In this section, we present some numerical results computed by the ParaMoM-MPP
code described in this thesis. The test targets in this study
are provided by the Electromagnetic Code Consortium. All numerical results are
checked against the measurements provided by the EMCC (see
references [82][81]). The incident angles are
defined in Appendix .
The parallel program requirements vary greatly and depend upon the EMCC benchmark target geometries, the frequency, and the model symmetries imposed. The numerical results are to provide a general idea of the accuracy obtained by using a parametric method of moments approach, especially the parallel code ParaMoM-MPP, in a MIMD with a distributed memory environment. In fact, the patch sampling rates used for the benchmark comparison runs are somewhat conservative and would be reasonable on a large problem. In most cases, the maximum edge length specified during target gridding was one eighth of a wavelength.
Depending upon the specific method used for RCS prediction, target
descriptions may vary greatly. Of particular concern for several
approaches, ParaMoM-MPP included, is the actual parametric model. Syracuse
Research Corporation provides these models. All of the EMCC benchmark test
cases indicated below were run on the NAS Paragon known as GRACE.
The RCS values for both horizontal (HH) and vertical (VV) polarizations are
plotted in dBSM the dB expression for the number of square meters that the RCS is as a function of the
azimuthal angle which is defined in Appendix .
The first example is the RCS from a flat plate in the shape of a wedge cylinder with gap. The incident
wave has a frequency of 5.9 GHz with horizontal polarization. The receiver's
polarization is also horizontal. The parametric patch model of the target
is shown in Figure 5.10, where the maximum edge length is
, and the number of triangular patches is 1008 with 553 nodes
and 1560 edges. The radius
of circular part of the wedge cylinder is
and the length of
its straight
side is two
.
The monostatic radar cross section for horizontal
transmitter and receiver polarizations (HH) is shown in Figure 5.11.
The result is obtained by
running the ParaMoM-MPP on a 80-node partition Intel NAS paragon. Six
Mbytes of memory are required to run this target.
There are 360 excitation vectors
to compute the monostatic radar cross section.
The computation takes 309 seconds by a wall clock. The RCS, in dB
, plotted against the azimuthal angle is shown
in Figure 5.11 in a
-elevation conical cut.
In Figure 5.12, NASA's half metallic almond parametric model is shown. The mathematical description for the NASA almond is as follows:
for and
for and
where
inches. The total length of the almond is 9.936 inches.
To take the symmetry into consideration, only half the NASA almond
is to be modeled as shown in Figure
5.12. The plane of symmetry
is the XY plane. The NASA's half almond is segmented by a curved grid with 1232
curved triangles,
1915 edges, and 684 nodes. The maximum edge length is
.
The monostatic radar cross section for both the HH and the vertical-vertical (VV) polarizations at 7 GHz are plotted in dBSM as functions of azimuthal angle in Figure 5.13 and Figure 5.14. The elevation angle is zero. There are 720 RHS vectors for this problem. This problem is run on a 32-node partition of the NAS Paragon. The memory required to run this problem in the system is 7 Mbytes per node. The wall clock time is 2817 seconds to complete the run.
A single metallic ogive's parametric patch model is shown in Figure 5.15, where three symmetry planes are used so one eighth of the body is actually modeled. The ogive is illuminated by an incident wave at 9 GHz. The ogival body is a classical test case for RCS. The analytical expression for the ogive is
and
where is greater than
inches and less than
inches, and
is greater then
and less than
. The single ogive has a half
angle of 22.62 degrees, a half-length of 5 inches, and a maximum radius of
1 inch.
The single ogive is 10 inches long physically and about 7.6 wavelengths
electrically. The single ogive is gridded into two different models. The
first model utilizes three symmetry planes, so only one eighth of the single ogive
needs to be modeled. This one-eighth single ogive is modeled by 2088 curved
triangles with 3208 edges and 1121 nodes. The maximum edge length is
. The second model uses no symmetry plane. It
consists of about 5000 curved patches, and 7332 edges.
The monostatic radar cross section characteristics for both HH and VV polarizations are plotted in dBSM as functions of the azimuthal angle in Figure 5.16 and Figure 5.17. The elevation angle is zero. In Figure 5.16 and Figure 5.17, and computed_without_symdenotes the model uses no symmetry planes, computed_with_symdenotes the target modeled with three symmetry planes. Although the number of the curved patches of the model without symmetry planes is much less dense than that with symmetry planes, Figure 5.16 and Figure 5.17 shows good agreement. This demonstrates that the parametric patch method of moments with fewer unknowns still obtains good accuracy. There are 360 RHS vectors for this problem. This problem is run on a 64-node partition of the NAS Paragon. The symmetry model requires 9 Mbytes per node and takes 9648 seconds to complete the run. The model without symmetry planes requires 21 Mbytes per node and takes 3918 seconds to complete the run.
