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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012
Explicit Time-Domain Finite-Element Method
Stabilized for an Arbitrarily Large Time Step
Qing He, Houle Gan
, Student Member, IEEE, and Dan Jiao
, Senior Member, IEEE
Abstract—The root cause of the instability is quantitatively iden-
tified for the explicit time-domain finite-element method that em-
ploys a time step beyond that allowed by the stability criterion.
With the identification of the root cause, an unconditionally stable
explicit time-domain finite-element method is successfully created,
which is stable and accurate for a time step solely determined by
accuracy regardless of how large the time step is. The proposed
method retains the strength of an explicit time-domain method
in avoiding solving a matrix equation while eliminating its short-
coming in time step. Numerical experiments have demonstrated its
superior performance in computational efficiency, as well as sta-
bility, compared with the conditionally stable explicit method and
the unconditionally stable implicit method. The essential idea of
the proposed method for making an explicit method stable for an
arbitrarily large time step irrespective of space step is also appli-
cable to other time domain methods.
Index Terms—Explicit time-domain methods, stability, time-do-
main finite-element methods, unconditionally stable methods.
I. INTRODUCTION
THE time-domain methods in computational electromag-
netics can be categorized into two classes. One is the ex-
plicit time-domain method; the other being the implicit time-do-
main method. In an explicit time-domain method, the field solu-
tion at each time step is evaluated from the field solutions at pre-
vious time steps; whereas in an implicit time-domain method,
the field solution at each time step involves the field solution
that is unknown. Explicit methods can avoid solving a matrix
equation, while implicit methods generally require a matrix so-
lution.
Despite its advantage of avoiding a matrix solution, an
explicit method requires the time step to be restricted by the
smallest space step for ensuring stability. For problems that
have fine features relative to working wavelength like on-chip
integrated circuits, explicit methods require a large number of
time steps to finish one simulation, which is computationally
expensive. In an unconditionally stable method, there is no
restriction between space step and time step for obtaining a
stable solution. It also permits the use of an arbitrarily large
time step without becoming unstable. Existing unconditionally
Manuscript received August 10, 2011; revised November 02, 2011; accepted
May 30, 2012. Date of publication July 10, 2012; date of current version October
26, 2012. This work was supported in part by a grant from Intel Corporation, in
part by a grant from the Office of Naval Research under award N00014-10-1-
0482, and in part by grants from the NSF under Awards 0747578, 0802178, and
1065318.
The authors are with the School of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN 47907 USA (e-mail: djiao@purdue.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAP.2012.2207666
stable methods are all implicit methods. In the FDTD based
methods, a family of implicit schemes [1]–[9] such as the
ADI (alternating direction implicit)-FDTD [1]–[3], the CN
(Crank–Nicolson)-FDTD [4], the LOD (locally 1-D)-FDTD
[5], the Laguerre-FDTD [6], and the split-step FDTD [8]
methods have been developed to achieve unconditional sta-
bility. In [7], it is shown that these unconditionally stable
implicit FDTD methods can be derived from a general formula-
tion based on generalized matrix operator equations pertaining
to some classical splitting formulae, from which a variety of
other unconditionally stable implicit schemes can further be
deduced. Similarly, a group of unconditionally stable time-do-
main finite-element methods (TDFEM) such as the Newmark
method [10], ADI-FETD [11], the CN-FETD [12], and the
backward difference have also been developed. These methods
require the solution of a weighted sum of the mass matrix and
the stiffness matrix, whereas the explicit TDFEM only requires
the solution of the mass matrix, which is either diagonal in
nature or can be diagonalized by the orthogonal vector basis
functions [13]–[15], hence eliminating the need for solving
a matrix equation. In summary, the large time step provided
by the existing unconditionally stable scheme is achieved by
resorting to implicit time-domain methods that sacrifice com-
putational efficiency. Moreover, late-time instability has also
been observed from implicit methods. In [16], a new FDTD
method using the alternating-direction explicit (ADE) method
was developed for efficient electromagnetic field simulation.
Though the method is explicit, yet it is not unconditionally
stable. In fluid dynamics, techniques have been developed to
filter unstable high frequency waves to increase the time step
of an explicit scheme [28]. However, the time step is only
extended by a limited amount. The filtering technique has also
been developed to extend the stability limit of the explicit
FDTD method [17] in electromagnetics. As to-date, no explicit
methods have been made unconditionally stable.
The research question considered in this work is: can an ex-
plicit method be made unconditionally stable so that its strength,
which lies in avoiding a matrix solution, is retained while elim-
inating its shortcoming in time step?
The contribution of this work is the successful development
of an explicit time-domain method that is unconditionally
stable, a capability that previously did not exist. It is stable for
any large time step without being restricted by the space step. If
the time step is chosen based on accuracy, the proposed method
produces not only stable but also accurate results no matter how
large the time step is. We have already conducted preliminary
research on the proposed method in [18], [19]. In this paper, we
complete it from both theoretical and numerical perspectives.
The paper is organized as follows. In Section II, we present
0018-926X/$31.00 © 2012 IEEE
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the background of a time-domain finite-element method.
In Section III, we describe the proposed theory for making
an explicit time-domain method unconditionally stable. In
Section IV, we propose an explicit time-domain finite-element
method that is unconditionally stable. The linear computational
complexity of the explicit method at each time step is preserved
by the proposed method. Section V demonstrates the stability,
accuracy, and efficiency of the proposed method. It is also
shown that the proposed unconditionally stable explicit method
outperforms both the conditionally stable explicit method and
the unconditionally stable implicit method in computational
efficiency as well as stability. Although the proposed method
is presented in the framework of a time-domain finite-element
method, the essential idea can be applied to other time domain
methods, hence, contributing to the removal of one major com-
putational bottleneck in time-domain electromagnetic analysis.
