9

,

ADVANCED ELECTROMAGNETIC MODELING USING HOBBIES

9.0 SUMMARY

In this chapter, the parallel electromagnetic (EM) computer code Higher Order Basis Based Integral Equation Solver (HOBBIES) presented in the earlier chapters is used to analyze challenging electromagnetic problems in the frequency domain. A unique feature of HOBBIES is the parallel out-of-core method of moments (MoM) based solver, as it can handle problems that cannot fit into the random access memory (RAM) of the computer systems, whether it be a laptop or a supercomputer. The parallel out-of-core solution feature of this code is a unique capability and is not readily available for most commercial codes. The application examples presented in this chapter illustrate the efficiency and the accuracy of HOBBIES, which can be used as a versatile tool for analyzing challenging, real-world electromagnetic problems.

At the beginning of this chapter, the computational results of a slotted waveguide array antenna using HOBBIES are compared with the experimental measurements. It is demonstrated that using this electromagnetic software, one can compute the radiation pattern that agrees well with measurements. A realistic 1712-element traveling wave slot antenna array is then analyzed.

The computational results of an antenna with some complex composite structures using HOBBIES are then compared with the experimental measurements. The complex composite structure to be studied consists of an L-band antenna array and a dielectric radome with seven large structural ribs for supporting the radome shell. It is demonstrated that using HOBBIES, one can compute the radiation patterns that are within several tenths of a decibel (dB) when compared with measurements for the grating lobe amplitudes. The difference between theory and experiment falls within the resolution of the measurements. The results demonstrate that HOBBIES can be used to predict accurately the interaction of an electrically large array with its surrounding dielectric structures.

In addition, a microstrip patch phased array is presented to demonstrate the advantage of the higher order basis functions utilized in HOBBIES compared with the traditional piecewise RWG (Rao–Wilton–Glisson) basis functions.

Regarding a radar cross section (RCS) calculation, three benchmark results are presented to demonstrate the accuracy of HOBBIES, which are followed by the applications including the formation of tanks, aircraft, and ships.

The advanced modeling of HOBBIES presented in this chapter represents the simulations of several real-life challenging models, which involve antenna analysis and RCS calculations. The numerical results clearly demonstrate that HOBBIES can be used to simulate and analyze challenging EM problems accurately.

The examples described in this chapter consist of realistic, complicated models, which far exceed the limitations of the academic version of HOBBIES. Hence, these projects are not included in the code of this book.

9.1 RADIATION ANALYSIS OF COMPLICATED ANTENNAS

In this section, numerical results are presented for slotted waveguide arrays, an L-band antenna with complex composite structures, and a microstrip patch array.

9.1.1 Radiation from Slotted Arrays

Low-profile, high-gain antennas can be configured easily using slotted waveguide arrays. Therefore, they are widely used in many high-performance radar and communication systems, such as airplane-to-ground communication systems.

9.1.1.1 Radiation from a 108-Slot Waveguide Antenna

To validate the accuracy and efficiency of the code, the computational and experimental results for a traveling waveguide array consisting of 108 narrow-wall slots are first presented in this section.

The simulated model is shown in Figure 9.1, and the experimental model is shown in Figure 9.2. Three waveguides are fabricated to reduce the machining error. The waveguide used is the WR-90 waveguide (X-band), with dimensions of 22.86 mm × 10.16 mm, and the wall thickness of 1.00 mm is assumed. The center frequency of the array is 9.375 GHz. The distance between the centers of any two adjacent slots is 15.5 mm. Three layers of dielectric with a thickness of λ_g for each layer are used as the matched load for the waveguide. A −30 dB Taylor distribution is used in the array design, and the power absorbed in the load is approximately 10%.

The HOBBIES software executes in the parallel in-core mode on a cluster containing 192 cores. The RAM (double precision) used and the central processing unit (CPU) time taken are listed in Table 9.1. The E-plane radiation patterns of the array obtained from simulation and experiment at 9.20 GHz and 9.50 GHz are given in Figure 9.3 and Figure 9.4, respectively. It can be seen that the simulation and experiment results agree with each other very well.

Figure 9.1. A 108-slot waveguide array.

Figure 9.2. The experimental model of the 108-slot waveguide array.

