# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 3 Sensitivity Analysis  
To guide the iteration process of the optimization algorithm, sensitivity analysis is necessary to determine which elements should be removed or kept to the next iteration. Sensitivity analysis can be achieved by deriving the objective formulation \(\text{f}(\rho_{\text{e}})\) with respect to the design variable. For most topology optimization algorithms, the design variable is material density \(\rho_{\text{e}}\). Based on the SIMP material model, the finite element elasticity matrix can be expressed by material density \(\rho_{\text{e}}(0\leq\rho_{\text{e}}\leq 1)\) with a power \(p(p\geq 1)\), as below  
\[\text{E}_{\text{e}}(\rho_{\text{e}})=\rho_{\text{e}}^{\text{P}}\text{E}_{ \text{e}}^{0} \tag{10}\]where \(\mathbf{E}_{\mathbf{e}}^{0}\) is the element elasticity matrix with fully solid material. The introduction of penalization power \(\mathbf{p}\) is to penalize the elements with intermediate density so that only the element with the density close to 1 will be left in the optimization result, benefitting from that, the optimization result becomes easy to interpret.  
The sensitivity of MPE variables can be deduced by:  
\[\frac{\partial\mathbf{K}}{\partial\rho_{\mathbf{e}}}\mathbf{u}_{\mathrm{A}}+ \mathbf{v}_{\mathrm{B}}^{\mathrm{T}}\mathbf{K}\frac{\partial\mathbf{u}_{ \mathrm{A}}}{\partial\rho_{\mathrm{e}}} \tag{11}\]  
Differentiating both sides of \(\mathbf{K}_{\mathrm{i}}\mathbf{v}_{\mathrm{B}}=\mathbf{f}_{\mathrm{B}}\), then substituting the result into the above equation, the sensitivity of MPE with respect to \(\rho_{\mathrm{e}}\) can be rewritten as:  
\[\frac{\partial\mathrm{MPE}}{\partial\rho_{\mathrm{e}}}=\mathbf{v}_{\mathrm{B}} ^{\mathrm{T}}\frac{\partial\mathbf{K}}{\partial\rho_{\mathrm{e}}}\mathbf{u}_{ \mathrm{A}}=\mathrm{p}\rho_{\mathrm{e}}^{\mathrm{(P-1)}}\mathbf{v}_{\mathrm{B} }^{\mathrm{T}}\mathbf{K}_{\mathrm{e}}^{0}\mathbf{u}_{\mathrm{A}} \tag{12}\]  
Similarly, the sensitivity of \(\lambda\) can be deduced by differentiating the equation with respect to \(\rho_{\mathrm{e}}\) and we obtain [35]:  
\[\frac{\partial\lambda}{\partial\rho_{\mathrm{e}}}=\varphi_{\mathrm{j}}^{ \mathrm{T}}\Big{(}\mathrm{p}\rho_{\mathrm{e}}^{\mathrm{(P-1)}}\mathrm{K}_{ \mathrm{e}}^{0}-\lambda_{\mathrm{j}}\mathrm{q}\rho_{\mathrm{e}}^{\mathrm{(q-1) }}\mathrm{M}_{\mathrm{e}}^{0}\Big{)}\varphi_{\mathrm{j}} \tag{13}\]  
\[\mathrm{E}=1,2,3\ldots,\mathrm{N}_{\mathrm{E}}.\]  
Finally, the sensitivity of synthesis weighting index (SWI) with respect to \(\rho_{\mathrm{e}}\) can be assembled and formulated as:  
\[\frac{\partial\mathrm{SWI}}{\partial\rho_{\mathrm{E}}}=-\frac{ \mathrm{p}\rho_{\mathrm{e}}^{\mathrm{(P-1)}}\sum_{i=1}^{\mathrm{n}}W_{\mathrm{ i}}\mathbf{v}_{\mathrm{B}}^{\mathrm{T}}\mathbf{K}_{\mathrm{e}}^{0}\mathbf{u}_{ \mathrm{A}}}{\mathrm{MPE}_{\mathrm{max}}}-\lambda_{\mathrm{min}}\sum_{i=1}^{ \mathrm{n}}\frac{W_{\mathrm{i}}\varphi_{\mathrm{j}}^{\mathrm{T}}\Big{(} \mathrm{p}\rho_{\mathrm{e}}^{\mathrm{(P-1)}}\mathbf{K}_{\mathrm{e}}^{0}- \lambda_{\mathrm{j}}\mathrm{q}\rho_{\mathrm{e}}^{\mathrm{(q-1)}}\mathbf{M}_{ \mathrm{e}}^{0}\Big{)}\varphi_{\mathrm{j}}}{\lambda_{1}^{2}} \tag{14}\] \[\mathrm{i}=1,\ldots,\mathrm{n}\]