II. BACKGROUND OF A TIME-DOMAIN
FINITE-ELEMENT METHOD
Consider the second-order vector wave equation
(1)
where , , , , and
are the electric field, free-space
permeability, relative permeability, permittivity, and current
density respectively. A time-domain finite-element based solu-
tion of (1) and its boundary conditions results in the following
system of linear equations [20]:
(2)
in which
is called a mass matrix, is called a stiffness ma-
trix, is the unknown field vector, and is a current excitation
vector. The
and are sparse and symmetric. Typically they
have only tens of nonzero elements in each row regardless of
the matrix size . These matrices can be assembled in linear
time and stored in linear computational resources from their el-
emental contributions as follows:
(3)
where
and
are the vector basis functions used to expand
while
denotes a volume integration in each element.
Compared to other time-domain methods, a time-domain finite-
element method possesses flexibility in both geometrical and
material modeling.
III. PROPOSED THEORY FOR MAKING AN EXPLICIT
TIME-DOMAIN METHOD UNCONDITIONALLY STABLE
A. Quantitative Analysis on the Root Cause of the Instability
When Using a Time Step Beyond Stability Criterion
In this section, we will use the time-domain finite-element
method as an example to develop a quantitative analysis on the
root cause of the instability associated with an explicit time-do-
main method, whenever a time step beyond the stability cri-
terion is used. However, the findings are equally applicable to
other time-domain methods.
In an explicit time-domain finite-element method, to maintain
stability, the time step is required to satisfy [21], [27]
(4)
where
denotes the spectral radius of
, which is
the largest eigenvalue of
. Since the largest eigenvalue of
that is supported by a numerical system is inversely pro-
portional to the smallest space resolution, like the CFL condition
in explicit FDTD-based methods [22], (4) also dictates that the
time step for a stable simulation is dependent on the smallest
space step. Since
is nonzero, apparently, there is no
obvious way to make an explicit scheme stable for any large
time step. However, from the following quantitative analysis on
the root cause of the instability, it will become clear that it is
feasible to make an explicit time-domain method uncondition-
ally stable.
The solution of (2) can be rigorously found by first solving
the following generalized eigenvalue problem:
(5)
The field solution vector is then expanded in the space formed
by all the eigenvectors of (5) followed by finding the corre-
sponding coefficients of each eigenvector in the field solution
of [23]. This approach is also known as modal superposition
method [24].
Let
be the solution to the generalized eigenvalue
problem shown in (5), in which the entries of diagonal matrix
are eigenvalues
, and the column vectors
of
are eigenvectors
. Physically speaking,
are the angular resonance frequencies of the
structure being simulated, which have the same unit as ;
while the eigenvectors of (5) represent the resonance modes
that can be intrinsically supported by the structure. Since
is symmetric positive definite and
being symmetric, the
eigenvectors of (5) are -and -orthogonal [25]. As a result,
we have
(6)
where is an identity matrix. The solution of (2) can then be
rigorously expanded in the eigenspace
(7)
where the unknown coefficient vector contains all the weights
of the eigenvectors in the field solution. From (7), it can be seen
that the field solution at each time instant in an arbitrary problem
is the superposition of the eigenvectors (modes). This is true for
1-D, 2-D, and 3-D problems. To obtain unknown coefficient ,
we substitute (7) into (2). Multiplying both sides of (2) by
,
and using the property shown in (6), we obtain
(8)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012
A central-difference based explicit solution to the above yields
(9)
where
. To analyze the stability of (9), we set the
excitation to be zero and perform a -transform of (9), obtaining
(10)
For an explicit time marching like (9) to be stable,
of (10)
should be bounded by 1. As a result, the following condition
must be satisfied:
(11)
where eigenvalue
is the -th entry of the diagonal matrix ,
and
is matrix size.
Due to the property of (positive definite) and (semi-pos-
itive definite), the eigenvalues of (5), , are nonnegative. The
smallest eigenvalue of (5) is zero, which is due to the null space
of the stiffness matrix . These zero eigenvalues always exist.
The corresponding eigenvector is called a DC mode. Despite
its zero eigenvalue, a DC mode can also have a complicated
field distribution such as the DC mode of an integrated circuit
made of multiple metallic wires immersed in dielectric mate-
rials. As for the largest eigenvalue of (5), although theoretically
speaking, the resonance frequency of a structure, and hence the
eigenvalues of (5), can be infinitely large, the largest eigen-
value that can be numerically found is limited by the smallest
space resolution. To be specific, the square root of the maximum
eigenvalue of (5),
(whose unit is angular frequency), is in-
versely proportional to the smallest space resolution
as the
following:
(12)
where is speed of light. This is because given the smallest
space step
, the minimum wavelength that can be captured
by the space discretization is
. The angular frequency cor-
responding to such a wavelength is
. From (12), it is
clear that the smaller the space step, the larger the maximum
eigenvalue that can be numerically identified from (5).
The meaning of (11) is significant. It demonstrates that when
an explicit method becomes unstable, among all the eigenvec-
tors
(modes) that are contained in the field solution shown
in (7), not every mode becomes unstable. Only a subset of the
modes, whose eigenvalues are so large that (11) is violated, is
unstable. The rest of the modes are stable. For example, the DC
modes, the eigenvalues of which are zero, are always stable irre-
spective of the choice of time step. Therefore, we conclude that
the set of modes that violate (11) in the field solution are the
root cause of the instability associated with an explicit time-do-
main method when a large time step is used. The
in (11) is
an angular resonance frequency of the system. When (11) is vi-
olated,
, where
is the frequency cor-
responding to . Therefore, given a time step
, the unstable
modes are also those modes which vary with space at such a high
frequency that it cannot be accurately captured by the given time
step based on sampling theorem. The number of these unstable
modes is also finite.
