# Form Finding of Asymmetric Cable Network Reflector with Flexible Ring Truss
## 2 Form-Finding Method for Asymmetric Cable Network Reflector
### The Optimal Design of the Pre-Stress in Cable Nets  
In this sub-section, we describe a procedure to obtain the proper pre-stress to form an asymmetric cable network reflector. Figure 5 shows the finite element model of the front reflector for a typical large cable network antenna. The surfaces are filled with triangular facets. Since the reflector is a central symmetric structure, the preload is also a central symmetric force. The unique cable nets are shown as Figure 6, which are one-twelfth those in Figure 5.  
From Equation (13), we can note that \(Q\) is the key component in obtaining the coordinates of the free nodes. The force density of each cable net is set as the design variable and an even distribution of the pre-stress is set as the objective function. The optimization problem for the form finding of the cable network reflector is formulated using the force density method. In this work, GA is used to find the optimal values of the force density. The GA [29; 30] is an optimization technique based on random search that is guided by the principles of evolution and natural genetics. Moreover, the GA has a large amount of implicit parallelism [31; 32], which is suitable for implementation. The optimization problem for the form finding of the cable network reflector can be expressed as:  
\[ \begin{array}{ll}\text{find}&q_{1},q_{2},q_{3},\cdots,q_{n}\\ \text{Min}&f=F_{Max}/F_{Min},\\ S.t.&C^{T}\cdot Q\cdot C\cdot x=f_{x},\\ &C^{T}\cdot Q\cdot C\cdot y=f_{y},\\ &z=\frac{x^{2}+y^{2}}{4f}-\eta,\\ &0<q_{i}\leq[q]\end{array} \]  
where, \(q_{1}\), \(q_{2}\), \(q_{3}\),..., \(q_{n}\) are the column vectors of the force densities for the cable net structure and \(f\) is max-min tension ratio of the cable net. The objective of the GA is to  
Figure 5: The finite element model for the antenna reflector. Figure 6: The unique cable nets. search for the appropriate force density values \(q_{1}\), \(q_{2}\), \(q_{3}\), \(\ldots\), \(q_{n}\), such that the max-min tension ratio is minimized. The obtained GA solution is substituted into Equation (12) to obtain the optimal reflector of the antenna. An element of the cable net structure is shown in Figure 7. Using the optimal force density values, we can obtain the pre-stress in the front reflector, given as:  
\[s_{i}=q_{i}^{*}l_{i} \tag{17}\]  
where \(q_{i}^{*}\) is the optimal force density in the cable \(i\) and \(l_{i}\) is the cable length. Then, from Equation (10), the equilibrium equations in the \(z\)-direction for the front reflector can be expressed as:  
\[ \begin{split}(C^{F})^{T}\cdot Q^{F}\cdot(C^{F}\cdot Z^{F}+C_{f} ^{F}\cdot Z^{F}{}_{f})+F^{f}{}_{z}=0\\ F_{z}^{F}=-(C^{F})^{T}\cdot Q^{F}\cdot(C^{F}\cdot Z^{F}+C_{f}^{F} \cdot Z^{F}{}_{f})\end{split}  \tag{18}\]  
where \(C^{F}\), \(C_{f}^{F}\), \(Q^{F}\), and \(Z^{F}\) are the topological relation matrix corresponding to the front reflector. \(F_{z}^{F}\) is equal to the force on the front reflector by the tension ties. The rear reflector only provides support for the front reflector and does not require high surface accuracy. Therefore, in order to reduce the quantity of calculations, the cable tension of the rear reflector can be given by:  
\[s_{j}^{\prime}=\frac{f_{2}l^{\prime}_{j}}{f_{1}l_{i}}s_{i} \tag{19}\]  
where \(l^{\prime}_{j}\) is the cable length in the rear reflector corresponding to the cable length \(l_{i}\) in the front reflector. Next, the static equilibrium of the cable nets in the rear reflector is verified. In this process, an arbitrary node \(i\) of the rear reflector is considered to be the object. The resultant force in the \(x\), \(y\)-directions for node \(i\) can be computed as:  
\[\left\{ \begin{array}{l}F_{xi}^{\prime}=\sum_{1}^{R}\frac{x_{i}^{\prime}-x_{ i}^{\prime}}{l^{\prime}_{i,k}}s^{\prime}{}_{i,k}\\ F_{yi}^{\prime}=\frac{R}{1}\frac{y_{k}^{\prime}-y_{i}^{\prime}}{l^{\prime}_{ i,k}}s^{\prime}{}_{i,k}\end{array} \right. \tag{20}\]  
Figure 7: An element of a cable net structure.