## 2 Topology Optimization for the Synthesis of Compliant Mechanisms
### Design Sensitivity Analysis via Adjoint Methodology
#### 2.3.4 Stress Constraint Sensitivity  
The sensitivities of the stress constraint function defined in Equation (27) are also derived by the same approach as in Section 2.3.1. Starting with the partial derivative of the stress constraint function  
\[\frac{\partial g_{\sigma}}{\partial\underline{\tilde{x}}}=\frac{1}{\sigma_{ \mathrm{allow}}}\frac{\partial\hat{\sigma}_{p}}{\partial\underline{\tilde{x}}}, \tag{47}\]  
where  
\[\frac{\partial\hat{\sigma}_{p}}{\partial\underline{\tilde{x}}_{l}}=c\left( \sum_{i}V_{i}\tilde{\sigma}_{i}^{p}\right)^{\frac{1-p}{p}}V_{l}\tilde{\sigma}_ {l}^{p-1}q\tilde{x}_{l}^{q-1}\sigma_{M,l}, \tag{48}\]  
with element index \(l\). The partial derivative with respect to the state variables \(\underline{u}\) is  
\[\frac{\partial g_{\sigma}}{\partial u}=\frac{1}{\sigma_{\mathrm{allow}}} \frac{\partial\hat{\sigma}_{p}}{\partial u}. \tag{49}\]Using this relation we write the linear adjoint system according to Equation (36) as  
\[\underline{\underline{K}}_{T}\underline{\lambda}=-\frac{\partial g_{\underline{c}} }{\partial\underline{u}}^{\mathsf{T}}. \tag{50}\]  
For solving this equation for the Lagrange multipliers, we need \(\frac{\partial\varphi_{p}}{\partial\underline{u}}\), which is defined by element via  
\[\frac{\partial\varphi_{p}}{\partial\underline{u}_{l}}=c\left(\sum_{i}V_{i} \widetilde{\sigma}_{i}^{p}\right)^{\frac{l-p}{p}}V_{i}\widetilde{\sigma}_{l}^ {p-1}\widetilde{\xi}_{l}^{q}\frac{1}{\sigma_{M,l}}c_{C,\underline{\underline{ \underline{\Delta}}}}^{\mathsf{T}}\underline{\underline{\underline{\Delta}}} \frac{\partial\varphi_{C,l}}{\partial\underline{u}_{l}}. \tag{51}\]  
The derivation of the sensitivities of the Cauchy stresses \(\frac{\partial\varphi_{C,l}}{\partial\underline{u}_{l}}\) is carried out by applying the product rule on Equation (7). The sensitivity with respect to the \(i\)th component of the \(l\)th elements nodal displacement vector \(\underline{u}_{l}\) is obtained as  
\[\frac{\partial\underline{\sigma}_{C,l}}{\partial u_{i,l}}=-\underbrace{\frac {1}{J_{l}^{2}}\frac{\partial J_{l}}{\partial u_{i,l}}\underline{\underline{F }}_{L}\underline{\underline{\sigma}}_{2\text{PJK},\underline{\underline{ \underline{\varepsilon}}}}\underline{\underline{\varepsilon}}^{\mathsf{T}}}_{( i)}+\underbrace{\frac{1}{J_{l}}\underline{\underline{\varepsilon}}_{L}\frac{ \partial\varphi_{2\text{PJK},\underline{\underline{\varepsilon}}}}{\partial u _{i,l}}\underline{\underline{F}}_{L}^{\mathsf{T}}}_{(ii)}+\underbrace{\frac{1} {J_{l}}\left(\frac{\partial\underline{\underline{\varepsilon}}_{L}}{\partial u _{i,l}}\underline{\underline{\sigma}}_{2\text{PJK},\underline{\underline{ \varepsilon}}}\underline{\underline{F}}_{l}^{\mathsf{T}}+\underline{\underline{ F}}_{L}\underline{\underline{\sigma}}_{2\text{PJK},\underline{\underline{\varepsilon}}} \left(\frac{\partial\underline{\underline{\varepsilon}}_{L}}{\partial u_{i,l} }\right)^{\mathsf{T}}\right)}_{(iii)}. \tag{52}\]  
Now generalizing Equation (52) for the entire element displacement vector \(\underline{u}_{l}\) and transferring it to Voigt notation, three core terms \(\underline{\underline{\Sigma}}_{1}\), \(\underline{\underline{\Upsilon}}_{2}\) and \(\underline{\underline{\Upsilon}}_{3}\) can be distinguished, corresponding to \((i)\), \((ii)\) and \((iii)\), respectively,  
\[\frac{\partial\underline{\sigma}_{C}}{\partial\underline{u}_{l}}=\underline{ \underline{\Upsilon}}_{1}+\underline{\underline{\Upsilon}}_{2}+\underline{ \underline{\Upsilon}}_{3}. \tag{53}\]  
For the derivation of the term \(\underline{\underline{T}}_{1}\), the only unknown is \(\frac{\partial J_{l}}{\partial\underline{u}_{l}}\).