 Original Article
 Open Access
Twostep Structural Design of Mesh Antennas for High Beam Pointing Accuracy
 Shuxin ZHANG^{1, 2},
 Jingli DU^{1},
 Wei WANG^{1}Email authorView ORCID ID profile,
 Xinghua ZHANG^{2} and
 Yali ZONG^{1}
https://doi.org/10.1007/s1003301701340
© The Author(s) 2017
 Received: 9 May 2016
 Accepted: 2 April 2017
 Published: 21 April 2017
Abstract
A welldesigned reflector surface with high beam pointing accuracy in electromagnetic performance is of practical significance to the space application of cable mesh reflector antennas. As for space requirements, circular polarizations are widely used in spaceborne antennas, which usually lead to a beam shift for offset reflectors and influence the beam pointing accuracy. A twostep structural design procedure is proposed to overcome the beam squint phenomenon for high beam pointing accuracy design of circularly polarized offset cable mesh reflectors. A simple structural optimal design and an integrated structural electromagnetic optimization are combined to alleviate the beam squint effect of circular polarizations. It is implemented by cable pretension design and adjustment to shape the offset cable mesh surface. Besides, in order to increase the efficiency of integrated optimization, an update BroydenFletcherGoldfarbShanno (BFGS) Hessian matrix is employed in the optimization iteration with sequential quadratic programming. A circularly polarized offset cable mesh reflector is utilized to show the feasibility and effectiveness of the proposed procedure. A high beam pointing accuracy in order of 0.0001º of electromagnetic performance is achieved.
Keywords
 Cable mesh reflector antennas
 Structural design
 Beam squint
 Beam pointing accuracy
1 Introduction
In recent years, the stringent requirements on large space reflectors become demanding for high electromagnetic performance [1]. As for space applications, circular polarizations are usually used in spaceborne antennas. With circularly polarized feeds, there exists a beam squint phenomenon in offset reflector antennas [2]. The squint angle, which is manifested by a small beam shift of the radiation pattern in the plane perpendicular to the principal offset plane, can significantly affect the beam pointing accuracy. As one of the most widely used space antennas, cable mesh reflector antenna has attracted much attention due to its advantages of large diameter, light weight, and reasonable cost [3]. Similarly with the smooth solid reflectors, the beam squint phenomenon can also be observed in offset cable mesh reflector antennas with reflecting mesh leakage [4, 5]. The beam squint angle should be taken into account for space applications such as satellite communications, deepspace telemetry, and radio astronomy [2], which concentrate more on beam pointing accuracy. With the stringent requirements on space reflector antennas, the compensation technology to overcome the antenna pattern degradation including beam squint to achieve high pointing accuracy becomes more demanding [6].
Since the simple formula which accurately predicts the squint angle in circularly polarized offset reflectors was proposed by ADATIA and RUDGE [7], the beam squint phenomenon and its compensation method have attracted many authors’ interests. A squint compensation method by properly tilting the feed to make the interpreted angle between the incident beam and the radiated beam zero is a natural choice for symmetrical reflectors with offfocus feeds [8]. A squint free approach for symmetrical dual reflector antennas is also proposed by properly choosing geometrical parameters [9]. Furthermore, XU and RAHMATSAMII [2] summarized the beam squint compensation methods, and proposed a compensation technology by optimally displacing circularly polarized feeds in the perpendicular plane to obtain high beam pointing accuracy. However, these methods in Refs. [2, 7–9] are presented from the simple electromagnetic disciplinary, and they are just practical for undistorted reflectors in the nominal state for preliminary design. In actual engineering, space reflectors including cable mesh antennas are easily susceptible to surface distortion under thermal load and other impacts, which enlarge the beam squint angle and seriously affect the beam pointing accuracy. Simply displacing and tilting the antenna feed cannot thoroughly compensate the distorted electromagnetic performance in actual engineering. Another consideration should be taken into account is that feed remains on focus with a satisfactory reflector surface is preferred due to the limited size in satellites. How to produce a cable mesh reflector with high beam pointing accuracy in electromagnetic performance is an urgent problem for space applications.
As for structural design of cable mesh reflectors, pretension design of cable nets is an important process to obtain the required reflector surface. Recently, there are several methods which investigate the formfinding analysis for cable mesh reflectors, such as the method presented by TANAKA, et al [10], optimal design method of initial surface in Ref. [11], simple technique in Ref. [12], numerical formfinding method proposed by MORTEROLLE, et al [13] to ensure uniform tension, formfinding analysis with PZT actuators [14] and pretension design under multiuncertainty [15]. These methods aim to design a surface profile with minimum or zero rootmeansquare (rms) error to ensure its surface accuracy. Although the reflector shape can be obtained with high surface accuracy by these methods, its beam pointing accuracy cannot be easily guaranteed, even for circularly polarized feeds. Thus, there rises a problem that is it possible to provide a pretension structural design considering electromagnetic performance to obtain high beam pointing accuracy for circularly polarized feeds? The integrated structural electromagnetic design concept [16, 17] inspires us with a combined procedure, which makes a pretension design from multidisciplinary viewpoint of structure and electromagnetism.
The main purpose of this paper is to present a twostep structural design technology for circularly polarized offset cable mesh reflectors with high beam pointing accuracy. The twostep pretension design combines a simple structural design and an integrated structural electromagnetic optimization. With this technology, high electromagnetic performance especially high beam pointing accuracy can be achieved in the antenna structural design. This technology not only can compensate the beam squint angle of circular polarizations, but also can produce a welldesigned cable mesh reflector with onfocus feeds. Comparing with the compensation methods proposed by electromagnetism designers, the limited weakness of aforementioned methods can be overcame.
This paper is organized as follows. Section 2 of this study outlines the procedure of the twostep structural design technology. In this technology, an update BroydenFletcherGoldfarbShanno (BFGS) Hessian matrix is employed to increase the efficiency of optimization iteration. In section 3, a circularly polarized offset cable mesh reflector is utilized to show the feasibility and effectiveness of this procedure with an onfocus feed to achieve high beam pointing accuracy in electromagnetic performance. The major achievements are summarized in section 4.
2 Twostep Structural Design Procedure
After determining the surface cable length l, the number division in radius can be obtained in preliminary design. Thus, with the required parabolic surface equation, the predesigned surface nodal positions in front and rear cable nets can be calculated.
In the following structural design, in order to obtain a circularly polarized cable mesh reflector with high beam pointing accuracy, the predesigned nodes are firstly assumed in the nominal undistorted state. As mentioned before, there exists a beam squint phenomenon for this offset antenna. Then, with the integrated structural electromagnetic optimization, the beam squint phenomenon will be compensated.
To determine the cable tension, the Singular Value Decomposition (SVD) is performed on the equilibrium matrix A, and the cable tensions can be expressed as the linear combination of the independent states of selfstress [22]. With optimizing the combination coefficients of multiple states of selfstress, the cable tensions can be obtained in this nominal state.
The derivation of G is based on two sensitivities  one is the electromagnetic sensitivity of boresight directivity with respect to surface nodal displacements, and the other is the structural sensitivity of surface nodal displacements with respect to cable dimensions. Its expression is illustrated in Refs. [17, 23]. The constraint gradient matrix G _{ t } is based on structural sensitivity analysis of cable tensions with respect to cable dimensions [23]. By using the nonlinear optimization function—quadprog in MATLAB, this optimization model can be solved.
 Step 1:

Provide the initial parameters of cable mesh reflector, including the diameter, focal length, offset height, mesh tension, working frequency, and feed polarization;
 Step 2:

Perform the preliminary design by the relationship between surface rms error and cable length;
 Step 3:

Obtain the equilibrium equation in the nominal state;
 Step 4:

Perform SVD operation to obtain the independent states of selfstress;
 Step 5:

Solve the pretension optimization model in (5);
 Step 6:

Perform structural and electromagnetic (EM) sensitivity analysis;
 Step 7:

Approximate Hessian matrix using BFGS update formula;
 Step 8:

Update cable dimensions;
 Step 9:

Obtain the structural and EM performance in the present state;
 Step 10:

Does the EM performance satisfy the convergence criterion? If no, go to Step 6, otherwise, export the optimum design.
It should be mentioned that the implementation from Step 3 to Step 5 belongs to the structural pretension design, and the procedure from Step 6 to Step 10 is a typical integrated structural electromagnetic optimization design. With this twostep structural design, an offset cable mesh reflector with high beam pointing accuracy under circular polarization will be obtained.
3 Simulation and Application
Cable mesh reflector specifications
Items  Value or character 

Reflector type  Single offset parabola 
Aperture diameter d/m  9.23 
Focal length f/m  6 
Offset height h/m  5 
Minimum distance between the front and rear cable nets h _{e}/m  0.2 
Young’s modulus of cables E/GPa  20 
Cable crosssectional area A/mm^{2}  3.14 
Mesh tension N _{m} /(N m^{−1})  2.0 
Frequency/GHz  2 
CosineQ feed Q _{ x } Q _{ y }  8.338 
Polarization  RCP 
Feed tilt angle ψ _{0}/(°)  41.64 
Feed position  On focus 
With the structural pretension design, a surface with uniform tie cable tension distribution can be obtained and all of the surface nodes are located at their nominal states. In the next, the electromagnetic performance is examined for this circular polarization. Beam squint occurs in this circularly polarized offset cable mesh reflector antenna. In the nominal state for RCP feed illumination, there exists a linear phase shift across the reflector aperture and the phase in the left side aperture region is lagging compared with the phase in the right side aperture region. The radiated lefthand circularly polarized (LCP) beam, which is launched from the RCP feed and reflected by this reflector, squints toward the right in xz plane and produces a negative squint angle.
Optimal results with different convergence criterions
Convergence criterion/ε  10^{–4}  10^{–5}  10^{–6}  Nominal state 

