# Form Finding of Asymmetric Cable Network Reflector with Flexible Ring Truss
## 4 Numerical Simulations  
In this section, to demonstrate the effectiveness of the proposed method, numerical examples for the design of an asymmetric cable network antenna are presented. A finite element model of the cable mesh antenna structure was built with the ANSYS software. Three views of the antenna structure are illustrated in Figure 9. The material parameters of the subject are given in Table 1 and the structural dimensions of the subject are as follows:  
The diameter of the aperture: 16 m.  
The focal length of the front reflector: 9.5 m.  
The focal length of the rear reflector: 304.2 m.  
Number of cable networks: 1555 (Front: 672, Rear: 672, Tie cables: 211).  
Figure 8: Flowchart of the proposed method using an iterative technique.  
Figure 10: The contour nodal displacement solution of the front reflector: (**a**) the initial iteration, (**b**) the first iteration, (**c**) the second iteration, and (**d**) the third iteration.  
Figure 11: Cont.  
Figure 10 shows the contour nodal displacement of the front reflector which was calculated in ANSYS during successive iterations. From the figures, we can see that the maximum nodal displacement in the second iteration is the smallest, and that the nodal displacement increases in the third iteration. Hence, we set the second iteration as the termination iteration.  
Figure 11 shows the contour nodal displacement of the supporting truss during successive iterations. From the figures, we can note that the deformation of the support truss decreases with successive iterations and the second iteration has the smallest nodal displacement.  
The differences between the free node positions of the objective shapes and that of the shapes at the equilibrium states are summarized in Table 2.  
From Table 2, it can be observed that the free nodal position error for the conventional design approach is large as it does not consider the ring truss. Further, the result of the conventional method does not satisfy the surface accuracy requirement. In the case of the proposed method, with finite iterative steps, the tension distribution is uniform. Further, there is an obvious improvement in reflector accuracy. Compared with the conventional design method, the proposed design method achieves an accuracy improvement of 71.69% and 89.95% for the first and second iteration, respectively. We can also observe that the free nodal displacement error was increased in the third iteration, and hence the algorithm was stopped.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multirow{2}{*}{**The Initial Reflector Data**} & \multicolumn{4}{c}{**Present Approach**} \\ \cline{3-5}  & & **1 st Iteration** & **2 nd Iteration** & **3 rd Iteration** \\ \hline Max-min distributed tension & 17.06/3.53 & 27.39/6.40 & 22.24/5.70 & 27.90/5.00 \\ in the front reflector (\(N\)) & & & & \\ Max-min tension ratio of & 4.82 & 4.28 & 3.90 & 5.58 \\ cable net & 7.56 \(\times\) 10\({}^{-3}\) & 2.14 \(\times\) 10\({}^{-3}\) & 0.76 \(\times\) 10\({}^{-3}\) & 1.95 \(\times\) 10\({}^{-3}\) \\ Improvement of & \multirow{2}{*}{—} & \multirow{2}{*}{71.69\%} & \multirow{2}{*}{89.95\%} & \multirow{2}{*}{74.21\%} \\ surface accuracy & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the design results using the proposed and the conventional method.  
Figure 11: The contour nodal displacement solution of the supporting truss: (**a**) the initial iteration, (**b**) the first iteration, (**c**) the second iteration, and (**d**) the third iteration.