Optimizing the Backscattering from Arrays of Perfectly Conducting Strips
Randy Haupt and You Chung Chung
Utah State University
Electrical and Computer Engineering
4120 Old Main Hill
Logan, UT 84322-4120
haupt@ieee.org
youchung@ieee.org
Introduction.
Numerical optimization is becoming increasingly important in the design of electromagnetic systems.
Deciding which numerical optimization approach to use can be confusing. We've taken two relatively
simple backscattering optimization problems and applied seven different techniques to see which ones
work best. Four approaches are known as local optimizers, because they head to the minimum close to
the starting point of the search. The other three have random components that allow them to search a
much broader area for a more "global" minimum.
Model.
Consider an array of perfectly conducting strips as shown in Figure 1. Each strip is .25λ wide and
infinitely long. The spacing between the strips is variable and limited to between 0 and .5λ.
φ
x1
0.25λ
xn
x
Figure 1. Diagram of an array of unequally spaced strips that are .25λ wide.
The currents induced by an incident field having the electric field parallel to the edge of the strips are
found using two methods. The first is given by the integral equation
e
jkx cosφ
k N
= ∑
4 n=1
bn
( )
∫ J ( x′) H ( k x − x′ ) dx ′
2
z
0
(1.1)
an
and the second by physical optics
J z ( x′) = 2sin φ e jkx′ cosφ
(1.2)
where
k = 2π /λ
φ = angle of incidence and observation
Jz(x ′) = surface current density
H0 (2) (*) = zeroth order Hankel function of the second kind
0.25λ = width of strip
N = number of strips
an , bn = strip end points (an − bn = 0.25λ)
The currents are found for each relevant angle and substituted into the expression for the
backscattering radar cross section (RCS)
k
σ (φ ) =
4
N
2
bn
∑ ∫ J (x ′ ) e
n =1
z
jkx′ cosφ
dx ′
(1.3)
an
The integral equation is solved using the method of moments (MOM).
The first cost function minimizes the maximum sidelobe level of the backscattering RCS pattern,
while the second cost function places a null in the highest sidelobe in the backscattering pattern. Both
objectives are achieved through optimizing the spacing between the strips. The maximum spacing is
limited to 0.5λ. Each numerical minimization method brings strengths and weaknesses to finding the
optimum solution. The Nelder Mead down-hill simplex method does not require the calculation of a
derivative. BFGS (Broyden-Fletcher-Goldfarb-Shannon), DFP (Davidon-Fletcher-Powell), and
steepest descent all require approximate gradient calculations. Steepest descent has fallen out of favor
in the past few decades, because certain minimization problems cause it to be extremely slow. The
"global" optimization methods used here are random search, binary genetic algorithm, and continuous
parameter genetic algorithm. These three approaches have random components, so they
simultaneously search many local minima. The random search algorithm just generated 1000 random
guesses of the solution and returned the best guess.
There are strategies to aid the various techniques in finding the minimum. For instance, all the local
optimizers can be run a second time using the optimum point found the first time. This approach is
very effective but not used here. We almost always finish a GA run by starting a local optimizer at the
optimum point found using the GA. This approach also works well but is not used here. Our goals in
this investigation were to see if 1. coupled variables, 2. number of variables, and 3. problem difficulty:
difficult (min max sidelobe level)/easy (null in sidelobe) effect convergence of various numerical
optimization approaches.
Results.
The seven minimization techniques mentioned in the previous section were applied to eight different
backscattering problems. The first problem optimized the spacing of 10 perfectly conducting strips
(0.25λ wide) in order to minimize the maximum relative sidelobe level. Results over twenty
independent runs for both method of moments and physical optics are shown in Table 1. The second
problem optimized the spacing of 20 perfectly conducting strips (0.25λ wide) in order to minimize the
maximum relative sidelobe level. Results over twenty independent runs for both method of moments
and physical optics are shown in Table 2. The third problem optimized the spacing of 10 perfectly
conducting strips (0.25λ wide) in order to place a null in the highest sidelobe at u=0.28. Results over
twenty independent runs for both method of moments and physical optics are shown in Table 3. The
final problem optimized the spacing of 20 perfectly conducting strips (0.25λ wide) in order to place a
null in the highest sidelobe at u=0.14. Results over twenty independent runs for both method of
moments and physical optics are shown in Table 4. The minimum value in each column of the four
tables is italicized.
Table 1. Optimize the spacing of 10 strips in order to minimize the maximum sidelobe level of
the backscattering pattern. These results are from twenty independent runs.
Nelder Mead
BFGS
DFP
Steepest descent
Random search
Binary GA
Continuous parameter GA
Method of moments
max
min
mean
-13.71
-15.77
-15.03
-13.31
-16.44
-14.80
-12.04
-16.18
-14.34
-12.61
-17.78
-15.35
-13.70
-15.53
-14.44
-14.68
-16.36
-15.64
-13.34
-15.97
-14.83
Max
-14.34
-13.97
-13.28
-13.60
-15.61
-15.68
-17.12
Physical optics
min
mean
-19.92
-16.79
-19.25
-17.35
-18.18
-15.79
-19.90
-16.83
-18.84
-16.65
-20.26
-19.15
-20.78
-18.97
Table 2. Optimize the spacing of 20 strips in order to minimize the maximum sidelobe level of
the backscattering pattern. These results are from twenty independent runs.
