The cable mesh reflector usually consists of front cables, rear cables, tie cables, reflective mesh, and ring truss. Its pretension design aims to find a reasonable cable tension distribution, which offers rigidity to form the required parabolic surface. The present twostep structural design technology incorporates a simple structural formfinding design and an integrated structural electromagnetic optimization to achieve high beam pointing accuracy. The simple structural formfinding design starts from the preliminary stage considering surface error requirement. As for the cable mesh reflector, the surface rms error can be expressed as [18]
$$ \delta_{{{\text{rms}},z}} = \frac{1}{{16\sqrt {15} }}\frac{{l^{2} }}{f}\left( {1 + 0.33\frac{{N_{\text{m}} l}}{T}} \right), $$
(1)
where δ
_{rms, z
} is the surface rms error in z direction, f is the focal length, l is the cable dimension, N
_{m} is the mesh tension, and T is the surface cable tension.
Usually, for a preliminary estimation, the surface cable tension T will be set equal to 10 times [19] the mesh tension N
_{m} multiplied by the side length l to suppress pillow deformation [20]. Such that, given the working wavelength (or frequency) and required surface rms error (usually smaller than 1/50 wavelength), the surface cable length can be determined as
$$ l \le \sqrt {{{16\sqrt {15} f\delta_{{{\text{rms}},z}} } \mathord{\left/ {\vphantom {{16\sqrt {15} f\delta_{{{\text{rms}},z}} } {1.033}}} \right. \kern0pt} {1.033}}} . $$
(2)
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.
Supposing that free node i is connected to node t by a cable, the equilibrium equation in z direction for node i can be derived as [21]
$$ \sum\limits_{t} {\frac{{T_{it} }}{{l_{it} }}} (z_{i}  z_{t} ) = 0, $$
(3)
where T
_{
it
} is the tension in element it and l
_{
it
} is the cable length of element it, z
_{
i
} and z
_{
t
} are the coordinates in z direction for node i and t, respectively.
Collecting the equilibrium equations for all free nodes in x, y, and z directions, a matrix form equation can be obtained as follows:
$$ \varvec{A}_{3n \times m} \varvec{T}_{m \times 1} = 0, $$
(4)
where A is the equilibrium matrix, T is the column vector of cable tensions, n is the number of free nodes and m is the number of cables. Usually, for cable net reflectors, the matrix form equilibrium Eq. (4) is statically indeterminate, which has many different cable tension distributions to satisfy this equation.
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 optimization model of this structural design can be written as
$$ \begin{array}{*{20}l} {\text{find}} \hfill & \varvec{\alpha}= (\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{p} )^{\text{T}} , \hfill \\ \hbox{min} \hfill & \left\ {\varvec{T}  \varvec{T}_{0} } \right\{\kern 1pt} {\kern 1pt} {\kern 1pt} , \hfill \\ {\text{s}} . {\text{t}} .\hfill & \varvec{A}_{3n \times m} \varvec{T}_{m \times 1} = 0{\kern 1pt} {\kern 1pt} , \hfill \\ {} \hfill & \varvec{T} = \varvec{V}_{m \times p} \cdot\varvec{\alpha}_{p \times 1} {\kern 1pt} {\kern 1pt} , \hfill \\ \hfill & \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{T} } \le \varvec{T} \le \bar{\varvec{T}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} , \hfill \\ \end{array} $$
(5)
where α is the column vector of combination coefficients, p is the number of independent states of selfstress, T
_{0} is the column vector of mean values of cable tensions, V is the matrix of independent states of selfstress, \( \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{T} } \), \( \bar{\varvec{T}} \) are the lower and upper limits of cable tensions, respectively. The object in this model is to find a uniform tension distribution for cable mesh reflectors. With this optimization, the pretension design of cable nets can be achieved. Other formfinding methods can also be employed in the first step.
The next step is to improve the electromagnetic performance with high beam pointing accuracy. In the beginning, all of the surface nodes are in nominal state, and poor beam pointing accuracy can be observed under circular polarizations. The high beam pointing accuracy is optimized by an integrated structural electromagnetic design, which directly chooses the electromagnetic performance as design object. The integrated implementation is accomplished by altering some cable lengths, usually tie cables, which shapes the front cable surface to obtain high beam pointing accuracy. This implementation is similar with the shape control concept [23], and beam pointing accuracy makes it different. The integrated structural electromagnetic optimization minimizing beam squint angle can be expressed as
$$ \begin{array}{*{20}l} {\text{find}}\hfill & \varvec{l} = (l_{1} ,l_{2} , \ldots ,l_{N} )^{\text{T}} , \hfill \\ \hbox{min} \hfill & \theta_{0} , \hfill \\ {\text{s}} . {\text{t}} .\hfill & D(\varvec{l}) \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{D} \;, \hfill \\ {} \hfill & \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{T} } \le \varvec{T} \le \bar{\varvec{T}}\;, \hfill \\ {} \hfill &\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{l} } \le \varvec{l} \le \bar{\varvec{l}}\;, \hfill \\ \end{array} $$
(6)
where l is the column vector of dimensions of altered tie cables, N is the number of tie cables, θ
_{0} is the beam squint angle represents beam pointing accuracy, D is the maximum directivity with its lower limit D, \( \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{l} } \), \( \bar{\varvec{l}} \) are the lower and upper limits of design variables, respectively. The directivity constraint is added in the optimization model to ensure better electromagnetic performance during iterations. Strictly speaking, other electromagnetic performance can also be added into the constraints in the optimization model.