Boresight directivity/dB  43.408 1  43.408 7  43.408 9  43.353 3 
Squint angle/(º)  −0.007 4  −0.002 9  −0.000 6  −0.075 1 
Iteration number  72  82  96  – 
Major parameters of far field patterns with RCP illumination
Items  Max Directivity/dB  HPBW^{a}/(º)  Left sidelobe Level/dB  Right Sidelobe level/dB  Beam Squint/(º) 

Exact Hessian  43.408 9  1.261  −28.754  −28.787  −0.0006 
BFGS Hessian  43.408 8  1.261  −28.899  −28.652  −0.0014 
Considering the far field patterns in Fig. 4 and the optimization model in Eq. (7), it can be seen that although the constraints of the other electromagnetic performance such as sidelobe levels and crosspolarization are not added in the optimization model, the simulation result in Fig. 4 shows very satisfactory far field patterns in sidelobes and crosspolarization pattern. This can be explained that the effects of surface error on boresight directivity and the other performance are harmonious; the nonuniform phase distribution will produce a lower directivity and higher sidelobe levels; as the phase distribution across the reflector aperture becomes more uniform, the antenna electromagnetic performance including boresight directivity, sidelobe levels and crosspolarization will become better.
From the application, it can be concluded that a welldesigned cable mesh reflector with high beam pointing accuracy in electromagnetic performance is obtained by a twostep structural design. This beam squint free technology is accomplished by structural design to shape surface with a uniform phase distribution in the aperture plane, and the linear phase shift caused on the polarized components of the incident field is thus reduced. This procedure benefits the radiation pattern with no need to displace feed position and orientation.
From the above comparative simulation between BFGS update Hessian matrix and exact Hessian matrix by secondorder derivative, it can be seen that the BFGS approximation matrix can provide less iteration time and a little worse electromagnetic performance with maximum directivity in accuracy of 0.000 1 dB than the exact one in the cable mesh reflector antenna design.
Compared with the previous compensation methods, this technology can not only compensate the beam squint angle with an onfocus circular polarized feed, but also provide a welldesigned surface with high beam pointing accuracy considering actual engineering. The mentioned pretension structural design can also be improved with considering electromagnetic performance. A statement should be addressed that the drawback of this method is that the procedure cannot handle both two circular polarizations simultaneously, which is also the drawback of other previous compensation methods.
4 Conclusions
 (1)
Less iteration time and a little worse electromagnetic performance with maximum directivity in accuracy of 0.000 1 dB than the exact one are provided by BFGS approximation Hessian matrix in the twostep structural design. A helpful guideline for the cable mesh reflector antennas design can be presented.
 (2)
A tiltlike surface deformation to achieve a uniform phase distribution in reflector aperture for circularly polarized offset cable mesh reflector antennas is provided in the optimal structural design, and the electromagnetic performance including boresight directivity, beam squint angle, sidelobe levels and crosspolarization approaches better as the phase distribution becomes uniform.
 (3)
Even though the other antenna electromagnetic performance besides boresight directivity is not added in the multidisciplinary optimization model, once the boresight directivity is optimized as its extremely maximum value with sufficiently small convergence criterion, the other performance will also be made as an acceptable value due to the electromagnetism property.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 HOFERER R A, RAHMATSAMII Y. Subreflector shaping for antenna distortion compensation: an efficient FourierJacobi expansion with GO/PO analysis[J]. IEEE Transactions on Antennas and Propagation, 2002, 50(12): 1676–1687.Google Scholar
 XU S, RAHMATSAMII Y. A novel beam squint compensation technique for circularly polarized conicsection reflector antennas[J] IEEE Transactions on Antennas and Propagation, 2010, 58(2): 307–317.Google Scholar
