View source for V&V Article 2: Computing Radar Cross Section Of Metallic Targets Using EM.Cube

{{projectinfo|V&V|Computing Radar Cross Section Of Metallic Targets Using EM.Cube|ART RCS title.png|In this article, metallic target structures of different geometrical shapes are simulated using EM.Tempo, EM.Libera and EM.Illumina, and the results are validated by the published data.|
*[[EM.Tempo]]
*[[EM.Libera]]
*[[EM.Illumina]]
*Perfect Electric Conductor
*Plane Wave Source
*Radar Cross Section
|All versions| None }}

== Introduction ==

In this verification & validation (V&V) article, we will model electromagnetic scattering from a number of canonical targets and compute their radar cross section (RCS) using [[EM.Cube]]&#39;s different simulation engines. Each scattering problem will be solved using different [[EM.Cube]] solvers such as [[EM.Tempo]]'s FDTD solver, [[EM.Libera]]'s surface MOM solver and [[EM.Illumina]]'s Physical Optics (PO) solver.  The simulation results are compared against analytical and published simulation or measurement data.

== Scattering from a Small Metal Plate ==

The first target to be simulated is a small thin metal (PEC) plate of dimensions 0.3&lambda;<sub>0</sub> &times; 0.3&lambda;<sub>0</sub>, where&nbsp;&lambda;<sub>0</sub> is the free space wavelength. The backscatter RCS of this target is computed first using [[EM.Libera]]'s Surface MoM simulation engine and then using [[EM.Tempo]]'s FDTD engine. To calculate the backscatter RCS as a function of elevation angle &theta;, we ran an angular sweep of &theta; in both [[EM.Libera]] and [[EM.Tempo]] from &theta; = 180&ordm; (corresponding to a downward, normally incident, plane wave source) to &theta; = 90&deg; (corresponding to a lateral, horizontally incident, plane wave source) with a step of 5 degrees. Figures 2 and 3 show the simulation results obtained using [[EM.Libera]] and [[EM.Tempo]] and compare them to the results obtained using the finite element method (FEM) given by Ref. [1]. Both theta (TMz) and phi (TEz) polarizations are considered, and the RCS results are plotted in Figures 2 and 3, respectively. Two sets of FDTD data are presented in these figures, one for a planar PEC square of zero thickness and one for a thin metal plate of thickness 0.01&lambda;<sub>0</sub>. It can be seen that the infinitesimally thin planar object in FDTD predicts slightly larger RCS than the actual values.

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[[Image:ART RCS1.png|thumb|left|360px|Figure 1: Geometry of a metal plate and its coordinate system..]]
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[[Image:ART RCS2.png|thumb|left|480px|Figure 2: Variation of normalized back-scatter RCS (&sigma;/&lambda;<sup>2</sup>) of a thin metal square plate of dimensions 0.3&lambda;<sub>0</sub> &times; 0.3&lambda;<sub>0</sub> as a function of elevation angle &theta; for the case of an incident TMz (E&theta;) polarization, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results with a zero-thickness plate, solid green line: [[EM.Tempo]] results with a 0.01&lambda;0 thick plate, magenta symbols: simulated data using the finite element method (FEM) presented by Ref. [1].]]
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[[Image:ART RCS3.png|thumb|left|480px|Figure 3: Variation of normalized back-scatter RCS (&sigma;/&lambda;<sup>2</sup>) of a thin metal square plate of dimensions 0.3&lambda;<sub>0</sub> &times; 0.3&lambda;<sub>0</sub> as a function of elevation angle &theta; for the case of an incident TEz (E&phi;) polarization, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results with a zero-thickness plate, solid green line: [[EM.Tempo]] results with a 0.01&lambda;0 thick plate, magenta symbols: simulated data using the finite element method (FEM) presented by Ref. [1].]]
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== Scattering from a Large Metal Plate ==

