Using FDTD to solve problem of scattering electromagnetic waves from flat surface with long groove.

1. Finite Difference Time Domain - Finite Element
Method Hybrid
Kimberly Kaplan, Megan Padula and Dr. John Fleming
Chatham University
June 07, 2010

2. Contents
1 Introduction 2
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The Two Method Hybrid 3
2.1 Finite Difference Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 The Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Solving with MATLab 4
3.1 MATLab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Problems We Encountered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2.2 Updating The Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Solving the MATLAB Problem 7
4.1 Mur’s Absorbing Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Combining Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 Results and Closing Thoughts 8
5.1 The Final Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2 Further Research, Work and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.3 Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
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3. (a) 3-dimensional rendering (b) 2-dimensional rendering
Figure 1: A flat surface with a long, narrow groove, virtually constant in the z-direction.
1 Introduction
We were assigned the task of creating a program that would model the scattering of electromagnetic
waves from a flat, metalic surface containing a single, long groove, see Figure 1a. If z-direction is much
larger in scale relative to the width of the cross section of the groove, then we can make the assumption
that our task is a two dimensional problem which we have thus dubbed the “two-dimensional cavity
problem”, see Figure 1b.
It is useful to study a problem of this nature due to the fact that scratches on large metalic
objects can be modelled by this type of geometry. By examining this problem, we can achieve a better
understanding of how radar signature is affected by a slight crack in a plane or boat, for example.
Surprisingly, even features which are small relative to an object’s overall size can significantly affect
the radar cross section.
1.1 Background
All electromagnetic waves obey Maxwell’s Equations∗ :
µ ∂H
×E=− (1)
c ∂t
ε ∂E
×H= (2)
c ∂t
·E=0 (3)
·H=0 (4)
∗ This version of Maxwell’s Equations assumes no current or charge, so it excludes the factor J, current density.
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4. Where:
H = Hx , Hy , Hz (5)
E = Ex , Ey , Ez (6)
†
With H representing the magnetizing field and E representing the electric field. Due to the assumption
that the geometry we are working with can be treated as a two dimensional probelem, we can also
assume transverse magnetic polarization, meaning no electric field in the direction of the propagating
wave. Therefore, we will only concern ourselves with the terms Hx , Hy , the magnetizing fields in
the x and y-direction respectively, and Ez , the electric field in the z-direction. Here µ represents the
permeability of the medium through which the wave is travelling, and ε represents the permitivity of
the medium. For simplicity, we will use the constants µ0 and ε0 , correlating to the medium free space,
for the remainder of this paper.
1.2 Approach
In order to solve the two-dimensional cavity problem, we will divide the geometry into a region outside
the cavity, Figure 2a, and a region inside the cavity, Figure 2b. The Finite Difference Time Domain
method will be applied to the outer region, while the Finite Element Method will be applied to the
inner region. We have combined the two approaches to form a hybrid solution method. Further details
will be presented below.
1.3 Outline
In Section 2 we will discuss the two different methods our group used to complete this task, Finite
Difference Time Domain and Finite Element Method, along with the hybrid solution we created. In
Section 3 we will describe the tool, MATLAB, which we used to write our program, and the problems
we encountered while trying to write our code. Section 4 will explain the methods we implemented to
solve the MATLAB problems. The last portion of our paper, Section 5, will display our final results
along with the possible present and future applications.
2 The Two Method Hybrid
2.1 Finite Difference Time Domain
The basic idea behind the Finite Difference Time Domain method, or FDTD, involves replacing the
partial derivatives in Maxwell’s Equations with finite difference approximations. We use the following
equations in order to compute the discrete values of Hx , Hy , and Ez with respect to position and
time:
t
H(n+1) = Hn −
x x Ez (7)
ε0 c y
t
H(n+1) = Hn +
y y Ez (8)
ε0 c x
(n+1)
t Hy Hx
E(n+1) = En +
z z + (9)
µ0 c x x y
† The magnetic field, B, can be represented by µH.
