# Bionic Design of the Vertical Bracket of Wide Angle Auroral Imager by Additive Manufacturing
## 3 Bionic Optimization Design of the Vertical Bracket of Wide Angle Auroral Imager
### Size Optimization  
In order to obtain the optimal model, size optimization has carried on for the bionic design model of the vertical bracket. Due to the good symmetry in geometric morphology, quarter of the vertical bracket is considered as the design object. Thicknesses of 11 zones (\(t_{i}\)), as shown in Figure 3c, are considered as the design variables in the size optimization. The initial values and the lower and upper bounds of all design variables are listed in Table 1. It should be mentioned that the increment of all design variables is 0.5 mm.  
To reduce the structural weight as much as possible, the structural mass \(m\) is chosen as the objective function in the size optimization procedure. \(U\) represents the global displacement, \(K\) represents the global stiffness matrix, and \(F\) represents the external load of the structure. The first-order fundamental frequency \(f_{e}\) of the bionic model should not be less than that of the original design. As for the performance constraints, the first-order fundamental frequency \(f_{e}\) is restricted [42]. In summary, the optimal design problem may be expressed as:  
Figure 4: Sketches of three typical animal lower limb bones. (**a**) Tiger. (**b**) Ostrich. (**c**) Human. (**d**) Sketch of human femur.  
\[t=\{t_{i}\},i=1,2,3,\cdots,11\] (4) Minimize: \[m\] (5) Subject to: \[KU=F\] \[f_{e}\leqslant\overline{f_{e}} \tag{6}\] \[t_{i}\leqslant t_{i}\leqslant\overline{t_{i}}\]  
It can be seen from the sectional profiles of the original design and the optimized design (as shown in Figure 5) that since a rotatable imaging system is installed in the middle of the tracking axis system, the upper part of the two design schemes adopts U-shaped structure, which is convenient for the rotation of the imaging system. The upper part of the original design adopts solid structure, the lower part adopts conical structure, and inclined stiffeners are used to strengthen the structural stiffness. The optimized design draws inspiration from the femur and refers to the shape of the femur. The upper structure is convex at both ends and concave in the middle. The upper and lower parts are wide and the middle is narrow. Further, according to the distribution of femur bone density, it is determined that the middle part is a solid structure, and the upper and lower parts are thin-walled structure. Then, the optimal size of each section is determined by size optimization, and the optimal design scheme is obtained. A comparison of the original and optimized designs is illustrated in Table 2. The mass of the original structure and the bionic optimized structure are 11.2 kg and 8.6 kg, respectively. Benefiting from structural optimization and bionic optimization design, the mass of the vertical bracket is reduced by 23% compared with the original design.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Zone** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline Lower bound \(t_{i}\) (mm) & 1 & 1 & 1 & 1 & 10 & 1 & 1 & 1 & 1 & 1 & 1 \\ Initial value (mm) & 2 & 2 & 5 & 5 & 10 & 1 & 1 & 1 & 1 & 1 & 1 \\ Upper bound \(\overline{t_{i}}\) (mm) & 10 & 10 & 15 & 15 & 20 & 10 & 10 & 10 & 10 & 10 \\ Optimized value (mm) & 4.5 & 4.5 & 4 & 4 & 15 & 5 & 5 & 4.5 & 6.5 & 4.5 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design variables in size optimization.  
Figure 5: The sectional profile. (**a**) The sectional profile of the original design. (**b**) The sectional profile of the optimized design.