IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 1, JANUARY 2008
113
Mixed Finite-Element Time-Domain Method
for Transient Maxwell Equations in
Doubly Dispersive Media
Burkay Donderici and Fernando L. Teixeira, Senior Member, IEEE
Abstract—We describe a mixed finite-element time-domain
algorithm to solve transient Maxwell equations in inhomogeneous
and doubly dispersive linear media where both the permittivity
and permeability are functions of frequency. The mixed finite-element time-domain algorithm is based on the simultaneous use of
both electric and magnetic field as state variables with a mix of
edge (Whitney 1-form) and face (Whitney 2-form) elements for
discretization of the coupled first-order Maxwell curl equations.
The constitutive relations are decoupled from the curl equations
and cast in terms of (auxiliary) ordinary differential equations involving time derivatives. Permittivity and permeability dispersion
models considered here are quite general and recover Lorentz,
Debye, and Drude models as special cases. The present finite-element time-domain algorithm also incorporates the perfectly
matched layer absorbing boundary conditions in a natural way.
Index Terms—Dispersive media, finite element time domain.
I. INTRODUCTION
INITE-ELEMENT time-domain methods have been successfully employed over the years for the simulation of
transient Maxwell equations in complex geometries [1]–[4].
In contrast to the finite-difference time-domain method [5],
the finite-element time-domain method requires the solution
of a sparse linear system at every time step; however, when
used in conjunction with simplicial grids, the latter is free from
staircasing error. Traditionally, the finite-element time-domain
method is based on the solution of the second-order vector
wave equation for the electric (or magnetic) field after the elimination of the magnetic (or electric) field [3]. This facilitates
the expansion of the unknown field by a single type of basis
function. For the electric field, the basis functions of choice are
typically edge elements. The choice of edge elements ensures
conformity to a discrete version of the de Rham complex
(exact sequence property) [6], [7]. This implies that discrete
solutions that are not divergence free necessarily correspond
to static fields (gradient-like eigenmodes at zero frequency),
which do not pollute the frequency spectrum. Edge elements
can be viewed as the natural interpolants of 1-forms and, hence,
F
Manuscript received April 29, 2007; revised August 14, 2007. This paper was
supported in part by the Air Force Office of Scientific Research under Grant
FA 9550-04-1-0359, by the National Science Foundation under Grant ECCS0347502, and by the Ohio Supercomputer Center under Grant PAS-0061 and
Grant PAS-0110.
The authors are with the ElectroScience Laboratory, Department of Electrical
and Computer Engineering The Ohio State University, Columbus, OH 43212
USA (e-mail: donderici.1@osu.edu; teixeira@ece.osu.edu).
Digital Object Identifier 10.1109/TMTT.2007.912217
provide the correct discrete representation for the electric field
intensity, which in its most general mathematical incarnation is
represented as a 1-form in a differential manifold [6]–[18].
In the frequency-domain second-order vector wave equation,
the divergence-free nature of the (source free) solutions is
enforced in a frequency-dependent fashion (more precisely,
quadratically with frequency). Hence, the divergence-free
condition becomes progressively weak in the zero-frequency
limit, leading to ill-conditioned matrices, which negatively
impacts accuracy and convergence [19]. This problem also
plagues adaptive mesh refinement because the field approaches
the static limit locally for elements in highly refined portions
of the grid. In the time domain, a different, but related, type
of problem arises: the source-free second-order wave equation
.
admits secular solutions of the form
These are spurious (nonphysical) modes in the null space of
the curl operator and in the null space of the second-order time
derivative. These spurious modes require elimination using,
for example, a tree–cotree decomposition (gauging) [20],
[21], grad–div regularization terms, or a posteriori filtering
approaches [22]. Alternatively, the finite-element time-domain
method can be based directly upon the first-order coupled
Maxwell curl equations. In this case, both electric and magnetic
fields are employed as unknowns, and mixed finite elements
are used. A mixed finite-element time-domain method emand magnetic field flux
ploying the electric field intensity
as simultaneous state variables have been considered, e.g.,
[23]–[27] where edge elements (Whitney 1-forms) are used
and face elements (Whitney 2-forms) are used for .