Next, we investigate the variation of RCS of a metal plate with its lateral size or frequency. For this purpose, we consider a thin metal (PEC) plate of dimensions a &times; a (same as shown in Figure 1), and compute its backscatter RCS as a function of k<sub>0</sub>a, where k<sub>0</sub> = 2 &pi; / &lambda;<sub>0</sub> = 2 &pi; f / c. A plot of RCS vs. k<sub>0</sub>a shows the variation of RCS with frequency when the plate's lateral size a is fixed, or alternatively, the variation of RCS with the lateral dimension at a fixed frequency. Figure 4 shows the variation of the backscatter RCS of the square PEC plate with k<sub>0</sub>a as simulated by [[EM.Libera]]. These results are compared with measured data reported by Ref. [2]. Excellent agreement is observed between the two sets of data. The figure also shows the backscatter RCS data computed by [[EM.Illumina]]'s Physical Optics (PO) solver. It is seen that the PO results do not agree well with the full-wave MoM results until the size of the plate grows to about a wavelength or more. This can also be interpreted as when frequency increases such that k<sub>0</sub>a &gt; 6, which justifies the physical optics approximation. 

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[[Image:ART RCS4.png|thumb|left|480px|Figure 4: Variation of normalized back-scatter RCS (&sigma;/&lambda;<sup>2</sup>) of a thin metal square plate of dimensions a &times; a&nbsp;as a function of k<sub>0</sub>a (normalized plate length or normalized frequency) for the case of a normally incident plane wave source, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Illumina]] (PO) results, magenta symbols: measured data referenced by Ref. [2].]]
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The next example involves a large square metal (PEC) plate of dimensions 5&lambda;<sub>0</sub> &times; 5&lambda;<sub>0</sub> illuminated by an obliquely incident, plane wave source with &theta; = 30&deg; measured from the zenith. For this electrically large plate, the physical optics method yields very good results at the main lobe and the first few side lobes. Figures 5 and 6 show the normalized bi-static RCS of the large plate as simulated by [[EM.Libera]] and [[EM.Illumina]]. The two figures correspond to the incident TMz and TEz polarizations, respectively. Note that the maximum RCS is observed at 30&deg; as one would expect. At the grazing angles, one can see significant discrepancies between the asymptotic PO and full-wave Surface MoM results. For comparison, Figure 7 shows a reproduction of the physical optics results given by Ref. [3], which have been calculated analytically using a simple PO approximation of uniform surface currents on the metal plate.

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[[Image:ART RCS5.png|thumb|left|480px|Figure 5: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a&nbsp;large metal square plate of dimensions 5&lambda;<sub>0</sub> &times; 5&lambda;<sub>0</sub> as a function of elevation angle &theta; for the case of an obliquely&nbsp;incident TMz polarization in the YZ plane (&phi;<sub>s</sub> = 90&deg;, 270&deg;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Illumina]] (PO) results.]]
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[[Image:ART RCS6.png|thumb|left|480px|Figure 6: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a&nbsp;large metal square plate of dimensions 5&lambda;<sub>0</sub> &times; 5&lambda;<sub>0</sub> as a function of elevation angle &theta; for the case of an obliquely&nbsp;incident TEz polarization in the YZ plane (&phi;<sub>s</sub> = 90&deg;, 270&deg;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Illumina]] (PO) results.]]
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[[Image:ART RCS7.png|thumb|left|480px|Figure 7: A reproduction of the results given by Ref. [3] for variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a large 5&lambda;<sub>0</sub> &times; 5&lambda;<sub>0</sub> metal square plate based on a simple physical optics approximation of uniform currents on the plate. The TMx and TEx polarizations in this figure correspond to the TMz and TEz polarizations in Figure 5 and 6, respectively.]]
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== Scattering from a Metal Cube ==

As the first volumetric target, we consider a metallic cube of dimensions a &times; a &times; a, illuminated by a normally incident, plane wave source. Similar to the metal plate case, we run a parametric sweep of [[EM.Libera]] and [[EM.Tempo]] with the cube length a defined as the sweep variable. We compute the backscatter RCS of the metallic cube as a function of k<sub>0</sub>a (normalized length or normalized frequency as you may interpret). Figure 8 shows the backscatter RCS results computed using [[EM.Libera]] and [[EM.Tempo]] and compares them with the published measured data given by Ref. [4]. Excellent agreement is observed among all three data sets.