The remaining question is why these unstable modes exist?
They exist because of the fine discretization as can be seen from
(12). A fine discretization cannot be avoided in problems having
fine feature sizes relative to working wavelength. The finer the
space discretization, the larger the maximum eigenvalue of (5).
Once these unstable modes are inherent in (5), even though the
right hand side does not have a projection onto them, the nu-
merical roundoff error will have a projection onto them. This
can be seen clearly from (9), which is a diagonal system of equa-
tions with the -th entry in vector , , representing the coeffi-
cient of the -th mode of (5). It is clear that even though
is zero, the roundoff error can make
nonzero. As a result, in
the field expansion shown in (7), the coefficients of the unstable
modes would not be zero, and hence the unstable modes exist
in the field solution at each time instant. Meanwhile their eigen-
values are so large, i.e., these modes vary with space at such a
high frequency that they cannot be accurately simulated by the
given time step, and hence instability occurs.
The above analysis also clearly shows why in cases, where
fine features do not exist and hence the space step can be solely
determined by accuracy, the time step suggested by the sta-
bility criterion has a good correlation with that required by accu-
racy. In this case, the frequency corresponding to the maximum
eigenvalue of (5) agrees well with the physically important max-
imum frequency to be captured. Therefore,
,
where
is the physically important maximum frequency to
be captured. As a result, the time step suggested by stability cri-
terion (4) has a good correlation with that dictated by accuracy,
for sampling an
-based system.
B. How to Make an Explicit Time-Domain Method
Unconditionally Stable
From the aforementioned root cause analysis, it becomes
clear how to make an explicit method unconditionally stable.
Given a time step
regardless of how large it is, one can cor-
respondingly remove those modes whose field variation with
space cannot be accurately simulated by the given time step
based on sampling theorem. For a time-domain finite-element
method, quantitatively, we remove the modes that violate (11),
i.e., those modes whose eigenvalues are greater than
,
from the numerical system. By doing so, an explicit method
can be made stable for any large time step. In the extreme case
that
, one can, also, make an explicit method stable
by simply keeping all the nullspace modes whose eigenvalues
are zero and removing the rest of the modes having nonzero
eigenvalues.
To completely remove the unstable modes, we expand the
field solution strictly in the space of stable modes; we also
project the numerical system onto the space of stable modes.
The detailed procedure is as follows.
Denoting the coefficient vector in (7) by
(13)
where
is a coefficient vector of unstable modes whose eigen-
values are greater than
, and is for the stable modes for
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the given time step
, denoted by
. We expand the field so-
lution in the space of stable modes, , as the following:
(14)
This is equivalent to setting
in (13) to be zero at each time
instant. Meanwhile, we project the numerical system (2) onto
the space of , which is achieved by multiplying both sides of
(2) by
. We thereby obtain
(15)
By using the property shown in (6), we have
and hence
(16)
where
is a diagonal matrix comprising the eigenvalues cor-
responding to stable modes, which are no greater than
.
The resultant explicit marching of (16) is stable for the given
time step
regardless of how large the time step is because
(11) is always satisfied.
It is worth mentioning that by removing unstable modes from
the field solution
only, we cannot make a time-domain scheme
stable for any large time step. This is because in this case, a
central-difference based explicit marching of (2) can be written
as
(17)
in which the unstable part of the field solution is excluded at
each time instant. Although now
is free of unstable modes,
constitutes a space that spans all the modes of (5) in-
cluding both stable and unstable modes for a given time step.
The roundoff error of
can always project
onto the unstable modes, causing instability. In other words, the
source of instability is contained in system matrix
instead
of field solution. Without changing the system matrix to a matrix
that only spans the space of stable modes, the source of insta-
bility still exists. That is why in this work, we find the space of
stable modes
for a given time step, expand the field solu-
tion in this space and, also, project the system matrix onto this
space, as shown by (15). The resultant numerical system (16) is
absolutely stable for the given time step no matter how large it
is.
C. How to Make an Explicit Time-Domain Method
Unconditionally Stable and Also Accurate
To satisfy accuracy criterion, the time step cannot be chosen
arbitrarily large, it has to satisfy sampling theorem, i.e.,
where
corresponds to the smallest wavelength, and hence
the maximum frequency of space variation that is physically
important in a system response. For good accuracy, the time step
is generally chosen as
(18)
In other words, in one wavelength, one should at least sample
10 points for achieving a good accuracy.
Based on the analysis given in previous section, for any given
, to make an explicit time-domain scheme stable, we should
remove the modes having eigenvalues greater than
. For
a time step given in (18) that is solely determined by accuracy,
the modes that are removed are also physically negligible. This
is because the removed modes have eigenvalues greater than
, by using (18), we have
(19)
In other words, the removed modes vary with space at a fre-
quency higher than
. Since
is the maximum frequency
of space variation that is physically important in a system re-
sponse, the beyond-
modes are physically negligible. As
a result, when
is chosen based on the accuracy criterion, if
we remove the modes having eigenvalues greater than
,
not only do we make the explicit time marching stable, but also
preserve the accuracy of the field solution. If
is chosen con-
servatively that
is larger than
, instead of only
removing the modes having eigenvalues greater than
, we
can remove more modes as long as their eigenvalues are greater
than
since the beyond-
modes are physically
negligible. In other words, the number of modes that need to
be kept for a stable and accurate simulation is bounded by the
number of modes whose resonance frequency is no greater than
. These modes are termed
physically important modes in
this paper.
