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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
Time-Domain Finite-Element Methods
Jin-Fa Lee, Member, IEEE, Robert Lee, Member, IEEE, and Andreas Cangellaris, Member, IEEE
Invited Review Paper
Abstract— In this paper, various time-domain finite-element
methods for the simulation of transient electromagnetic wave phenomena are discussed. Detailed descriptions of test/trial spaces,
explicit and implicit formulations, nodal and edge/facet element
basis functions are given, along with the numerical stability properties of the different methods. The advantages and disadvantages
of mass lumping are examined. Finally, the various formulations are compared on the basis of their numerical dispersion
performance.
Index Terms— Electromagnetic transient analysis, finiteelement methods.
I. INTRODUCTION
N
UMEROUS time-domain finite-element methods (TDFEM’s) have been proposed for electromagnetic modeling in the past ten years; however, the popularity of TDFEM’s
is very small compared to that for the finite-difference timedomain (FDTD) method.
There are several reasons why the FDTD method is so popular. The most important ones are its programming simplicity,
its minimum bookkeeping complexity, and the simplicity of
its numerical integration algorithm. Because of the simplicity
of the method, a novice can develop a three-dimensional (3-D)
FDTD code in a relatively short amount of time.
TDFEM offers some important advantages over the standard
FDTD method. The most obvious one has to do with its
use of unstructured grids. Since the grid is unstructured, it
offers superior versatility in modeling complex geometries.
Furthermore, the Faedo–Galerkin procedure, used for the
development of the weak statement, provides for a very natural
way for handling field and flux continuity conditions at material interfaces, thus further enhancing modeling accuracy. In
addition, the use of Galerkin formulation provides a unifying
approach for the exploitation of a variety of choices of trial
functions, some of them more appropriate for specific types
of geometries than others.
Despite the aforementioned advantages there are some major reasons why TDFEM’s have not been used widely. The
most important is unstructured 3-D mesh generation. It requires a significant amount of effort to develop a proficiency
Manuscript received May 8, 1996; revised October 2, 1996.
J.-F. Lee is with EMCAD Laboratory, Department of Electrical and
Computer Engineering, Worcester Polytechnical Institute, Worcester, MA
01609 USA.
R. Lee is with the ElectroScience Laboratory, Electrical Engineering
Department, The Ohio State University, Columbus, OH 43210 USA.
A. Cangellaris is with the Electromagnetics Laboratory, Department of
Electrical and Computer Engineering, University of Arizona, Tucson, AZ
85721 USA.
Publisher Item Identifier S 0018-926X(97)02292-8.
at using 3-D mesh generator packages. Also, it is not always
possible to get a high-quality mesh for a given geometry.
Another difficulty is the relative complexity of TDFEM’s
compared to the standard FDTD method. The time needed
to learn TDFEM’s and write a usable 3-D code is an order of
magnitude greater than that for the standard FDTD method.
From the above discussion, one is led to the conclusion
that the standard FDTD method is the method of choice when
modeling geometries of low complexity, while TDFEM’s are
most appropriate when complicated geometries need to be
modeled for which application of the FDTD method would
require a very dense grid to resolve accurately the variation
in geometric features. Since TDFEM’s have not received as
much attention as the FDTD method, they are lacking in both
maturity and variety of applications. Therefore, a comparison
of TDFEM’s with the FDTD method on the basis of current
capablities would be misleading. Instead, the goal of this paper
is to present TDFEM’s in a general framework which unifies
much of the previous work done in this area. The merits of the
various TDFEM’s are discussed, and some future directions
are proposed.
II. FAEDO–GALERKIN FORMULATIONS
In this section, we discuss the Faedo–Galerkin formulation
as the general procedure for the development of semi-discrete
approximations to Maxwell’s time-dependent system. Formulations based on both the hyperbolic system of the two curl
equations and the vector wave equation are discussed. The
derived formulations will serve as a unifying framework for
the discussion of specific TDFEM’s in later sections. For the
purpose of generality, time discretization is not discussed in
this section. Instead, it is considered in conjunction with the
specific TDFEM schemes in Sections III and IV.
A. Formulation Based Upon the Two First-Order Equations
Let , and
be the position-dependent, relative permittivity and permeability, respectively, in a volume , and let
denote an applied current density inside this volume. The
initial value problem (IVP) of interest is described by
0018–926X/97$10.00  1997 IEEE
(1)
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inside
431
B. Formulation Based on the Second-Order
Vector Wave Equation
, subject to boundary conditions of the form
on
To derive TDFEM’s in terms of the
field, we start with
the following initial value problem (IVP) description:
on
where
,
are perfect electric and magnetic surfaces,
respectively, inside , and
is the unit normal on these
surfaces. For the case of bounded domains, the tangential
components of or , or an impedance relationship between
and
on the bounding surface must be specified for
. For unbounded regions, Sommerfeld’s radiation
all
condition must hold. Finally, for a unique solution, the initial
state for all fields inside at
must be known.
