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Study of Scattering using a 2-D Total Field/Scattered Field Perfectly
Matched Layer
Research · May 2015
DOI: 10.13140/RG.2.1.1970.5762
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EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
Study of Scattering using a 2-D Total Field/Scattered
Field Perfectly Matched Layer Ananth Saran Yalamarthy, ME10B148
Abstract—This report starts with the FDTD Method in
Cylindrical Co-ordinates. We discuss the termination of the
Cylindrical Grid by means of reflective, Bayliss-Turkel ABC, and
a split field Perfectly Matched Layer Technique. The PML is
validated by comparing the solution with a theoretical
Monochromatic Cylindrical wave. This is followed by a
description of a Total field/Scattered Field Perfectly Matched
Layer implemented in Cartesian coordinates.
Index Terms—FDTD, PML, Cylindrical,Total/Scattered Field
I. INTRODUCTION
The field of Computational Electromagnetics hinges around
two popular methods: Frequency domain methods and Time
domain methods. FDTD belongs to the latter class, and is
easily the most popular method in recent times, due to its ease
of implementation and inherent simplicity. The basic FDTD
algorithm is the Yee algorithm [1], which was described for
Cartesian grids. A Cartesian grid, however, may not be the
best co-ordinate system to describe a physical problem. A
good example is the scattering of electromagnetic waves from
a cylindrical object. The basic Yee algorithm can be easily
extended to a cylindrical co-ordinate system; however, this
extension produces several difficulties that are not seen in
standard Cartesian grids. Nevertheless, the implementation of
the FDTD algorithm in a cylindrical grid offers some distinct
advantages with respect to termination, because we only need
to terminate along the radial axis, as opposed to multiple axes
on the Cartesian system. In this report, we develop, from the
basic Maxwell Equations, a thorough implementation of the
FDTD method in cylindrical co-ordinates, and follow it up
with a discussion on using a PML to study scattering from
metallic objects.
This report is organized as follows:
Section II contains the theoretical description of the FDTD implementation in Cylindrical Coordinates
Section III discusses the computational grid, the discretization scheme and the boundary conditions.
Section IV discusses simulation results from various boundary
Section V discusses a verification mechanism to test the ‘goodness’ of the PML
Section VI discusses the formulation of the PML TF/SF approach
Section VII describes a the results from a scattering experiment
Section VIII concludes the work and hints at possible future extensions
II. THEORETICAL DESCRIPTION
We discuss the theoretical implementation of the FDTD
algorithm in cylindrical coordinates for the configuration.
The vacuum Maxwell equations to be solved are:
)
)
These equations can be discretized by means of the staggered
leapfrog method presented in III. The mere implementation of
these equations for solving a problem is not enough; we need
to find some way to terminate the grid without spurious wave
reflection from the boundary. The simplest kind of boundary
condition is the PEC boundary condition, which simply sets
the field at the outermost edge (in this case, the outermost
ring) to zero. This boundary condition is totally reflective, and
hence not useful. The problem of boundary conditions was
addressed by Mur[2], Engquist and Majda[3], Bayliss and
Turkel[4], and most lately by Berenger[5]. The approaches
presented by [2], [3] and [4] try to terminate the grid by
fooling the field at the periphery into thinking that it is
actually propagating in an infinite dimension. The Balyiss-
Turkel condition constructs a series of annihilation operators
[6] to prevent reflection to increasing orders of accuracy. The
condition can be expressed as:
(
)
The best approach, however to terminating the grid is to
construct a Perfectly Matched Layer. The PML layer allows
the incident radiation to pass through into an attenuating
medium without reflection. The wave then attenuates in the
PML and reflects off the edge, and further attenuates. But the
time it returns to the simulation space, its magnitude is so
small that the system can be considered to be reflectionless, in
theory. We can formulate the PML equations using either the
stretched co-ordinate approach or the split-field approach. We
discuss the latter here. Unlike the Cartesian formulation which
requires termination in both and directions, we only need
to terminate the grid in the radial direction. The electric field
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
is split into two components and . Thus we only need a
and a to terminate the grid. Note that applies to
the tangential magnetic field , which propagates in the
radial direction. The split-field PML equations are:
)
The implementation of these equations is discussed in later
sections.
III. COMPUTATIONAL GRID
The basic computational grid for FDTD implementation in
cylindrical coordinates is depicted in Figure 1.
Fig. 1. Grid scheme for cylindrical FDTD[6]
The entire grid is divided into circles and azimuthal
sections creating trapezoidal simulation cells. The
electric field is at the center of the trapezoidal cells,
surrounded by and . Note the lone electric field at the
center of the grid, unlike a Cartesian scheme, which has to be
handled specially.