Why there exists a maximum frequency of space variation
that is physically important in a system response? In other
words, why removing beyond-
modes does not affect the
accuracy of the field solution? This can be understood from the
following theoretical analysis.
Equation (7) shows that the field solution is a superposition of
all the
eigenmodes of (5). However, given an input pulse that
is band limited, the number of modes that make nontrivial con-
tributions to the field solution is also limited. To see how many
vectors in should be included in the field solution (7), we can
convert (2) to frequency domain for a quantitative analysis. In
frequency domain, (2) becomes
(20)
From (5), the solution to the above can be written as
(21)
which is the superposition of all the eigenmodes [23]. Al-
though the eigenvectors (modes) do not depend on frequency,
their weights in the field solution do vary with frequency. As
can be seen from (21), the weight of each mode
in
is
. Clearly, given a frequency
or a band of
frequencies, not all of the modes make important contributions
in the field solution for the given spectrum. Only those modes
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that have a large weight are important, whereas the modes,
whose eigenvalues are so far away from the working frequency,
can be truncated based on prescribed accuracy. Thus, (21) can
be computed as
(22)
with controlled accuracy , where
is composed of
eigenmodes whose relative weights in the field solution are
greater than for the given frequency or the given spectrum.
The frequency corresponding to the maximum eigenvalue of
these
modes represents the maximum frequency of space
variation that is physically important in a system response,
which is
. It is clear that by removing beyond-
modes,
the accuracy of the field solution is not affected. This is true for
the field solution at any point in the computational domain, no
matter how far or close the point is from the source since the
weights of beyond-
modes are negligible due to the large
gap between their eigenvalues and working frequency square
.
In addition, as can be seen from (12), if the space step
is chosen based on the accuracy requirement, i.e., capturing
a space variation up to
frequency, then the maximum
eigenvalue
of (5) does not go beyond that associated
with
. Then, no beyond-
modes exist in the numerical
system. Once beyond-
modes exist, that is because the
space step is finer than that required by accuracy for discretizing
an
-based system. Therefore, they are not necessary for
the required accuracy, and hence can be eliminated without
affecting accuracy.
IV. PROPOSED EXPLICIT TIME DOMAIN FINITE ELEMENT
METHOD THAT IS UNCONDITIONALLY STABLE
The proposed explicit time-domain finite-element method
that is stable for any large time step has two steps. The first
step is a preprocessing step for building a complete and also
accurate space that spans all the stable modes for a given time
step
. The second step is to perform a march-on in time with
the given time step in an explicit time-domain method without
violating stability. Both steps retain the strength of an explicit
method in avoiding a matrix solution, and hence achieving sta-
bility for any time step without sacrificing the linear (optimal)
complexity of an explicit method based simulation.
A. Preprocessing for Building a Complete and Accurate Space
That Spans all the Stable Modes for a Given Time Step
From Section III, it is clear that one straightforward approach
for finding the stable eigenmodes is to solve the generalized
eigenvalue problem shown in (5). After obtaining all the eigen-
values and eigenvectors of (5), one can identify those eigen-
vectors whose eigenvalues are no greater than
. The cor-
responding eigenvectors,
, can then be used to form the
space
that spans all the stable modes for the given time step,
which can be written as
(23)
The number of stable modes, i.e., the column dimension of , is
between 1 and the number of physically important modes whose
resonance frequency is no greater than
, and hence having
eigenvalues no greater than
. We do not need to keep
the remaining modes with higher eigenvalues even though they
can be simulated in a stable manner by the given time step
as these modes are physically negligible. When
is chosen
based on accuracy, the
is also the union of physically impor-
tant modes. The disadvantage of the aforementioned approach
is that it requires an efficient solution of a generalized eigen-
value problem of size
.
To bypass the large-scale eigenvalue solution of
, in this
work, we develop a time-domain solution based fast eigenvalue
solution of
, where is the number of physically important
modes of (5). In general, is orders of magnitude smaller than
in problems where the space discretization is much finer than
that required by accuracy for sampling an
-based system
because many eigenvalues higher than
will be generated.
As will be shown in the numerical results, could be as small
as 3 while
is large. In these problems, the time step issue of
an explicit method is also the most critical. The details of this
method are given below.
1) Transforming the Original Generalized Eigenvalue
Problem of
to a Reduced Eigenvalue Problem of
by Field Solutions Obtained at a Small Number of Time Steps:
We first employ a conventional explicit time-domain method
to solve (2) at a small number of time steps (terminating
criterion will be discussed in Section IV-A-2). By doing so,
we take advantage of the strength of an explicit method in
avoiding a matrix solution. Meanwhile, we do not suffer from
the shortcoming of an explicit method in requiring many time
steps for finishing one simulation. This is because compared to
the total number of time steps required by the explicit method
for finishing the entire simulation, the number of time steps
to be simulated is small for revealing the physically important
eigenvalues and eigenvectors from time-domain solutions.
When we solve (2), we store the solution vector in
and
also orthogonalize
to ensure its column vectors are linearly
independent of each other. The orthogonalization is done when-
ever a new is added in . Since is generally less than ,
we do not need to store the field solution at each time instant
to construct . Instead, we can select a subset of the obtained
field solutions to form . We develop the following method to
quantitatively judge whether
is complete enough to find all
the physically important modes.
We expand the field solution in the space of as
(24)
with being the unknown coefficient vector. Substituting (24)
into (2) and multiplying (2) by
, we obtain
(25)
where
(26)
Assume the matrix system in (2) is of size
and there are
vectors in , then
is an
matrix, where
. As a
result, both
and are a small
matrix. Thus, instead of
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solving the
eigenvalue problem shown in (5), we only
need to solve a reduced
eigenvalue problem as
(27)
Denoting the union of the eigenvectors of (27) by
, and the
eigenvalue matrix by
. The solution of (25) can be expanded
in the space of
. We therefore have
(28)
where contains the time-variant weights of the eigenvectors
, which is similar to in (7) but with a significantly reduced
size . Since
is symmetric positive definite and
is sym-
metric, we have [25]
(29)
Substituting (28) into (25), multiplying both sides of (25) by
, and using the property (29), we obtain
(30)
where can be solved via a central-difference based scheme
like (9).
