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{{projectinfo|V&V|Computing Radar Cross Section Of Metallic Targets Using EM.Cube|ART RCS title.png|In this projectarticle, you will construct a center‐fed resonant dipole antennametallic target structures of different geometrical shapes are simulated using EM.Tempo, analyze it EM.Libera and visualize its near EM.Illumina, and far field characteristicsthe results are validated by the published data.|*PEC Objects[[EM.Tempo]]*Lumped Source[[EM.Libera]]*Port Definition[[EM.Illumina]]*Mesh DensityPerfect Electric Conductor*S/Z/Y ParametersPlane Wave Source*Radiation Pattern*Field Sensor Observable*EM.Grid *Cartesian and Polar GraphsRadar Cross Section|All versions|{{download|http://www.emagtech.com/downloads/ProjectRepo/EMTempo_Lesson1.zip EMTempo_Lesson1}} None }}

[[Image:ART RCS2.png|thumb|left|480px|Figure 2: Variation of normalized back-scatter RCS (σ/λ²) of a thin metal square plate of dimensions 0.3λ₀ × 0.3λ₀ as a function of elevation angle θ for the case of an incident TMz (Eθ) 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 0.01λ0 thick plate, red magenta symbols: simulated data using the finite element method (FEM) presented by Ref. [1].]]

[[Image:ART RCS3.png|thumb|left|480px|Figure 3: Variation of normalized back-scatter RCS (σ/λ²) of a thin metal square plate of dimensions 0.3λ₀ × 0.3λ₀ as a function of elevation angle θ for the case of an incident TEz (Eφ) 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 0.01λ0 thick plate, red magenta symbols: simulated data using the finite element method (FEM) presented by Ref. [1].]]

[[Image:ART RCS4.png|thumb|left|480px|Figure 4: Variation of normalized back-scatter RCS (σ/λ²) of a thin metal square plate of dimensions a × a as a function of k₀a (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, red magenta symbols: measured data referenced by Ref. [2].]]

The next example involves a large square metal (PEC) plate of dimensions 5λ₀ × 5λ₀ illuminated by an obliquely incident, plane wave source with θ = 30° measured from the zenith. For this electrically large plate, the physical optics method yields very good results at the main laobe 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° 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.

[[Image:ART RCS6RCS7.png|thumb|left|480px|Figure 7: A reproduction of the results given by Ref. [3] for variation of normalized bistatic RCS (σ/λ²) of a large 5λ₀ × 5λ₀ metal square plate based on a simple physical optis 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.]]

[[Image:ART RCS8.png|thumb|left|480px|Figure 8: Variation of normalized back-scatter RCS (σ/λ²) of a metal cube of dimensions a × a × a as a function of k₀a (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, red magenta symbols: measured data given by Ref. [4].]]

Next, we examine scattering from a large metallic sphere. For this case, we consider a PEC sphere of radius 477.465 mm corresponding to k₀a = 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/λ₀² has been used for this mesh. Figure Figures 14 and 15 show the bistatic RCS of the metallic sphere as a function of the elevation angle θ 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 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].

Figure 17 16 shows the geometry of a metallic cylindrical rod with rounded (hemispherical) ends. The total end-to-end length of the rod is 1.5λ₀ , and its diameter is 0.4λ₀. This structure was simulated using [[EM.CubeLibera]]'s Surface MoM and FDTD simulators[[EM.Tempo]]. Figures 18 17 and 19 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/λ₀² was used, while the FDTD module used a mesh density of 40 25 cells/λ_0eff along each linear dimension with high precision adaptive mesh contour settings to better capture the curved surface of the target.

<table><tr><td>[[Image:ART RCS16.png|thumb|left|240px|Figure 1716: Geometry of a metallic cylindrical rod with rounded ends.]]</td><td>[[Image:ART RCS17.png|thumb|left|240px|Figure 17: Surface triangular mesh of a metallic cylindrical rod with rounded ends.]]</td><td>[[Image:ART RCS18.png|thumb|left|240px|Figure 18: FDTD mesh of a metallic cylindrical rod with rounded ends.]]</td></tr></table>

Figure 18: Surface triangular mesh Figures 19 and 20 show the computed bistatic RCS of a metallic 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 rounded endsthose reported in Ref. [7] based on a method of moments (MOM) formulation of bodies of revolution (BOR).</div>

