## 3 Frame Optimization Design  
### Introduction to Topology Optimization Method  
Topology is a way to improve the spread of materials in the design space under the constraints of the initial conditions and internal structural performance parameters. Among them, the variable density method is widely used in topology. It takes the unit density of each module in the design space as the design variable. In density-based topology optimization, the continuous density field \(\rho\):\(\Omega\rightarrow\left[0,1\right]\) is replaced by the constant density distribution of the element caused by the finite element discretization. Topological design is within the design domain \(\rho=1\). Void material \(\rho=0\) is described by the continuous variation of fictional unit density [33], but in reality, intermediate-density units cannot exist and cannot be manufactured. Therefore, it is necessary to avoid the generation of intermediate-density units as much as possible and reduce the number of intermediate-density units. In this case, it is necessary to penalize the intermediate density values that appear in the design variables. The most popular material interpolation model method currently is the SIMP method, which is represented by the formula:  
\[\mathrm{E}\left(\mathrm{x_{i}}\right)=\mathrm{E_{min}}+\mathrm{x_{i}}^{ \mathrm{P}}\left(\mathrm{E_{0}}-\mathrm{E_{min}}\right) \tag{7}\]  
In the equation:  
\(\mathrm{E(x_{i})}\)--Elastic modulus after interpolation.  
\(\mathrm{E_{0}}\)--Elastic modulus of material in the solid part.  
\(\mathrm{E_{min}}\)--Elastic modulus of material in the porous part.  
\(\mathrm{x_{i}}\)--Unit relative density, with a value of 1 indicating the presence of material and 0 indicating the absence of material or holes.  
\(\mathrm{p--Penalty}\) factor.  
The higher the unit density, the more critical it is that the material at this location be retained, as it is responsible for bearing the load. Reduce the use of materials with low unit density, achieving lightweight design and rational utilization of materials.  
The mathematical model of topology optimization can be expressed as [32]:  
Minimize:  
\[\mathrm{f}\left(\mathrm{X}\right)=\mathrm{f}\left(\mathrm{x_{1}},\mathrm{x_{2 }},\dots,\mathrm{x_{n}}\right) \tag{8}\]  
Constraints:  
\[\mathrm{g_{j}}\left(\mathrm{X}\right)\leq 0\,\mathrm{j}=1,2,\dots,\mathrm{m} \tag{9}\]  
\[\mathrm{h_{k}}\left(\mathrm{X}\right)=0\,\mathrm{k}=1,2,\dots,\mathrm{m} \tag{10}\]  
\[\mathrm{x_{i}}^{\mathrm{L}}\leq\mathrm{x_{i}}\leq\mathrm{x_{i}}^{\mathrm{U}} i=1,2,\dots,n \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Modal Order** & **First Order** & **Second Order** & **Third Order** & **Fourth Order** & **Fifth Order** & **Sixth Order** \\ \hline Natural frequency & 36 Hz & 45 Hz & 71 Hz & 79 Hz & 90 Hz & 118 Hz \\ \hline \hline \end{tabular}
\end{table}
Table 2: Frame-free modal analysis results.  
In the function, X represents the design variable, f (X) represents the objective function, g\({}_{\text{j}}\) (X) represents inequality constraint functions, h\({}_{\text{k}}\) (X) represents the equality constraint function, L represents the upper limit of the constraint, and U represents the lower limit of the constraint.