2) Identify Physically Important Eigenvalues and Eigen-
vectors of the Original System From the Reduced Eigenvalue
Problem: One important fact is that the eigenvalues of physi-
cally important modes computed from the original system (5)
will also satisfy the reduced system (27) as long as the space
contains the information of these physically important modes.
This is true because
is formed by a set of solutions of (2) that
are nothing but the superposition of the physically important
modes. As a result, we can solve a
system shown in (27)
to obtain the physically important modes of (5). A theoretical
proof of this is given in the appendix.
During the time marching process, whenever we add a so-
lution vector in space , we compute the eigenvalues from the
reduced system (27). If the size of (27) is , we obtain eigen-
values. However, only a subset of the eigenvalues belongs to
the set of physically important eigenvalues of (5). We devel-
oped the following procedure to quantitatively identify the
physically important modes.
Our strategy is to monitor the weights of the eigenmodes
in the time-marching process to identify physically important
modes. The weight of the -th mode is nothing but the -th entry
of vector shown in (28). At the early time, very large eigen-
values are observed from (27). They correspond to the largest
eigenvalues that are supported by the numerical system. These
large eigenvalues can be observed at an early time because the
frequency carried by the early-time response is the highest com-
pared to the frequency carried by the system response in other
time. As can be seen from (21), the field solution for a given
frequency is dominated by eigenmodes whose eigenvalues are
the closest to the given frequency as their weights in the field
solution are the largest.
When the early time is passed and dominant frequency com-
ponents that are no greater than
set in, a set of eigenvec-
tors whose eigenvalues are smaller than
start to appear.
Although
can be estimated from the input spectrum, one
may not know
quantitatively in advance. The proposed
method does not require users to quantitatively know
ei-
ther because
can be numerically identified in the procedure
of finding physically important modes. Without knowing the
exact
, what one observes is that after early time is passed,
eigenvalues smaller than those observed in early time start to
appear. When one enlarges the size of space
by adding a new
solution vector from time to time, one can observe that a set of
common eigenvalues reappear from time to time. When this set
of eigenvalues starts to have its weights significantly larger
than those of the rest of the eigenvalues which are larger, the
corresponding eigenvectors are ready to be sampled as physi-
cally important eigenmodes. This is because once the weights of
the modes having large eigenvalues become significantly small,
in future time steps, the weights of these modes, instead of be-
coming larger, can only become smaller because the frequency
carried by the later time response can only be lower than higher.
In our implementation, we use the following condition to sys-
tematically identify physically important eigenmodes:
(31)
where
is a small parameter defined based on prescribed accu-
racy, is the weight associated with the common eigenvalues
that reappear from time to time, and
is that associated with
larger eigenvalues. After identifying the physically important
eigenmodes, the
can be quantitatively determined from the
largest eigenvalue among the set of physically important eigen-
values. As a result, we do not need to pre-assume
based
on empirical knowledge.
When the number of physically important eigenvalues does
not increase in the time marching process, the space
con-
structed can be considered complete. To obtain a complete as
well as accurate space that spans all the physically important
modes, we further apply the following accuracy requirement to
each physically important eigenvalue :
(32)
where the superscript denotes the time index, is a small pa-
rameter defined based on an accuracy requirement, and is the
number of . We select the eigenvectors of (27) corresponding
to these ,
, to form the space
(33)
which is the space that spans all the physically important modes.
It is clear that the space
that is formed by
field solution
vectors cannot be used directly to construct
because not all
the eigenvalues and eigenvectors contained in
belong to the
space. We have to select only those eigenvalues that are the
physically important eigenvalues of (5). This is accomplished
by multiplying
from right by
. When both criteria (31)
and (32) are satisfied, the
is complete as well as accurate.
3) Form
That Spans all the Stable Modes for a Given Time
Step: With
obtained, the preprocessing can be terminated.
For a given time step
, from the physically important modes,
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012
we select those modes whose eigenvalues are no greater than
to form
. Thus
where
are the physically important eigenvectors of
(27) whose eigenvalues are no greater than
. When
is
chosen based on accuracy requirement, is equal to , i.e., all
the physically important modes will be included in
. Hence,
the simulation is not only stable but also accurate.
It is also worth mentioning that we store
by separately
storing
and
, which has a linear cost. We do
not need to multiply them together since their direct product is
not required in the computation, which will be seen very clearly
from Section IV-B.
B. Explicit Time Marching Stable for Any Given Time Step
With
, the space formed by stable modes for a given time
step
, obtained systematically, we can simulate (2) in a stable
fashion for the given
regardless of how large
is.
At each time step, we solve (2). We first expand in the space
of
, which is the same as the union of the
eigenvectors of (5) corresponding to the stable modes for the
given time step. Thus, we obtain
(34)
Substituting (34) into (2) and multiplying
on both
sides of (2), we obtain
(35)
where
and
are the same as (26). Because of (29), we have
(36)
in which
is a diagonal matrix that contains the eigenvalues
corresponding to stable modes. A central-difference based dis-
cretization of (35) thus yields
(37)
After a time marching of (37) at all the time steps, if the field
solution all over the structure is needed, it can be obtained by
(38)
Since
and
are time independent, at each time step, we
only need to update and, also, from the reduced system shown
in (37), the cost of which is
, and hence negligible.