Assuming the existence of a unique solution to (1), we can
formulate two weak forms using the Galerkin or weighted
residual method. The first approach tests (1) by vector functions
, viz. square integrable vector functions in
(fields with finite energy). The weak form obtained is thus
in
on
on
(6)
on
for
is the truncation boundary. The formulation in
where
can be derived in a similar
terms of the magnetic field
fashion. We have also adopted, for simplicity, the first-order
to truncate the
absorbing boundary condition (ABC) on
infinite domain into a finite region.
The Galerkin form of the IVP (6) can be stated as
find
such that
(2)
. By using this weak form, we seek the fields
and
, where
on
on
(3)
In the second approach [1], we multiply the first of (1) by
and integrate over . Similarly, multiplying the
, integrating over
and
second of (1) by
integrating the curl term by parts, we obtain a weak form as
(7)
, and
, where
and
are the trial and
test spaces, respectively. In general, in the weighted residual
method, the trial and test spaces are not necessarily the same.
One special case is the collocation method, where the test
space is a finite number of delta functions located at some
chosen collocation points . However, for the collocation
method to work, the trial function must be smooth enough
such that the integrand in the
at the collocation point
bilinear form is bounded. The challenges in the development
of a collocation method for unstructured grids are mainly
the proper choice of the basis functions (explicitly and/or
implicitly) and the collocation points. In particular, in the
collocation FEM’s, the trial function must satisfy
(8)
(4)
In this case, the admissible trial spaces are
and
. Notice that a
natural boundary condition comes with this second weak form
from the variational principle, namely
on
.
(5)
Equation (8) requires to be a smoother function than what
is usually needed for tangential continuity across element
boundaries. The continuity requirements for the trial function
in the collocation FEM’s are
(9)
(10)
Equations (9) and (10) are also required in the standard weak
Galerkin formulation. However, in the collocation FEM’s, (10)
has to be a strong boundary condition (essential boundary
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
condition) instead of the weak boundary condition obtained
through the natural boundary condition in the weak Galerkin
formulation. We also comment here that the collocation TDFEM approach, when properly formulated, offers great potential benefits such as superior accuracy (superconvergence) [2]
if optimal collocation points are chosen and explicit time stepping schemes for higher order (more than linear) interpolation.
However, this is a subject which receives virtually no attention
currently in TDFEM for solving transient wave equation. For
readers who are interested in pursuing this topic further, we
recommend [2] as a starting point.
A more common approach to the derivation of TDFEM’s is
to balance the smoothness requirements between test and trial
functions through the use of Green’s theorem. In particular, a
weak form of the IVP (6) can be stated as
(12)
(15)
The finite-element process is to seek the best solution
of (11) in a finite-dimensional subspace
with a finite number of degrees of freedom. Consequently, the
application of the Faedo–Galerkin process results in a system
of ordinary differential equations (ODE’s)
(13)
where
A. Two First-Order Equations with Point-Matching
.
is admissible in this bilinear form if
(11)
, and
A vector function
and only if
III. TDFEM’S USING NODAL ELEMENTS
Among the variety of approaches used for the extension
of Yee’s FDTD method to nonorthogonal grids, the method
of point-matched finite elements was probably the first one
to propose the use of finite-element interpolation practices
for the development of an explicit time-domain integration
scheme for Maxwell’s curl equations [3], [4]. The following
is a concise description of the application of the method to
the discretization of Maxwell’s hyperbolic system, presented
in the light of the Faedo–Galerkin process.
The point-matched finite-element method maintains the
staggering of the electric and magnetic fields in space;
however, in contrast to the FDTD method, all three
components of the electric field are placed at the same
node. The same is true for the magnetic field. Thus, two
complementary grids are used inside
for the development
of the numerical approximation. The two grids are constructed
in such a manner that every electric field node is contained
within the volume of an element in the magnetic field grid, and
every magnetic field node is contained within the volume of an
element in the electric field grid. Using nodal finite elements in
conjunction with these two grids leads to the finite-dimensional
subspaces that are used for the finite-element approximation
of the fields (see Section II-A). More specifically, the fields
are approximated as follows:
such that
find
are the vector basis functions that span the finitedimensional subspace
. Note that we have used bold face
letters to indicate matrices.
is the coefficient vector of
and
(14)
,
are the number of electric and magnetic field
where
nodes in the two grids, and
,
are the scalar basis
functions used for effecting the finite-element approximations
in the electric and magnetic grids, respectively. Clearly, the
degrees of freedom of the approximation are the temporal
variations
,
at the nodes of the electric and
magnetic grids, respectively.
