integration of PO surface currents were shown to agree with the geometrical optics
aperture fields on the aperture plane to within the local plane wave approximation. It was
concluded that the accuracy obtained by the two methods is comparable.
C. Geometrical Theory of Diffraction
The approximations in both physical optics and geometrical optics are based on the following
assumptions [194]:
•
The current density is zero on the shadow side of the reflector
•
The discontinuity of the current density over the rim of the reflector is neglected
•
Direct radiation from the feed and aperture blockage by the feed are neglected.
Both PO and GO ignore the edge diffractions which are highly dependent on the whether the
edges of the reflector are flared, sharp, absorber lined or serrated. Thus, they cannot accurately
predict the far fields beyond the first few side lobes. For predicting the patterns more accurately
in all regions, geometrical diffraction techniques are required.
As an extension of GO, the Geometrical Theory of Diffraction (GTD) overcomes the
limitations of GO by introducing a diffraction mechanism [203]. The diffracted field is
determined at the points on the surface where there is a discontinuity in the incident and reflected
field. The value of the diffracted field is evaluated at these points with the aid of an appropriate
diffraction coefficient. Usually, the coefficient is determined from asymptotic solutions of simple
boundary-value problems with so called canonical geometries, such as a conducting wedge,
cylinder or sphere. Since the solutions of these canonical problems are known, the object under
investigation can be partitioned into smaller components, so that each component represents a
canonical geometry. The ultimate solution is a superposition of the contributions from each
component [193].
Two major advantages of GTD over other high frequency asymptotic techniques are that it
provides insight into the radiation and scattering mechanisms from the various parts of the
structure, and it can yield more accurate results. The method has attracted increasing attention;
especially for applications to reflector antennas [204-209]. Unfortunately, GTD fails in the
transition region adjacent to the shadow boundary, at caustics (points through which all the rays
27
of a wave pass), or in close proximity to the surface of the scatterer. In these zones, the field
cannot be treated as a plane wave. Thus, ray techniques become invalid. To deal with this
problem, a number of alternative approaches have been proposed: uniform solutions [211-212],
methods for dealing with caustic curves [213-215], physical theory of diffraction (PTD) [216],
and the spectral theory of diffraction (STD) [217-218]. A comprehensive introduction to these
methods can be found in [210].
D. Uniform Theory of Diffraction
The Uniform Theory of Diffraction (UTD) is a uniform version of the geometrical theory of
diffraction. It was initially proposed [219] to deal with the problem that GTD produces
inaccurate results at the shadow boundaries. The uniform theory of diffraction approximates near
electromagnetic fields as quasi-optical and uses ray diffraction to determine diffraction
coefficients for each diffracting object-source combination. These coefficients are then used to
calculate the field strength and phase for each direction away from the diffracting point.
VIII.
HYBRID METHODS
Many practical problems are too complicated to be solved accurately by a single numerical or
asymptotic method. It is often advantageous to combine two numerical modeling techniques in a
single field solver in order to take advantage of the strengths of each technique to solve problems
that neither technique alone could model efficiently. For example, a finite element method can be
combined with a boundary element method to form a more powerful hybrid numerical technique
for analyzing both open-region problems and complex inhomogeneous objects. A full-wave
numerical technique can also be combined with an asymptotic method to model very large
objects with small features that require detailed analysis, such as an antenna mounted on an
airplane.
Hybrid methods are not simply two separate modeling codes with a common user interface. A
hybrid method generally divides a problem into two parts and applies a different technique to
each part while matching the currents or fields at the boundary to ensure a unique solution. Some
hybrid codes solve one part first, and then use the boundary fields as the sources when solving
the second part. Other hybrid codes solve both parts simultaneously allowing the solution of each
part to influence the solution of the other.
Formulations for many hybrid techniques have been developed and reported in the literature.
Some of the more common (and useful) hybrid techniques are described in the following
sections.
A. Hybrid FEM/BEM
Finite element methods excel at modeling complex volumetric structures, but are weak when
it comes to modeling thin wires and unbounded radiation problems. Boundary element methods
excel at modeling wires and unbounded geometries, but do not model complex structures that
include a variety of materials well. The complementary strengths of these two methods make
them ideal candidates for hybridization. Full-wave FEM/BEM (also called FEM/MOM, FE-BE
and FE-BI) techniques have been successfully used to model many problems that could not be
modeled effectively using either of the two techniques alone [220-227].
28
Hybrid FEM/BEM techniques introduce a fictitious surface (which may or may not coincide
with an actual material surface) that separates an interior volume from an exterior volume. The
interior region is analyzed using a finite element method with unknown electric or magnetic
surface currents establishing the boundary condition on the outer surface. The exterior region is
analyzed using a boundary element method, with unknown electric or magnetic currents on the
fictitious surface. Two sets of matrix equations are developed that share unknowns on the
boundary between the interior and exterior volumes. By forcing the fields on both sides of the
fictitious surface to be consistent with each other, the two matrix equations can be combined into
one larger equation with a unique solution.
In practice, it is relatively inefficient to generate one large matrix that is partly dense and
partly sparse. Inward looking techniques repeatedly solve the finite element portion of the
problem, while populating the boundary element method matrix. Outward looking techniques
repeatedly solve the boundary element method portion while populating the finite element
matrix. Choosing the right hybrid technique for a particular application greatly increases the
efficiency of a hybrid FEM/BEM approach [228].
B. Hybrid MOM/GTD, MOM/PO
Asymptotic methods can deal with objects whose overall dimensions are large in terms of the
wavelength. However, if large objects contain features that are too fine to be analyzed by an
asymptotic method, it becomes necessary to employ hybrid methods that combine asymptotic
techniques with numerically rigorous methods. Techniques that combine a moment method with
an asymptotic method can be broadly categorized as either ray-based or current-based. Raybased methods, such as MOM/GTD, provide a considerable speed advantage, but can be difficult
to implement for arbitrary and complex objects [237].
Hybrid MOM/GTD techniques were first described in the 1970s [229-230]. Since GTD fails
in regions where the field cannot be approximated by a local plane wave, uniform solutions have
been developed that overcome some of the limitations of GTD. Hybrid approaches combining
MOM and UTD are discussed in [231-233].
Physical optics is a current-based asymptotic method. The hybridization of PO with MOM
has two advantages over the combination of ray-based methods and MOM. First, since both
MOM and PO are current-based, they can be easily blended on the same surface. Second,
MOM/PO is relatively general in that there are no specific restrictions on the geometries that can
be modeled [234]. Consequently, this hybrid technique has received considerable attention in the
literature [234-241]. It has also been implemented in a commercial numerical code [242].
C. Hybrid FEM/PO
Hybrid methods that combine high-frequency asymptotic techniques with the method of
moments are not suitable for solving problems with inhomogeneous or anisotropic materials.
These types of problems are better suited for analysis by a hybrid FEM/PO technique. The
hybridization of FEM and asymptotic techniques is described in [243-244].
29
IX.
COMMERCIAL CEM SOFTWARE SURVEY
As part of this project, lists of free and commercial CEM modeling codes were developed and
published on the web at the following URLs,
Free CEM Codes:
Commercial CEM Codes:
http://www.cvel.clemson.edu/modeling/EMAG/free-codes.html
http://www.cvel.clemson.edu/modeling/EMAG/csoft.html.
The companies or contacts associated with each code in these lists were asked to complete a brief
online survey describing the general features, capabilities and costs of their products. Their
responses are summarized in the following tables. In some cases, the responses received on the
survey forms were inconsistent with the information on the company’s web site or with the first
hand experience of the authors. We’ve done our best to insure that the information provided in
the tables below is accurate. Any errors or omissions can be reported to CVEL-L@clemson.edu.
Reported corrections will be made to the information posted on the web site.
A. 3D Full-Wave Codes
The main emphasis of this survey was on general purpose 3D full-wave codes. Twelve of the
codes included in the survey responses were placed in this category. Note that some of the codes
in the “hybrid” category can also be used as general purpose full-wave codes.
Table 8: General Purpose 3D Full-Wave Codes
Software Name
Company
Technique
Cost
Agilent
FDTD
between
AMDS Antenna
$10,000
Modeling Design
and
System
$50,000
Applied Simulation FDTD
between
ApsimFDTD
Technology
$10,000
and
$50,000
Computer
FDTD
between
CST Microwave
$10,000
Studio – Transient Simulation
Technology
and
Solver
$50,000
Computer
FEM
between
CST Microwave
Simulation
$10,000
Studio –
Technology
and
Frequency
$50,000
Domain Solver
Comsol
FEM
between
Comsol
$10,000
Multiphysics – RF
and
Module
$50,000
Comment
for wireless appliance
design and modeling
part of a suite of tools
for IC and package
analysis
part of a suite of tools
part of a suite of tools
part of a suite of tools
30
EMA3D
Electro Magnetic
Applications
FDTD
EMDS-for-ADS
Agilent
Technologies
FEM
emGine
Environment
Petr Lorenz
TLM
EMPIRE XCcel
Empire
FDTD
EZ-EMC
EMS-Plus
FDTD
EZ-FDTD
EMS-Plus
FDTD
EZNEC Pro
EZNEC
BEM
Fidelity
Zeland Software
FDTD
GEMS
Computer and
Communication
Unlimited
FDTD
HFSS
Ansys/Ansoft
FEM
IE3D
Zeland Software
BEM
LC
MaX-1
LC
John Wiley & Sons
FDTD
GMT
MEEP
MIT
FDTD
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$1000 and
$10,000
between
$1000 and
$10,000
between
$200 and
$1000
not
reported
between
$10,000
and
$50,000
greater
than
$50,000
not
reported
FREE
between
$1000 and
$10,000
FREE
EMDS was integrated
into Agilent ADS in
Aug. 2008.