TABLE 9.1. Simulation Parameters for the 108-Slot Waveguide Array.

Figure 9.3. E-plane radiation pattern of the array at 9.20 GHz (0° starts from the z-axis in the yOz-plane).

Figure 9.4. E-plane radiation pattern of the array at 9.50 GHz (0° starts from z-axis in the yOz-plane).

9.1.1.2 Radiation from a Two-Dimensional Slot Antenna Array with 100 Slots

In this section, a 10 × 10 waveguide array with narrow-walled slots is considered and its radiation pattern is characterized with all the mutual coupling effects included [1]. The dimensions and notations of the full model are depicted in Figure 9.5 and Figure 9.6. The waveguide is chosen as the WR-90 waveguide (X-band), with dimensions of 22.86 mm × 10.16 mm, and a wall thickness of 1.00 mm. The center frequency of the array is 9.375 GHz. The plate behind the radiating waveguides is used to fix the array in place. The model used for measurement is shown in Figure 9.7.

We use a single-layer dielectric with a thickness of λ_g as the matched loads for the model. A −20 dB Taylor distribution is used in the array design, and the power absorbed in the loads is approximately 25%. The voltage standing wave ratio (VSWR) is 1.05 for the simulation and experiment.

The array can scan in the H-plane by adjusting the phase of the excitation at the feed point of each radiating waveguide. The normalized radiation pattern of the array is shown in Figure 9.8. In comparison, the computed result by HOBBIES software agrees well with the measured one in the mainlobe region. There are relatively large errors for the sidelobes and backlobes due to the shielding provided by both the support frame of the array and the turntable. A phase difference in the feeds of the adjacent radiating waveguides of about 0.9π has been introduced so that the direction of the major lobe in the H-plane is offset from 0° by about 5°.

Figure 9.5. Top view of the 100-slot waveguide array.

Figure 9.6. Perspective view of the 100-slot waveguide array.

Figure 9.7. Model of the waveguide array used in the measurement.

Figure 9.8. H-plane far-field pattern of the waveguide array (0° starts from the z-axis in the yOz-plane).

The inner and outer surfaces of the array are discretized into 4,840 bilinear patches, and HOBBIES uses two wires for the feed, corresponding to a total of 34,090 unknowns, as shown in Figure 9.9 (a). In contrast, for the RWG basis functions (RWGs) with the discretization shown in Figure 9.9 (b), the number of unknowns is 149,790 (using the λ/10 criterion for the size of a side of a triangle, where λ is the free-space wavelength). This results in about 4.4 times as many unknowns as that for the higher-order basis functions (HOBs) used in HOBBIES. The RWGs require not only more memory than the HOBs, but also the computation time can be up to 85 times longer for the same accuracy of the current when LU decomposition with a complexity of O(N³) is used as the matrix equation solver for both types of basis functions, where N is the number of unknowns. More accurately, HOBBIES with HOBs needs approximately 17 GB of memory (double precision), while with RWGs it needs more than 334 GB of memory. Hence, the solution of practical problems that require large computer resources when using the piecewise RWGs can be executed on desktops using HOBBIES without loss of accuracy.

Figure 9.9. Mesh of narrow-wall slots: (a) bilinear patches, (b) planar triangular patches.

9.1.1.3 Radiation from a Large Elliptical Slot Antenna Array with 1712 Slots

An elliptical waveguide array with 1712 narrow-wall slots is simulated, and its radiation pattern is characterized with all the mutual coupling effects included. The perspective view of the model is shown in Figure 9.10. The waveguide chosen has the dimensions 56.90 mm × 28.45 mm (S band), and a wall thickness of 10.0 mm is assumed. The center frequency of the array is 3.5 GHz. We use five layers of dielectric with a thickness of λ_g for each layer as the matched loads for the model. A −35 dB Taylor distribution is used in the array design.

The normalized radiation pattern of the array is shown in Figure 9.11. In comparison, the computed result by HOBBIES agrees well with the measured one in the mainlobe region. There are large errors for the sidelobes and backlobes because of the background noise in the antenna measuring environment.

Figure 9.10. A 1712-slot waveguide array.

Figure 9.11. E-plane radiation pattern of the 1712-slot waveguide array (0° starts from the x-axis in the xOy-plane).