C. Summary of the Overall Procedure and Cost Analysis
The overall procedure is as follows.
Step I: Preprocessing for building a complete and accurate
space
that spans all the stable modes for any given time
step irrespective of its size:
(I-1). Use the conventional explicit time-domain
method to solve (2), and march on in time by one step.
This can be performed in linear complexity.
(I-2). At selected time instants, the number of which
is
(which instants to select does not affect the
accuracy because the criteria (31) and (32) will ensure
accuracy):
Add field solution vector in . Orthogonalize the new
solution vector with respect to the other vectors that
have been already stored in . The cost is linear for
orthogonalizing
vectors of length .
Solve a reduced eigenvalue problem of size
shown
in (27). Solve the weight from (30). Check whether
(31) and (32) are satisfied. If not, go back to substep
(I-1); if yes, stop, and then determine eigenvalues and
eigenvectors of the stable modes. The cost of this step
is negligible because of the reduced system size.
Step II: Explicit time-marching stable for any given time
step
(II-1) Perform time-marching. Compute the coefficient
of each mode from the reduced system (37) of size
at each time step. The cost is
, where is the
number of stable modes.
(II-2) After is obtained at all time steps, the field solu-
tion can be recovered from (38) at each time instant.
If only selected field solutions are of interest, we can
select the
rows corresponding to the selected loca-
tions to compute , the cost is
.
V. NUMERICAL RESULTS
We have simulated a number of examples of both m- and
millimeter-scales to validate the unconditional stability, accu-
racy, efficiency, and late-time stability of the proposed method.
A. Demonstration of Unconditional Stability
First, we demonstrate the fact that the proposed method is
stable regardless of the space step and the choice of time step.
The example considered was a parallel plate structure that has
an analytical solution. The fill-in material was air. The height
(along ), width (along ), and length (along ) were set to
be 1 m, 5 m, and 900 m, respectively. The space reso-
lution along , , and
was 1 m, 1 m, and 100 m, re-
spectively. A current source was launched from bottom plate
to top plate at the near end while the voltages were extracted
between the two plates at the near and far ends. The compu-
tational domain was truncated by a PEC (perfect electrically
conducting) boundary condition on the top and at the bottom
planes ( -planes), a PMC (perfect magnetically conducting)
boundary condition at the left and right boundaries ( -plane
boundaries), and the first-order absorbing boundary condition
at the front and back ends. The current source was the deriva-
tive of a Gaussian pulse with
,
where
and
s. For this example, the conven-
tional explicit scheme has to use a time step of
s to main-
tain time-domain stability because of small space step. In con-
trast, as shown in Fig. 1, the proposed method permits the use
of any large time step such as 0.0001 s, 0.001 s, 0.01 s, and 0.1
s without becoming unstable. As described in Section III, the
proposed method achieves stability for any large time step cour-
tesy a representation of field solution strictly by stable modes for
the given time step. In this simulation, by keeping only the DC
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5247
Fig. 1. Illustration of the unconditional stability of the proposed method.
mode whose eigenvalue is zero from the reduced eigenvalue so-
lution (27), we allow for the use of any large time step without
making the simulation unstable. The same applies to other ex-
amples if one would like to use a time step that is infinitely large
to examine stability. In addition, in this example, since the time
step of 0.0001 s, 0.001 s, and 0.01 s also satisfies accuracy re-
quirements for the given input source, the results shown in Fig. 1
for these three time steps are not only stable but also accurate,
which reveal that the voltages at the near and far end of the par-
allel plate operating in the input spectrum are nothing but the in-
tegration of the input current divided by a capacitive parameter
of the structure. With 0.1 s time step, the proposed method still
generates a stable result. But the result is not accurate because
the time step of 0.1 s is greater than that required by accuracy.
B. Demonstration of Unconditional Stability, Accuracy, and
Efficiency
1) Three-Dimensional On-Chip Interconnect: The first ex-
ample for demonstrating the unconditional stability as well as
accuracy of the proposed method was a 600- m-long test-chip
interconnect structure with 3 metal layers and 4 dielectric
layers provided by Intel Corporation. The current source was
again the derivative of a Gaussian pulse but with
and
s. The maximum input frequency of the source
was 12.8 GHz, at which the magnitude of the source was 0.1%
smaller than the maximum magnitude in the spectrum of the
input signal. Because of the fine feature size of the structure,
which was at 0.1 m-level, the time step allowed by the con-
ventional central-difference based explicit method was only
s, whereas the proposed explicit method was able to
use a time step of
s to generate accurate and stable
results. The parameters
and
used in (31) and (32) were
both chosen to be
. Three physically important modes
were detected from 14,800 time steps simulated in the prepro-
cessing. Their eigenvalues were
,
,
and
rad s respectively. These values
agreed very well with those obtained from (5) directly, with
the maximum error being 0.3%. To simulate for 0.5 ns in time,
the original central-difference scheme required 5 million steps
to complete the simulation, whereas the proposed method only
Fig. 2. Voltage waveforms of an on-chip interconnect.
Fig. 3. Simulation of a millimeter-scale waveguide with thin films. (a) Struc-
ture. (b). Voltage waveforms.
need 14,800 steps in preprocessing and the cost after prepro-
cessing is negligible. Thus, the speedup is 330. Fig. 2 shows
an excellent agreement between the proposed method and the
conventional central-difference based method.
2) Millimeter-Scale Waveguide With Thin Films: Next
example was a millimeter-scale waveguide with thin films as
shown in Fig. 3(a). The PEC boundary condition was applied
on the top plane, bottom plane, and the thin film. The first-order
absorbing boundary condition was applied at the two ends of the
waveguide. In order to accurately capture the geometry of the
thin film and slit, a fine space discretization as small as 0.03 mm
was used in the
plane. The waveguide was discretized to
seven layers along the 35-mm length and the film occupied
one layer. The structure was excited by a Gaussian’s derivative
current source with
s and
from bottom
plate to top plate at the near end of the waveguide. The voltages
were extracted between the two plates at the near and far ends.