To complete the Faedo–Galerkin process, the test functions
need to be defined. Different choices of test functions are
possible and lead to either implicit or explicit integration
schemes. The most direct way to the development of an
explicit scheme is through point-matching or collocation. The
collocation points are taken to be the nodes in the two
complementary grids. In other words, delta functions
,
associated with the electric field nodes and
delta functions
,
associated with
the magnetic field nodes are used as test functions. Substituting
(15) in (2) and testing the first of (2) with the delta functions
associated with the electric field nodes and the second of (2)
LEE et al.: TIME-DOMAIN FINITE-ELEMENT METHODS
433
with the delta functions associated with the magnetic field
nodes yields
field nodes inside . The permittivity matrix with elements
,
, and the permeability matrix
with elements
,
.
The notation
is used to denote the 3 3 unit matrix.
Using the aforementioned matrices and vectors, the state
equations (16) assume the compact matrix form
(20)
(16)
The above system of state equations may be cast in a
compact matrix form as follows. First, we introduce the vectors
This step completes the development of the semi-discrete
approximation of Maxwell’s curl equations in the context
of point-matched finite elements. Use of a central difference
scheme for the approximation of the time derivatives, with
and
discretized in time in equal intervals
and, in such
a way that the temporal nodes of
are shifted one-half time
step with respect to those of , results in the simple leapfrog
numerical integration algorithm for the state equations. This
algorithm is conditionally stable. The stability condition is
derived using the procedure of Section V and is
(17)
(21)
where it is
and the superscript
denotes matrix transposition. Clearly,
the dimension of
and
is
, while the dimension of
is
. Furthermore, we observe that the cross-product
operations in (16) may be written in matrix form as follows:
(18)
for
, and
(19)
for
. The superscripts
in the 3
3 array
and
in the 3
3 array
are used
to indicate that the elements of these arrays are calculated
at points ,
, and ,
,
respectively. These definitions lead to the introduction of two
supermatrices, supermatrix
with elements
,
,
, and supermatrix with
,
,
.
elements
Finally, we introduce two diagonal supermatrices to describe
the electromagnetic properties at the electric and magnetic
where
denotes the maximum eigenvalue of the
matrix defined by
.
It is important to point out that if standard finite-element
interpolation functions are used for the approximation of the
spatial variation of the fields, the matrices and
are very
sparse due to the compact support of the basis functions,
and the numerical integration algorithm is very efficient.
Even though the use of two complementary grids appears
cumbersome at first sight, it is actually quite straightforward
if quadrilateral elements in two dimensions and hexahedron
elements in three dimensions are used. The major drawback
of this method comes from the fact that all three components
for each of the fields are placed at the same node. This
leads to complications in enforcing the appropriate boundary
conditions at surfaces where the electromagnetic properties of
the media exhibit abrupt changes, in the sense that special
treatment is needed for updating the nodes on such surfaces
[3], [4].
To alleviate this difficulty, the use of vector finite elements
has been proposed for the spatial interpolation of the fields [5].
Hybrid finite-element interpolation schemes, implementing
different types of basis functions for the approximation of
the different field components and/or associated fluxes have
been proposed also [6]. As a matter of fact, this is an area of
continuing research that, one might argue, was motivated by
the pioneering work of Bossavit [7]. This work is discussed
in Section IV.
B. Second-Order Equation with Integral Lumping
In this section, we provide a brief review of the Lynch
and Paulsen [8] formulation for a time-domain nodal finiteelement method applied to the second-order equation with
integral lumping. Because of the integral lumping, an explicit
time integration scheme can be obtained. The second-order
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
equation under consideration is somewhat different than (6)
because of Lynch and Paulsen’s concerns about generating
spurious solutions. From the analysis in [9], it is argued that
no spurious modes are produced if the following equation is
used:
in
(22)
with the same boundary conditions as in (6). The final weak
form of (22) can be written as
There are several good features about this formulation.
First, it is explicit, so no matrix solution is required. Second,
a traditional frequency domain FEM code based on nodal
elements can be easily adapted to this time-domain method.
However, there are several things that one must be careful
about in using this method. Because it is a node-based method,
material discontinuities require special treatment (see [8] for
details). Furthermore, the presence of perfectly conducting
edges and corners can produce large errors in the solution. The
errors are usually not localized near the edge or corner, but are
present throughout the computation domain. Finally, there is
some concern about the accuracy of the method. The numerical
dispersion error for this FEM formulation is compared with
several others in a later section of this paper.