GUI is open source.
Binaries are free for
non-commercial use.
wire and wire-grid
modeling
did not respond to
survey
Did not respond to
survey
package includes 2D
and 3D FDTD solvers
Free software under the
GNU GPL.
31
numerical TLM
simulation engines for
2D problems are also
included.
wire grid modeling plus
simple plates modeled
by point matching, text
file input and output
U.S. export controlled
MEFiSTo-3D Pro
Faustus Scientific
Corporation
TLM
between
$1000 and
$10,000
NEC2
Lawrence
Livermore Lab
BEM
FREE
NEC4
Lawrence
Livermore Lab
BEM
PAM-CEM
ESI Group
FDTD
PhysPack
Physware
BEM
SEMCAD X
Schmid & Partner
Engineering
FDTD
Toy
FDTD
TLM
FREE
for educational use
WIPL-D Pro
The CEMTACH
Group
The CEMTACH
Group
WIPL-D
between
$200 and
$1000
between
$10,000
and
$50,000
greater
than
$50,000
between
$10,000
and
$50,000
FREE
BEM
metallic and dielectric
3D structures with
wires and plates
XFDTD
Remcom
FDTD
between
$10,000
and
$50,000
between
$10,000
and
$50,000
ToyTLM
chip-package-board
simulation
package includes quasistatic solvers
for educational use
B. 3D Quasi-Static Codes
In many situations, it is better to use a quasi-static modeling code to model components that are
small relative to the wavelengths of interest, even when these components are used at RF or
microwave frequencies. Generally, quasi-static modeling codes are more powerful and more
efficient for modeling complex electrically small geometries than full-wave codes. The codes
listed in the Table 9 are electrostatic and/or magnetostatic modeling codes.
32
Table 9: 3D Quasi-Static Codes
Software Name
Company
Field Precision
AMaze - HiPhi
Technique
FEM
AMaze Magnum
Field Precision
FEM
Amperes
Integrated Engineering
Software
BEM
Coulomb
Integrated Engineering
Software
BEM
Comsol
Multiphysics –
AC/DC module
Comsol
FEM
EMS
ElectromagneticWorks
FEM
EMPLab
EM Photonics
FDTD
Faraday
Integrated Engineering
Software
BEM
Flux3D
Magsoft
FEM
Opera 3D
Vector Fields Inc.
FEM
MagNet
Infolytica Corporation
FEM
Maxwell
Ansys/Ansoft
FEM
Cost
between
$1000 and
$10,000
between
$1000 and
$10,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$1000 and
$10,000
between
$1000 and
$10,000
between
$10,000
and
$50,000
between
$1000 and
$10,000
between
$1000 and
$10,000
between
$10,000
and
$50,000
not reported
Comment
electrostatic (part
of a suite of tools)
magnetostatic (part
of a suite of tools)
magnetostatic
electrostatic
electrostatic,
magnetostatic suite
of tools
electrostatic,
magnetostatic suite
of tools
MATLAB based
software
magnetostatic
electrostatic and
magnetostatic tools
electrostatic and
magnetostatic tools
magnetostatic
2D/3D
electrostatic /
magnetostatic
(includes 2D tool)
33
ElecNet
Infolytica Corporation
FEM
PCBMod
Simlab GmbH
2D BEM or
3D PEEC
Q3D Extractor
Ansys/Ansoft
FEM
between
$10,000
and
$50,000
between
$10,000
and
$50,000
greater than
$50,000
electrostatic 2D/3D
electrostatic
electrostatic and
magnetostatic tools
C. 2D High-Frequency Codes
Great insight into wave propagation, reflection, diffraction, shielding, etc. can be obtained from
2D high-frequency modeling codes. 2D codes use a fraction of the computational resources
required by 3D codes; and it is much simpler to generate 2D inputs and visualize 2D outputs.
Table 10: 2D High-Frequency Codes
Software Name
Company
Faustus Scientific
MEFiSTo-3D
Corporation
Pro
Momentum
Agilent
TriComp WaveSim
Field Precision
Technique
Cost
TLM
between
$1000 and
$10,000
BEM
between
$10,000 and
$50,000
FEM
between
$1000 and
$10,000
Comment
2D simulation
engines packaged
with 3D tool
2.5D code for planar
circuits
part of a suite of
tools
34
D. 2D Quasi-Static Codes
2D quasi-static codes are excellent tools for visualizing electric and magnetic field distributions.
Generally, these tools have a relatively low cost and do not require a lot of computer resources.
Table 11: 2D Quasi-Static Codes
Software Name
Company
Field Precision
TriComp - EStat
TriComp - PerMag
Field Precision
Finite Element
Method Magnetics
(FEMM)
Electro
Dr. David Meeker
Flux2D
Magneto
Maxwell
Oersted
Technique
Cost
Comment
FEM
between $200 electrostatic
and $1000
FEM
between $200 magnetostatic
and $1000
FEM
FREE
electrostatic /
magnetostatic
Integrated
Engineering
Software
Magsoft
BEM
Integrated
Engineering
Software
Ansys/Ansoft
FEM /
BEM
Integrated
Engineering
Software
FEM /
BEM
FEM
FEM
pdnmesh
Q2D
Ansys/Ansoft
FEM
FEM
QuickField
Tera Analysis
FEM
between
$10,000 and
$50,000
between
$1000 and
$10,000
between
$10,000 and
$50,000
not reported
between
$10,000 and
$50,000
FREE
greater than
$50,000
not reported
electrostatic
electrostatic,
magnetostatic suite of
tools
includes BEM and
FEM solvers
electrostatic /
magnetostatic
(includes 3D tool)
includes BEM and
FEM solvers
electrostatic
bundled with Q3D
Extractor
did not respond to
survey
35
E. Hybrid Codes
Hybrid codes combine the features of two different modeling techniques in order to be able to
model a wider range of problem geometries. Hybrid codes are not suites of tools that share a
common user interface. Hybrid codes are capable of simultaneously applying two different
solvers to different regions of a problem geometry. Each of the codes below is also capable of
employing just a single technique and therefore could have been included with the 3D full-wave
codes listed in Table 8.
Table 12: Hybrid Codes
Software
Company
Name
CONCEPT- Technical
University of
II
Hamburg-Harburg
Applied Research
GEMACS
Associates
FEKO
Singula
SuperNEC
efield
EMAP5
Technique
Hybrid BEM and
PO
Cost
between
$1000 and
$10,000
BEM-FDFD-UTD
between
$200 and
$1000
EM Software and BEM-FEM-PO
between
Systems
$10,000 and
$50,000
Integrated
BEM-PO
between
Engineering
$10,000 and
Software
$50,000
Poynting Antennas BEM -PO
between
$1000 and
$10,000
Efield AB
hybrid FDTD-FEM- greater than
TD hybrid MoM$50,000
MLFMM-PO
Clemson
BEM-FEM
FREE
University
Comment
large wire and thin
plate structures
large wire and thin
plate structures, with
dielectric capability
two different hybrid
tools
F. Special Purpose Codes
The codes below employ numerical electromagnetic modeling techniques, but are optimized for
specific applications.
36
Table 13: Special Purpose Codes
Software
Company
Name
Field Precision
AMaze OmniTrak
Technique
FEM
Analyst
Simulation Technology
& Applied Research
FEM-TD,
FEM
Antenna
Model
Teri Software
BEM
AXIEM
AWR
BEM
CableMod
Simlab GmbH
BEM
Compliance
Quantic EMC
BEM
EM Explorer EM Explorer
FDTD
EM3DS
MEM Research
BEM
EMFlex
Weidingler Associates
Inc.
FEM-TD
EZPowerPlane
EMS-Plus
Cavity
Resonance
FlexPDE
PDE Solutions
FEM-TD
Cost
between
$1000 and
$10,000
between
$10,000
and
$50,000
less than
$200
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$1000 and
$10,000
between
$1000 and
$10,000
between
$10,000
and
$50,000
between
$1000 and
$10,000
between
$1000 and
$10,000
Comment
3D charged particle
beam analysis
suite of tools for
designing microwave
and accelerator
components
antenna model uses the
MININEC code, used
for the analysis of wire
antennas
for modeling planar
circuits
2D electrostatic solver
for calculating
transmission line
parameters and
crosstalk
calculates currents on
circuit traces (2D
BEM), then calculates
radiated fields in 3D
solver for scattering
problems of periodic
structures
MMIC modeling
optical modeling
modeling power bus
noise in printed circuit
boards
general PDE solver can be 1D, 2D or 3D
37
GSolver
Grating Solver
Development Company
between
$1000 and
$10,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
FREE
HFWorks
ElectromagneticWorks
FEM
JCMsuite
JCMwave GmbH
FEM
Open FMM
Open FMM
BEM
OptEM
Inspector
OptEM Engineering
not reported
OptEM
Cable
Designer
OptEM Engineering
not reported
ScatLab
ScatLab
FREE
Speed2000
Sigrity
Mie theory,
T-Matrix
method
FDTD
Trace
Analyzer
Trace Analyzer
BEM or
MOM
between
$200 and
$1000
TriComp Trak
Field Precision
FEM
XGtd
Remcom
UTD/GTD
between
$1000 and
$10,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
between
$10,000
and
$50,000
not
reported
periodic (grating)
boundaries
RF, Microwave and
MMwave component
analysis
For optical modeling
Large scattering
problems
RC extraction software
tool for deep
submicron (DSM) IC
designs
For modeling multiconductor flex,
unshielded twistedpair and twisted-pair
cables
Large Scattering
Problems
for modeling
interactions in multilayer chip packages
and printed circuit
boards
Electrostatic solution
to calculate the [C]
matrix, and the R/L/G
matrices are derived
from [C]
2D charged particle
beam analysis
3D high-frequency ray
tracing code.