9.1.2 Radiation from a L-Band Antenna with Complex Composite Structures

In this example, the numerical results from HOBBIES are compared with the measured data for a complex composite structure. The complex structure consists of a dielectric radome containing seven large structural ribs that support an aircraft radome shell along with an L-band antenna array. The goal of this study is to examine the suppression of radome-induced grating lobes as illustrated in [2]. The grating lobes are generated by seven large structural ribs that support an aircraft radome shell in front of an L-band antenna array. The structural ribs form a diffraction grating that introduces grating lobes in the array radiation pattern [2]. The goal here is to demonstrate that using HOBBIES, one can compute results for the radiation pattern that are within several tenths of a dB when compared with measurements for the grating lobe amplitudes. The difference between theory and experiment falls within the resolution of the measurements [2].

The complex composite structure considered here consists of the seven structural ribs bounding the L-band array, as shown in Figure 9.12, which is taken from Figure 3 of [2].

Figure 9.12. L-band antenna array with seven ribs.

The L-band array feeds a diffraction grating comprising seven large ribs. The center rib is the largest, while the rest become progressively smaller in a symmetric fashion about the center rib. These ribs generate grating lobes in the radiation pattern. The tapering of the diffraction grating has an effect on the grating lobes. The grating lobes appear because the distance between consecutive ribs is larger than the wavelength. A rib of a certain size, however, scatters a wave differently from a rib of another size. Thus, each rib has its own element pattern that influences the location and shape of a grating lobe. These structural ribs are fiber-reinforced composite sandwich structures using a honeycomb core. An effective dielectric constant for these structures was computed so that they could be modeled as homogenous structures in HOBBIES.

The main-beam-first-grating-lobe-amplitude ratio is the metric of interest in this study. The main-beam-first-grating-lobe-amplitude ratio computed by HOBBIES was within 0.2 dB for all cases evaluated [2]. The measured and calculated patterns were nearly indistinguishable down to levels nearly 30 dB below the peak of the mainbeam. Figure 9.13 shows an example of the excellent match that was achieved.

Figure 9.13. Comparison of the measured array pattern and HOBBIES calculation.

9.1.3 Radiation from a Microstrip Patch Phased Array

An 11 × 11 microstrip patch phased array is printed on a substrate ε_r = 2.67 and μ_r = 1.0 and is housed in a 520 mm × 580 mm × 7 mm cavity in a ground plane [3], as illustrated in Figure 9.14. The feeding line for each patch has the radius r = 0.48 mm. The dimensions of each patch element are 30 mm × 35.6 mm, and the gaps between any two neighboring elements are 14.0 mm along both the length and width directions.

The entire array is discretized into 121 wires for the feeds and 6,490 bilinear patches for the microstrip surfaces, corresponding to a total of 14,956 unknowns. In contrast, for the RWG basis functions, the number of unknowns required is 75,465 (using the λ/10 mesh criterion). This results in about 5 times as many unknowns as that for the higher order basis functions used in HOBBIES. The memory requirement for the RWG basis functions can be up to 25 times more than that for the HOBBIES, and the simulation time can be up to 125 times longer.

Figure 9.14. Perspective view of the 11 × 11 microstrip patch phased array.

Figure 9.15 shows the radiation patterns of the microstrip patch phased array when the array elements are fed with three different progressive phase shifts, which result in a scan toward the x-direction. The phase shifts of 0°, −15°, and −30° corresponding to the scan angles of about 0°, 7.2°, and 14.5° in the xoz-plane are considered. It is seen that the results from the HOBBIES and a code using the RWG basis functions agree with each other very well. To show the scanning effect more clearly, Figure 9.16 depicts the three-dimensional radiation patterns of the three different phase shifts together with the array. The array can also scan toward the y-direction when it is fed with a proper phase shift along the y-direction.

Figure 9.15. Radiation patterns in the xoz-plane of the microstrip patch phased array, (a) 0° phase shift toward the x-direction, (b) −15° phase shift toward the x-direction, (c) −30° phase shift toward the x-direction (0° starts from the x-axis in the xOz-plane).

Figure 9.16. 3D radiation patterns of the microstrip patch phased array. (a) 0° phase shift toward the x-direction, (b) −15° phase shift toward the x-direction, (c) −30° phase shift toward the x-direction.

