## 3 Cell Sensitivity Analysis  
In order to solve the problem that the local properties of the constraints that appear in the optimization problem model cannot be described by a clear mathematical analytic formula, this chapter derives the element sensitivity and analyzes and calculates BESO as the basic method. In order to solve this problem using BESO method, the Lagrange multiplier needs to be introduced, namely \(\lambda\). The stress constraint and volume constraint are added to the objective function, while the BESO method takes the structural volume as the evolution parameter. So, the volume constraint is easy to meet. So, the objective function justly introduces a Lagrange multiplier, and the objective function can be described as:  
\[f_{1}=V+\lambda(\sigma_{PN}-\sigma_{vm}^{*}) \tag{19}\]  
\(\sigma_{vm}^{*}\) is considered to be a given stress constraint value and is a constant value. Therefore, the derivative of this term is 0 during the sensitivity analysis. In sensitivity analysis, the objective function \(f_{1}\) is equivalent to the following formula:  
\[f=V+\lambda\sigma_{PN} \tag{20}\]  
The element sensitivity calculation formula is derived. The partial derivative of the objective function \(f\) to the design variable \(x_{i}\) is obtained as follows:  
\[\frac{\partial f}{\partial x_{i}}=\frac{\partial C}{\partial x_{i}}+\lambda \frac{\partial\sigma_{PN}}{\partial x_{i}} \tag{21}\]  
According to the chain rule, it is easy to obtain \(\sigma_{PN}\) with respect to the design variables in Formula (21). The sensitivity can be described as:  
\[\frac{\partial\sigma_{PN}}{\partial x_{i}}=\frac{\partial\sigma_{PN}}{ \partial\sigma_{vm,i}}\frac{\partial\sigma_{vm,i}}{\partial\sigma_{i}}\frac{ \partial\sigma_{i}}{\partial x_{i}} \tag{22}\]  
It can be seen from the above formula that to calculate the sensitivity of the global stress cohesion function, the derivative of the stress cohesion function to the Mises stress, the derivative of the Mises stress to the stress component, and the derivative of the stress component to the design variable must be calculated first. The following three items are calculated separately. (1) The derivative of the stress condensation function to the Mises stress  
For the Mises stress of the P-norm in Formula (22) with respect to each element, we should strive to derive:  
\[\frac{\partial\sigma_{PN}}{\partial\sigma_{vm,i}}=(\sum_{j=1}^{n}\sigma_{vm,i }^{p})^{(\frac{1}{p}-1)}\cdot(\sigma_{vm,i})^{p-1} \tag{23}\]  
(2) The derivative of the Mises stress to the stress component  
The Mises stress in Formula (22) is derived from the stress vector of each element:  
\[\frac{\partial\sigma_{vm,i}}{\partial\sigma_{i}}=\sigma_{vm,i}^{-1}\sigma_{i }^{T}\textbf{V} \tag{24}\]  
(3) The derivative of the stress vector to the design variable  
The stress vector in Formula (22) is derived from the design variable \(x_{i}\), as follows:  
\[\frac{\partial\sigma_{i}}{\partial x_{i}}=\textbf{D}_{0}\textbf{B}_{i}\frac{ \partial\textbf{u}_{i}}{\partial x_{i}} \tag{25}\]In the structural dynamics analysis, the stress produced varies with time. The dynamic stress is calculated according to the element, which is the same as the stress in the static finite element analysis. That is, the stress at the node of the element or the stress at any point on the element can be calculated through the element stress calculation formula. The dynamic sensitivity can also be calculated according to the element; that is, the element stress sensitivity can be obtained by calculating the partial derivative of the design variable with the element stress calculation formula. As the dynamic stress changes with time, the dynamic stress sensitivity also changes with time. If the solution time step is n in the dynamic finite element analysis, the stress at n time points can be obtained, and the stress sensitivity at n time points can also be obtained.