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To simulate this example, the conventional explicit scheme re-
quires a time step of
s to maintain stability. Because
of this small time step, over ten thousand steps were needed
to finish the simulation. With the proposed explicit method,
we were able to use a time step of
s solely determined
by accuracy (because
) to generate accurate
and stable results within 1,000 steps. We collected the field
solution every 50 steps during the preprocessing. The solver
automatically simulated for 3,000 time steps in preprocessing
from which 60 vectors were sampled, among which 47 orthog-
onal vectors were constructed to form
. Using the space ,
the original large eigenvalue problem was transformed to a
small eigenvalue problem of size 47, from which 33 physically
important modes were identified with the first two eigenvalues
found to be
and
rad s, with error
being less than 1%. Since the conventional explicit method
needs
steps and the proposed method requires
only 3,000 steps in preprocessing, the speedup of the proposed
method is 6. In this millimeter-scale example, because the space
discretization is not significantly smaller than that required by
accuracy, the gap between the time step allowed by accuracy
and that permitted by the stability criterion is not that large.
Therefore, the speedup of the proposed scheme is not as signifi-
cant as that observed in previous example that involves a space
resolution much smaller than that required by accuracy due to
the existence of fine features relative to working wavelength.
In Fig. 3(b), we plot the voltages sampled at the near and far
ends of the waveguide simulated by the proposed method in
comparison with those generated by the conventional explicit
method as well as implicit Newmark-based unconditionally
stable scheme. Excellent agreement is observed.
C. Comparison With Unconditionally Stable Implicit Method
and Conditionally Stable Explicit Method
In addition to comparing the performance of the proposed
method with that of the conditionally stable explicit method, we
have also compared the performance of the proposed method
with the unconditionally stable implicit method. The example
considered was a 3-D on-chip bus with three parallel buses in
M2 layer, one metal layer on the top, and the other at the bottom,
as shown in Fig. 4(a). The width of each bus was 3 m as well
as the spacing between buses. The thickness of each dielectric
and metal layer was 0.3 m. There were 4 dielectric layers. The
dielectric constant in the two layers adjacent to M2 layer was 4,
and that for the other two dielectric layers was 8. The structure
was excited by a current source
with
and
s. We simulated a suite
of such 3-D bus structures, the discretization of which resulted
in 23,677; 45,427; 88,927; 175,927; 349,927; and 1,089,427
unknowns, respectively. The total time interval simulated was
s. We compared the CPU time of the proposed ex-
plicit method in comparison with the latest linear-complexity
conditionally stable explicit method reported in [15] for inte-
grated circuit simulation and the unconditionally stable implicit
Newmark method [10] that used a state-of-the-art multifrontal
based sparse matrix solver [26]. In Fig. 4(b), we plot the total
CPU time cost by the three methods versus . The advan-
tage of the proposed method can be clearly seen. The proposed
Fig. 4. Simulation of an on-chip bus. (a) Illustration of the structure and mate-
rial. (b) Total CPU time comparison between 3 methods.
Fig. 5. Comparison of two methods in late-time stability.
explicit method and the explicit method in [15] both employ
orthogonal prism vector bases and hence exhibit linear com-
plexity. However, the proposed method was able to use 3000
steps to finish the entire simulation with a time step of
s whereas the conditionally stable explicit method in [15]
required 300,000 steps to finish the simulation. As for the im-
plicit Newmark method, although it permitted the same large
time step as that used by the proposed method, it failed to fac-
torize the matrix for large problem sizes.
D. Examination of Late-Time Stability
For all the examples simulated in this paper, we also per-
formed simulations to very late time. No late-time instability
is observed from the proposed method. This is well understood
because at each time step, the field solution in the proposed
method is strictly obtained from a space that spans the stable
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5249
modes only. In contrast, we do observe late-time instability of
implicit methods in some of the examples we simulated. For
example, when simulating the millimeter-scale waveguide ex-
ample described in Section V-B-2), we observed late-time in-
stability from the Newmark-based implicit scheme, as shown in
Fig. 5, whereas the proposed method is stable.
VI. CONCLUSION
In this work, we propose an unconditionally stable explicit
time-domain finite-element method that removes the depen-
dence of the time step on the space step. It is stable for an
arbitrarily large time step. If the time step is chosen based
on accuracy, the proposed method is not only stable but also
accurate no matter how large the time step is. The method is
applicable to general 3-D problems with inhomogeneous mate-
rials and irregular structures. Different from previous methods
for achieving such a capability, in which the source that is
responsible for the instability still remains in the numerical
system, and hence one has to resort to implicit methods to
bound the error amplification factor by 1 to ensure stability,
the proposed method completely eliminates the root cause that
is responsible for the instability associated with an explicit
time-domain method. As a result, an explicit method can also
be made unconditionally stable, allowing the time step to be
solely determined by accuracy requirements.
The complete removal of the unstable modes for a given time
step is achieved by strictly expanding the field solution in the
space of stable modes, and also projecting the numerical system
onto the space of the stable modes. By doing so, the original
system matrix that spans both stable and unstable modes is
transformed to a system of stable modes only. The stable modes
are found in the preprocessing step by a time-domain solution
based fast eigenvalue solution of
, with
. This also
allows the explicit marching to be efficiently performed in a
reduced system of
.