Nelder Mead
BFGS
DFP
Steepest descent
Random search
Binary GA
Continuous parameter GA
Method of moments
max
min
mean
-14.71
-18.72
-16.23
-13.27
-16.80
-15.17
-12.67
-16.56
-15.07
-12.66
-16.94
-14.99
-15.31
-17.63
-15.93
-16.84
-18.25
-17.71
-16.18
-19.05
-17.43
Physical optics
max
min
mean
-15.93
-19.26
-17.43
-15.66
-19.06
-17.19
-15.30
-21.81
-17.27
-14.75
-18.87
-16.93
-16.56
-19.06
-17.37
-18.00
-21.21
-19.90
-17.85
-21.07
-19.24
Table 3. Optimize the spacing of 10 strips in order to put a null in the backscattering pattern at
u=0.28. These results are from twenty independent runs. Median is used instead of mean due to
the possibility of a -∞ null.
Nelder Mead
BFGS
DFP
Steepest descent
Random search
Binary GA
Continuous parameter GA
Method of moments
max
min
median
-11.46
-303.58
-∞
-8.64
-171.76 -100.59
1.21
-173.95
-25.40
-12.74
-15.42
-14.10
-21.11
-38.64
-26.41
-34.09
-54.24
-44.43
-32.85
-56.36
-43.51
max
-7.17
0.00
0.79
-0.32
-13.31
-47.69
-41.77
Physical optics
min
median
-328.65 -205.32
-119.14
-9.43
-65.49
-10.70
-195.59
-8.26
-27.21
-20.92
-67.21
-55.14
-67.67
-52.66
Table 4. Optimize the spacing of 20 strips in order to put a null in the backscattering pattern at
u=0.14. These results are from twenty independent runs. Median is used instead of mean due to
the possibility of a -∞ null.
Nelder Mead
BFGS
DFP
Steepest descent
Random search
Binary GA
Continuous parameter GA
Method of moments
max
min
median
-9.86
-303.10
-∞
-1.91
-117.59
-19.17
2.5723
-103.29
-8.08
1.11
-188.54
-168.3
-16.83
-37.02
-23.72
-29.36
-57.25
-38.86
-27.67
-61.24
-38.77
Physical optics
max
min
median
-10.08
-166.71 -152.05
-6.58
-74.82
-21.22
-4.72
-86.82
-22.03
-3.07
-197.67 -169.33
-23.98
-44.86
-29.84
-43.73
-62.03
-51.10
-35.57
-59.71
-47.01
The population size and mutation rate for the GA used to generate the results presented here were
found by running the GA for various population sizes and mutation rates and picking the best
performers. Results for the GA were averaged over ten runs. Table 5 gives the best population size
and mutation rate for the various approaches to finding the minimum maximum sidelobe level of a 10
strip array.
Table 5. Best population size and mutation rate of GAs for low sidelobe level of the
backscattering pattern.
Binary GA
Continuous parameter GA
Population
Mutation
Population
Mutation
Physical optics
16
3%
8
8%
Method of moments
16
3%
12
12%
Figure 1 shows the minimum maximum backscattering sidelobe level for the four local optimizers as
the number of function evaluations increases. In this case, steepest descent converged very fast, but
BFGS won out in the long run. BFGS and DFP have large oscillations in their convergence curves due
to the singular matrix calculations when approximating the derivatives. Similar results for the GAs are
shown in Figure 2.
Figure 3 is an example of an optimized backscattering sidelobe level for 10 strips. The strip spacing
does not decrease the absolute sidelobe level from the strips but does increase the absolute level of the
main beam. Thus, the relative sidelobe level goes down. Figure 4 is an example of placing a null in the
quiescent RCS pattern (no spacing between the 10 strips) at the peak of the maximum sidelobe at u =
0.28.
Figure 1. Minimum maximum backscattering sidelobe level for the four local optimizers versus
the number of function calls.
Figure 2. Minimum maximum backscattering sidelobe level for the GAs versus the number of
function calls.
Figure 3. Example of an optimized low sidelobe backscattering RCS patterns obtained using
continuous parameter and binary GAs.
Figure 4. Example of a null placed in the highest sidelobe of a backscattering RCS pattern using
continuous parameter and binary GAs.
Conclusions.
One surprising result is that the Nelder Mead down-hill simplex algorithm out-performed the other
local optimizers. Perhaps the fact that a gradient was not supplied hurt the three gradient based
approaches. Another surprise is that steepest descent did very well and sometimes out-performed some
of the other techniques.
The genetic algorithms consistently performed very well. These algorithms were superior performers
for the more difficult problem of finding the spacing that optimizes the sidelobe level. In addition, a
GA always had the best worst solution out of the twenty random trial runs. In other words on any
given run, a GA would always produce a good solution. Some of the other techniques sometimes
produce very poor solutions.
The strengths of the various algorithms encourages the use of a hybrid algorithm, such as a GA/local
optimizer.