for
This choice satisfies a discrete version of the de Rham diagram
as well. Other desirable characteristics of this approach are:
1) it is free of secular solutions with linear growth; 2) it produces energy-conserving (symplectic) algorithms [28] under
an appropriate choice for the time integration scheme; 3) it
provides a natural path for hybridization with the finite-difference time-domain method since the latter can be formulated
in terms of edge and face elements (now hexahedral) as well;
and 4) it is more easily extended to complex media (i.e., with
frequency dispersion and/or anisotropy). Although this mixed
– finite-element time-domain method utilizes two fields as
unknowns, its computer time and memory costs are comparable
to those of the second-order vector wave equation formulation.
This is because the former requires discretization of first-order
time derivatives, while the latter requires second-order derivatives. As a result, only one past time-step electric/magnetic field
0018-9480/$25.00 © 2007 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 1, JANUARY 2008
value is necessary during the time update, as opposed to two
past field values.1 In addition, the sparse linear system solution
in the mixed finite-element time-domain method is required
only for the electric field update, and not for the magnetic
field update. Consequently, the resulting linear system solution
has the same number of unknowns as in the finite-element
formulation based on the second-order wave equation.
The use of finite-element time-domain methods to simulate
linear dispersive dielectrics has been considered before in
[29]–[33]. These approaches were based on the second-order
wave equation and included dispersion only in the permittivity.
In this study, we take advantage of property 4) above to construct a mixed – finite-element time-domain method for
inhomogeneous and doubly dispersive media where both the
permittivity and permeability are functions of frequency. This
type of behavior is found, for example, in many metamaterials
[34], [35]. The dispersion models that we consider here are
very general and recover Debye, Lorentz, and Drude models
as special cases. The algorithm also incorporates the perfectly
matched layers [5] in a natural way.
II. MIXED
–
FINITE-ELEMENT TIME-DOMAIN METHOD
column indices respectively, the discrete Hodge matrices
(size
by ) and
(size
by
) in (2) are given by
the following integrals [24]:
(4)
(5)
where is the computation domain, and
and
are permittivity and permeability tensors at location , respectively.
The integrals in (4) and (5) are volume integrals in three dimensions and area integrals in two dimensions.
In nondispersive media and using leap-frog time discretization, the finite-element time-domain update equations are given
by [24]
(6)
(7)
The Hodge (mass) matrix
is sparse, but not diagonal, and
the solution of the associated linear system is the most computationally intensive part of the update.
A. 3-D Case
B. 2-D
We expand the electric field intensity in terms of Whitney
,
, and the magnetic field flux
edge elements
,
density in terms of Whitney face elements
as [24]
In the 2-D
case considered in the numerical examples
that follow, is still a 1-form and is still a 2-form, thus, all
the above formulas still apply. Note, however, that in two-dibecomes a -directed function,
mensions, the Whitney form
which is uniform over each face (2-D volume form).
case (not considered
On the other hand, in the 2-D
here), is a 0-form and, hence, should be expanded in terms
of Whitney 0-forms (nodal elements), while is a 1-form and,
hence, should be expanded in terms of Whitney 1-forms (edge
elements).
A complete classification table with the degrees of the differential forms representing each field for different dimensions
and polarizations is presented in [17]. Moreover, a discussion
on the appropriate expansions (based on such classification) for
the various dimensions/polarizations is provided in [25].
(1)
and
where and are the unknown of the problem, and
are the number of interior (or free) edges and faces, respectively.
The expansion above refers to the 3-D case, where is a 1-form
(edge elements) and
expanded in terms of Whitney 1-forms
is a 2-form expanded in terms of Whitney 2-forms
(face
elements), as expressed in (1).