38
REFERENCES
[1] Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Philosophical
Transactions of the Royal Society of London 155, pp. 459-512, 1865. (This article
accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
[2] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s
equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. 14, no. 4, pp. 302-307,
1966.
[3] R. F. Harrington, “Matrix Methods for Field Problems,” Proc. IEEE, vol. 55, no. 2,
pp. 136-149, Feb. 1967.
[4] T. Hubing, “Survey of Numerical Electromagnetic Modeling Techniques,” University of
Missouri-Rolla, Department of Electrical Engineering, TR91-1-001.3, Sep. 1991.
[5] R. F. Harrington, Field Computation by Moment Methods, Macmillan Publishing Company,
New York, 1968. (reprinted by IEEE Press in 1993.)
[6] R. F. Harrington, Time Harmonic Electromagnetic Fields, New York, McGraw Hill, 1961,
Ch. 3.
[7] R. E. Collin, Field Theory of Guided Waves, 2nd ed., New York, IEEE Press, 1991, Ch. 2.
[8] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of
arbitrary shape," IEEE Trans. Antennas Propagat., vol. 30, pp. 409–418, May 1982.
[9] D. C. Chang and J. X. Zheng, “Electromagnetic modeling of passive circuit elements in
MMIC,” IEEE Trans. on Microwave Theory and Tech., vol. 40, no. 9, pp. 1741 – 1747,
Sep. 1992.
[10] P. Silvester, “High-order finite element waveguide analysis (Program Descriptions),” IEEE
Trans. on Microwave Theory and Tech., vol. 17, no. 8, pp. 651-652, Aug. 1969.
[11] O. C. Zienkiewicz, A. K. Bahrani, and P. L Arlett, “Numerical solution of 3-dimensional
field problems,” Proc. IEE (London), vol. 115, pp. 367-369, Feb. 1968.
[12] B. H. McDonald and A. Wexler, “Finite-element solution of unbounded field problems,”
IEEE Trans. on Microwave Theory and Tech., vol. 20, no. 12, pp. 841-847, Dec. 1972.
[13] D. R. Lynch and K. D. Paulsen, “Origin of Vector Parasites in Numerical Maxwell
Solutions,” IEEE Trans. on Microwave Theory and Techniques, vol. 39, no. 3, pp. 383-394,
Mar. 1991.
[14] D. R. Lynch and K. D. Paulsen, “Elimination of Vector Parasites in Numerical Maxwell
Solutions,” IEEE Trans. on Microwave Theory and Techniques, vol. 39, no. 3, pp. 395-404,
Mar. 1991.
[15] Y. Li and Z. Cendes, “High-accuracy absorbing boundary condition,” IEEE Trans. on
Magnetics, vol. 31, no. 3, pp. 1524-1529, May 1995.
[16] A. Bayliss, E. Turkel, "Radiation boundary conditions for wave-like equations,"
Communications on Pure and Applied Mathematics, vol. 33, pp. 707--725, 1980.
39
[17] A. Bayliss, M. Gunzburger, E. Turkel, "Boundary conditions for the numerical solution of
elliptic equations in exterior regions," SIAM Journal of Applied Mathematics, vol. 42, pp.
430-451, 1982.
[18] T.G. Moore, J.G. Blaschak, A. Taflove, G.A. Kriegsmann, "Theory and application of
radiation boundary operators," IEEE Trans. Antennas Propagat., vol. 36, pp. 1797-1812,
1988.
[19] J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J.
Comput. Phys. 114, pp. 185-200, 1994.
[20] J. M. Jin, and W. C. Chew, “Combining PML and ABC for finite element analysis of
scattering problems,” Micro. Opt. Tech. Lett., vol. 12, pp. 192—197, 1996.
[21] Wu J-Y, Kingsland DM, Lee J-F, Lee R. , “A Comparison of anisotropic PML to
Berenger's PML and its application to the finite-element method for EM scattering,” IEEE
Trans. on Antennas and Propagat., vol. 45, no.1, pp. 40-50, 1997.
[22] R. Mittra and M. Kuzuoglu, "A review of some recent advances in perfectly-matched
absorbers for mesh truncation in FEM," IEEE Antennas and Propagation Society
International Symposium, 1997 Digest, vol. 2, pp.1302 -1305, 1997.
[23] M. Ali, T. H. Hubing, and J. Drewniak, “A hybrid FEM/MOM technique for
electromagnetic scattering and radiation from dielectric objects with attached wires,” IEEE
Trans. on Electromag. Compat., vol. 39, no. 4, pp. 304-314, Nov. 1997.
[24] B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of
waves,” Mathematics of Computation, 31, pp. 629-651, 1977.
[25] E. L. Lindman, “‘Free-space’ boundary conditions for the time dependent wave equation,”
Journal of Computational Physics, 18, pp. 67-78, 1975.
[26] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the
time-domain electromagnetic-field equations,” IEEE Trans. on Electromag. Compat., vol.
23, no. 4, pp. 377-382, Nov. 1981.
[27] Z. P. Liao, H. L. Wong, B.-P. Yang, and Y.-F. Yuan, “A transmitting boundary for transient
wave analysis,” Science Sinica, Series A, 27, 10, pp. 1063-1076, 1984.
[28] R. G. Keys, “Absorbing boundary conditions for acoustic media,” Geophysics, 50, 6,
pp. 892-902, 1985.
[29] R. L. Higdon, “Numerical absorbing boundary conditions for the wave equation,”
Mathematics of Computation, 49, 179, pp. 65-90, 1987.
[30] C. Rappaport and L. Bahrmasel, “An absorbing boundary condition based on anechoic
absorber for EM scattering computation,” Journal of Electromagnetic Waves and
Applications, 6, 12, pp. 1621-1634, 1992.
[31] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetics waves,”
Journal of Computational Physics, 1 14, 1, pp. 185-200, 1994.
40
[32] W. Andrew, C. Balanis, and P. Tirkas, “A comparison of the Berenger perfectly matched
layer and the Lindman higher-order ABCs for the FDTD method,” IEEE Microwave and
Guided Wave Letters, 5, 6, pp. 192-194, 1995.
[33] C. J. Railton and E. M. Daniel, “Comparison of the effect of discretisation on absorbing
boundary algorithms in finite difference time domain method,” Electronics Letters, 28, 20,
pp. 1891-1893, 1992.
[34] V. Betz and R. Mittra, “Comparison and evaluation of boundary conditions for the
absorption of guided waves in an FDTD simulation,” IEEE Microwave and Guided Wave
Letters, 2, 12, pp. 499-501, 1992.
[35] R. Diaz and I. Scherbatko, “A new radiation boundary condition for FDTD based on selfteleportation of fields,” 2003 IEEE Microwave Symposium Digest, 3, 8, pp. 2073-2076,
2003.
[36] A.C. Cangellaris and D.B. Wright, “Analysis of the numerical error caused by the stairstepped approximation of a conducting boundary in FDTD simulations of electromagnetic
phenomena,” IEEE Trans. Antennas Propagat., vol. 39, no. 10, pp. 1518-1525, 1991.
[37] R. Holland, “Pitfalls of staircase meshing,” IEEE Trans. on Electromag. Compat., vol. 35,
no. 4, pp. 434-439, Nov. 1993.
[38] K. K. Mei, A. Cangellaris, and D. J. Angelakos, “Conformal time domain finite difference
method,” Radio Science, 19, 5, pp. 1145-1 147, 1984.
[39] K. S. Yee, J. S. Chen, and A. H. Chang, “Conformal finite difference time-domain (FDTD)
with overlapping grids,” IEEE Trans. Antennas Propagat., AP-40, 9, pp. 1068- 1075, 1992.
[40] T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference timedomain modeling of curved surfaces,” IEEE Trans. Antennas Propagat., AP-40, 4, pp. 357366, 1992.
[41] T. G. Jurgens and A. Taflove, “Three-dimensional contour FDTD modeling of scattering
from single and multiple bodies,” IEEE Trans. Antennas Propagat., AP-41, 12, pp. 17031708, 1993.
[42] S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD)
algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microwave
and Guided Wave Letters, vol. 7, no. 9, pp. 273–275, Sep. 1997.
[43] W. Yu and R. Mittra, “A Conformal Finite Difference Time Domain Technique for
Modeling Curved Dielectric Surfaces,” IEEE Microwave and Guided Wave Letters, vol. 11,
no. 1, Jan. 2001.
[44] I. A. Zagorodnov, R. Schuhmann, and T.Weiland, “A uniformly stable conformal FDTDmethod in cartesian grids,” Int. J. Numer. Modelling- Electron. Networks Devices Fields,
vol. 16, pp. 127–141, 2003.
[45] W. Sha, X. Wu, Z. Huang, and M. Chen, “ A new FDTD (2,4) scheme for modeling threedimensional curved perfectly conducting objects,” IEEE Microwave and Wireless
Components Letters, 18, 3, pp. 149-151, 2008.