Figure 19: FDTD mesh of a metallic cylindrical rod with rounded ends.</divtable> Figures&nbsp;20 and&nbsp;21 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&ordm;)&nbsp;and ZX (&phi; = 0&ordm;), respectively. These figure compare the results simulated by [[EM.Cube]]&#39;s&nbsp;Surface MoM and FDTD&nbsp;engines with those reported in Ref. [7] based on a MoM formulation of bodies of revolution (BOR). Figure 20: Variation of normalized bistatic RCS (&sigma;/&lambda;<suptr>2</suptd>) of a metallic cylindrical rod with rounded ends as a function of elevation angle &theta; in YZ plane (&phi; = 90&ordm;), , solid red line: [[EM.Libera]] results, solid blue lineImage: [[EMART RCS19.Tempo]](FDTD) results, red symbols: simulated data using a BOR-MoM method presented by Ref. [7]. png|thumb|left|480px|Figure 2119: Variation of normalized bistatic RCS (&sigma;/&lambda;²) of a metallic cylindrical rod with rounded ends as a function of elevation angle &theta; in ZX YZ plane (&phi; = 090&ordmdeg;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]](FDTD) results, red magenta symbols: simulated data using a BOR-MoM method presented by Ref. [7].  == Scattering from a Metal Cone with Spherical Caps == Figure&nbsp;22 shows the geometry of a metallic cone target&nbsp;with rounded&nbsp;(spherical) caps. The total end-to-end length of the target is 2&lambda;₀&nbsp;and the diameter of the bottom hemisphere is 0.94&lambda;₀. This structure was simulated using [[EM.Cube]]&#39;s Surface MoM and FDTD simulators. Figure&nbsp;23 shows the triangular surface mesh of the target with a mesh density of&nbsp;225 samples/&lambda;₀². Figure 24 shows the FDTD mesh of the same structure with a mesh&nbsp;density of&nbsp;40 cells/&lambda;₀&nbsp;along each linear dimension. Figure 22: Geometry of a metallic cone with spherical caps. Figure 23: Surface triangular mesh of a metallic cone with spherical caps. Figure 24: FDTD mesh of a metallic cone with spherical caps. This target is illuminated by a normally incident plane wave source first from the bottom and then from the top. The bottom-up illumination is considered first. Figure 25 shows the 3D&nbsp;total bistatic RCS pattern&nbsp;of the metallic target&nbsp;computed by [[EM.Cube]]&#39;s Surface MoM simulator. Figure 26 shows the four-port view of the current distribution over the surface of the conical metal target computed by [[EM.Cube]]&#39;s Surface MoM simulator.&nbsp;One can clearly see&nbsp;the surface current excited by&nbsp;the incident plane wave source&nbsp;over the top spherical cap, which falls into the shadow region of the target.&nbsp;To better appreciate these results, Figure 27 shows the four-port view of the current distribution over the surface of the conical metal target computed by [[EM.Cube]]&#39;s Physical Optics (PO) simulator. Here you can see the dark region over top spherical cap.   Figure 25: 3D RCS pattern of the metallic cone with spherical caps illuminated from bottom up. Figure 26: Four-port view of surface current distribution on the metallic cone with spherical caps illuminated from bottom up computed by [[EM.Cube]]&#39;s Surface MoM solver. Figure 27: Four-port view of surface current distribution on the metallic cone with spherical caps illuminated from bottom up computed by [[EM.Cube]]&#39;s PO solver.  Figures 28 and 29 show the bistatic RCS results for the metallic cone with spherical caps illuminated from bottom up. They correspond to the YZ plane(&phi; = 90&ordm;), and ZX plane (&phi; = 0&ordm;), respectively. Both figures show the results generated by [[EM.Cube]]&#39;s Surface MoM, FDTD and Physical Optics simulation engines and compare them with those reported in Ref. [7] based on a MoM formulation of bodies of revolution (BOR). Figure 30: 3D RCS pattern of the metallic cone with spherical caps illuminated from top down.</divtd> Figure 29: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</suptr>) of a metallic cone with spherical caps illuminated from the bottom by a normally incident plane source as a function of elevation angle &theta; in ZX plane (&phi; = 0&ordm;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]](FDTD) results, solid green line: [[EM.Illumina]] (PO) results, red symbols: simulated data using a BOR-MoM method presented by Ref. [7]. Figure 28: Variation of normalized bistatic RCS (&sigma;/&lambda;<sup>2</suptable>) of a metallic cone with spherical caps illuminated from the bottom by a normally incident plane source as a function of elevation angle &theta; in YZ plane (&phi; = 90&ordm;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]](FDTD) results, solid green line: [[EM.Illumina]] (PO) results, red symbols: simulated data using a BOR-MoM method presented by Ref. [7]. Next, we consider the same metallic conical target of Figure 22, but this time illuminated by a normally incident plane wave source from top down. Figure&nbsp;30 shows the 3D&nbsp;total bistatic RCS pattern&nbsp;of the metallic target&nbsp;computed by [[EM.Cube]]&#39;s Surface MoM simulator. Figure&nbsp;31 shows the four-port view of the current distribution over the surface of the conical metal target computed by [[EM.Cube]]&#39;s Surface MoM simulator.&nbsp;You can clearly see&nbsp;the surface current excited by&nbsp;the incident plane wave source&nbsp;over the bottom spherical cap, which&nbsp;represents the shadow region of the target.&nbsp;To better appreciate these results, Figure&nbsp;32 shows the four-port view of the current distribution over the surface of the conical metal target computed by [[EM.Cube]]&#39;s Physical Optics (PO) simulator. Here, too,&nbsp;you can see the dark region over the entire bottom spherical cap. Figure 34: Variation of normalized bistatic RCS (&sigma;/&lambda;²) of a metallic cone with spherical caps illuminated from the top by a normally incident plane source as a function of elevation angle &theta; in ZX plane(&phi; = 0&ordm;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]](FDTD) results, solid green line: [[EM.Illumina]] (PO) results, red symbols: simulated data using a BOR-MoM method presented by Ref. [7]. Figure 33: Variation of normalized bistatic RCS (&sigma;/&lambda;²) of a metallic cone with spherical caps illuminated from the top by a normally incident plane source as a function of elevation angle &theta; in YZ plane (&phi; = 90&ordm;), solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]](FDTD) results, solid green line: [[EM.Illumina]] (PO) results, red symbols: simulated data using a BOR-MoM method presented by Ref. [7]. Figure 32: Four-port view of surface current distribution on the metallic cone with spherical caps illuminated from top down computed by [[EM.Cube]]&#39;s PO solver. Figure 31: Four-port view of surface current distribution on the metallic cone with spherical caps illuminated from top down computed by [[EM.Cube]]&#39;s Surface MoM solver.  Figures&nbsp;33 and 34 show the bistatic RCS results for the metallic cone with spherical caps illuminated from top down. They correspond to the YZ plane(&phi; = 90&ordm;), and ZX plane (&phi; = 0&ordm;), respectively. Both figures show the results generated by [[EM.Cube]]&#39;s Surface MoM, FDTD and Physical Optics simulation engines and compare them with those reported in Ref. [7] based on a MoM formulation of bodies of revolution (BOR).