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5. FDTD is most effectively applied to pulses or waves in a time domain, specifically those which
are scattered from rectangular geometries; however, the size of the wavelength must be comparable
to the size of the object that is scattering the wave. In order to calculate the updated magnetizing
and electric fields of a given region, one must determine the appropriate cell size, time step, and field
parameters of a that region.
Some advantages of FDTD include an ability to compute either 1, 2, or 3 dimensional problems, and
a flexibility to work with problems of various frequencies, sources, and geometries. The applications
of FDTD range from biomedical imaging, such as MRIs, to radar and antenna systems in military
facilities.
2.2 Finite Element Method
Finite Element Method‡ , or FEM, provides an integral based approach for solving differential equations.
The main benefit of this alternate approach is the capability of solving non-rectangular geometries in
a natural manner. FEM has been used to create visualizations of the bends, twists, stresses and
displacements in a given structure.
2.3 The Hybrid
FEM requires us to construct and solve a matrix equation, and because the size of the matrix is
determined by the number of points in our grid, it is better to keep this matrix small otheriwse it
will be time consuming to build and solve. Since FDTD does not require a matrix and the solution is
simply updated point by point, we will use FDTD in the large outer region to order to keep the FEM
portion contained to the smaller cavity region, saving us time and computational resources.
Our group divided into two subgroups: one subgroup modelled the wave scattering in both regions
with FDTD, treating the entire shape as rectangular, while the other subgroup modelled the wave
scattering from just the inner region with FEM, treating the shape as arbitrary. This paper focuses
on the work of the former two subgroups. We then collaborated to combine the two methods into one
working hybrid model.
3 Solving with MATLab
3.1 MATLab
MATLAB, or Matrix Laboratory, is a programming language which contains visualization tools to
illustrate the results, and it is relatively intuitive, making it easy for us to learn in a short period of
time, thus the appropriate choice for our task.
3.1.1 The Set-Up
In order to set up our two-dimensional cavity problem in MATLAB to work with FDTD, we first set
the boundary conditions for a large rectangular region to represent the flat surface and the surrounding
empty space, labeled here as Region 1, see Figure 2a. For simplicity, we constructed a square with
arbitrary numerical dimensions of 1000 by 1000, and chose the value 2 for our cell size. We then
added a second, smaller rectangular region to represent the groove, or in this two-dimensional case,
to represent a cavity, labeled here as Region 2, see Figure 2b. Again, for simplicity we constructed a
square with arbitrary numerical dimensions of 50 by 50 and the value 1 for our cell size. The main
point being to emphasize that Region 1 is several orders of magnitudes larger than Region 2. The
combined regions bare a two-dimensional resemblance to the surface we desired.
‡ Further details can be found in a separate report filed by our partners at Duquesne University, Kimberly Daley and
Lindsey Chesmar, whose focus was on the FEM portion of our group’s overall task.
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6. (a) surface and surrounding empty space (b) small cavity
(c) combined boundaries
Figure 2: A flat surface with a long, narrow groove, virtually constant in the z-direction.
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7. We chose to place a test source field, again something arbitrary such as a cosine function, to begin
its propagation from point P1 in the center of Region 1, and continue to pulsate out in circular form
so that the wave will simultaneously contact all four of Region 1’s walls; see Figure 2c. But this simple
model will need some modifications to truly represent the two-dimensional cavity problem previously
described.
3.2 Problems We Encountered
3.2.1 Boundary Conditions
Region 1 is meant to not only represent the flat surface, but also the infinitely surrounding empty
space; however, the space in our diagram is obviously bounded, 1000 square units to be precise. The
initial run of our preliminary MATLAB code revealed that the program treated all the boundaries as
solid walls. This result was not surprising, but it did prompt us to search for a simple solution to
create special boundary conditions that would imitate transparent boundaries on three of the edges,
displayed in purple on Figure 3.