The double ogive consists of two different-size half ogives. One half ogive has a half angle of 46.4 degrees at the tip, a half length of 2.5 inches, and a maximum radius of 1 inch. The other half ogive has a half angle of 22.62 degrees at the tip, a half length of 5 inches, and a maximum radius of 1 inch. The parametric patch model of the double ogive target is shown in Figure 5.18. The double ogive is 7.5 inches long or about 5.7 wavelengths long under the illumination of 9 GHz incident waves. The target surface is modeled by a set of 3822 curved triangles with 5772 edges and 1950 nodes.
The monostatic radar cross sections for both HH and VV polarizations are plotted in dBSM as functions of the azimuthal angle in Figure 5.19 and Figure 5.20. The elevation angle is zero. There are 360 RHS vectors for this problem. This problem is run on a 64-node partition of NAS Paragon. The memory required to run this problem on that system is 17 Mbytes per node. The wall clock time is 2320 seconds to complete the run.
The metallic cone-sphere has a half angle of 7 degrees, and a sphere radius of 2.947 inches. The length of the cone part is 23.821 inches, and the side of the cone is tangent to the sphere, to provide a smooth transition and minimum diffraction at the joint. The cone-sphere can be described as:
for inches
inches and
,
for 0 inch 3.306 inches and
,
The cone-sphere is 27.127 inches
long or about 10.33
The monostatic RCS for both HH and
VV polarizations are plotted in dBSM, as functions of the azimuthal angle
in
Figure 5.22 and Figure 5.23. The elevation angle is
zero. There are 360 RHS
vectors for this problem. This problem is run on a 64-node partition of NAS
Paragon. The memory required to run this problem on that system is 11 Mbytes
per node. The wall clock time is 6357 seconds to complete the run.
The metallic cone-sphere with gap target is the same as the cone-sphere
target, except for a gap next to the cone-sphere joint. The gap is
The discretized cone-sphere with gap surface is shown in
Figure 5.24.
As with the cone-sphere without gap, only a quarter of the surface is
modeled to compute the radar cross section. The quarter of the
surface is modeled by a set of 2576 curved triangles with 3970 edges and
1395 nodes. The maximum edge length is
The monostatic radar cross sections for both HH and
VV polarizations are plotted against azimuthal angle in
Figure 5.25 and Figure 5.26. The elevation angle is
zero. There are 360 RHS
vectors for this problem. This problem is run on a 64-node partition of NAS
Paragon. The memory required to run this problem in the system is 11 Mbytes
per node. The wall clock time is 6639 second to complete the run.
The RCS of a metallic rectangular parallelpiped is computed for the purpose of
testing the accuracy of the ParaMoM-MPP code. The rectangular parallelpiped is
illuminated by incident waves with the frequency at 5GHz. The electrical size
of the rectangular parallelpiped is
The monostatic radar cross sections for both HH and
VV polarizations are plotted as functions of the azimuthal angle in
Figure 5.28 and Figure 5.29. There are 360 RHS
vectors for this problem. This problem is run on a 32-node partition of NAS
Paragon. The memory required to run this problem on that system is 9 Mbytes
per node. The wall clock time is 3813 seconds to complete the run.
A perfectly conducting sphere with radius 0.8 meter is covered by a
dielectric
material. Where the thickness of the coating
is 0.2
meter with the permittivity
The bistatic radar cross section for the HH polarizations is ploted in
dBSM as function
of azimuthal angle
in Figure 5.30.
There are three results shown in Figure
5.30, a Mie theory
result, the modified ParaMoM code result, and the ParaMoM-MPP result. All
the
results are agreed. On a 32 nodes CM-5, the ParaMoM-MPP code spent 11
seconds
to fill the matrix, 1.24 seconds to do the LU/Solve, and the wall clock
time
28 seconds. On a SUN 10 workstation, the modified ParaMoM code spent 261
seconds
to fill the matrix, 22 seconds to do the LU/Solve, and the wall clock time
287
seconds.
at 4.5 GHz.
The parametric patch model of the cone-sphere is shown in
.
inch deep. The difference between the cone-sphere with and
without gap is only from
inch, where it is defined by
where
.
. There are two symmetry planes which are the XY and XZ planes. A
quarter of the surface is gridded into a set of 1693 curved triangles with
2596 edges and 903 nodes. The maximum edge length is
.
The quarter of the rectangular parallelpiped surface which is gridded is shown in
Figure 5.27.
and the permeability
, wh
ere
and
are free space permittivity and permeability, respectively. The
incide
nt plane wave has a frequency of 47.77 MHz with horizontal polarization.
The
receiver's polarization is also horizonal. The parametric patch model of
the sph
ere has 182 triangular patches with 104 nodes and 286 edges.
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