41
[46] R. Holland and L. Simpson, “Finite-difference analysis EMP coupling to thin struts and
wires,’’ IEEE Trans. on Electromag. Compat., vol. 23,no. 2, pp. 88-97, 1981.
[47] K. S. Yee, “Numerical solution to Maxwell’s equations with non-orthogonal grids,” Tech.
Rep. UCRL-93268, Lawrence Livermore National Laboratory, 1987.
[48] A. Taflove, K. R. Umashankar, B. Beker, F. Harfoush, and K. S. Yee, “Detailed FD-TD
analysis of electromagnetic fields penetrating narrow slots and lapped joints in thick
conducting screens,” IEEE Trans. Antennas Propagat., vol. 36, no. 2, pp. 47-257, 1988.
[49] K. R. Demarest, “A finite difference-time domain technique for modeling narrow apertures
in conducting scatterers,” IEEE Trans. Antennas Propagat., vol. 35, no. 7, pp. 826-831,
1987.
[50] C. D. Turner and L. D. Bacon, “Evaluation of a thin-slot formalism for finite-difference
time-domain electromagnetics codes,” IEEE Trans. on Electromag. Compat., vol. 30, no. 4,
pp. 523-528, 1988.
[51] D. J. Riley and C. D. Turner, “The inclusion of wall loss in finite-difference time-domain
thin-slot algorithms,” IEEE Trans. on Electromag. Compat., vol. 33, no. 4, pp. 304-311,
1991.
[52] J. H. Oates and R. T. Shin, “Small aperture modeling for EM1 applications using the finitedifference time-domain technique,” Journal of Electromagnetic Waves and Applications, 9,
112, pp. 37-69. 1995.
[53] B.-Z. Wang, “Enhanced thin-slot formalism for the FDTD analysis of thin-slot
penetration,” IEEE Microwave and Guided Wave Letters, 5, 5, pp. 142-143, 1995.
[54] D. Hockanson, J. Drewniak, T. Hubing and T. Van Doren, “FDTD modeling of commonmode radiation from cables,” IEEE Trans. on Electromag. Compat., vol. 38, no. 3, pp. 376387, Aug. 1996.
[55] K.-P. Ma, M. Li, J. Drewniak, T. Hubing and T. Van Doren, “Comparison of FDTD
algorithms for subcellular modeling of slots in shielding enclosures,” IEEE Trans. on
Electromag. Compat., vol. 39, no. 2, pp. 147-155, May 1997.
[56] R. J. Luebbers, F. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A
frequency-dependent finite-difference time-domain formulation for dispersive materials,”
IEEE Trans. on Electromag. Compat., vol. 32, no. 3, pp. 22-221, Aug. 1990.
[57] T. Kashiwa, N. Yoshida, and I. Fukai, “A treatment by the finite-difference time-domain
method of the dispersive characteristics associated with orientation polarization,”
Transactions IEICE, E73, 8, pp. 1326-1328, 1990.
[58] T. Kashiwa and I. Fukai, “A treatment by the FD-TD method of the dispersive
characteristics associated with electronic polarization,” Microw. Opt. Technol. Lett.,, 3, 6,
pp. 203- 205, 1990.
[59] T. Kashiwa, Y. Ohtomo, and I. Fukai, “A finite-difference time-domain formulation for
transient propagation in dispersive media associated with Cole-Cole’s circular ARC law,”
Microw. Opt. Technol. Lett., 3, 12, pp. 416-419, 1990.
42
[60] R. Pontalti, L. Cristoforetti, R. Antolini, and L. Cescatti, “A multi-relaxation (FD)2-TD
method for modeling dispersion in biological tissues,” IEEE Trans. on Microwave Theory
Techniques, vol. 42, no. 3, pp. 526-527, 1994.
[61] P. G. Petropoulos, “Stability and phase error analysis of FDTD in dispersive dielectrics,”
IEEE Trans. Antennas Propagat., vol. 42, no. 1, pp. 62-69, 1994.
[62] J. G. Maloney and G. S. Smith, “The use of surface impedance concepts in the finitedifference time-domain method,” IEEE Trans. Antennas Propagat., vol. 40, no. 1, pp. 3848, 1992.
[63] K. S. Yee, K. Shlager, and A. H. Chang, “An algorithm to implement a surface impedance
boundary condition for FDTD,” IEEE Trans. Antennas Propagat., vol. 40, no. 7, pp. 833837, 1992.
[64] S. Kellali, B. Jecko, and A. Reineix, “Implementation of a surface impedance formalism at
oblique incidence in FDTD method,” IEEE Trans. on Electromag. Compat., vol. 35, no. 3,
pp. 347-356, 1993.
[65] C. J. Railton and J. P. McGeehan, “An analysis of microstrip with rectangular and
trapezoidal conductor cross sections,” IEEE Trans. on Microwave Theory Techniques, vol.
38, no. 8, pp. 1017-1022, 1990.
[66] J. G. Maloney and G. S. Smith, “The efficient modeling of thin material sheets in the finitedifference time-domain (FDTD) method,” IEEE Trans. Antennas Propagat., vol. 40, no. 3,
pp. 323-330, 1990.
[67] J. G. Maloney and G. S. Smith, “A comparison of methods for modeling electrically thin
dielectric and conducting sheets in the finite-difference time-domain (FDTD) method,”
IEEE Trans. Antennas Propagat., vol. 41, no. 5, pp. 690-694, 1993.
[68] A. Taflove, “Review of the formulation and applications of the finite-difference timedomain method for numerical modeling of electromagnetic wave interactions with arbitrary
structures,” Wave Motion, 10, 6, pp. 547-582, 1988.
[69] I. S. Kim and W. J. R. Hoefer, “Numerical dispersion characteristics and stability factor for
the TD-FD method,” Electronics Letters, 26, 7, pp. 485-487, 1990.
[70] S. L. Ray, “Numerical dispersion and stability characteristics of time-domain methods on
nonorthogonal meshes,” IEEE Trans. Antennas Propagat., vol. 41, no. 2, pp. 233-235,
1993.
[71] K. L. Shlager, J. G. Maloney, S. L. Ray, and A. F. Peterson, “Relative accuracy of several
finite-difference time-domain methods in two and three dimensions,” IEEE Trans.
Antennas Propagat., vol. 41, no. 12, pp. 1732-1737, 1993.
[72] A. Zhao, “Analysis of the numerical dispersion of the 2D alternating-direction implicit
FDTD method,” IEEE Trans. on Microwave Theory and Tech., vol. 50, no. 4, pp. 11561164, 2002
[73] T. Kashiwa, Y. Sendo, K. Taguchi, T. Ohtani, and Y. Kanai, “Phase velocity errors of the
nonstandard FDTD method and comparison with other high-accuracy FDTD methods,”
IEEE Trans. on Magnetics, vol. 39, no. 4, pp. 2125-2128, 2003
43
[74] J. Shibayama, M. Muraki, R. Takahashi, J. Yamauchi, and H. Nakano, “Performance
evaluation of several implicit FDTD methods for optical waveguide analyses,” Journal of
Lightwave Tech., vol. 24, no. 6, pp. 2465–2472, 2006.
[75] V. E. Nascimento, B. H. V. Borges, and F. L. Teixeira, “Split-field PML implementations
for the unconditionally stable LOD–FDTD method,” IEEE Microwave and Wireless
Components Letters, vol. 16, no. 7, pp. 398–400, Jul. 2006.
[76] E. Li, I. Ahmed, and R. Vahldieck, “Numerical dispersion analysis with an improved LODFDTD method,” IEEE Microwave and Wireless Components Letters, vol. 17, no. 5,
pp. 319-321, 2007.
[77] T. Weiland, “A discretization method for the solution of Maxwell's equations for sixcomponent fields”, Electronics and Communication (AEÜ), vol. 31, p. 116, 1977.
[78] T. Weiland, “Time Domain Electromagnetic Field Computation with Finite Difference
Methods”, International Journal of Numerical Modelling, vol. 9, pp. 295-319, 1996.
[79] B. Krietenstein, R. Schuhmann, P. Thoma, and T. Weiland, “The perfect boundary
approximation technique facing the challenge of high precision field computation”, in
Proceedings of the XIX International Linear Accelerator Conference (LINAC’98), Chicago,
USA, 1998, pp. 860-862.
[80] M. Walter, I. Munteanu, “FIT for EMC,” Proc. of the 17th International Zurich Symposium
on Electromagnetic Compatibility, 2006, pp. 15-17.
[81] A. Ruehli, “Equivalent circuit models for three dimensional multiconductor systems,” IEEE
Trans. on Microwave Theory and Tech., vol. 22, no. 3, pp. 216-221, Mar. 1974.
[82] A. Ruehli and H. Heeb, “Circuit models for three-dimensional geometries including
dielectrics,” IEEE Trans. on Microwave Theory and Tech., vol. 40, no. 7, pp. 1507-1516,
July 1992.
[83] C. Wollenberg and A. Gurisch, “Analysis of 3-D interconnect structures with PEEC using
SPICE,” IEEE Trans. on Electromag. Compat., vol. 41, no. 4, pp. 412-417, Nov. 1999.
[84] Y. Cao, Z.-F. Li, J.-F Mao, J.-F Mao, “A PEEC with a new capacitance model for circuit
simulation of interconnects and packaging structures,” IEEE Trans. on Microwave Theory
and Tech., vol. 48, no. 9, pp. 1435-1442, Sep. 2000.