The last metallic volumetric target to be considered is a long conesphere as shown in Figure 3521.&nbsp;The diameter of the sphere&nbsp;is 0.94&lambda;₀, 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;₀. This structure was simulated using [[EM.CubeLibera|EM.LIbera]]&#39;s Surface MoM and FDTD simulators[[EM.Tempo]]. Figure 36 22 shows the triangular surface mesh of the target with a mesh density of&nbsp;225 samples/&lambda;₀². Figure&nbsp;37 23 shows the FDTD mesh of the same structure with a mesh&nbsp;density of&nbsp;40 25 cells/&lambda;_0eff&nbsp;along each linear dimensiontogether with high precision adaptive mesh contour settings.

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 3726 shows the corresponding results for the same target given by Ref. [6], where three data sets are compared: FDTD mesh of a long metallic cone sphereone 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.

<table><tr><td>[[Image:ART RCS24.png|thumb|left|480px|Figure 3624: Surface triangular mesh Variation of normalized bistatic RCS (&sigma;/&lambda;²) of a long metallic cone sphereconesphere 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.]]</td></tr></table>

<table><tr><td>[[Image:ART RCS25.png|thumb|left|480px|Figure 3525: Geometry Variation of normalized bistatic RCS (&sigma;/&lambda;²) of a long metallic cone sphereconesphere 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.]]</td></tr></table>

 Figures 38 and 39 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.Cube]]&#39;s Surface MoM and FDTD simulation engines. To avoid overlapping figures, Figure 40 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]]&#39;s results. Figure 39: Variation of normalized bistatic RCS (&sigma;/&lambda;<suptable>2</suptr>) of a long metallic conesphere&nbsp;illuminated by a normally incident plane source with TMz (vertical) polarization&nbsp;as a function of elevation angle &theta;, solid red line: [[EM.Libera]] results, solid blue line: [[EM.Tempo]] (FDTD) results. Figure 38: Variation of normalized bistatic RCS (&sigma;/&lambda;<suptd>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 lineImage: [[EMART RCS26.Tempo]] (FDTD) results. png|thumb|left|640px|Figure 4026: A reproduction of the results given by Ref. [6] for&nbsp;the variation of normalized bistatic RCS (&sigma;/&lambda;²) of a long metallic conesphere based on three techniques: standard MoM, hybrid PO/MoM and hybrid Fock/MoM. The left an and right graphs correspond to vertical (TMz) and horizontal (TEz) polarizations, respectively.]]</td></tr></table>