9.1.4 Radiation from a Helical Antenna Mounted on a Satellite

The OSTM satellite [4] is simulated, which has two reflector antennas and a helical antenna, as shown in Figure 9.17. The dimensions of the satellite are 6.98 m × 2.01 m × 3.82 m. The helical antenna is simulated at 2.065 GHz and it is fed by a coaxial line. Figure 9.18 shows the computed current distribution and Figure 9.19 plots the three-dimensional (3D) radiation pattern. The number of unknowns of this simulation is 44,577. It took 73 seconds for filling the matrix and 194 seconds for solving the equation using 592 cores on a DAWNING blade system.

Figure 9.17. Simulation model of OSTM satellite: (a) the full model, (b) the helical antenna.

Figure 9.18. Current distribution on the satellite.

Figure 9.19. 3D radiation pattern (in dB) of the helical antenna on the satellite.

9.2 RADAR CROSS SECTION (RCS) CALCULATION OF COMPLEX TARGETS

9.2.1 RCS of Three Benchmarks

In this section, numerical results for the NASA almond, double ogive and that of the truncated cone are presented and compared with the measured results to demonstrate the accuracy of HOBBIES when dealing with scattering problems.

9.2.1.1 RCS of NASA Almond

The monostatic analysis of the NASA almond is considered first. The parametric equations that define the geometry of the NASA almond are well known and available in the literature [5]. The Non-Uniform Rationale B-Spline (NURBS) model and the meshed model are shown in Figure 9.20. The comparison between the computed result and the measurement for 9.92 GHz is shown in Figure 9.21.

9.2.1.2 RCS of a Double Ogive

The monostatic analysis of the metallic double ogive is considered [5]. The NURBS model and the meshed model are shown in Figure 9.22. The comparison between the computed results and the measurements for the double ogive at 9 GHz are shown in Figure 9.23.

Figure 9.20. NASA almond: (a) the NURBS model, (b) the meshed model.

Figure 9.21. VV polarized monostatic RCS of NASA almond at 9.92 GHz (0° starts from the x-axis in the xOy-plane).

Figure 9.22. A double ogive: (a) the NURBS model, (b) the meshed model.

Figure 9.23. Monostatic RCS of the double ogive at 9 GHz: (a) HH polarized, (b) VV polarized (0° starts from the x-axis in the xOy-plane).

9.2.1.3 RCS of a Truncated Cone

This example is an end-capped truncated cone oriented along the z-axis and centered in the plane z = 0 (Figure 9.24). The height of the target is 200 mm. The major diameter is 200 mm, and the minor one is 100 mm [6]. There are several interesting points in this target. First, it shows the RCS response of targets with single curvature (common in structural parts of aircraft, such as the fuselage). It is also important to know the diffraction mechanism along curved edges. Reflection from planar surfaces with curved edges can also be observed. Therefore, this target is especially suitable for the validation of the prediction of objects with flat surfaces delimited by curved edges and for evaluation of curved edges contribution.

This model has been simulated at 7 GHz. The incident direction is perpendicular to the generatrix. Figure 9.25 shows the monostatic RCS pattern of the truncated cone for HH polarization. Three main lobes are clearly defined. Two of them correspond to the specular reflection from the two bases. The minor one corresponds to θ = 0° and the major one to θ = 180°. The different levels are expected due to the different areas of the corresponding bases. The other main lobe corresponds to the angle at which the generatrix is perpendicular to the incident direction. Diffraction from the curved edges becomes important in the intermediate region between the main lobes. The RCS pattern is compared with the measurement result [6], and a good agreement can be found.

Figure 9.24. The NURBS model of the truncated cone.

9.2.2 RCS of a Tank and a Squadron of Tanks

In this example, simulation of the RCS of several tanks is performed. The dimensions of the tanks are 9.9 m × 3.4 m × 2.2 m, as shown in Figure 9.26. The arrows show the plane wave propagating direction and its polarization. The current distribution over the tank surfaces at 200 MHz is given in Figure 9.27. Three near-field cut-planes are plotted in Figure 9.28. The 3D RCS is given in Figure 9.29.

Figure 9.25. HH polarized monostatic RCS pattern of the truncated cone at 7 GHz (0° starts from the z-axis in the xOz-plane).