IV. TDFEM’S USING EDGE/FACET ELEMENTS
A. An Explicit WETD Method
(23)
where
is approximated in terms of its nodal values
by
(24)
The traditional central difference in time is used to approximate the second-order time derivative. Substituting this
approximation into (23), we obtain the following form for the
left-hand side of (23)
(25)
is the electric field at node and time step .
where
The resulting finite-element equation is implicit, requiring the
solution of a matrix equation.
Lynch and Paulsen apply the concept of integral lumping
(see the discussion on integral lumping later in this paper)
to obtain an explicit time-domain formulation. Thus, (25)
becomes
(26)
It was long advocated by Bossavit [7] that in solving
Maxwell’s equations, Whitney 1-forms (edge elements) should
and , whereas Whitney 2be used to approximate fields
forms (facet elements) should be used to model the fluxes
and . Since then, many variants (with various interesting
mutations) using this basic notion have been proposed and implemented. Among them, we mention Mahadevan and Mittra
[10], Wong et al.. [11], Lee and Sacks [12], Chan et al. [13],
and Feliziani and Maradei [5]. For Whitney 1-forms, the basis
functions are well known by now. For example, for
it is
(28)
where is the Lagrange interpolation polynomial at vertex
[14]. The circulation of
is one along
and zero
along all other edges. Any square integrable vector field, , can
now be interpolated by a combination of Whitney 1-forms as
(29)
is the circulation of along
. Similarly, the
where
vector field for Whitney 2-forms associated with a particular
can be written as
(30)
We can substitute (26) into (23) to obtain
is
It can be shown that the normal component of
continuous across element boundaries, and its flux is one
through
and zero through all other facets. Such
fluxes relate to the degrees of freedom of a tetrahedral element
by
(31)
(27)
is the flux of vector through
.
where
We shall now present a very interesting dual-field TDFEM
formulation which is an explicit scheme using edge and facet
LEE et al.: TIME-DOMAIN FINITE-ELEMENT METHODS
435
elements. The scheme was first proposed by Bossavit in [7].
It starts with
They are
(38)
(32)
Using collocation, (38) becomes
and
(39)
where
(33)
(40)
Substituting (32) and (33) into Maxwell’s equations, we have
(34)
and
and
are understood to be the numbers
corresponding to the unknowns on
and
,
respectively.
Approximation of the time derivatives in (37) can be done in
several ways. Here we choose the central difference scheme
and stagger
and
in time, evaluating them one
half of a time step apart. Thus
To make (34) operational, we apply collocation method. For
example, on
we have
(41)
Equation (41) is now fully discretized and can be used to
update the electromagnetic field distribution in a leapfrog
fashion.
B. General Implicit Two-Step Recurrence Algorithms
(35)
To formulate implicit TDFEM’s that involve the solution
of a matrix equation, we start with the semi-discrete equation
derived previously
(42)
From the properties of Whitney 1-forms and 2-forms, (35)
reduce to the simple expressions
(36)
These can be expressed in matrix form as
(37)
where indicates a column vector and
is the circulation
matrix whose entries are either zero or one [clearly, only add or
subtract operations are involved in performing (37)]. Finally,
to relate fields to fluxes we employ the constitutive relations.
Also,
and
are assumed to be given
either from the initial excitation or from previous iteration.
Even though the implicit schemes involve solving a matrix
equation for every time step, it is possible to derive implicit
TDFEM’s that are unconditionally stable. The unconditional
stability is very desirable when solving problems with very
small features that lead to large variations in element size
across the problem domain. Without loss of generality, we
shall concentrate on deriving general two-step algorithms to
obtain
using the weighted residual principle.
The approach begins by approximating the function
,
by a second-order polynomial expansion
(43)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
From (43), the first- and second-time derivatives are approximated as
A. Stability Condition for the Explicit TDFEM
For the purpose of deriving the stability condition, let us
rewrite (41), assuming
in the following form:
(44)
Substituting (43) and (44) into the semi-discrete equation, (42)
results in
(48)
This can be expressed in a more compact matrix notation as
(49)
(45)
Weighting the residual (45) by a test function
defining
where
, and
and
(50)
A necessary condition for (50) to be stable is [15]
(51)
where
is the th eigenvalue of the amplification matrix . Consequently, the stability condition for the explicit
TDFEM of (48) is
(52)
(46)
where
is the largest eigenvalue (in terms of magnitude) of matrix
. In practical computation, this largest
eigenvalue can be estimated efficiently either by the power
method or the Arnoldi method [16]. The stability condition
for the point-matched TDFEM is derived in a similar fashion
and is given in (21).
yields
(47)
has been obtained from the above equation, a new
Once
time step can be started. Implicit algorithms of various kinds
can be obtained from (47) by employing different
and
values. Here, we recommend a particular pair
and
that results in an unconditionally stable implicit
algorithm with truncation error
.