3.2.2 Updating The Wave
Electromagnetic waves propagate because a changing electric field induces a changing magnetic field,
which induces a slightly different changing electric field, which in turn induces a slightly different
magnetic field, and so on. The information needed to update the constantly changing electric and
magnetic fields of our arbitrary electromagnetic wave is not the same within the two regions, requiring
us to update the electromagnetic wave separately for the two separate regions. However, the two
separate Regions 1 and 2 share information in the space where they are connected, which will require
slightly different methods from Equations 7, 8 and 9 in order to update. The shared space is depicted
in green on Figure 3.
Figure 3: Issues with initial MATLAB code highlighted in purple and green.
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8. 4 Solving the MATLAB Problem
4.1 Mur’s Absorbing Boundary Condition
The boundaries representing the flat, metalic surface and long, thin groove can be treated as perfect
electrical conductors, for which we use the simple formula:
Ez = 0 (10)
For the boundaries representing empty space, however, we need to use Mur’s absorbing boundary
conditions§ :
δEz 1 δEz
=− (11)
δt c δR
With R representing x, -x, or y, correlating to the right, left, and top boundaries of Figure 3 respectively.
4.2 Combining Regions
Recalling Equations 7, 8 and 9, one can see that the initial condition values can be plotted in points
on a grid to represent Hx , Hy and Ez , see Figure 4. However, because we are attempting to create
a hybrid method with both FDTD and FEM, we must evaluate Regions 1 and 2 separately, and thus
include a second set of these variables, Hx2 , Hy 2 and Ez 2 . The main trick in updating the shared
space, marked in red on Figure 4, is by simply remembering that in order to update this shared border,
we must use one variable from each region.
Figure 4: A small scale rendering of the grid model used with FDTD for our cavity problem.
§ This version of Mur’s absorbing boundary conditions is only of the 1st order.
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9. 5 Results and Closing Thoughts
5.1 The Final Code
With the adjustments, our new code created the expected model for geometry of our problem. A
depiction of the result is displayed below, Figure 5.¶
5.2 Further Research, Work and Applications
Although our current model is incomplete in the sense that it can be improved upon and altered for even
more specific cases. By applying higher orders of Mur’s absorbing boundary conditions, the scattered
waves from the boundaries representing empty space can be diminished even further. Another area
for improvement involves finding the best cell size for a particular wave, and then inserting real values
from worldly, physical situations rather than arbitary values.
One significant application derived from research like ours is the ability to observe how the radar
signatures of large, metalic objects such as planes or boats are affected by cracks or scratches in those
objects. Radar signatures are especially important when trying to either cloak or detect these large,
metalic objects.
5.3 Experience
This internship allowed four students from two different schools, Chatham University and Duquesne
University, who would have otherwise never met, to collaborate on a single project that gave new
insight and knowledge to all those involved. Not only did students receive a better understanding
of the physics involved in electromagnetic waves, but also new expertise in various coding softwares
and languages. We travelled together as a group to Utah in order to present our results at Brigham
Young University’s Center for Undergraduate Research in Mathematics (CURM) conference, the first
formal presentation for every student in our group. We built new friendships, acquired new skills and
expanded our network, all for which we are grateful.
Figure 5: A still image from the visualization of the model produced by our code.
¶ This is the final code for only the FDTD portion of our model, before the addition of the FEM portion.
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10. Acknowledgements
We would like to thank Dr. Michael Dorff at Brigham Young University and the Center of Undergraduate
Research in Mathematics (CURM) for funding our research through the NSF grant DMS-0636648. We would
also like to thank Dr. John Fleming at Duquesne University for his guidance and patience in advising us.
References
Kunz, Karl S., and Raymond J. Luebbers. The Finite Difference Time Domain Method for Electromagnetics.
Boca Raton: CRC, 1993. Print.
Purcell, Edward M. Electricity and Magnetism: Berkeley Physics Course . Volume 2. New York: McGraw .
Hill Book Company, 1965.
Tipler, Paul A., and Ralph A. Llewellyn. Modern Physics. New York: W.H. Freeman and Company, 2008.
Weaver, William. Matrix Analysis of Framed Structures. Boston: Kluwer Academic, 2001. Print.
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