By denoting arrays of unknowns (column vectors) as
and
, the semidiscrete
Maxwell equations in a source-free region can be written as
(2)
(3)
where
and
are (metric free) sparse incidence matrices on the primal (finite-element mesh) and dual grid, revalues [9],
spectively, whose elements assume only
holds, in general, up to
[18]. The identity
boundary terms. The column vectors
and
represent the electric field flux
density and the magnetic field intensity, respectively. The and
arrays are not associated with the finite-element mesh directly,
but with dual mesh instead [18]. If we denote and as row and
1These numbers refer to a first-order time discretization in dispersionless
media only, but a similar comparison can be made for high-order time discretizations and for dispersive media.
and
Cases
III. INHOMOGENEOUS AND DOUBLY DISPERSIVE MEDIA
In inhomogeneous media, the material tensors are assumed
and
,
uniform within each face, i.e.,
where is the face index. Material tensors could alternatively
have been associated with edges; however, since an implementation of (4) requires knowledge of the material parameters pointwise, an interpolation of the material parameters would have
been necessary. Moreover, an edge-based definition for the material tensors in inhomogeneous media would lead to nonsymmetric Hodge matrices.
problem in Cartesian coordinates
We consider a 2-D
with inhomogeneous and dispersive diagonal material tensors
given as
(8)
(9)
DONDERICI AND TEIXEIRA: MIXED FINITE-ELEMENT TIME-DOMAIN METHOD FOR TRANSIENT MAXWELL EQUATIONS
115
with
(10)
Fig. 1. Permittivity values associated with the two parent faces of a given edge
j . Note that the labeling order of faces is arbitrary.
and similarly for
and
. A basic requirement
on these tensors is that they should respect Kramers–Kronig
relations, a requirement from (primitive) causality [36]. Even
though the above tensors are specific for two dimensions, a 3-D
formulation can be derived following similar steps noted below.
with faces indices
and
. As a result, each element
the array can be written as
of
A. Electric Field Constitutive Equation Update
Here, we decompose in (2) in a form suitable for time disas the area of the th face, (4) can
cretization. If we denote
be rewritten as a sum of face contributions as
(11)
Substituting (8) into (11) and separating terms for different axis
components, we get
(16)
We can further write
(17)
where
(18)
(19)
(12)
where we have dropped the frequency dependence for simplicity. Since material parameters are assumed uniform inside
each element, (12) can be further simplified to
(13)
with
(14)
and similarly for
written as
. Substituting (13) into (2),
is
(15)
Although the summation above includes all faces, the innerand
are nonzero for two
product integrals
faces only, viz., the two faces that touch edge , denoted here as
and
, respectively (see Fig. 1). Consequently, the summation over faces in (15) reduces to a summation of two terms
and similarly for
,
,
,
,
, and
with
.
We note that the form of (16) resembles that of the finite difference time domain in dispersive and inhomogeneous media,
where the field variables multiply a dispersion function that depends on each grid point. However, four different functions per
edge, viz.,
,
,
, and
, are used here,
as opposed to a single one in the finite-difference time-domain
algorithm. This comes from the fact that each edge contacts two
faces with possibly independent dispersion characteristics and
each of these two faces may have independent dispersion characteristics along each of the two axis and in two dimensions.
In the time domain, (18) becomes an ordinary differential
equation and a time discretization can be performed to obtain
update equations. For example, if we denote
and
as
and
, respectively, and
the th-order time derivatives of
substitute (10) into the dispersion relation (18), we obtain
(20)
and apply
We discretize the above at discrete time steps
the following finite-difference approximation recursively:
(21)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 1, JANUARY 2008
and similarly for
can then be written as
. The update equation for
(22)
(23)
,
, and
are constants that dewhere
and
. These conpend on material parameters
stants are given in the Appendix for a dispersion model with
. The auxiliary variable
depends only on values
and
from previous time steps. Update equations for
of
,
, and
are obtained following similar steps.
and similarly
By denoting
, we can combine (19) and (22) in matrix form as
for
(24)
where
for
,
, and
and
,
and
. From (17), we have
similarly
(25)
and similarly for
.