B. Stability Condition for the Implicit TDFEM’s
We shall follow the general approach outlined in
Zienkiewicz and Taylor [17] for the second-order wave
equation. Assuming a modal decomposition of the system
of (13) with no excitation, it suffices to examine the scalar
equation
(53)
V. STABILITY CONDITIONS
In the linear theory of finite methods for the numerical
solution of ordinary and partial differential equations, the
success of numerical schemes is summed up in the wellknown Lax equivalence theorem [2]: “consistency and stability
imply convergence.” For general TDFEM’s, the consistency is
self-evident and assured by their formulation. Subsequently,
the question of stability is of paramount importance. In this
section, we present two approaches for deriving the stability
condition for general TDFEM’s.
is the amplitude for mode
in the system. We
where
remark also that
and
, since is positive
definite and and are semi-definite matrices. The two-step
recurrence algorithm for
takes the form
(54)
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437
The general solution can be written as
(55)
is the growth factor for mode . Substituting (55)
where
into (54) provides the second-order characteristic polynomial
for
(56)
For stability, it is necessary that the modulus of the growth
factor be bounded by one for all modes, viz.
.
To derive the stability limits, we shall use the so-called
transformation. We introduce the mapping
(57)
are in general complex numbers. It
where as well as
is easy to show that the aforementioned stability requirement
is equivalent to demanding the real part of to be negative.
Equation (56) is now rewritten in terms of as
(58)
Furthermore, through the Routh–Hurwicz condition [17] it can
be shown that in order for the real part of to be negative,
the following conditions must hold
(59)
The inequalities (59), as well as the fact that
, lead to the following conditions on
and
this mismatch can be characterized in terms of an error in the
wave number where the numerical wave number is not
a linear function of frequency. Thus, an artificial numerical
dispersion is present in the solution.
For time-harmonic waves, the effects of numerical dispersion is to produce a phase error in the solution. The phase
error is cumulative, so as a wave propagates, the phase error
increases. Consequently, the error increases with the distance
a wave propagates in the computation domain. The phase error
is especially significant for electrically large problems and
high
resonators. A discussion of this phase error for the
FDTD method [18] is given in [19]. Its extension to TDFEM
is straightforward if the mesh is assumed to be regular. In
this section, we compare the phase error for several TDFEM’s
through a two-dimensional dispersion analysis of square elements. The analysis does not truly model the phase error
for arbitrary unstructured meshes; however, it provides insight
into the relative performances of the various methods. In most
finite elements, there is always an accumulation of phase error
since the sign of the error is always the same for all angles
of incidence. Thus, although a nonorthogonal or unstructured
mesh does not produce the same phase error as a mesh
composed of square elements, its overall characteristics does
not change unless the elements is very badly deformed. The
one exception for frequency-domain methods is the tetrahedral
edge elements. For this element, the sign of the phase error
changes depending on the incidence angle. The phase error
may cancel especially for unstructured grids [20]. In the time
domain, the phase cancellation can be obtained for methods
which have implicit time integration, as we will see later in
this section.
To perform the dispersion analysis, let us consider a TE
plane wave propagating in the – plane through an infinite
mesh composed of square elements with sides of length . The
incidence angle of the plane wave is defined such that
0
represent a
propagating wave, and
90 represents a
propagating wave. The numerical solution is
and
:
(61)
and
(60)
for (47) to be unconditionally stable.
VI. NUMERICAL DISPERSION ANALYSIS
The accurate modeling of electromagnetic phenomena with
FEM is strongly dependent on both the shape and electrical
size of the elements. Even one poorly shaped element (elements with large interior angles) can cause large solution errors
throughout the mesh. The element size is also very important;
the elements must be small enough to model rapid field
variations due to such things as geometrical discontinuities and
highly conductive media. There is an additional source of error
due to the incorrect modeling of the sinusoidal behavior of the
propagating wave. The piecewise polynomial approximation of
FEM does not exactly match a sine function. The error due to
where we use the discretization
,
, and
.
The fields in (61) can be substituted into the finite-element
equations to obtain a transcendental equation for . Examples
of such an equation in the frequency domain are given in [21]
and [22]. For the purposes of comparison, we also consider
the dispersion analysis for the FDTD method.