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[[Image:ART RCS8.png|thumb|left|480px|Figure 8: Variation of normalized back-scatter RCS (&sigma;/&lambda;<sup>2</sup>) of a metal cube of dimensions a &times; a &times; a&nbsp;as a function of k<sub>0</sub>a (normalized cube dimension or normalized frequency) for the case of a normally incident plane wave source, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results, magenta symbols: measured data given by Ref. [4].]]
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== Scattering from a  Finite Metal Cylinder ==

The next volumteric target to be considered is a finite metallic cylinder with an equal height and diameter of 0.6&lambda;<sub>0</sub>. Figures 9 and 10 show the triangular surface mesh and Yee voxel mesh of this target illuminated by a normally incident plane wave source. The backscatter RCS of this target is computed first using [[EM.Libera]]&#39;s surface MOM simulation engine and then using [[EM.Tempo]]&#39;s FDTD solver. Figures 11 and 12 show the simulation results obtained using [[EM.Libera]] and [[EM.Tempo]] and compare them to the results obtained using the finite element method (FEM) given by Ref. [5]. The figures correspond to the theta (TMz) and phi (TEz) polarizations, respectively. Excellent agreement is observed among the three data sets for both polarizations. 

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[[Image:ART RCS9.png|thumb|left|360px|Figure 9: Surface triangular mesh of a metallic cylindrical rod with rounded ends.]]
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[[Image:ART RCS10.png|thumb|left|360px|Figure 10: FDTD mesh of a metallic cylindrical rod with rounded ends.]]
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[[Image:ART RCS11.png|thumb|left|480px|Figure 11: Variation of normalized back-scatter RCS (&sigma;/&lambda;<sup>2</sup>) of a finite metallic cylinder&nbsp;with an&nbsp;equal height and diameter &nbsp;square plate of 0.6&lambda;<sub>0 </sub>as a function of elevation angle &theta; for the case of an incident TMz (E&theta;) polarization, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results, magenta symbols: simulated data using the finite element method (FEM) presented by Ref. [5].]]
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[[Image:ART RCS12.png|thumb|left|480px|Figure 12: Variation of normalized back-scatter RCS (&sigma;/&lambda;<sup>2</sup>) of a finite metallic cylinder&nbsp;with an&nbsp;equal height and diameter &nbsp;square plate of 0.6&lambda;<sub>0 </sub>as a function of elevation angle &theta; for the case of an incident TEz (E&phi;) polarization, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results, magenta symbols: simulated data using the finite element method (FEM) presented by Ref. [5].]]
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== Scattering from a Large Metal Sphere ==

Next, we examine scattering from a large metallic sphere. For this case, we consider a PEC sphere of radius 477.465 mm corresponding to k<sub>0</sub>a = 10, at the frequency f = 1GHz. Figure 13 shows the triangular surface mesh of this sphere generated by the [[EM.Libera]] or [[EM.Illumina]] mesh generators. A mesh density of 100 samples/&lambda;<sub>0</sub><sup>2</sup> has been used for this mesh. Figures 14 and 15 show the bistatic RCS of the metallic sphere as a function of the elevation angle &theta; for the two cases of TMz and TEz polarizations, respectively. The two figures compare the results computed by [[EM.Libera]]'s surface MOM solver and [[EM.Illumina]]'s Physical Optics (PO) solverand compare them with the simulated results given by Ref. [6], which presents two sets of data, one based&nbsp;on the method of moments (MoM) and the other based on a hybrid PO/MoM/Fock technique. The two data sets in Ref. [6] are almost identical. Like in the previous example, physical optics predicts the RCS over the main beam (or maximum RCS angles) adequately; however, its accuracy degrades over the side lobes.[[EM.Libera]]'s results almost exactly match those of Ref. [6].