[85] D. Gope, A. Ruehli, V. Jandhyala, “Solving low-frequency EM-CKT problems using the
PEEC Method,” IEEE Trans. on Adv. Packaging, vol. 30, no. 2, pp. 313-320, may 2007.
[86] G. Antonini, “PEEC capacitance extraction of 3-D interconnects,” Science, Measurement &
Technology, IET, vol. 1, no. 4, pp. 201-209, July 2007.
[87] P. B. Johns and R. L. Beurle, “Numerical solution of 2-dimensional scattering problems
using a transmission-line matrix,” Proc. Inst. Elec. Eng., vol. 118, no. 9, pp. 1203-1208,
1971.
[88] P.B. Johns, “A symmetrical condensed node for the TLM method,” IEEE Trans. on
Microwave Theory and Tech., vol. 35, no. 4, pp. 370-377, 1987.
44
[89] N. R. S. Simons and E. Bridges, “Method for modelling free space boundaries in TLM
simulations,” Electronics Letters, vol. 26, no. 7. pp. 453-155, Mar., 1990.
[90] N. R. S. Simons and E. Bridges, “Application of the TLM method to two-dimensional
scattering problems.” Int. J. Num. Modelling: Electron. Network Devices, Fields, vol. 5.
pp. 93-119, 1992.
[91] P. Saguet, “TLM method for the three dimensional analysis of microwave and mm-wave
structures.” Paper presented at Int. Workshop of the German IEEE M7T/AP Chapter on
CAD Oriented Numerical Techniques for the Analysis of Microwave and MM-Wave
Transmission-Line Discontinuities and Junctions, Stuttgart, Germany, Sept. 13, 1991.
[92] J. A. Morente, J. A. Porte, and M. Khalladi, “Absorbing boundary conditions for the TLM
method,” IEEE Trans. on Microwave Theory and Tech., vol. 40, no. 11, pp. 2095-2099,
1992.
[93] Z. Chen, M. M. Ney, and W. J. R. Hoefer, “Absorbing and connecting boundary conditions
for the TLM method,” IEEE Trans. on Microwave Theory and Tech., vol. 41, pp. 2016–
2024, Nov. 1993.
[94] N. Kukutsu and R. Konno, “Super absorption boundary condition for guided waves in the
3-D TLM simulation,” IEEE Microwave Guided Wave Lett., vol. 5, pp. 299–301, Sep.
1995.
[95] C. Eswarappa, and W.J.R. Hoefer, “Implementation of Berenger absorbing boundary
conditions in TLM by interfacing FDTD perfectly matched layers,” Electronics Letters, vol.
31, no. 15, pp. 1264-1266, 1995.
[96] N. Pena, and M. M. Ney, “ Absorbing-boundary conditions using perfectly matched-layer
(PML) technique for three-dimensional TLM simulations,” IEEE Trans. on Microwave
Theory and Tech., vol. 45, no. 10 pp. 1749-1755, 1997.
[97] Z. Shao, W. Hong, and H. Wu, “A Z-transform-based absorbing boundary conditions for 3D TLM-SCN method,” IEEE Trans. on Microwave Theory and Tech., vol. 50, no. 1,
pp. 222-225, 2002.
[98] L. de Menezes and W. J. R. Hoefer, “Modeling nonlinear dispersive media in 2D TLM,”
Proc. 24th European Microwave Conf., pp. 1739–1744, Sept. 1994.
[99] L. de Menezes and W. J. R. Hoefer, “Modeling frequency dependent dielectrics in TLM,”
1994 IEEE APS Symp. and URSI Meet. Dig., Seattle, WA, pp. 1140-1143, Jun. 1994.
[100] L. de Menezes and W. J. R. Hoefer, “Modeling of general constitutive relationships using
SCN TLM,” IEEE Trans. on Microwave Theory and Tech., vol. 44, pp. 854–861, June
1996.
[101] S. Hein, “Synthesis of TLM algorithms in the propagator integral framework,” 2nd Int.
Workshop Transmission-Line Matrix (TLM) Modeling Theory Applicat., Tech. Univ.,
Munich, Germany, pp. 1–11, Oct. 1997.
[102] S. Hein, “TLM numerical solution of Bloch’s equations for magnetized gyrotropic
media,” Appl. Math. Modeling, vol. 21, pp. 221–229, Apr. 1997.
45
[103] J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM part 1: Materials with frequency-dependent properties,” IEEE Trans. Antennas Propagat.,
vol. 47, pp. 1528–1534, Oct. 1999.
[104] J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM part 2: Materials with anisotropic properties,” IEEE Trans. Antennas Propagat., vol. 47, pp.
1535–1542, Oct. 1999.
[105] J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM part 3: Materials with nonlinear properties,” IEEE Trans. Antennas Propagat., vol. 50, pp.
997–1004, Jul. 2002.
[106] N. K. Madsen, R.W. Ziolkowski, "A three-dimensional modified finite volume technique
for Maxwell's equations", Electromagnetics, vol. 10, pp. 147-161, 1990.
[107] V. Shankar, A.H. Mohammadian, W.F. Hall, "A time-domain, finite-volume treatment
for the Maxwell equations", Electromagnetics, vol. 10, pp. 127-145, 1990.
[108] P. Bonnet, X. Ferrières, B. Michielsen, and P. Klotz, Time Domain Electromagnetics, S.
M. Rao, Ed., Academic Press, 1997, ch. 9, pp. 307–367.
[109] F. Bonnet and F. Poupaud, “Berenger absorbing boundary condition with time finitevolume scheme for triangular meshes,” Appl. Numer. Math., vol. 25, no. 4, pp. 333–354,
Dec. 1997.
[110] K. Sankaran, C. Fumeaux, and R. Vahldieck, “Cell-centered finite-volume-based
perfectly matched layer for time-domain Maxwell system,” IEEE Trans. on Microwave
Theory and Tech., vol. 54, no. 3, Mar. 2006.
[111] K. Sankaran, C. Fumeaux, and R. Vahldieck, “Finite-volume Maxwellian absorber on
unstructured grid,” IEEE MTT-S Int. Microw. Symp. Dig., pp. 169–172, Jun. 2006.
[112] K. Sankaran, C. Fumeaux, and R. Vahldieck, “Uniaxial and radial anisotropy models for
finite-volume Maxwellian absorber,” IEEE Trans. on Microwave Theory and Tech., vol.
54, no. 12, pp. 4297 – 4304, Dec. 2006.
[113] C. Fumeaux, K. Sankaran, and R. Vahldieck, “Spherical perfectly matched absorber for
finite-volume 3-D domain truncation,” IEEE Trans. on Microwave Theory and Tech., vol.
55, no.12, Dec. 2007
[114] D. Pinto and S. S. A. Obayya, “Accurate perfectly matched layer finite-volume timedomain method for photonic bandgap devices,” IEEE Photonics Technology Letters, vol.
20, no. 5, Mar. 2008.
[115] R. Holland, V.P. Cable, and L.C. Wilson, “Finite-volume time-domain (FVTD)
techniques for EM scattering,” IEEE Trans. on Electromag. Compat., vol. 33, no. 4, pp.
281-294, Nov. 1991.
[116] P. Bonnet, X. Ferrieres, F. Paladian, J. Grando, J. Alliot, and J. Fontaine,
“Electromagnetic wave diffraction using a finite volume method,” Electronics Letters, vol.
33, no. 1, pp. 31-32, 1997.
46
[117] C. Fumeaux, D. Baumann, and R. Vahldieck, “Advanced FVTD simulations of dielectric
resonator antennas and feed structures,” Journal of the Applied Computational
Electromagnetics Society, vol. 19, pp. 155-164. 2004.
[118] D. Baumann, C. Fumeaux, P. Leuchtmann, and R. Vahldieck, “Finite volume timedomain (FVTD) modeling of a broad-band double-ridged horn antenna,” Int. J. Numer.
Modeling, vol. 17, no. 3, pp. 285-298, 2004.
[119] P. Bonnet, X. Ferrieres, F. Paladian, J. Grando, J.C. Alliot and J. Fontaine,
“Electromagnetic wave diffraction using a finite volume method,” Electronics Letters, vol.
33, no. 1, pp. 31-32, Jan. 1997.
[120] D. Lacour, X. Ferrieres, P. Bonnet, V. Gobin and J.C. Alliot, “Application of multidomain decomposition method to solve EMC problem on an aeroplane,” Electronics
Letters, vol. 33, no. 23, pp.1932-1933, Nov. 1997.
[121] D. Baumann, C. Fumeaux, and R. Vahldieck, “A novel wave-separation scheme for the
extraction of S-parameters in non-TEM waveguides for the FVTD method,” IEEE MTT-S
Int. Microwave Symp. Dig., Long Beach, CA, Jun. 2005.
[122] D. Baumann, C. Fumeaux, and R. Vahldieck, “Field-Based Scattering-Matrix Extraction
Scheme for the FVTD Method Exploiting a Flux-Splitting Algorithm,” IEEE Trans. on
Microwave Theory and Tech., vol. 53, no. 11, Nov. 2005.
[123] C. Fumeaux, D. Baumann, P. Leuchtmann, R. Vahldieck, “A generalized local time-step
scheme for efficient FVTD simulations in strongly inhomogeneous meshes,” IEEE Trans.
on Microwave Theory and Tech., vol. 52, no. 3, pp. 1067-1076, Mar. 2004.
[124] A. Cangellaris, C. Lin, and K. Mei, “Point-matched time-domain finite element methods
for electromagnetic radiation and scattering,” IEEE Trans. Antennas Propagat., vol. 35, no.