The FDTD implementation hinges on two crucial factors:
stability and dispersion, which is undesirable. Dispersion
becomes significant the moment the cell size becomes
comparable to the wavelength. The usual procedure that is
followed is to ensure that the maximum cell size is , and
this gives us a way of estimating the maximum cell size. The
stability criterion or the courant condition can be written as:
√ ) )
The critical condition occurs for the cell of the smallest size,
which in our case is the triangular cell with at the center.
This allows us to fix the time step required for simulation.
Because of the peculiar nature of the cylindrical grid with
varying cell sizes, the time step often turns out to be very
small, leading to a high computational cost. A common
solution is to use a variable time-stepping scheme, by having a
small time step till radius and using a larger time
step from , where marks the end of the
computational regime. The procedure in this case is to update
the fields in the domain first, perform ‘n’ updates in the
domain so that the fields as the interface match. The
process can then be repeated to solve the fields over the
computational domain.
Notice that the equation to update ( ) for the first
ring requires the electric field at the center (r=0) and
(r=1.5 ). These electric fields are at unequal distance
from , thus we do an inverse weighting of the electric fields
based on the distance to update for the inner loop.
We now list the update equations for the Perfectly Matched
Layer; which, upon setting 0, produce the standard
vacuum update equations. Let the mesh be characterized by
radial increment and the angular increment . Any spatial
location in the mesh can be specified by the set ),
and the time step is specified as . The fields used are
and . The FDTD equations are:
)
(
)
( )
(
)
(
)
)
)
)
(
)
)
(
)
(
)
(
)
(
)
( )
(
)
(
)
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
(
)
)
)
(
)
( )
)
)
(
)
(
)
We must use and corresponding to the grid location,
although this is not obvious from the above set of equations.
Discussion of the results from this implementation is in IV.
We had earlier discusses the Bayliss-Turkel termination
conditions in differential form. We present the discretized
version of this equation, applied for :
)
[ ) )
]
[ ) )
]
[ ) )
]
The implementation of the Bayliss-Turkel condition is
discussed in III.
In the Cartesian formulation, the discretized dispersion
relation reduces to , for small increments. This is
not true of the cylindrical formulation; in fact, substituting the
plane wave solution into the discretized vacuum Maxwell
cylindrical co-ordinate equations, we can formulate the
following dispersion relation for small increments:
[
]
Thus, even for small increments, we do not reach the ideal
dispersion relation. Also note that as the cell curvature ( )
tends to infinity/high frequencies, we get back to the standard
dispersion relation. Of course, using a wave with a high
frequency can be very costly given the condition that the
maximum cell size must not exceed 0.1 .
IV. SIMULATION RESULTS
We use to the above formalism to create a simulation
experiment on python. The source is a sinusoidal excitation of
the node at the center of the grid, producing waves of
wavelength 1.55 microns. The simulation space is divided into
50 circles and 160 azimuthal sections. The radial increment
= 7.55 m. The azimuthal increment is
The time step used
, which can be shown
to satisfy the limiting condition previously described. The
results from the implementation of a Bayliss Turkel condition
are shown in Figure 2, below. Note the significant amount of
reflection observed from the boundary.
Fig. 2. Resutls from the Bayliss- Turkel boundary condition
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
We now proceed to implement a Perfectly Matched Layer
(PML), based on the previous description. We choose a PML
thickness of 20 radial layers, with a conductivity profile
described by:
The results from the PML formulation are depicted in Figure
3, which shows the contour plot of the simulation region for
various instants of time.
Fig. 3. Resutls from the Perfectly-Matched Layer boundary condition
V. VERIFICATION OF SIMULATION RESULTS
In order to test the simulated solutions, we need to compare
the solution at a particular spatial location with an analytical
solution for a time period greater than the time taken for the
wave to hit the boundary of the simulation region. The
analytical solution for our initial source excitation condition is
a monochromatic cylindrical wave, which can be written as:
√ )
The √ dependence can be understood from the Poynting
Power theorem, which states that the total power at any radial
cross section is the same. The power through a particular cross
section can be written as: Since this is constant for
every cross section, we can write:
√
In order to test the PML, we plot the electric field at along the
radius for , at a large time instant, well after the wave
reaches the edge of the simulation region, and compare it with
the
√ envelope. This is shown in Figure 4, below. Not that
the envelope does not tally with the solution after radial
distance 30, for the simple reason that the PML damps the
solution beyond =30. Thus, the efficacy of the PML is
demonstrated.
Fig.4. Verification of PML with analytical envelope
In order to further test the PML, we plot the expected
analytical and simulated field, as shown in Figure 5. Notice
again, the agreement between the two fields in the region
before the PML. Figure 5 also shows similar performance
graphs for the Bayliss-Turkel and Perfectly Reflecting
boundary conditions, both recorded at simulation time t=700.