for
The update for from (31) is derived in a similar fashion as
the update for . The main difference is that while the support
consists of two faces, the support of
of each edge element
consists of only face and, as such,
each face (cell) element
there is no need for the field splitting such as done in (17). Furis diagonal in the 2-D
case. From
thermore, matrix
(31) and after application of similar steps to those in (20)–(26),
one arrives at the following equation:
(32)
and
are analogous to
and
, respectively,
where
is a (diagonal) matrix
now for the permeability. The matrix
given by the product of two diagonal matrices
(33)
where
version is trivial (explicit
. Since
update).
is diagonal, its in-
C. Curl Equations and Complete Update
The update equations for and are both explicit and follow
directly from Maxwell curl (3) as
(34)
. Using (24) and (25), we finally arrive at
(26)
where
(35)
A complete time-step update consists of application of (35),
(32), (34), and (26), in this sequence.
IV. PERFECTLY MATCHED LAYER
(27)
The above constitutes the update equation for given . As
mentioned before, the solution of the sparse linear system assoconstitutes the most computationally intensive
ciated with
part of the update.
B. Magnetic Field Constitutive Equation Update
Following the same methodology of part
can be written as
trix
, the Hodge ma-
(28)
A rectangular (Cartesian) perfectly matched layer [5] implementation is used to truncate the computational domain and simulate an open-domain problem. The implementation described
here is based on the first-order Maxwell equations and it is a particular case of the double-dispersive anisotropic material modeling discussed above. We refer the reader to [38] for an implementation of the rectangular perfectly matched layer in finite-element time-domain simulations based on the second-order wave
equation.
Given an anisotropic and dispersive interior media with conand
stitutive tensors and , the associated tensors
to achieve reflectionless absorption are given by [39]
(36)
with
(37)
(29)
(38)
if faces
otherwise
(30)
Substituting (30) and (28) into (2), we obtain the following
equation for :
(31)
where , , and are complex stretching variables [37]. For
diagonal tensors , , the above simplifies to
(39)
(40)
DONDERICI AND TEIXEIRA: MIXED FINITE-ELEMENT TIME-DOMAIN METHOD FOR TRANSIENT MAXWELL EQUATIONS
117
with
, For the 2-D
cases considered
in the examples that follow, and using conventional stretching
, and similarly for ,
variables of the form
the perfectly matched layer tensor elements reduce to
(41)
(42)
(43)
If the background material parameters in (41)–(43), i.e.,
,
, and
, are modeled by second-order
polynomials in , the tensors
and
can be realized by
.
a fourth-order polynomial
V. RESULTS
We consider a 2-D
problem with a broadband soft
magnetic point source. The time-domain excitation given
by
for
and
otherwise (Blackman–Harris pulse derivative). This excitation
is used in all test cases unless stated otherwise. Here,
is the
free-space wavelength associated with the central frequency,
and is the speed of light. The time step is chosen according
to length of the shortest edge of the mesh (
) and given by
. The Courant number is chosen as
.
In all simulations below, a mesh generation algorithm with a
maximum area constraint
for all elements
is used, where
is the mesh resolution in terms of an edge
length. A sparse incomplete Cholesky factorization with a drop
tolerance value 10
is used for solving (26).
A. Validation: Cylindrical Scatterer
In order to verify the accuracy of the scheme, we simulate the
fields due to a point source in the vicinity of a frequency-dispersive circular cylinder, as depicted in Fig. 2. A computational
domain with size 0.56 0.56 m is used with average mesh resolution
m. The (free-space) central wavelength is
m and the dispersive cylinder has radius
m. The
field is sampled at the center of the cylinder (where it is more
sensitive to the dispersive properties) and the point source is located at 0.4 m away from the receiver. Perfectly matched layers
are used for mesh termination (as detailed above). Two scenarios
are considered, which are: 1) a cylinder filled with a dielectric
dispersive material and 2) a cylinder filled with a doubly dispersive material. The two-pole Debye model described as follows
is used for the dispersive permittivity and permeability (a static
conductivity term is also included as an additional parameter to
this model):
(44)
Fig. 2. Finite-element mesh for the dispersive cylindrical scatterer. The source
location is indicated by the and the probe location by the .
