## 2 The Design Requirement of the Aerospace Bracket  
The original aerospace bracket is a connecting piece in aerospace equipment, the structure of which is shown in Figure 1. The schematic diagram of the assembly of the original bracket and the fatigue testing platform are shown in Figure 2. From Figure 2 where it can be seen that the vertical plate and the bottom plate of the original bracket structure have a number of through-holes, which are all connected to the fixture with bolts, as well as being subjected to a sine load in the vertical direction together with other components. In response to the design requirement of a lightweight aerospace bracket, the material of the bracket is high-strength alloy TC4. The chemical composition of TC4 alloy is shown in Table 1. The new bracket should satisfy engineering specification while being as light as possible. This demands that the designed bracket is required to meet the requirement of more than 10,000 cycles without failure under a maximum load of 50 kN and a minimum load of 5 kN vertical cycle load on a fatigue test machine. Prior to new bracket design, static analysis is conducted with the commercial finite element software COMSOL to determine its mechanical properties. In finite element analysis, it is assumed that all the bolts of the bracket are well fixed during the loading process, which means that the restraint should be set to be fixed near the bolt holes of the bottom and the lower part of the three bolt holes above the vertical plate which is subjected to a total vertical direction load of 50 kN. The restraint boundary and load condition are shown in Figure 3. The lower part of the three bolt holes above the vertical plate is subjected to a total vertical downward load of 50 kN and the numerical simulation of the result is shown in Figure 4.  
Figure 1: Original structure of the aerospace bracket. (**a**) represents the angle-specific view of original structure; (**b**) represents front view of original structure; (**c**) represents top view of original structure.  
Figure 4: Von-Mises stress distribution of the original bracket.  
Figure 3: Boundary condition and load condition setting.  
Figure 2: The schematic diagram of the assembly of the original bracket and the fatigue testing platform.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Elements**} & \multicolumn{5}{c}{**Impurity Elements, Max**} \\ \hline \multirow{2}{*}{**Al**} & \multirow{2}{*}{**V**} & \multirow{2}{*}{**Ti**} & \multirow{2}{*}{**Fe**} & \multirow{2}{*}{**C**} & \multirow{2}{*}{**N**} & \multirow{2}{*}{**H**} & \multirow{2}{*}{**O**} & \multicolumn{2}{c}{**Other Elements**} \\ \cline{5-5} \cline{7-10}  & & & & & & & & **Each** & **All** \\ \hline \(5.5\sim 6.75\) & \(3.5\sim 4.5\) & The rest & 0.3 & 0.08 & 0.05 & 0.015 & 0.2 & 0.1 & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Chemical composition of TC4 alloy (weight%).  
From the result of the numerical simulation, the maximum von-Mises stress value of the original bracket structure is 2550 MPa and the von-Mises stress value at the connection between the vertical and horizontal plates is about 900 MPa. According to the material properties of TC4, the tensile strength of TC4 is 1075 MPa and the yield strength is 965 MPa, which can be found in the Electro Optical Systems (EOS) company official website. The stress of the original bracket during the loading process is close to the yield limit of the material and the structure may fail due to fatigue during the loading process of the fixture. For the purpose of reasonable stress distribution in the bracket and removal of excess material, this article utilizes topology optimization to redistribute the material in the design domain to achieve this goal.