In this solution procedure, in order to avoid the computation of directivity values at different radiation angles during iterations to determine the beam pointing direction, the optimization model in Eq. (6) can be converted to a new procedure which maximizes the boresight directivity at boresight direction (0, 0). It is rewritten as
$$ \begin{array}{*{20}l} {\text{find}}\hfill & \varvec{l} = (l_{1} ,l_{2} , \ldots ,l_{N} )^{\text{T}} , \hfill \\ \hbox{min} \hfill &  D(0,0)\;, \hfill \\ {\text{s}} . {\text{t}} .\hfill & D(0,0) \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{D} \;, \hfill \\ {} \hfill & \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{T} } \le \varvec{T} \le \bar{\varvec{T}}\;, \hfill \\ {} \hfill & \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{l} } \le \varvec{l} \le \bar{\varvec{l}}\;, \hfill \\ \end{array} $$
(7)
where D(0, 0) is the boresight directivity and the boresight direction is defined at (0, 0). The equivalence of the optimization model in Eqs. (6) and (7) is easily understood from the viewpoint of electromagnetism and it will be shown in section 3.
In order to solve this optimization model in Eq. (7), sensitivity analysis is employed in this implementation. The optimization model can be converted into a new one by expanding the object into a secondorder Taylor series and the constraint into a firstorder Taylor series. The new optimization model in the ith iteration can be illustrated as
$$ \begin{array}{*{20}l} {\text{find}}\hfill & \Delta \varvec{l} = (\Delta l_{1} ,\Delta l_{2} , \ldots ,\Delta l_{N} )^{\text{T}} , \hfill \\ \hbox{min} \hfill &  D(0,0) =  D^{(i)}  \varvec{G}^{\text{T}} \Delta \varvec{l}  \frac{1}{2}\Delta \varvec{l}^{\text{T}} \varvec{B}\Delta \varvec{l}\;, \hfill \\ {\text{s}} . {\text{t}} .\hfill & D^{(i)} + \varvec{G}^{\text{T}} \Delta \varvec{l}  \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{D} \ge 0\;\;, \hfill \\ {} \hfill & \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{T} } \le \varvec{G}_{t}^{\text{T}} \Delta \varvec{l} + \varvec{T}^{(i)} \le \bar{\varvec{T}}\;, \hfill \\ {} \hfill & \varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{l} }  \varvec{l}^{(i)} \le \Delta \varvec{l} \le \bar{\varvec{l}}  \varvec{l}^{(i)} \;, \hfill \\ \end{array} $$
(8)
where ∆l is the increment column vector of cable dimensions, l
^{(i)} is the vector of cable dimensions in the ith iteration, D
^{(i)} is the boresight directivity in the present state, T
^{(i)} is the vector of cable tensions in the present state, G is the gradient vector of boresight directivity with respect to cable dimensions, B is the BFGS update Hessian matrix, G
_{
t
} is the gradient matrix of cable tensions with respect to cable dimensions.
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.
Besides, in order to increase the efficiency of this optimization and avoid the direct computation of Hessian matrix, an update BFGS formula [24] is employed in the iteration. The approximation to Hessian matrix in the ith iteration is denoted by B
_{
i
}, and the wellknown BFGS formula is defined as [24]
$$ \varvec{B}_{i + 1} = \varvec{B}_{i}  \frac{{\varvec{B}_{i} \varvec{P}_{i} \varvec{P}_{i}^{\text{T}} \varvec{B}_{i} }}{{\varvec{P}_{i}^{\text{T}} \varvec{B}_{i} \varvec{P}_{i} }} + \frac{{\varvec{Y}_{i} \varvec{Y}_{i}^{\text{T}} }}{{\varvec{Y}_{i}^{\text{T}} \varvec{P}_{i} }}, $$
(9)
where P
_{
i
} is the difference of design variables between last two iterations, and Y
_{
i
} is the difference of the gradient vectors of object function between last two iterations.
$$ \varvec{P}_{i} = \Delta \varvec{l}^{(i)} , $$
(10)
$$ \varvec{Y}_{i} = \varvec{G}^{(i + 1)} {}\varvec{G}^{(i)} , $$
(11)
where ∆l
^{(i)} is the difference of cable dimensions between last two iterations, G
^{(i)} is the gradient vector of boresight directivity with respect to cable dimensions in the ith iteration. The gradient vector is updated by sensitivity analysis during the iterations [25].
By adding the solution of the optimization model in Eq. (8) to the cable dimensions in the present state, the updated column vector of cable dimensions in the next iteration can be obtained as
$$ \varvec{l}^{(i + 1)} = \varvec{l}^{(i)} + \Delta \varvec{l}^{(i)} . $$
(12)
The procedure of this twostep structural design technology is shown in Fig. 1. It is described as follows.
 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.