Note the improvement in performance over these three
termination schemes.
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
Fig.5. Comparison of simulated and analytical electric field for the PML,
Bayliss Turkel and Reflective Boundary (PEC) conditions
We also plot the electric field at a particular space point
location: , and see the evolution of the electric
field with time. This is depicted in Figure 8 below. The PML’s
goodness is easily observed by noting that the electric field
amplitude in the domain of interest does not change over time,
well after the wave hits the boundary and enters the PML
region.
Fig.6. The electric field at a particular point in space
Now that we have evaluated the performance of the Perfectly
Matched Layer, we proceed to study scattering from objects
using the PML formulation.
VI. A TF/SF PML FOR SCATTERING STUDIES
The study of scattering of electromagnetic waves from
physical objects is a problem of considerable practical interest.
An interesting example of where such studies find application
is in the development of combat aircraft. Combat aircraft are
often designed to efficiently scatter incoming electromagnetic
waves, so that an enemy radar cannot trace its location. In
order to study scattering from an object, we need to satisfy
certain requirement, listed below:
A mechanism to create/inject electromagnetic waves
into the simulation space
A method to measure the scattered wave,
independent of the incident wave
A scheme by which the scattered wave and the
incident wave are not reflected back into the
simulation space, so that we mimic an infinite
boundary
The first two requirements are satisfied by the TF/SF
approach, whereas the last approach stems from the PML we
have discussed in earlier sections. Scattering studies are often
done by using a plane wave as the incident source. The
formulation of a plane wave as the incident source is easier in
Cartesian co-ordinates, and is adopted as the co-ordinate
system for this section. The basic experimental setup to study
scattering is depicted in Figure 7. We use a Cartesian grid of
size 200 200, with 20 grids on all edges reserved for the
PML region. The dotted white line represents the TF/SF
boundary, with the region inside it representing the total field
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
and the region outside it representing the scattered field The
total field is thus defined as the sum of the scattered and the
incident fields, and we thus have (where can represents the
electric and magnetic fields).
Fig.7. A TF/SF PML setup
Notice that the electric field is split into and , in
accordance with the standard split-field PML approach. The
update equations for this PML formulation are similar to the
ones for cylindrical co-ordinates presented earlier, except that
we now have two conductivities and to attenuate the
fields in the x and y directions. The update equations can be
written as ( is the magnetic conductivity,=
):
[
]
[
]
[
]
[
]
)
[
]
[
])
[
]
[
])
Notice that in order to update the total electric fields and
at the boundary, we only have information on the
scattered fields and , making the FDTD update
equations inconsistent. The fix is to add an incident field to
these scattered magnetic fields. This incident field needs to be
specified by the user for every time instant during the
simulation and this will help us define the plane wave that is
incident on the scattering object later on. The update equations
for the Front, Back, Left and Right faces for the electric field
are as follows[7] (Coefficients are listed in the footer):
Front face:
[
]
Back face:
[
]
Left face:
[
]
Right face:
[
]
Similarly, the and fields just outside the total field
region update using the total electric field. Since only a
scattered field is required, an incident field is subtracted
from these electric fields. The definition of these incident
fields must be consistent with the definition of the incident
magnetic fields defined above. The complete incident and
electric field set that is specified by the user at all points of
time helps us define the incident plane wave. The update
equations for scattered magnetic fields at the Front, Back,
Left and Right faces are as follows:
Front face:
[
]
Back face:
[
]
Left face:
[
]
𝐶𝑏𝑦𝑃𝑀𝐿
𝑡
𝑦 𝜎𝑦 𝑡
𝐶𝑏𝑥
𝑃𝑀𝐿 𝑡
𝑥 𝜎𝑥 𝑡
3 𝐷𝑏𝑦
𝑃𝑀𝐿 𝑡
𝜇 𝑦 𝜎𝑦
𝑡
𝜇 𝐷𝑏𝑥
𝑃𝑀𝐿 𝑡
𝜇 𝑥 𝜎𝑥
𝑡
𝜇
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
Right face:
[
]
Now that we have a complete set of consistent equations, the
next task is to define the incident fields so that a plane wave
can be generated.
The method to generate the plane wave described here is
based on the Incident Field Array (IFA) technique, described
by Taflove[8]. The basic premise of this technique is to
generate a plane wave by interpolating the incident field on
the 2-D FDTD TF/SF boundary using a 1-D FDTD grid.
Fig.8. Creating a plane wave using a 1-D FDTD grid[9]
In Figure 8, incident fields need to be defined in the vicinity of
the black dots on the TF/SF boundary, such that a plane wave
travelling at -45 with the X axis is produced. The line marked
IFA represents the 1-D FDTD grid, whose one end is excited
manually (defined as a hard source). The other end is often
terminated by using either a 1-D PML or MUR absorbing
boundary condition. Thus, the hard source is the only point
where the user defines a field for every time instant during the
simulation. The 1-D update equations along the IFA
automatically update the E and H fields for every time instant.
