# Topology Optimization with Explicit Components Considering Stress Constraints
## 4 Sensitivity Analysis  
In this paper, the method of moving asymptotes (MMA) is selected as the nonlinear optimization solver. Since the design variables in the components can be accurately described, the sensitivity values of the objective function and the volume constraint function are similar to those solved in the literature [5,27,29,33,34,35], and are not discussed in detail in this paper. The corresponding variable sensitivity value of \(g(y^{\prime})\) in the constructed TDF (taking \(l_{l1}<l_{l2}\) as an example) can be expressed as follows:  
\[\frac{\partial g(y^{\prime})}{\partial x_{0}}=-\frac{\sin\theta_{i}(y^{\prime }-y^{\prime}_{0i})}{\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}} \tag{18}\]  
\[\frac{\partial g(y^{\prime})}{\partial y_{0}}=\frac{\cos\theta_{i}(y^{\prime }-y^{\prime}_{0i})}{\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}} \tag{19}\]  
\[\frac{\partial g(y^{\prime})}{\partial l_{1}}=\frac{l_{l1}l_{l2}(x^{\prime}_{0 i}+l_{l2})}{(l_{l2}-l_{l1})^{2}\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}}+ \frac{l_{l1}l_{l2}}{(l_{l2}-l_{l1})^{2}} \tag{20}\]  
\[\frac{\partial g(y^{\prime})}{\partial l_{2}}=\frac{(x^{\prime}_{0i}+l_{l2}) (1-\frac{l_{0i}^{2}}{2(l_{l2}-l_{l1})^{2}})+l_{l2}(t_{l1}+y^{\prime})}{\sqrt{ r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}}-\frac{l_{l1}^{2}}{2(l_{l2}-l_{l1})^{2}} \tag{21}\]  
\[\frac{\partial g(y^{\prime})}{\partial t_{1}}=\frac{y^{\prime}_{0i}+t_{l1}(y ^{\prime}-y^{2}_{0i}+1))}{\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}} \tag{22}\]  
\[\frac{\partial g(y^{\prime})}{\partial\theta}=\frac{(y^{\prime}-y^{\prime}_{0 i})[\cos\theta_{i}(x-x_{0i})+\sin\theta_{i}(y-y_{0i})]}{\sqrt{r_{i}^{2}-(y^{ \prime}-y^{\prime}_{0i})^{2}}} \tag{23}\]  
The derivation process of the variable sensitivity value corresponding to the global stress constraint is as follows:  
\[\frac{\partial\sigma_{\text{VM}}}{\partial\sigma}=\frac{1}{2}(\sigma^{\text {T}}\text{V}\sigma)^{-\frac{1}{2}}2\sigma^{\text{T}}\text{V}=\sigma_{\text{VM }}^{-1}\sigma^{\text{T}}\text{V}, \tag{24}\]  
\[\frac{\partial\sigma_{VM}}{\partial d}=\frac{\partial\sigma_{VM}}{\partial \sigma}\frac{\partial\sigma}{\partial d}=\sigma_{VM}^{-1}\sigma^{T}V\frac{ \partial\sigma}{\partial d} \tag{25}\]  
\[\frac{\partial\sigma_{VM}}{\partial d}=\frac{1}{4}\sigma_{VM}^{-1}\sigma^{T} VEBL\sum_{e=1}^{n}\sum_{i=1}^{4}H(\theta^{e}_{i})\frac{\partial\text{U}}{ \partial d}, \tag{26}\]In Equation (26), \(L\) is the selection vector used to extract the corresponding displacement vector of the cell from the global displacement vector. The term \(\frac{\partial\mathrm{U}}{\partial d}\) can be derived from the balance equation \(F=KU\) by taking a partial derivative with respect to \(d\). \[\frac{\partial\mathrm{U}}{\partial\mathrm{d}}=-\ \mathrm{K}^{-1} \frac{\partial\mathrm{K}}{\partial\mathrm{d}}\mathrm{U}, \tag{27}\] \[\frac{\partial g(y^{\prime})}{\partial y_{0}}=\frac{\cos\theta_{i}(y^{\prime }-y^{\prime}_{0i})}{\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}} \tag{19}\]0  
In the above equation, the expression for the global stiffness matrix \(K\) is calculated. \(k_{e}\) represents the element stiffness matrix, and \(E_{e}\) denotes the Young's modulus of the element, as determined in Equation (10).