Figure 9.26. A tank illuminated by a plane wave.

Figure 9.27. Current distribution over the tank surfaces.

Figure 9.28. Near-field cut-planes around the tank model.

Figure 9.29. 3D RCS with the tank model.

The RCS from a tank formation, which consists of five tanks shown in Figure 9.30, is also simulated, and the results are plotted in Figure 9.31 and Figure 9.32. Each tank is staggered 30.0 m behind and 30.0 m to the side of the tank in front of it in the formation. Note that the tanks are considered to be in free space without a ground plane for demonstration purposes.

Figure 9.30. The perspective view of the tank formation.

Figure 9.31. Bistatic RCS in the xOz-plane (0° starts from the x-axis in the xOz-plane).

Figure 9.32. Bistatic RCS in the xOy-plane (0° starts from the x-axis in the xOy-plane).

9.2.3 RCS of an Aircraft and a Formation of Aircrafts

The scattering from a full-size aircraft is simulated here. The aircraft structure is 17.32 m in length, 11.4 m in width, and 3.7 m in height. It is modeled as a perfect electric conductor (PEC) surface. The plane wave is incident from the minus y-axis direction and is polarized along the z-axis. The bistatic RCS of the full-scale airplane is simulated at 1.0 GHz. The wavelength at 1.0 GHz is 0.3 m, and hence, the structure is about 57.7λ long, 38λ wide, and 12.3λ high. The number of unknowns required to analyze this problem is 179,472. It took 23,226.42 seconds for filling the matrix and 147,413.3 seconds for solving the equation using 20 single-core CPUs on a DELL (www.DELL.com) blade computer system (www.em-hobbies.com/platforms.html). Figure 9.33 shows the airplane structure. The induced current over the surface of the airplane is shown in Figure 9.34. Three dimensional RCS is plotted in Figure 9.35 with the aircraft.

Figure 9.33. Model of the aircraft created in HOBBIES.

Figure 9.34. Current distribution over the aircraft surfaces.

Figure 9.35. 3D bistatic RCS of the aircraft.

The RCS of a “V” shape aircraft formation is calculated next. The plane wave is incident from the minus y-axis direction and is polarized along the z-axis. The distances between any two neighboring aircrafts is Δy = 20.0 m along the head direction, Δx = 20.0 m along the wing direction, and Δz = 0.0 m along the height direction. The aircraft formation is shown in Figure 9.36. The RCS is calculated at 1.0 GHz. The number of unknowns for the aircraft formation is 897,360. It took 350,175.86 seconds for filling the matrix and 555,515.51 seconds for solving the equation (totally about 10.5 days) using 456 cores on a HP system (www.em-hobbies.com/platforms.html). A three-dimensional bistatic RCS is plotted in Figure 9.37 with the aircraft. The bistatic RCS of a single aircraft and the aircraft formation in the yOz-plane and the xOy-plane are plotted in Figure 9.38.

9.2.4 RCS Simulation of an Aircraft using a Million Unknowns

To validate the robustness and the stability of the code, the bistatic RCS of a single aircraft is calculated at 6.15 GHz. The full-size aircraft with the mesh is shown in Figure 9.39. The corresponding electrical size of the aircraft is about 237.8 λ × 143.5 λ × 59.86 λ. The aircraft is illuminated by a plane wave incident from the x-axis and is polarized along the z-axis. The numbers of unknowns in this case is 954,618 (approximately one million unknowns) when the structure is meshed at a frequency of 5.95 GHz. The bistatic RCS is given in Figure 9.40. Hence, for this problem approximately, a million by a million complex matrix is calculated and solved using a LU-decomposition using a parallel out-of-core matrix equation solver. This simulation demonstrates that the code is stable and can be used for even more unknowns as long as sufficient hard disk space is available.

To investigate the stability of HOBBIES, this model was simulated twice. The first run took 637,962 seconds, while the second run took 638,365 seconds. The second run took more than about 7 minutes longer than the first run. The relative difference is less than 0.07%, which means that the code is very stable and the results can be repeated even if this project used almost all the hard disk space resources.

Figure 9.36. Aircraft flying in a V formation.

Figure 9.37. 3D bistatic RCS of the aircraft formation.