10, pp. 1160–1173, Oct. 1987.
[125] C. Chan, J. Elson, and H. Sangani, “An explicit finite-difference time-domain method
using Whitney elements,” Proc. IEEE APS Int. Symp. Dig., vol. 3, pp. 1768–1771, Jul.
1994.
[126] M. F. Wong, O. Picon, and V. Hanna, “A finite element method based on Whitney forms
to solve Maxwell equations in the time domain,” IEEE Trans. on Magnetics, vol. 31, no. 3,
pp. 1618–1621, May 1995.
[127] K. Choi, S. J. Salon, K. A. Connor, L. F. Libelo, and S. Y. Hahn, “Time domain finite
analysis of high power microwave aperture antennas,” IEEE Trans. on Magnetics, vol. 31,
no. 5, pp. 1622–1625, May 1995.
[128] J. T. Elson, H. Sangani, and C. H. Chan, “An explicit time-domain method using threedimensional Whitney elements,” Microw. Opt. Technol. Lett., vol. 7, pp. 607–610, Sep.
1994.
[129] M. Feliziani and F. Maradei, “Point matched finite element-time domain method using
vector elements,” IEEE Trans. on Magnetics, vol. 30, no. 5, pt. 2, pp. 3184–3187, Sep.
1994.
47
[130] M. Feliziani and F. Maradei, “An explicit-implicit solution scheme to analyze fast
transients by finite elements,” IEEE Trans. on Magnetics, vol. 33, no. 2, pp. 1452–1455,
Mar. 1997.
[131] T. Yamada and K. Tani, “Finite element time domain method using hexahedral
elements,” IEEE Trans. on Magnetics, vol. 33, no. 2, pp. 1476-1479, Mar. 1997.
[132] T. Yioultsis, N. Kantartzis, C. Antonopoulos, and T. Tsiboukis, “A fully explicit Whitney
element-time domain scheme with higher order vector finite elements for three-dimensional
high frequency problems,” IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 3288-3291, Sep.
1998.
[133] W. Artuzi, “Finite element time domain method using piecewise constant basis
functions,” Proc. IEEE MTT-S Int. Microw. Optoelectr. Conf., vol. 2, pp. 1029–1032, 2003.
[134] O. Zienkiewicz and R. Taylor, The Finite Element Method, 4th ed., New York: McGrawHill, 1988.
[135] R. Fernades and G. Fairweather, “An alternating direction Galerkin method for a class of
second order hyperbolic equations in two space dimensions,” SIAM J. Numer. Anal., vol.
28, no. 5, pp. 1265–1281, 1991.
[136] G. Mur, “The finite-element modeling of three-dimensional time-domain electromagnetic
fields in strongly inhomogeneous media,” IEEE Trans. on Magnetics, vol. 28, no. 3,
pp. 1130–1133, Mar. 1992.
[137] J. F. Lee and Z. Sacks, “Whitney elements time domain (WETD) methods,” IEEE Trans.
on Magnetics, vol. 31, no. 5, pp. 1325–1329, May 1995.
[138] S. D. Gedney and U. Navsariwala, “An unconditionally stable finite-element timedomain solution of the vector wave equation,” IEEE Microw. Guided Wave Lett., vol. 5, no.
5, pp. 332–334, May 1995.
[139] J. Lee, R. Lee, and A. Cangellaris, “Time-domain finite-element methods,” IEEE Trans.
Antennas Propagat., vol. 45, no. 3, pp. 430–442, Mar. 1997.
[140] W. P. Capers, Jr., L. Pichon, and A. Razek, “A 3-D finite element method for the
modeling of bounded and unbounded electromagnetic problems in the time domain,” Int. J.
Numer. Modeling, vol. 13, pp. 527–540, 2000.
[141] J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed., New York: Wiley,
2002.
[142] F. Edelvik, G. Ledfelt, P. Lotstedt, and D. Riley, “An unconditionally stable subcell
model for arbitrarily oriented thin wires in the FETD method,” IEEE Trans. Antennas
Propagat., vol. 51, no. 8, pp. 1797–1805, Aug. 2003.
[143] D. Jiao, J. M. Jin, E. Michielssen, and D. Riley, “Time-domain finite-element simulation
of three-dimensional scattering and radiation problems using perfectly matched layers,”
IEEE Trans. Antennas Propagat., vol. 51, no. 2, pp. 296–305, Feb. 2003.
48
[144] T. Rylander and J. M. Jin, “Perfectly matched layers in three dimensions for the timedomain finite element method,” Proc. IEEE APS Int. Symp. Dig., Monterey, CA, Jun. 2004,
vol. 4, pp. 3473–3476.
[145] Z. Lou and J. M. Jin, “Modeling and simulation of broadband antennas using the timedomain finite element method,” IEEE Trans. Antennas Propagat., vol. 53, no. 12, pp.
4099–4110, Dec. 2005.
[146] A. Peterson and J. M. Jin, “A three-dimensional time-domain finite element formulation
for periodic structures,” IEEE Trans. Antennas Propagat., vol. 54, no. 1, pp. 12–19, Jan.
2006.
[147] Z. Lou and J. M. Jin, “A novel dual-field time-domain finite-element domaindecomposition method for computational electromagnetics,” IEEE Trans. Antennas
Propagat., vol. 54, no. 6, pp. 1850–1862, Jun. 2006.
[148] Z. Lou and J. M. Jin, “A new explicit time-domain finite-element method based on
element-level decomposition,” IEEE Trans. Antennas Propagat., vol. 54, no. 10, pp. 2990–
2999, Oct. 2006.
[149] H. Ali and G. Costache, “Finite-element time-domain analysis of axisymmetrical
radiators,” IEEE Trans. Antennas Propagat., vol. 42, pp. 272–275, Feb. 1994.
[150] S. Caorsi and G. Cevini, “Assessment of the performances of first- and second-order
time-domain ABC’s for the truncation of finite element grids,” Microwave Opt. Technol.
Lett., vol. 38, no. 1, pp. 11–16, Jul. 2003.
[151] D. Jiao, M. Lu, E. Michielssen, and J. M. Jin, “A fast time-domain finite elementboundary integral method for electromagnetic analysis,” IEEE Trans. Antennas Propagat.,
vol. 49, no. 10, pp. 1453–1461, Oct. 2001.
[152] M. Kuzuoglu and R. Mittra, “Investigation of nonplanar perfectly matched absorbers for
finite-element mesh truncation,” IEEE Trans. Antennas Propagat., vol. 45, no. 3, Mar.
1997.
[153] V. Mathis, “An anisotropic perfectly matched layer-absorbing medium in finite element
time domain method for Maxwell’s equations,” Proc. IEEE Antennas and Propagation Soc.
Int. Symp., vol. 2, pp. 680–683, Jul. 1997.
[154] D. Jiao and J. M. Jin, “An effective algorithm for implementing perfectly matched layers
in time-domain finite-element simulation of open-region EM problems,” IEEE Trans.
Antennas Propagat., vol. 50, pp. 1615–1623, Nov. 2002.
[155] D. Jiao, J. M. Jin, E. Michielssen, and D. J. Riley, “Time-domain finite element
simulation of three-dimensional scattering and radiation problems using perfectly matched
layers,” IEEE Trans. Antennas Propagat., vol. 51, no. 2, pp. 296–305, Feb. 2003.
[156] H. P. Tsai, Y. Wang, and T. Itoh, “An unconditionally stable extended (USE) finiteelement time-domain solution of active nonlinear microwave circuits using perfectly
matched layers,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 10, pp. 2226–2232,
Oct. 2002.
49
[157] T. Rylander and J.-M. Jin, “Conformal perfectly matched layers for the time domain
finite element method,” IEEE AP-S Int. Symp. Digest, Jun. 2003, vol. 1, pp. 698–701.
[158] T. Rylander and J.-M. Jin, “Perfectly matched layer for the time domain finite element
method applied to Maxwell’s equations,” J. Comp. Phys., vol. 200, pp. 238–250, May
2004.
[159] T. Rylander and J.-M. Jin, “Perfectly matched layer in three dimensions for the timedomain finite element method applied to radiation problems,” IEEE Trans. Antennas
Propagat., vol. 53, no. 4, pp. 1489–1499, Apr. 2005.
[160] S. Wang, R. Lee, and F. L. Teixeira, “Anisotropic-medium PML for vector FETD with
modified basis functions,” IEEE Trans. Antennas Propagat., vol. 54, no. 1, pp. 20–27, Jan.
2006.
[161] S. M. Rao, T. K. Sarkar, and S. A. Dianat, “A novel technique to the solution of transient
electromagnetic scattering from thin wires,” IEEE Trans. Antennas Propagat., vol. 34, pp.
630–634, May 1986.
[162] A. R. Bretones, R. G. Martin, and A. Salinas, “DOTIG1, a time-domain numerical code
for the study of the interaction of electromagnetic pulses with thin-wire structures,”
COMPEL, vol. 8, no. 1, pp. 39–61, 1989.
[163] S. M. Rao and D. R. Wilton, “Transient scattering by conducting surfaces of arbitrary
shape,” IEEE Trans. Antennas Propagat., vol. 39, no. 1, pp. 56–61, Jan. 1991.
[164] B. P. Rynne, “Time domain scattering from arbitrary surfaces using the electric field
integral equation,” J. Electromagn. Waves Appl., vol. 5, pp. 93–112, 1991.
[165] D. A. Vechinski and S. M. Rao, “A stable procedure to calculate the transient scattering
by conducting surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol. 40, pp.