In writing out the dispersion relation for the various methods, we use certain abbreviations. They are
,
, and
. The numerical wave number
of the FDTD method is given by
(62)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
For the point-matched time-domain node-based FEM [23], the
dispersion relation is
(63)
For the time-domain node-based FEM with integral lumping
[8], we have
(64)
Finally, the dispersion relation for the implicit edge-based
time-domain method in (47) for the lossless case (
) is
considered [12]. The numerical wave number, for
, is
the solution to
(65)
, and
where
(66)
The variable
in (66) can be chosen to obtain a range of
different time difference schemes. For
the scheme is
unconditionally stable.
The phase error for each wavelength that the wave propagates is given by
Phase Error (degrees)
(67)
In Fig. 1, the phase error per wavelength ( ) is plotted as a
function of angle of incidence from 0 to 90 for
,
, and
. Due to symmetry, the phase error
is periodic with period 90 , so the range that is plotted fully
describes the error for all angles of incidence. Fig. 1 reveals
some interesting characteristics of various finite methods. The
edge-based FEM shows the least phase error and, in fact, is
the only one with a negative phase error. The point-matched
FEM has the worst error, with the integral lumped nodal
FEM being the second worst. The poor performance of the
two node-based FEM methods compared to the edge-based
method might appear surprising in view of the fact that the
node-based FEM approximates the field to a complete first
order, while the edge-based FEM only represents the field to an
incomplete first order. However, it should be pointed out that
the stability condition for the point-matched FEM on a uniform
grid is given by
, independent of the number of spatial
dimensions and, as discussed in [4], the algorithm should be
run with as close as possible to one for minimum dispersion.
Fig. 1. Plots of phase error per wavelength as a function of the angle of
incidence for the FDTD method and three FEM methods (point-matched
nodal FEM, nodal FEM with integral lumping, and edge-based FEM) with
h =20, p = 0:5, and = 23 .
=
Consequently, the choice
results in a point-matched
FEM dispersion error much larger than what can be achieved
on a uniform grid for values of close to one. However, it
is the objective of TDFEM’s to allow for unstructured grids
and, for general applications with considerable variability in
grid size, the numerical integration for large portions of the
grid will be done at Courant numbers smaller than one. Thus,
the choice
appears to be a reasonable one for the
purposes of demonstrating the dispersion error that should
be expected from the application of point-matched FEM’s in
the modeling of realistic structures. Finally, we mention that
integral lumping negatively impacts the phase error since other
researchers have observed this difference when comparing
formulations with and without integral lumping [24].
Another important feature of numerical dispersion is its
anisotropy with respect to the incidence angle. A measure of
anisotropy is the difference in the maximum and minimum
phase errors over all incidence angles. Let be this difference.
From Fig. 1, we see that FDTD, node-based FEM with integral
lumping, and edge-based FEM have approximately the same
anisotropy (
0.75 ), while the anisotropy associated with
the point-matched node-based FEM is greater (
6.7 ). The
anisotropy of the phase error is important because techniques
are currently being developed to minimize the phase error for
finite method [25]. These techniques can shift the numerical
wave number, so that the average phase error with respect to
is zero; however, the anisotropy is still present.
In frequency-domain FEM methods, square bilinear firstorder elements usually produce the least phase error at 45
and the most at 0 and 90 , since the field is approximated
by a quadratic function at 45 and a linear function at 0 and
90 [21]. From Fig. 1, we see that the effect of point matching
and integral lumping is to exactly reverse the points of least
and greatest phase error.
To see how the phase error compares for a different choice
of time step, let us choose
with
and
.
The results are shown in Fig. 2. The phase error increases for
LEE et al.: TIME-DOMAIN FINITE-ELEMENT METHODS
Fig. 2. Plots of phase error per wavelength as a function of incidence angle
for FDTD and three FEM methods (point-matched nodal FEM, nodal FEM
=20, p = 0:2, and
with integral lumping, and edge-based FEM) with h
= 23 .
=
439
Fig. 4. Plot of phase error per wavelength as a function of incidence angle
for the edge based FEM with h = =20, p = 0:5. The variable is varied
from 0.2–0.8 in increments of 0.2.
shift the level of the phase-error curve without changing the
anisotropy. From these curves, we can deduce that the best
choice of to minimize the maximum phase error is between
0.4 and 0.6.
Increasing the order of approximation within an element
decreases both the phase error and the anisotropy. In [21], it
has been shown that increasing the element order is more efficient than decreasing the element size for frequency-domain
FEM, and this behavior is expected to hold in the time domain
as well. This behavior provides yet another argument against
mass lumping since, as discussed in the next section, mass
lumping cannot be used in conjunction with higher order
approximations.
