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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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