661–665, June 1992.
[166] P. J. Davies, “Time-Marching numerical schemes for the electric field integral equation
on a straight thin wire,” Advances in Computational Mathematics, 2, 1994.
[167] G. Manara, A. Monorchio, R. Reggiannini, “A space-time discretization criterion for a
stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas
Propagat., vol. 45, no. 3, March 1997.
[168] G. Cerri, R. de Leo, R. de Rentiis, V.M. Primiani, “ESD field penetration through slots
into shielded enclosures: a time domain approach,” IEEE Trans. on Electromag. Compat.,
vol. 39, no. 4, pp. 377-386, Nov. 1997.
[169] M. J. Bluck and S. P. Walker, “Time-domain BIE analysis of large three-dimensional
electromagnetic scattering problems,” IEEE Trans. Antennas Propagat., vol. 45, pp. 894–
901, May 1997.
[170] S. M. Rao and T. K. Sarkar , “Time domain modeling of two dimensional conducting
cylinders utilizing an implicit scheme,” Microwave Opt. Technol. Lett., vol. 15, pp. 342–
347, Aug. 1997.
50
[171] S. M. Rao and T. K. Sarkar, “Transient analysis of electromagnetic scattering from wire
structures utilizing an implicit time domain integral equation technique,” Microwave Opt.
Technol. Lett., vol. 16, pp. 66–69, Jan. 1998.
[172] S. M. Rao and T. K. Sarkar, “An efficient method to evaluate the time-domain scattering
from arbitrarily shaped conducting bodies,” Microwave Opt. Technol. Lett., vol. 17, pp.
321–325, Apr. 1998.
[173] M. D. Pocock, M. J. Bluck, and S. P. Walker, “Electromagnetic scattering from 3-D
curved dielectric bodies using time-domain integral equations,” IEEE Trans. Antennas
Propagat., vol. 46, pp. 1212–1219, Aug. 1998.
[174] S. M. Rao, D. A. Vechinski, and T. K. Sarkar, “Transient scattering by conducting
cylinders—Implicit solution for transverse electric case,” Microwave Opt. Technol. Lett.,
vol. 21, pp. 129–134, Apr. 1999.
[175] S. M. Rao and T. K. Sarkar, “Implicit solution of time domain integral equations for
arbitrarily shaped dielectric bodies,” Microwave Opt. Technol. Lett., vol. 21, pp. 201–205,
May 1999.
[176] D. S. Weile, B. Shanker, E. Michielssen, “An accurate scheme for the numerical solution
of the time domain electric field integral equation,” Proc. of IEEE Int. Symp. on Antennas
and Propagation, vol. 4, pp. 516-519, July 2001.
[177] P. Argus, H. Singer, “New basis functions for connections between wire and surface
structures for time domain method of moments,” Proc. IEEE Int. Symp. on Electromagnetic
Compatibility, vol. 1, pp. 463 – 468, Aug. 2002.
[178] S. M. Rao and T. K. Sarkar, “Numerical solution of time domain integral equations for
arbitrary shaped conductor/dielectric composite bodies,” IEEE Trans. Antennas Propagat.,
vol. 50, no. 12, pp. 1831–1837, Dec. 2002.
[179] B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient
electromagnetic scattering phenomena using the multilevel plane wave time domain
algorithm,” IEEE Trans. Antennas Propagat., vol. 51, no. 3, pp. 628-641, March 2003.
[180] R.A. Wildman, G. Pisharody, D.S. Weile, S. Balasubramaniam, and E. Michielssen, “An
accurate scheme for the solution of the time-domain Integral equations of electromagnetics
using higher order vector bases and band limited extrapolation,” IEEE Trans. Antennas
Propagat., vol. 52, no. 11, pp. 2973-2984, Nov. 2004.
[181] Y.S. Chung, T.K. Sarkar, B.H. Jung, M. S. Palma, Z. Ji, S. Jang, and K. Kim, “Solution
of time domain electric field Integral equation using the Laguerre polynomials,” IEEE
Trans. Antennas Propagat., vol. 52, no. 9, pp. 2319-2328, Sep. 2004.
[182] D.S. Weile, G. Pisharody, N.W. Chen, B. Shanker, E. Michielssen, “ A novel scheme for
the solution of the time-domain integral equations of electromagnetics,” IEEE Trans.
Antennas Propagat., vol. 52, no. 1, pp. 283-295, Jan. 2004.
[183] Z. Ji, T.K. Sarkar, B.H. Jung, Y.S. Chung, M.S. Palma, and M. Yuan, “A stable solution
of time domain electric field Integral equation for thin-wire antennas using the Laguerre
polynomials,” IEEE Trans. Antennas Propagat., vol. 52, no. 10, pp. 2641-2649, Oct. 2004.
51
[184] B. Jung, T. Sarkar, Y. Chung, M. Palma, Z. Ji, S. Jang, and K. Kim, “Transient
electromagnetic scattering from dielectric objects using the electric field Integral equation
with Laguerre polynomials as temporal basis functions,” IEEE Trans. Antennas Propagat.,
vol. 52, no. 9, pp. 2329-2340, Sep. 2004.
[185] S. Wang, S. Chai, J. Mao, “A method to accelerate the MOT arithmetic in TD-MOM
EFIE,” Microwave Conference Proceedings, 2005. APMC 2005. Asia-Pacific Conference
Proceedings, vol. 4, Dec. 2005.
[186] R.A. Lemdiasov, A.A. Obi, and R. Ludwig, “Time domain formulation of the method of
moments for inhomogeneous conductive bodies at low frequencies,” IEEE Trans. Antennas
Propagat., vol. 54, no. 2, pp. 706-714, Feb. 2006.
[187] G. Guarnieri, S. Selleri, G. Pelosi, C. Dedeban and C. Pichot, “Innovative basis and
weight functions for wire junctions in time domain moment method,” IEE Proceedings Microwaves, Antennas and Propagation, vol. 153, no. 1, pp. 61-66, Feb. 2006.
[188] F.P. Andriulli, H. Bagsi, F. Vipiana, G. Vecchi, and E. Michielssen, “ A marching-on-intime hierarchical scheme for the solution of the time domain electric field integral
equation,” IEEE Trans. Antennas Propagat., vol. 55, no. 12, pp. 3734-3738, Dec. 2007.
[189] L.J. Li, L. Li, C. Gao, X. Gu, and Z. Zhao, “Numerical Treatment for Time Domain
Integral Equations of Thin Wire Structures in Half-Space Configuration,” IEEE Trans. on
Magnetics, vol. 44, no. 6, pp. 774-777, June 2008.
[190] G. Pisharody, and D.S. Weile, “An accurate solution to time domain integral equations
for inhomogeneous dielectric bodies using higher-order volume bases,” Proc. of IEEE Int.
Symp.on Antennas and Propagation, pp. 2805 – 2808, July 2006.
[191] Christian Hafner, The Generalized
Electromagnetics, Artech House, 1990.
Multipole
Technique
for
Computational
[192] W. V. T. Rusch and P. D. Potter, Analysis of Reflector Antennas, New York: Academic,
1970, pp. 92-93.
[193] C. A. Balanis, Antenna Theory Analysis and Design, Third edition, John Wiley & Sons,
Inc., 2005.
[194] A. D. Yaghjian, “Equivalence of surface current and aperture field integrations for
reflector antennas,” IEEE Trans. Antennas Propagat., vol. 32, no. 12, Dec. 1984.
[195] J. F. Kauffman, W. F. Croswell, and L. J. Jowers, “Analysis of the radiation patterns of
reflector antennas,” IEEE Trans. Antennas Propagat., vol. 24, no. 1, pp. 53-65, Jan. 1976.
[196] Y. Rahmat-Samii and V. Galindo, "Shaped reflector antenna analysis using the JacobiBessel series," IEEE Trans. Antennas Propagat., vol. 28, pp. 425-435, July 1980.
[197] T. F. Eibert, “Modeling and design of offset parabolic reflector antennas using physical
optics and multilevel fast multipole method accelerated method of moments,” Multi-optic
Conference, 2006. INMIC '06, IEEE, pp. 377-382, Dec. 2006.
[198] A. Boag, “A fast physical optics (FPO) algorithm for high frequency scattering,” IEEE
Trans. Antenna Propagat., vol. 52, no. 1, Jan. 2004.
52
[199] E. Y. Sun and W. V. T. Rusch, “Time-domain physical-optics,” IEEE Trans. Antennas
Propagat., vol. 42, pp. 9-15, Jan. 1994.
[200] R. Mittra, Y. Rahmat-Samii, V. Galindo-Israel, R. Norman, "An efficient technique for
the computation of vector secondary patterns of offset reflectors," IEEE Trans. Antennas
Propagat., vol. 27, pp. 294-304, May 1979.
[201] Y. Rahmat-Samii. "A comparison between go/aperture-field and physical optics methods
for offset reflectors," IEEE Trans. Antennas and Propagat., vol. 32, pp. 301-306, Mar.
1984.
[202] G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE, vol. 60, pp. 10221035, Sept. 1972.
[203] J. B. Keller, “Geometrical theory of diffraction,” Journal of Optical Society of America,
52(2), 1962.
[204] D. L. Hutchins, “Asymptotic series describing the diffraction of a plane wave by a twodimensional wedge of arbitrary angle,” Ph.D. Dissertation, The Ohio State University,
Dept. of Electrical Engineering, 1967.