VII. THE MASS LUMPING PROCESS:
TO LUMP OR NOT TO LUMP
Fig. 3. Plots of phase error per wavelength as a function of incidence angle
for FDTD and three FEM methods (point-matched nodal FEM, nodal FEM
with integral lumping, and edge-based FEM) with h = =10, p = 0:2, and
= 23 .
all methods, but the amount of anisotropy remains basically
case. Thus, the effect of
the same as compared to the
changing the time step is to shift the level of the phase error
curves but not its shape. Finally, let us consider a coarser
mesh. The phase error is plotted in Fig. 3 for the case where
,
, and
. The effect of the coarser mesh
is to increase both the phase error and the anisotropy. Note
that for this grid spacing, the edge-based FEM has slightly
2.8 ) as compared to FDTD and the
smaller anisotropy (
3.2 ).
node-based FEM with integral lumping (
Up until this point, the results have been generated with
for the edge-based FEM. In Fig. 4, the phase error for
the edge-based FEM is plotted for values of from 0.2–0.8
and
in increments of 0.2. The other parameters are
. Similar to changing , the effect of changing is to
Formulations of TDFEM’s based on the second-order wave
equation using Galerkin’s method usually result in implicit
time-marching schemes. Motivated by the successes of finitedifference methods on regular grids, various approaches have
been proposed to formulate explicit TDFEM’s which do not involve inversion of matrix equation. As mentioned earlier, one
possibility is to employ collocation methods, an approach that
has not received substantial attention within the computational
electromagnetics community yet. Another popular approach is
to lump the mass matrix into a diagonal matrix, thus reducing
the algorithm to an explicit TDFEM. In this section, we
outline the lumping process for tetrahedral edge elements and
comment on its implications in practical implementations. In
general, these comments are also applicable to other types of
finite elements.
The mass lumping is typically achieved by using reduced
integration. In particular, for a single tetrahedral edge element
(see Fig. 5), we require the following mass integration to hold:
(68)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
We conclude this section with a brief list the most common
problems with mass lumping.
1) It only works for low-order (at most linear) finite elements.
2) It results in worse numerical dispersion error than the
conventional Galerkin’s method.
3) It sometimes generates a diagonal matrix with zero
and/or negative entries.
In light of these problems as well as the potential benefits
from the use of collocation methods, we do not recommend the
use of mass lumping for the derivation of explicit TDFEM’s.
VIII. CONCLUDING REMARKS
Fig. 5. A good shaped tetrahedral element.
where
is the th diagonal entry in the final diagonal
matrix. Thus, essentially, lumping is achieved by the proper
determination of the diagonal entries
such that (68) will
. To proceed
be exact for any constant field
further, we first note that the unknown coefficient for edge
i is
(69)
where it is assumed that edge is the edge from
to
. Substituting (69) into (68), and requiring that
it holds true for any arbitrary choice of
, we obtain the
following six simultaneous equations for
:
(70)
is the volume of the tetrahedron.
where
To demonstrate one of the major undesirable characteristics
of mass lumping, let us investigate the element shown in
Fig. 5. After some algebraic manipulations, it can be shown
that the diagonal entries are
and
. Note that in this case the lumping process
results in a singular mass matrix. In practical applications,
where many tetrahedral elements are involved (some could be
very bad shaped), zero and/or negative diagonal entries from
mass lumping are not uncommon. This imposes a very severe
problem: no time step can be found to make the numerical
algorithm stable.
In this paper, TDFEM’s were presented for the secondorder vector wave equation and Maxwell’s hyperbolic system.
Both node-based and edge-based elements were considered.
Because of the importance of stability for time-domain methods, the underlying theory for the development of the stability
condition for a general TDFEM formulation was presented in
some detail. In addition, the numerical dispersion (or phase)
error performance of various TDFEM’s was compared with
that of the standard FDTD method. Finally, the advantages
and disadvantages of mass lumping were considered.
Progress in the advancement of TDFEM’s has been very
slow in comparison to the standard FDTD method. However,
there has been an increased interest in the last few years for
the development of transient electromagnetic simulators with
geometric modeling versatility superior to the one offered by
the FDTD method. This demand has stimulated a significant
activity toward the development of various methodologies
for enhancing the standard FDTD method to facilitate the
modeling of nonrectangular geometries without the typical
staircase approximation. While many of the proposed numerical approximations appear to be, at first sight, more general
implementations of the standard rectangular FDTD method,
their development is also possible through the Faedo–Galerkin
formulation discussed in this paper. Thus, many of these
methods may be viewed as special examples from the general
class of TDFEM’s. The success of these methods is strongly
dependent on the availability of reliable and robust mesh
generators. As mesh generation techniques mature and reliable
and user-friendly mesh generation packages start becoming
available, research work in development of TDFEM’s and their
application to transient electromagnetic modeling is expected
to increase significantly.
