## 3 Numerical Examples
### Numerical Example  
_Path-Generation Mechanism Design_  
We introduce the novel case of the addition of stress constraints to the topology optimization of path-generating compliant mechanisms and show results for a generic example mechanism. Moreover, we apply the introduced extension of multiresolution topology optimization to stress constrained problems to this example and compare the results (problem type II). The complete optimization problem is formulated as  
\[ \begin{split}&\min_{\underline{x}}\Bigl{\{}f_{\text{path}}\Bigr{\}}\\ \text{where}& f_{\text{path}}(\underline{x})=\sum_{i=0 }^{n}\alpha_{i}\sum_{j=1}^{m}\sum_{k=1}^{l}(\underline{L}_{\text{tr}}^{\text{ T}}\underline{u}_{i,j}-u_{k,j}^{*})^{2}\\ \text{subject to}& g_{V}(\underline{x})=\frac{\varphi( \underline{x})}{\varphi_{\text{allow}}}-1\leq 0\\ & g_{\sigma}(\underline{x})=\frac{\hat{\sigma}_{p,k}(\underline{x} )}{\sigma_{\text{allow}}}-1\leq 0\\ & 0\leq x_{i}\leq 1\\ \text{governed by}&\underline{R}=\underline{P}_{\text{ int}}-\underline{P}_{\text{ext}}=\underline{0}.\end{split}  \tag{65}\]  
Due to multiple load cases and deformation states considered in the path-generation approach, stress constraints can be considered for each case. The focus in this work is the general applicability of the stress constrained formulation to this problem type. Therefore, only the stress state in the last precision point for the zero counter load case \(j=3\) is constrained. An example for a path-generation mechanism problem is defined in Figure 5, where the gray area again represents the available design space and the black areas represent prescribed solid regions. A mechanism is to be found that most closely transforms an input motion \(u_{\text{in}}\) in positive \(x\)-direction into a circular output motion \(u_{\text{out}}\) as prescribed in Table 4. This prescribed output path must be followed under each of the load cases specified in Table 5. For this example the input node is allowed to deform freely in \(y\)-direction. The parameters \(t\), \(E_{\text{min}}\) and the filter radius \(r_{\text{min}}\) are selected according to Table 2 equal to the prior example. The domain length is \(l=100\) mm and a volume fraction constraint of \(\varphi_{\text{allow}}=0.2\) is selected. Bilinear quadrilateral finite elements including four Gauss integration points and the linear elastic material defined by Table 1 are used for the structural analysis (same as previous example). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Solution Topology** & **Validation Analysis Type** & \(u_{\text{out}}\) [mm] & **max**\(\widetilde{\sigma}\) [MPa] \\ \hline a (Figure 3a) & linear & 9.76 & 528.28 \\  & nonlinear & 7.23 & 624.65 \\ \hline b (Figure 3b) & linear & 7.55 & 49.37 \\  & nonlinear & 5.53 & 173.74 \\ \hline c (Figure 3c) & nonlinear & 7.84 & 571.28 \\ \hline d (Figure 3d) & nonlinear & 4.27 & 49.95 \\ \hline e (Figure 4) & nonlinear & 4.50 & 51.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Output displacement and maximum stress values for the gripper mechanism topologies in Figures 3 and 4. Figure 6 shows the results for the output path-generation mechanism example. Above, the solution was obtained by volume constrained topology optimization. Below, a stress constraint was additionally applied. From top to bottom, the solutions' deformation states and stress distributions for the different precision points \(j\) are shown. The corresponding load case is the case of no counter load. For the depicted case, both results approximate the prescribed path closely. Moreover, it is observed that the stress constraints again successfully reduce the maximum stress level. This is achieved at the cost of a slightly higher objective function value of \(f=0.224\) mm\({}^{2}\) compared to \(f=0.155\) mm\({}^{2}\) for the volume constrained solution. Looking closer to the stress fields, it can be further seen that the maximum stress in the volume constrained topology exceeds the material strength for each precision point.