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[[Image:ART RCS13.png|thumb|left|360px|Figure 13: Surface triangular mesh of a large metallic sphere with k<sub>0</sub>a = 10.]]
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[[Image:ART RCS14.png|thumb|left|480px|Figure 14: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a large metallic sphere of radius a (k<sub>0</sub>a = 10) with elevation angle &theta; for the case of an incident TMz (vertical) polarization, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Illumina]] (PO) results, magenta symbols: simulated data using a hybrid PO/MoM/Fock method presented by Ref. [6].]]
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[[Image:ART RCS15.png|thumb|left|480px|Figure 15: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a large metallic sphere of radius a (k<sub>0</sub>a = 10) with elevation angle &theta; for the case of an incident TEz (horizontal) polarization, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Illumina]] (PO) results, magenta symbols: simulated data using a hybrid PO/MoM/Fock method presented by Ref. [6].]]
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== Scattering from a Metal Cylindrical Rod with Rounded Ends ==

Figure 16 shows the geometry of a metallic cylindrical rod with rounded (hemispherical) ends. The total end-to-end length of the rod is 1.5&lambda;<sub>0</sub>, and its diameter is 0.4&lambda;<sub>0</sub>. This structure was simulated using [[EM.Libera]] and [[EM.Tempo]]. Figures 17 and 18 show the triangular surface mesh and the FDTD mesh of the same structure, respectively. For the surface mesh generation, a mesh density of 225 samples/&lambda;<sub>0</sub><sup>2</sup> was used, while the FDTD module used a mesh density of 25 cells/&lambda;<sub>eff</sub> along each linear dimension with high precision adaptive mesh contour settings to better capture the curved surface of the target.

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[[Image:ART RCS16.png|thumb|left|240px|Figure 16: Geometry of a metallic cylindrical rod with rounded ends.]]
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[[Image:ART RCS17.png|thumb|left|240px|Figure 17: Surface triangular mesh of a metallic cylindrical rod with rounded ends.]]
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[[Image:ART RCS18.png|thumb|left|240px|Figure 18: FDTD mesh of a metallic cylindrical rod with rounded ends.]]
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Figures 19 and 20 show the computed bistatic RCS of the cylindrical rod as a function of the elevation angle when the target is illuminated from the bottom by a normally incident plane wave source. The two figures correspond to the bistatic RCS in the two principal planes YZ (&phi; = 90&deg;) and ZX (&phi; = 0&deg;), respectively. These figures compare the results simulated by [[EM.Libera]] and [[EM.Tempo]] with those reported in Ref. [7] based on a method of moments (MOM) formulation of bodies of revolution (BOR).

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[[Image:ART RCS19.png|thumb|left|480px|Figure 19: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a metallic cylindrical rod with rounded ends as a function of elevation angle &theta; in YZ plane (&phi; = 90&deg;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results, magenta symbols: simulated data using a BOR-MoM method presented by Ref. [7].]]
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[[Image:ART RCS20.png|thumb|left|480px|Figure 20: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a metallic cylindrical rod with rounded ends as a function of elevation angle &theta; in ZX plane (&phi; = 0&deg;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results, magenta symbols: simulated data using a BOR-MoM method presented by Ref. [7].]]
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== Scattering from a Long Metal Conesphere ==

The last metallic volumetric target to be considered is a long conesphere as shown in Figure 21. The diameter of the sphere is 0.94&lambda;<sub>0</sub>, and the total end-to-end length of the target from the apex of the cone to the bottom of the sphere is 3.137&lambda;<sub>0</sub>. This structure was simulated using [[EM.Libera|EM.LIbera]] and [[EM.Tempo]]. Figure 22 shows the triangular surface mesh of the target with a mesh density of 225 samples/&lambda;<sub>0</sub><sup>2</sup>. Figure 23 shows the FDTD mesh of the same structure with a mesh density of 25 cells/&lambda;<sub>eff</sub> along each linear dimension together with high precision adaptive mesh contour settings.