In the proposed method, the strength of an explicit method
in avoiding computationally intensive matrix solutions is re-
tained, while its shortcoming of requiring a small time step is
overcome for problems having fine features relative to working
wavelength. Numerical experiments have demonstrated that the
proposed unconditionally stable explicit method outperforms
both the conditionally stable explicit method and the uncondi-
tionally stable implicit method in efficiency. The method is also
shown to be stable at late time, while late-time instability is ob-
served in unconditionally stable implicit methods. Although the
proposed method is presented in a time-domain finite-element
method, the essential idea can be applied to other time domain
methods.
APPENDIX
Here, we prove that the eigenvalues of the large system (5)
can be found from the reduced system (27) as long as the space
used to reduce (5) to (27) contains the information of the
eigenvectors corresponding to these eigenvalues.
Consider an eigenpair
of (5). It satisfies
(A-1)
If the space
contains the information of , the
can be
expanded in the space
as the following:
(A-2)
where is a coefficient vector. Substituting (A-2) into (A-1) and
testing both sides of (A-1) by
, we obtain
(A-3)
From (26), the above can be further written as
(A-4)
As a result, the eigenpair
is the solution of (27). There-
fore, the eigenvalues that satisfy (5) also satisfy (27) as long
as the
used to reduce (5) to (27) contains the information
of the eigenvectors corresponding to these eigenvalues. In ad-
dition, by front multiplying the eigenvector obtained from (27)
corresponding to
by , one can obtain the eigenvector of (5)
as can be seen from (A-2).
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Qing He received the B.S. degree in electronic and
information engineering from Zhejiang University,
Hangzhou, China, in 2006, and was an M.S. student
at the Graduate School of the Chinese Academy
of Sciences from 2006 to 2007. He is currently
pursuing the Ph.D. degree at the School of Electrical
and Computer Engineering, Purdue University, West
Lafayette, IN.
He was a Research Assistant with the Center
for Space Science and Applied Research, Chinese
Academy of Sciences, from 2006 to 2007. He is
currently with the On-Chip Electromagnetics Group, Purdue University. His
current research interests are computational electromagnetics, high-perfor-
mance VLSI CAD, and fast- and high-capacity numerical methods.
Houle Gan (S’08) received the B.S. and M.S.
degrees in information science and electronic engi-
neering from Zhejiang University, Hangzhou, China,
in 2003 and 2006, respectively, and the Ph.D. degree
in electrical engineering from Purdue University,
West Lafayette, IN, in 2010.
In 2006, he was a System Engineer with Re-
alsil Microelectronics, Inc., Suzhou, China. From
September 2006 to August 2010, he was a research
assistant with the On-Chip Electromagnetics Group,
Purdue University. In August 2010, he joined Intel
Corporation as a Senior Engineer. His current research interests include com-
putational electromagnetics for large-scale high-frequency integrated circuit
design, signal integrity, and power integrity.
Dr. Gan was one of the three recipients of the IEEE Antennas and Propagation
Society Ph.D. Research Award for 2008–2009.
Dan Jiao (S’00–M’02–SM’06) received the Ph.D.
degree in electrical engineering from the University
of Illinois at Urbana-Champaign, Urbana, in 2001.
She was with the Technology Computer-Aided
Design (CAD) Division, Intel Corporation, until
September 2005, as a Senior CAD Engineer, Staff
Engineer, and Senior Staff Engineer. In September
2005, she joined Purdue University, West Lafayette,
IN, as an Assistant Professor with the School of
Electrical and Computer Engineering, where she
is now a tenured Associate Professor. She has
authored two book chapters and over 170 papers in refereed journals and
international conferences. Her current research interests include computa-
tional electromagnetics, high-frequency digital, analog, mixed-signal, and RF
integrated circuit (IC) design and analysis, high-performance VLSI CAD,
modeling of microscale and nanoscale circuits, applied electromagnetics, fast
and high-capacity numerical methods, fast time-domain analysis, scattering
and antenna analysis, RF, microwave, and millimeter-wave circuits, wireless
communication, and bio-electromagnetics.
Dr. Jiao has served as the reviewer for many IEEE journals and conferences.
She is an Associate Editor for the IEEE TRANSACTIONS ON COMPONENTS,
PACKAGING, AND MANUFACTURING TECHNOLOGY. She was among the
85 engineers selected throughout the nation for the National Academy of
Engineering’s 2011 U.S. Frontiers of Engineering Symposium. She was the
2010 recipient of the Ruth and Joel Spira Outstanding Teaching Award, the
2008 National Science Foundation (NSF) CAREER Award, the 2006 Jack
and Cathie Kozik Faculty Start up Award (which recognizes an outstanding
new faculty member of the School of Electrical and Computer Engineering,
Purdue University), a 2006 Office of Naval Research (ONR) Award under
the Young Investigator Program, the 2004 Best Paper Award presented at
the Intel Corporation’s annual corporate-wide technology conference (De-
sign and Test Technology Conference) for her work on generic broadband
model of high-speed circuits, the 2003 Intel Corporation’s Logic Technology
Development (LTD) Divisional Achievement Award in recognition of her
work on the industry-leading BroadSpice modeling/simulation capability
for designing high-speed microprocessors, packages, and circuit boards, the
Intel Corporation’s Technology CAD Divisional Achievement Award for the
development of innovative full-wave solvers for high frequency IC design,
the 2002 Intel Corporation’s Components Research the Intel Hero Award
(Intel-wide she was the tenth recipient) for the timely and accurate 2-D and 3-D
full-wave simulations, the Intel Corporation’s LTD Team Quality Award for
her outstanding contribution to the development of the measurement capability
and simulation tools for high frequency on-chip crosstalk, and the 2000 Raj
Mittra Outstanding Research Award presented by the University of Illinois at
Urbana-Champaign.