+
2
San Antonio clay loam parameters [40] are used for the dispersive permittivity model, given by
,
,
,
ns,
ns,
m S/m,
and
. For a doubly dispersive cylinder, we consider the
same permittivity together with a two-pole Debye permeability
with
,
,
,
ns, and
ns.
Finite-element time-domain results are compared against
finite-difference time-domain results and an analytical solution.
The analytical solution is obtained from the Green’s function for this problem, which can be expressed in terms of a
Hankel–Bessel series over the azimuth index [41, pp. 574–667]
with the first 50 terms included. The time-domain source
excitation is first converted to frequency domain by a Fourier
transformation. Zero padding with a length that is ten times
the total number of time steps is used in this operation to
ensure good enough frequency resolution. At each frequency,
the source spectrum is multiplied by the analytical solution
at that frequency. The result is inverse Fourier transformed
to yield the time-domain response. The total number of cells
of the uniform finite-difference time-domain grid is equal to
the number of faces in the finite-element mesh. Figs. 3(a) and
4(a) show the calculated magnetic field values as a function of
time at the probe position. Figs. 3(b) and 4(b) show the relative
errors against the analytical result, indicating that the proposed
method can accurately simulate materials with dispersive
and doubly dispersive characteristics. The finite-difference
time-domain method produces a higher residual error for this
geometry and mesh resolution mainly because of staircasing
approximations.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 1, JANUARY 2008
Fig. 3. Calculated field and normalized residual error produced by the mixed
finite-element time-domain and finite-difference time-domain methods for
the San Antonio clay loam cylinder model. The peak residual error of the
finite-element time-domain method is approximately 50 dB. (a) Field values.
(b) Residual error.
Fig. 4. Calculated field and normalized residual error produced by the mixed
finite-element time-domain and finite-difference time-domain methods for the
doubly dispersive cylinder model. The peak residual error of finite-element
time-domain method is approximately 54 dB. (a) Field values (b) Residual
error.
B. Perfectly Matched Layer Performance
in the perfectly matched layer case, the cavity case, and reference case, respectively. The residual reflection coefficient from
the perfectly matched layer is calculated by
0
In order to test the perfectly matched layer performance, a domain with size
and mesh resolution
is used.
The domain is extruded by
and
perfectly
matched layer cells, where a polynomial profile with exponent
is used for the conductivity [5]. A planarly layered
mesh with rectangular tiles made of two triangular cells each
is used in the perfectly matched layer. The edge length of each
tile is chosen approximately equal to
. The reference finite-element time-domain result for subtraction to calculate the
perfectly matched layer reflection coefficient is obtained with a
larger mesh constructed by extruding the same mesh by a large
number of free-space layers. The same exact rectangular tile geometry is used in the extrusion region to avoid any differences in
the numerical dispersion effects and isolate the residual reflections from the perfectly matched layer. For reflection coefficient
normalization purposes, a third mesh is constructed by terminating the original mesh by perfect electrical conducting walls.
In the numerical tests, the source is placed at the center of the domain at
and
. The receiver is placed at
and
. We denote
,
, and
as the magnetic field at grid cell (face)
at time-step
0
(45)
The reflection coefficient for different perfectly matched layer
thickness in terms of number of cells is plotted in Fig. 5. A
reflection level no larger than approximately 55 dB is obtained
for eight cells. Lower reflection levels are obtained by further
increasing the perfectly matched layer thickness.
C. Zero-Index Lens Example
Zero-index metamaterials are doubly dispersive media that
exhibit zero permittivity and zero permeability at a specified frequency. Interestingly, the electromagnetic fields inside such materials (at the critical frequency) have a spatial distribution akin
to that of a static field (i.e., infinitely long wavelength) while remaining dynamic (oscillatory) in time [34]. Here, we consider
a lens object made of a homogeneous zero-index metamaterial
in free space, as shown in Fig. 6. The zero-index material is
DONDERICI AND TEIXEIRA: MIXED FINITE-ELEMENT TIME-DOMAIN METHOD FOR TRANSIENT MAXWELL EQUATIONS
119
and sine frequency . The source is located at the center of the
top circular disk associated with the lens geometry.