Figure 9.38. Bistatic RCS of the aircraft and its formation: (a) yOz-plane (0° starts from the y-axis in the yOz-plane), (b) xOy-plane (0° starts from the x-axis in the xOy-plane).

Figure 9.39. The meshed airplane model (mesh frequency is 5.95 GHz) simulation with approximately a million unknowns.

Figure 9.40. Bistatic RCS at ϕ cut-plane (0° starts from the x-axis in the xOz-plane).

9.2.5 RCS of a Ship and a Formation of Ships

In this section, the RCS of a ship and the formation of ships are calculated. The ship structure is 153 m in length and 16.5 m in width. It is modeled as a PEC surface, as shown in Figure 9.41. The plane wave is incident from the x-axis direction and is polarized along the y-axis.

The bistatic RCS of the full-scale ship is simulated at 100 MHz. The formation of five ships is given in Figure 9.42. The structure is considered in free space (not over an ocean surface) for demonstration purposes.

The bistatic RCS of a single ship and the ship formation in the xOz-plane and the xOy-plane are plotted in Figure 9.43 and Figure 9.44, respectively. The current distribution over the surface of a single ship is given in Figure 9.45, and the 3D RCS is given in Figure 9.46, respectively. The current distribution, the near-field around the ships, and the 3D RCS of the ship formation are plotted in Figure 9.47, Figure 9.48, and Figure 9.49, respectively.

Figure 9.41. The meshed ship model.

Figure 9.42. The meshed ship formation model.

Figure 9.43. Bistatic RCS in the xOz-plane (0° starts from the x-axis in the xOz-plane).

Figure 9.44. Bistatic RCS in the xOy-plane (0° starts from the x-axis in the xOy-plane).

Figure 9.45. Current distribution over the surfaces of a single ship.

Figure 9.46. 3D RCS of a single ship.

Figure 9.47. Current distribution over the surfaces of the ships.

Figure 9.48. Near-field distribution around the ships.

Figure 9.49. 3D RCS of the ship formation.

9.3 CONCLUSION

In this chapter, several examples have been presented to provide an overview of various applications of the code, HOBBIES. Numerical results from these examples, along with the comparison between HOBBIES and experimental measurements, show that the parallel in-core and out-of-core integral equation solver provides a very accurate and cost-effective solution for analyzing challenging real-life EM problems in the frequency domain, using modest computational resources, particularly when using the parallel out-of-core solver.

REFERENCES

[1] X.-W. Zhao, Y. Zhang, T. K. Sarkar, S.-W. Ting, and C.-H. Liang. “Analysis of a Traveling-Wave Waveguide Array with Narrow-Wall Slots Using Higher-Order Basis Functions in Method of Moments,” IEEE Antennas and Wireless Propagation Letters, Vol. 8, pp. 1390–1393, 2009.

[2] S. N. Tabet, J. S. Asvestas, and O. E. Allen. “Suppression of Radome Induced Grating Lobes,” 24th Annual Review of Progress in Applied Computational Electromagnetics, Niagara Falls, Canada, pp. 714–718, Apr. 2008.

[3] J.-M. Jin, Z. Lou, Y.-J. Li, N. W. Riley, and D. J. Riley, “Finite Element Analysis of Complex Antennas and Arrays,” IEEE Transactions on Antennas and Propagation, Vol. 56, No. 8, pp. 2222–2240, Aug. 2008.

[4] C. L. Parkinson, A. Ward, and M. D. King, Earth Science Reference Handbook, National Aeronautics and Space Administration, Washington, DC, 2006.

[5] A. Woo, H. Wang, M. Schuh, and M. Sanders, “EM Programmer's Notebook-Benchmark Radar Targets for the Validation of Computational Electromagnetics Programs,” IEEE Antennas and Propagation Magazine, Vol. 35, No. 1, pp. 84–89, Feb. 1993.

[6] R. Fernandez-Recio, A. Jurado-Lucena, B. Errasti-Alcala, D. Poyatos-Martinez, D. Escot-Bocanegra, I. Montiel-Sanchez, “RCS Measurements and Predictions of Different Targets for Radar Benchmark Purpose”, International Conference on Electromagnetics in Advanced Applications, Torino, Italy, pp. 443-446, Sept. 14-18 2009.