## 3 Cell Sensitivity Analysis
The calculation formula of dynamic displacement sensitivity can be obtained by synthesizing (28)-(31):  
\[\frac{\partial\mathbf{u}^{t+\Delta t}}{\partial x_{i}}=(\mathbf{K}+\alpha_{0} \mathbf{M}+\alpha_{1}\mathbf{C})^{-1}\left[\begin{array}{ll}\frac{\partial \mathbf{F}^{t+\Delta t}}{\partial x_{i}}+\frac{\partial\mathbf{M}}{\partial x _{i}}(\alpha_{0}\mathbf{u}^{t}+\alpha_{2}\mathbf{u}^{t}+\alpha_{3}\mathbf{u} ^{t})\\ +\frac{\partial\mathbf{C}}{\partial x_{i}}(\alpha_{1}\mathbf{u}^{t}+\alpha_{4} \mathbf{u}^{t}+\alpha_{5}\mathbf{u}^{t})\\ +\mathbf{M}(\alpha_{0}\frac{\partial\mathbf{u}^{t}}{\partial x_{i}}+\alpha_{ 2}\frac{\partial\mathbf{u}^{t}  
the current time by solving the dynamic displacement response at the current time and combining the displacement, velocity, and acceleration sensitivity at the previous time. The displacement sensitivity of the whole time domain can be obtained by iteration. According to the initial conditions, it is easy to determine that the sensitivity of displacement, velocity, and acceleration at time \(t=0\) is 0; F is the external load vector, independent of the design variable \(x_{i}\); so, the partial derivative of this term is 0. Therefore, the displacement sensitivity at time \(\Delta t\) is:  
\[\frac{\partial\textbf{u}^{\Delta t}}{\partial x_{i}}=-(\textbf{K}+\alpha_{0} \textbf{M}+\alpha_{1}\textbf{C})^{-1}(\frac{\partial\textbf{K}}{\partial x_{ i}}+\alpha_{0}\frac{\partial\textbf{M}}{\partial x_{i}}+\alpha_{1}\frac{ \partial\textbf{C}}{\partial x_{i}})\textbf{u}_{i}^{\Delta t} \tag{33}\]  
Finally, the expression of dynamic stress sensitivity at time \(t\) can be obtained by introducing Equations (23), (24), (26) and (33) into Equation (22), as shown in Equation (34):  
\[\frac{\partial f}{\partial x_{i}}=V_{i}-\lambda\sum\limits_{j=1}^{n}\Bigl{[} (\sigma_{vm,j}^{\Delta t})^{p}\Bigr{]}^{\left(\frac{1}{p}-1\right)}(\sigma_{vm,i}^{\Delta t})^{p-2}(\sigma_{i}^{\Delta t})^{T}\textbf{VD}_{0}\textbf{B}_{i }\textbf{L}_{i}\tilde{\textbf{K}}^{-1}(\frac{\partial\textbf{K}}{\partial x_ {i}}+\alpha_{0}\frac{\partial\textbf{M}}{\partial x_{i}}+\alpha_{1}\frac{ \partial\textbf{C}}{\partial x_{i}})\textbf{u}_{i}^{\Delta t} \tag{34}\]  
When a uniform grid is used (i.e., the cells have the same volume), the relative ranking of the sensitivities of each cell can be defined by the following sensitivities:  
\[\alpha_{i}=\sum\limits_{j=1}^{n}\Bigl{[}(\sigma_{vm,j}^{\Delta t})^{p}\Bigr{]} ^{\left(\frac{1}{p}-1\right)}(\sigma_{vm,i}^{\Delta t})^{p-2}(\sigma_{i}^{ \Delta t})^{T}\textbf{VD}_{0}\textbf{B}_{i}\textbf{L}_{i}\tilde{\textbf{K}}^{ -1}(\frac{\partial\textbf{K}}{\partial x_{i}}+\alpha_{0}\frac{\partial\textbf {M}}{\partial x_{i}}+\alpha_{1}\frac{\partial\textbf{C}}{\partial x_{i}}) \textbf{u}_{i}^{\Delta t} \tag{35}\]  
In the whole process of structural optimization, a series of displacement values in the time domain will be output in detail after each finite element analysis. If these displacement values are processed in parallel, the whole calculation will be more complicated and difficult, and the time cost will increase. Moreover, in the optimization iteration step under a fixed mode determined by the structure, even if the displacement is always changing, the displacement is only related to the time history of the load; so, the series of displacement values is the same in the physical sense. Therefore, the element sensitivity at only one time point is sufficient to meet the requirements.