In order to find the incident electric or magnetic field at any
point in the vicinity of the TF/SF boundary, the procedure
followed is:
First, define a reference point for specifying the
position vector of each of the TF/SF boundary points.
Calculate the projection of each of the TF/SF
boundary points on the IFA line.
Find the two closest field nodes (electric field nodes,
if we are trying to excite an electric field, and vice-
versa).The incident field at the desired point to be the
average of the fields at the closest two nodes.
Repeat this procedure for all incident fields at every
time step. and can easily be obtained by
using the equations, ) and
), where is the angle of the incident
plane wave.
Using these incident fields in the update equations for the TF-
SF boundary, we can easily generate the required plane wave
at any angle of incidence for scattering studies.
VII. A SCATTERING EXPERIMENT
We study the scattering of plane waves from a 2-D square
rectangular box. The basic dimensions of the experiment of
the grid are described in Figure 7. The outer grid is size 200
200, with 20 grids on all sides reserved for a PML. The TF/SF
is a square box of side 140 placed in the center of the grid, and
its corner points have the co-ordinates (29, 29) and (169,169).
A sinusoidal hard source with a wavelength 1.55 microns is
used to excite the 1-D FDTD grid, which is terminated using a
1-D perfectly matched layer, whose conductivity profile is
exponential. The field along the 1-D array is shown in Figure
9. Note that the conductivity along the 1-D grid is zero inside
the TF region. This is because we do not want the plane wave
to attenuate in the region of the scattering experiment, i.e. the
TF region. The incident field is at an angle of 45 for this
experiment. A plot of the plane wave generated using this
procedure is shown in Figure 8. Note how the PML sucks
away all the scattered fields travelling towards the edges.
Fig.9. Field from 1-D FDTD grid. Note the effect of the PML at the right end
EE5472- Computational Electromagnetics, Final Term Paper, 4th December 2013
Fig.9. Plane wave at 45 created inside the TF region
In the absence of a scatterer, the total scattered field is ideally
zero. This can be seen in Figure 9 by noting the intensity of
the field in the SF region. Note that the field is not perfectly
zero because of numerical errors. We now create a metallic
square object of side 20, placed at the center of the grid, and
study the total field pattern after the plane wave strikes it. We
use a PEC boundary condition on the edges of the square
object. Since the edges of the square object are sharp, there is
a profound magnification of the field due to charge
accumulation in these regions. The results from this
experiment are shown in Figure 10.
VIII. CONCULSIONS/FUTURE WORK
In this report, we discussed the formulation of a PML in
cylindrical co-ordinates, although the scattering experiment
was done in Cartesian coordinates. This is because the
generation of a plane wave is much more difficult in the
cylindrical system, because we have to resolve into
and , which is complicated and time consuming because
these directions are not fixed. A formulation in the cylindrical
system might be useful to study waves of a different kind
being scattered from an object. The PML TF/SF approach can
also be used to study the scattering of plane waves from
infinite objects, like an infinite triangular wedge, for example.
In this, case we need embed some of the TF/SF boundaries
into the PML region, and calculate the incident fields after
taking into account the loss in field due the PML region. This
can serve a good continuation of this work.
Fig.10. Plane wave at 45 interacting with a sqare scatterer embedded at the center of the TF region.
IX. REFERENCES
[1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, no. 3, pp. 302-307, May 1966.
[2] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat., vol. EMC-23, pp. 377-382, Nov. 1981
[3] B. Engquist and A. Majda, “Absorbing boundary conditions for numerical computation of waves,” Math. Comput., vol. 31, pp. 639-651, July 1977.
[4] A. Bayliss and E. ’hrkel, “Radiation boundary conditions for wavelike equations,” Commun. Pure Appl. Math., vol. 33, pp. 707-725, 1980.
[5] JP Berenger "A perfectly matched layer for the absorption of electromagnetic waves" J Computational Physics, 1994.
[6] Fusco, M., "FDTD algorithm in curvilinear coordinates [EM scattering]," Antennas and Propagation, IEEE Transactions on , vol.38, no.1, pp.76,89, Jan 1990
[7] Anantha, V.; Taflove, Allen, "Efficient modeling of infinite scatterers using a generalized total-field/scattered-field FDTD boundary partially embedded within PML," Antennas and Propagation, IEEE Transactions on , vol.50, no.10, pp.1337,1349, Oct 2002
[8] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1995, pp. 107–144.
[9] Oğuz, Uğur,Interpolation techniques to improve the accuracy of the plane wave excitations in the finite difference time domain method.’97
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