As mentioned in the text, there are several issues in TDFEM’s that need to be researched further, and the potential
is there for the development of very sophisticated and versatile transient electromagnetic simulators. Among them we
identified the use of collocation methods and the possibility of
using different types of trial functions in different regions of
the geometry of interest, appropriately selected so that they are
consistent with the anticipated smoothness of the fields in the
various regions. The use of more accurate spatial interpolation
schemes for the reduction of dispersion error have started
making their appearance in the computational electromagnetics
community both in the context of spectral approximations
LEE et al.: TIME-DOMAIN FINITE-ELEMENT METHODS
[26], [27] and in the context of wavelet expansions [28]. The
use of Galerkin’s method in the development of TDFEM’s
offers the ideal framework for the implementation of such
diverse types of spatial approximations over different regions
of the geometry under study in a consistent manner, maintaining the appropriate field continuity conditions across region
boundaries.
In addition to the aforementioned “hybridization” of the
trial functions, there are some more topics associated with
TDFEM’s that have not been addressed in this paper and
are the subject of on-going research. One of them is the
development of radiation boundary conditions for TDFEM’s.
Research into accurate ABC’s is very sparse. To our knowledge, only the first-order ABC’s have been implemented in
conjunction with TDFEM’s. We believe that advancement in
this area will benefit from the extensive work on ABC’s done
for the FDTD method. More specifically, we expect truncation
boundary conditions based on the perfectly matched layer
(PML) [29], [30] to find significant application in TDFEM’s
also. They have been shown to work well for both the FDTD
and frequency domain FEM, and the recent work in the
reformulation of the PML theory in terms of time-dependent
sources [31] is expected to facilitate their incorporation into
TDFEM’s.
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Jin-Fa Lee (S’85–M’88) was born in Taipei, Taiwan, in 1960. He received
the B.S. degree from National Taiwan University, in 1982, and the M.S. and
Ph.D. degrees from Carnegie-Mellon University, Pittsburgh, PA, in 1986 and
1989, respectively, all in electrical engineering.
From 1988 to 1990, he was with ANSOFT Corporation, Pittsburgh, PA,
where he developed several CAD/CAE finite-element programs for modeling
three-dimensional microwave and millimeter-wave circuits. From 1990 to
1991, he was a Postdoctoral Fellow at the University of Illinois at UrbanaChampaign. Currently, he is an Associate Professor at the Department
of Electrical and Computer Engineering, Worcester Polytechnic Institute,
Worcester, MA. His current research interests include analyses of numerical
methods, coupling of active and passive components in high-speed electronic circuits, three-dimensional mesh generation, and nonlinear optic fiber
modelings.
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 3, MARCH 1997
Robert Lee (S’82–M’90) received the B.S.E.E. degree from Lehigh University, Bethlehem, PA, in 1983, and the M.S.E.E. and Ph.D. degrees from the
University of Arizona, Tucson, in 1988 and 1990, respectively.
From 1983 to 1984, he worked for Microwave Semiconductor Corporation,
Somerset, NJ, as a microwave engineer. From 1984 to 1986 he was a Member
of the Technical Staff at Hughes Aircraft Company, Tucson, AZ. From
1986 to 1990 he was a Research Assistant at the University of Arizona.
In addition, during the summers of 1987 through 1989, he worked at Sandia
National Laboratories, Albuquerque, NM. Since 1990, he has been at The
Ohio State University, where he is currently an Associate Professor. His major
research interests are in the analysis and development of finite methods for
electromagnetics.
Dr. Lee is a member of the International Union of Radio Science (URSI)
and was a recipient of the URSI Young Scientist Award in 1996.
Andreas Cangellaris (M’86), received the Dipl. degree in electrical engineering from the Aristotle University of Thessaloniki, Greece, in 1981, and
the M.S. and Ph.D. degrees in electrical engineering from the University of
California, Berkeley, in 1983 and 1985, respectively.
From 1985 to 1987, he was with the Electronics Department of General Motors Research Laboratories, Warren, MI. In 1987 he joined the
Department of Electrical and Computer Engineering at the University of
Arizona, Tucson, where he is now an Associate Professor. His research
interests include computational electromagnetics, microwave engineering, and
algorithm development for high-speed digital circuit analysis. In the area
of computational electromagnetics his research has emphasized differential
equation-based techniques, both in the frequency and time domain. He has
published more than 50 journal papers in the aforementioned areas of research.
Dr. Cangellaris is a member of the International Union of Radio Science
(URSI). In 1993 he received the International Union of Radio Science (URSI)
Young Scientist Award. He is an active member of the following IEEE
Societies: Antennas and Propagation, Microwave Theory and Techniques, and
Components, Packaging, and Manufacturing Technology.