 ENTEKHABI D, NJOKU E G, O’NEILL, et al. The soil moisture active passive (SMAP) mission[J]. Proceedings of the IEEE, 2010, 98(5): 704–716.Google Scholar
 RAHMATSAMII Y, LEE S W. Vector diffraction analysis of reflector antennas with mesh surfaces[J]. IEEE Transactions on Antennas and Propagation, 1985, 33(1): 76–90.Google Scholar
 MIURA A, RAHMATSAMII Y. Spaceborne mesh reflector antennas using complex weaves: extended PO/periodicMoM analysis[J]. IEEE Transactions on Antennas and Propagation, 2007, 55(4): 1022–1029.Google Scholar
 XU S, RAHMATSAMII Y, IMBRIALE W A. Subreflectarrays for reflector surface distortion compensation[J]. IEEE Transactions on Antennas and Propagation, 2009, 57(2): 364–372.Google Scholar
 ADATIA N A, RUDGE A W. Beam squint in circularly polarized offset reflector antennas[J]. Electronic Letter, 1975, 11(21): 513–515.Google Scholar
 DUAN D W, RAHMATSAMII Y. Beam squint determination in conicsection reflector antennas with circularly polarized feeds[J]. IEEE Transactions on Antennas and Propagation, 1991, 39(5): 612–619.Google Scholar
 EIHARDT K, WOHLLEBEN R, FIEBIG D. Compensation of the beam squint in axially symmetric, large dual reflector antennas with largeranging laterally displaced feeds[J]. IEEE Transactions on Antennas and Propagation, 1990, 38(8): 1141–1149.Google Scholar
 TANAKA H, SHIMOZONO N, and NATORI M C. A design method for cable network structures considering the flexibility of supporting structures[J]. Trans. Japan Soc. Aero. Space Sci., 2008, 50(170): 267–273.Google Scholar
 YANG B, SHI H, THOMSON M, et al. Optimal design of initial surface profile of deployable mesh reflectors via static modeling and Quadratic programming[C]//50th AIAA/ASME/ASCE/AHS/ASC/ Structures, Structural Dynamics, and Materials conference, Palm Springs, California, AIAA 20092173, May 4–7 2009: 1–9.Google Scholar
 LIW W, LI D, YU X, JIANG J. Exact mesh shape design of large cablenetwork antenna reflectors with flexible ring truss supports[J]. Acta Mechanica Sinica, 2014, 30(2): 198–205.Google Scholar
 MORTEROLLE S, MAURIN B, QUIRANT J, et al. Numerical formfinding of geotensoid tension truss for mesh reflector[J]. Acta Astronautica, 2012, 76: 154–163.Google Scholar
 WANG Z, LI T, DENG H. Formfinding analysis and active shape adjustment of cable net reflectors with PZT actuators[J]. Journal of Aerospace Engineering, 2014, 27: 575–586.Google Scholar
 DENG H, LI T, WANG Z. Pretension design for space deployable mesh reflectors under multiuncertainty[J]. Acta Astronautica, 2015, 115, 270–276.Google Scholar
 PADULA S L, ADELMAN H M, BAILEY M C, et al. Integrated structural electromagnetic shape control of large space antenna reflectors[J]. AIAA Journal, 1989, 27(6): 814–819.Google Scholar
 ZHANG S, DU J, DUAN B, YANG G,, et al. Integrated structuralelectromagnetic shape control of cable mesh reflector antennas[J]. AIAA Journal, 2015, 53 (5): 1395–1398.Google Scholar
 HEDGEPETH J M. Accuracy potentials for large space antenna reflectors with passive structure[J]. Journal of Spacecraft and Rockets, 1982, 19(3): 211–217.Google Scholar
 TIBERT A G, and PELLEGRINO S. Deployable tensegrity reflectors for small satellites[J]. Journal of Spacecraft and Rockets, 2002, 39(5): 701–709.Google Scholar
 DATASHVILI L, BAIER H, SCHIMITSCHEK J, et al. High precision large deployable space reflector based on pilloweffectfree technology[C]//48th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii, 23–26 April 2007: 1–10.Google Scholar
 TIBERT A G. Optimal design of tension truss antennas[C]//44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Norfolk, VA, 7–10 April, 2003: 1–11.Google Scholar
 TRAN H C, LEE J. Formfinding of tensegrity structures with multiple states of selfstress[J]. Acta Mech., 2011, 222, 131–147.Google Scholar
 DU J, ZONG Y, BAO H. Shape adjustment of cable mesh antennas using sequential quadratic programming[J]. Aerospace Science and Technology, 2013, 30: 26–33.Google Scholar
 ZHANG L., GUO F., LI Y. et al. Global dynamic modeling of electrohydraulic 3UPS/S parallel stabilized platform by bond graph [J]. Chinese Journal of Mechanical Engineering, 2016, 29 (6): 1176–1185.Google Scholar
 LIU S., DAO J., LI A., et al. Analysis of frequency characteristics and sensitivity of compliant mechanisms[J]. Chinese Journal of Mechanical Engineering, 2016, 29 (4): 680693.Google Scholar