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[[Image:ART RCS21.png|thumb|left|240px|Figure 21: Geometry of a long metallic conesphere.
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[[Image:ART RCS22.png|thumb|left|240px|Figure 22: Surface triangular mesh of a long metallic conesphere.]]
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[[Image:ART RCS23.png|thumb|left|240px|Figure 23: FDTD mesh of a long metallic conesphere.]]
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Figures 24 and 25 show the bistatic RCS results for the metallic conesphere illuminated from the top (apex) by a normally incident plane wave source with TMz (vertical) and TEz (horizontal) polarizations, respectively. Both figures show and compare the results generated by [[EM.Libera]] and [[EM.Tempo]]. To avoid overlapping figures, Figure 26 shows the corresponding results for the same target given by Ref. [6], where three data sets are compared: one using the standard MoM (similar to [[EM.Libera]]), plus two other hybrid PO/MoM and Fock/MoM methods. Very good agreement is observed among the published data and [[EM.Cube]]'s results.

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[[Image:ART RCS24.png|thumb|left|480px|Figure 24: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a long metallic conesphere illuminated by a normally incident plane source with TMz (vertical) polarization as a function of elevation angle &theta;, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results.]]
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[[Image:ART RCS25.png|thumb|left|480px|Figure 25: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a long metallic conesphere illuminated by a normally incident plane source with TEz (horizontal) polarization as a function of elevation angle &theta;, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] results.]]
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[[Image:ART RCS26.png|thumb|left|640px|Figure 26: A reproduction of the results given by Ref. [6] for the variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</sup>) of a long metallic conesphere based on three techniques: standard MoM, hybrid PO/MoM and hybrid Fock/MoM. The left and right graphs correspond to vertical (TMz) and horizontal (TEz) polarizations, respectively.]]
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== References ==

[1] J. Jin, The Finite Element Method in Electromagnetics. Chapter 8, pp. 275-277, John Wiley &amp; Sons, 1993.

[2] S.M. Rao, D.R. Wilton,&nbsp;and A.W. Glisson, &ldquo;Electromagnetic scattering by surface of arbitrary shape,&rdquo; IEEE Trans. on Antennas &amp; Propagat, Vol. 30, No. 3, pp. 107-116, 1982.

[3] C.A. Balanis, Advanced Engineering&nbsp;Electromagnetics. Chapter 11, pp. 591-599, John Wiley &amp; Sons, 2012.

[4] R.P. Penno, G.A. Thiele, and K.M. Pasala,&nbsp;&ldquo;Scattering from a perfectly conducting cube,&rdquo; Proc. IEEE, Vol. 77, pp. 815-823, 1989.

[5] A. Chatterjee, J.M. Jin and J. L. Volakis, &quot;Edge-based finite elements and vector ABC&#39;s applied to 3-D scattering,&quot; IEEE Trans. on Antennas &amp; Propagat, Vol. 41, No. 2, pp.221-226, 1993.&nbsp;&nbsp;

[6] L.N. Medgyesi-Mitschang and D-S Wang, &ldquo;Hybrid solutions for&nbsp;scattering from perfectly conducting bodies of revolution,&rdquo; IEEE Trans. on Antennas &amp; Propagat, Vol. 31, No. 4, pp. 570-583, 1983.

[7] M.G. Andreasen, &ldquo;Scattering from bodies of revolution,&rdquo; IEEE Trans. on Antennas &amp; Propagat, Vol. 13, No. 2, pp. 303-310, 1965.

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