Fig. 6 shows a snapshot of magnetic field values taken at
s, where it is seen that the cylindrical wave
originated from the source above the lens is focused below the
lens (wavefront reshaping property). The uniform field values
inside the lens indicate the static-like spatial distribution of the
field inside the zero-index lens.
VI. SUMMARY AND CONCLUSION
Fig. 5. Perfectly matched layer numerical reflection coefficient for various
number of layer cells. A reflection level no larger than 55 dB is obtained for
eight layers for the source excitation considered.
0
We have described an – mixed finite-element time-domain method to simulate Maxwell equations in inhomogeneous
and doubly dispersive linear media. The proposed method provides a more straightforward approach to incorporate complex
frequency dispersive characteristics simultaneously in the permittivity and in the permeability because it factors out the update of Maxwell curl equations (involving the spatial derivatives) from the update of the constitutive equations.
The finite-element time-domain update equations have a form
somewhat similar to that of the finite-difference time-domain
update equations in (doubly) dispersive media, except for the
sparse linear solve required in the electric field update equation. Perfectly matched layers can also be included in a simple
fashion.
APPENDIX
,
, and
in (22) and
The constants
and
below for a
(23) are given in terms of
. The face index is the same for all
dispersion model with
equations below and is omitted for simplicity. Expressions for
,
, and
can be obtained by replacing
by as follows:
(47)
(48)
(49)
2
(50)
Fig. 6. Magnetic field at t = 8:8151
10
s for the zero-index lens
problem. The lens is made of a homogeneous doubly dispersive material. The
field originating from a point source is refocused below the lens, demonstrating
a wavefront reshaping property of this lens.
(51)
(52)
(53)
realized by the following (isotropic) doubly dispersive Drude
model [34]:
(54)
(55)
(46)
The zero-index frequency is given by
with
. The lens has a double-concave semicircular geometry with 12-cm width and 7.4-cm height. The radii
of curvature of the semicircular regions are 12.2 and 21.8 cm. A
18 cm 23.5 cm domain is used for the finite-element time-domain simulation with average mesh resolution
cm. A
ramped-sine source [42] is used with half-period ramp length
(56)
(57)
where
and
.
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Burkay Donderici was born in Ankara, Turkey, in 1980. He received the B.S.
degree from Bilkent University, Bilkent, Turkey, in 2002, the M.S. degree from
The Ohio State University (OSU), Columbus, in 2004, and is currently working
toward the Ph.D. degree at The Ohio State University.
Since 2002, he has been a Graduate Research Associate with the ElectroScience Laboratory, OSU. During Summer 2007, he was with SciberQuest Inc.,
where he was involved with mesh stitching (subgridding) algorithms for electromagnetic particle-in-cell simulations. His research interests include computational electrodynamics and inverse scattering.
Fernando L. Teixeira (S’89–M’93–SM’04) received the B.S. and M.S. degrees
from the Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil,
in 1991 and 1995, respectively, and the Ph.D. degree from the University of
Illinois at Urbana-Champaign, in 1999, all in electrical engineering.
From 1999 to 2000, he was a Post-Doctoral Research Associate with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT).
Since 2000, he has been with the Department of Electrical and Computer Engineering and the ElectroScience Laboratory, The Ohio State University (OSU),
Columbus, where he is currently an Associate Professor. He edited one book and
has authored over 70 journals papers and book chapters. His current research
interests include modeling of wave propagation, scattering, and transport phenomena for communications, sensing, and device applications.
Dr. Teixeira is a member of Commissions B and F of URSI. He was the recipient of numerous awards including the 2004 National Science Foundation
(NSF) CAREER Award and the triennial Henry Booker Fellowship presented
by the USNC/URSI in 2005.