[205] C. A. Balanis and L. Peters, Jr., “Analysis of aperture radiation from an axially slotted
circular conducting cylinder using GTD,” IEEE Trans. Antennas Propagat., vol. 17, no. 1,
pp. 93-97, Jan. 1969.
[206] G. L. James and V. Kerdemelidis, “Reflector antenna radiation pattern analysis by
equivalent edge currents,” IEEE Trans. Antenna Propagat., vol. 21, no. 1, pp. 19-24, Jan.
1973.
[207] C. A. Mentzer and L. Peter, Jr., “A GTD analysis of far-out sidelobes of Cassegrain
antennas,” IEEE Trans. Antenna Propagat., vol. 23, no. 5, pp. 702-709, Sep. 1975.
[208] P .H. Pathak and R. G. Kouyoumjian, “An analysis of the radiation from aperture on
curved surfaces by the geometrical theory of diffraction,” Proc. IEEE, vol. 62, no. 11, pp.
1438-1447, Nov. 1974.
[209] A. W. Rudge, “Offset-parabolic-reflector antennas: A review,” Proc. IEEE, vol. 66, no.
12, pp. 1592-1618, Dec. 1978.
[210] D. P. Bouche, F. A. Molinet and R. Mittra, “Asymptotic and hybrid techniques for
electromagnetic scattering,” Proc. IEEE, vol. 81, no. 12, pp. 1658-1684, Dec. 1993.
[211] R. C. Kouyoumjian and P. H. Pathak, “A geometrical theory of diffraction for an edge in
a perfectly conducting surface,” Proc. IEEE, vol. 62, pp. 1448-1461, Nov. 1974.
[212] S. W. Lee and G. A. Deschamps, “A uniform asymptotic theory of electromagnetic
diffraction by a curved wedge,” IEEE Trans. Antennas Propagat., vol. 24, pp. 25-34, Jan.
1976.
[213] R. W. Ziolkowski and G. A. Deschamps, “Asymptotic evaluation of high frequency
fields near a caustic: An introduction to Maslov’s method,” Radio Sci., vol. 19, no. 4, pp.
1001-1025, Jul.-Aug. 1984.
53
[214] Y. Kravtsov and Y. Orlov, “Caustics, catastrophes and wave fields,” Sov. Phys.-Usp., vol.
26, pp. 1039-1058, Dec. 1983.
[215] H. Ikuno and L. B. Felsen, “Complex ray interpretation of reflection from concaveconvex surfaces,” IEEE Trans. Antennas Propagat., vol. 36, pp. 1206-1211, 1988.
[216] P. Y. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,”
Electromagnetics, vol. 11, pp. 125-160, 1991.
[217] R. Mittra, Y Rahmat-Samii, and W. L. Ko, “Spectral Theory of Diffraction,” Appl. Phys.,
vol. 10, pp. 1-13, 1976.
[218] Y. Rahmat-Samii and R. Mittra, “A Spectral Domain interpretation of High-Frequency
Diffraction Phenomena,” IEEE Trans. Antennas Propagat., vol. 25, pp. 676-687, Sept.
1977.
[219] R. G. Kouyoumjian and P. H. Pathak, “A Uniform Geometrical Theory of Diffraction for
an Edge in A Perfectly Conducting Surface,” Proc. IEEE, vol. 62, pp. 1448–1461, Nov.
1974.
[220] T.-K. Lee, S.-Y. Lee and J.-W. Ra, “A hybrid finite element-boundary element method
and its application to inverse scattering,” Proc. IEEE Antennas and Prop. Symp., vol. 3,
pp. 1628-1631, June 1989.
[221] W. E. Boyse and A. A. Seidl, “A hybrid finite element and moment method for
electromagnetic scattering from inhomogeneous objects,” Proc. 7th Ann. Rev. Progress
Appl. Computat. Electromag., Mar. 1991.
[222] J. M. Jin and John L. Volakis, “Electromagnetic scattering by and transmission through a
three dimensional slot in a thick conducting plane,” IEEE Trans. Antennas Propagat., vol.
39, pp. 543-550, April 1991.
[223] K. D. Paulsen, D. R. Lynch, and J. W. Strohbehn, “Three-dimensional finite, boundary,
and hybrid element solutions of the Maxwell’s equations for lossy dielectric media,” IEEE
Trans. on Microwave Theory and Tech., vol. 36, pp. 682-693, April 1988.
[224] J. Shen and A. Kost, “Hybrid FE-BE method for EMC problems in power cable
systems,” IEEE Trans. on Magnetics, vol. 32, no. 3, pp. 1493-1496, May 1996.
[225] Y. Ji, J. Chen, T.H. Hubing, J.L. Drewniak, “Application of a hybrid FEM/MOM
method to a canonical PCB problem,” Proc. 1999 IEEE Int. Symp. Electromagn. Compat.
Seattle, WA, August 1999, pp. 91-96.
[226] Y. Ji, T. H. Hubing, “EMAP5: A 3D hybrid FEM/MOM code,” Journal of the Applied
Computational Electromagnetics Society, vol. 15, no. 1, March 2000, pp. 1-12.
[227] K. Zhao, M. Vouvakis and J.-F. Lee, “Solving electromagnetic problems using a novel
symmetric FEM-BEM approach,” IEEE Trans. on Magnetics, vol. 42, no. 4, pp. 583-586,
April 2006.
[228] Y. Ji, H. Wang and T. Hubing, “A novel preconditioning technique and comparison of
three formulations for hybrid FEM/MOM methods,” Journal of the Applied Computational
Electromagnetics Society, vol. 15, no. 2, Aug. 2000, pp. 103-114.
54
[229] W. D. Burnside, C. L. Yu, and R. J. Marhefka, "A technique to combine the geometrical
theory of diffraction and the moment method", IEEE Trans. Antennas Propagat., vol. 23,
no. 4, pp. 551-558, July 1975.
[230] G. A. Thiele and T. H. Newhouse, “A hybrid technique for combining moment methods
with the geometrical theory of diffraction,” IEEE Trans. Antennas Propagat., vol. 23, no. 1,
pp. 62–69, Jan. 1975.
[231] U. Jakobus and F. M. Landstorfer, “A combination of current-and ray-based techniques
for the efficient analysis of electrically large scattering problems,” Proc. 13th Annu. Rev.
Progress Appl. Computat. Electromagn., Monterey, CA, Mar. 1997, pp. 748–755.
[232] I. P. Theron, D. B. Davidson, and U. Jakobus, “Extensions to the hybrid method of
moments/uniform GTD Formulation for sources located close to a smooth convex surface,”
IEEE Trans. Antennas Propagat., vol. 48, no. 6, pp. 940 – 945, June 2000.
[233] I. P. Theron, U. Jakobus, I. B. Davidson, and F. J. Meyer, “Recent progress on moment
method / UTD hybridization,” Proc. of PIERS98 - Progress in Electromagnetics Research
Symposium, Nantes, France, vol. 1, p. 459, July 1998.
[234] H. J. Bilow, “Scattering by an infinite wedge with tensor impedance boundary conditions
- a moment method/physical optics solution for currents,” IEEE Trans. Antennas Propagat.,
vol. 39, pp. 761-773, June 1991.
[235] L. N. Medgyesi-Mitschang and D.-S. Wang, “Hybrid solutions for scattering from
perfectly conducting bodies of revolution,” IEEE Trans. Antennas Propagat., vol. 31,
p. 570, 1983.
[236] M. Kaye, P. K. Murthy, and G. A. Thiele, “An iterative method for solving scattering
problems,” IEEE Trans. Antennas Propagat., vol. 33, pp. 1272-1279, 1985.
[237] D. S. Wang, “Current-based hybrid analysis for surface wave effects on large scatterers,”
IEEE Trans. Antennas Propagat., vol. 39, pp. 839-850, June 1991.
[238] C. S. Kim and Y. Rahmat-Samii, “Low profile antenna study using the physical optics
hybrid method (POHM),” Proc. IEEE Int. Symp. Antennas Propagat. Soc. Meet., London,
ON, Canada, June 1991.
[239] U. Jakobus and F. M. Landstorfer, “Improvement of the PO-MM hybrid method by
accounting for effects of perfectly conducting wedges,” IEEE Trans. Antennas Propagat.,
vol. 43, pp. 1123-1129, Oct. 1995.
[240] R. E. Hodges and Y. Rahmat, “An iterative current-based hybrid method for complex
structures,” IEEE Trans. Antennas Propagat., vol. 45, pp. 265-276, Feb. 1997.
[241] Johan Edlund, “A parallel, iterative method of moments and physical optics hybrid solver
for arbitrary surfaces,” UPPSALA University, Department of Information Technology,
August, 2001.
[242] D. B. Davidson, I. P. Theron, U. Jakobus, F. M. Landstorfer, F. J. C. Meyer, J. Mostert, J.
J. Van Tonder, “Recent progress on the antenna simulation program FEKO”, Proc. of the
1998 South African Symposium on Communications and Signal Processing, 1998. COMSIG
'98, pp. 427-430, 7-8 Sept. 1998.
55
[243] J. M. Jin, S. Ni, and S. W. Lee, “Hybridization of SBR and FEM for scattering by large
bodies with cracks and cavities,” IEEE Trans. Antennas Propagat., vol. 43, pp. 1130-1139,
Oct. 1995.
[244] Dong-Ho Han, A. C. Polycarpou, C. A. Balanis, “FEM-based hybrid methods for the
analysis of antennas on electrically large structures,” Radio and Wireless Conference, 2000.
RAWCON 2000, pp. 59-61.
56