## 3 Sensitivity Analysis and Optimization Implementation  
### Sensitivity Analysis  
Sensitivity analysis is usually implemented as an essential step in structural optimization for the following reasons: to provide the information about how sensitive the objective function is with respect to design variables; to solve optimization problems via efficient gradient-based topology algorithms; to carry out the sensitivity filtering scheme, and so on.  
The sensitivity of the objective function \(J\) defined in Equation (13) can be computed by solving its partial differentiation to the design variable \(x_{e}\) as  
\[\begin{array}{l}\frac{\partial I}{\partial x_{e}}&=-\sum\limits_{i=1}^{n} \omega_{i}\frac{\partial u_{Oi}}{\partial x_{e}}+\sum\limits_{i=1}^{n}w_{II} \frac{\partial C_{li}}{\partial x_{e}}+\sum\limits_{i=1}^{n}w_{Oi}\frac{ \partial C_{Oi}}{\partial x_{e}}\\ &=\sum\limits_{i=1}^{n}\omega_{i}\textbf{V}_{OI}^{T}\frac{\partial\textbf{K} }{\partial x_{e}}\textbf{U}_{Ii}-\sum\limits_{i=1}^{n}w_{II}\textbf{V}_{Ii}^{T} \frac{\partial\textbf{K}}{\partial x_{e}}\textbf{V}_{Ii}-\sum\limits_{i=1}^{n} w_{Oi}\textbf{V}_{OI}^{T}\frac{\partial\textbf{K}}{\partial x_{e}}\textbf{V}_{Oi}\\ &=px_{e}^{p-1}(E_{0}-E_{\text{min}})\left[\sum\limits_{i=1}^{n}\omega_{i} \textbf{V}_{Oi}^{T}\textbf{k}\textbf{U}_{Ii}-\sum\limits_{i=1}^{n}w_{II} \textbf{V}_{Ii}^{T}\textbf{k}\textbf{V}_{Ii}-\sum\limits_{i=1}^{n}w_{Oi} \textbf{V}_{Oi}^{T}\textbf{k}\textbf{V}_{Oi}\right]\end{array} \tag{14}\]  
It is worth mentioning that weighting factors \(\omega_{i}\), \(w_{ii}\) and \(w_{Oi}\) remain constant in the current optimization iteration, so their sensitivities are not involved in the above equation.  
The sensitivity of the output coupling constraint function \(\delta_{j,i}\) with respect to \(x_{e}\) could be calculated as  
\[\begin{split}\frac{\partial\tilde{\epsilon}_{j,i}}{\partial x_{e}}& =\partial\left(\frac{\widetilde{u}_{0,ij}}{u_{Oi}}\right)^{2}/ \partial x_{e}\\ &=2\frac{\widetilde{u}_{0,ij}}{u_{Oi}}\left(\frac{\widetilde{u} _{0,ij}}{\widetilde{u}_{0,ij}}u_{Oi}-\widetilde{u}_{0,ij}\frac{\widetilde{u} _{0,ij}}{\widetilde{u}_{0,i}}\right)\\ &=2\frac{\
abla_{0,ij}^{V}\mathbf{K}\mathbf{u}_{li}}{V_{Oi}^{\prime}\mathbf{ K}\mathbf{u}_{li}}\left(\frac{-V_{Oj,i}\frac{\partial\mathbf{K}}{\partial x_{e}}\mathbf{u}_{li} \cdot V_{Oi}^{\prime}\mathbf{K}\mathbf{u}_{li}-V_{Oj,i}^{\prime}\mathbf{K}\mathbf{u}_{li}\cdot \left(-V_{Oi}^{T}\frac{\partial\mathbf{K}}{\partial x_{e}}\mathbf{u}_{li}\right)}{ \left(V_{Oi}^{T}\mathbf{K}\mathbf{u}_{li}\right)^{2}}\right)\\ &=2 px_{e}^{p-1}(E_{0}-E_{\min})\frac{V_{Oj,i}^{\prime}\mathbf{K}\mathbf{u }_{li}\left(V_{Oj,i}^{\prime}\mathbf{K}\mathbf{u}_{li}\cdot V_{Ok}^{\prime}\mathbf{u}_{li} -V_{Oj,i}\mathbf{K}\mathbf{u}_{li}\cdot V_{Ok}^{\prime}\mathbf{u}_{li}\cdot V_{Ok}^{\prime }\mathbf{K}\mathbf{u}_{li}\right)}{\left(V_{Ok}^{\prime}\mathbf{K}\mathbf{u}_{li}\right)^{3}} \end{split} \tag{15}\]  
Similarly, the sensitivity of the input coupling constraint function \(\gamma_{j,i}\) can be derived as  
\[\begin{split}\frac{\partial\gamma_{j,i}}{\partial x_{e}}& =\partial\left(\frac{\widetilde{u}_{l,i}}{u_{Oi}}\right)^{2}/ \partial x_{e}\\ &=2\frac{\widetilde{u}_{l,j}}{u_{Oi}}\left(\frac{\widetilde{u}_{ l,i}}{\widetilde{u}_{0,i}}\frac{\widetilde{u}_{l,i}}{u_{Oi}^{2}}-\widetilde{u}_{l,j} \frac{\widetilde{u}_{0,i}}{\widetilde{u}_{0,i}}\right)\\ &=2 px_{e}^{p-1}(E_{0}-E_{\min})\frac{\
abla_{l,j}^{\prime}\mathbf{K} \mathbf{u}_{li}\left(\
abla_{l,j}^{\prime}\mathbf{K}\mathbf{u}_{li}\cdot\
abla_{l}^{ \prime}\mathbf{K}\mathbf{u}_{li}-V_{l,j}\mathbf{K}\mathbf{u}_{li}\cdot V_{Ok}^{\prime}\mathbf{u}_{ li}\right)}{\left(V_{Oi}^{\prime}\mathbf{K}\mathbf{u}_{li}\right)^{3}}\end{split} \tag{16}\]