## 3 Numerical Examples
### Numerical Example
In contrast, the maximum stress of the stress constraint reaches the prescribed threshold value only in the last precision point, where also the largest deformations occur. The example is also demonstrated with the stress field of a smoothed solution topology. Therefore, the stress constrained solution in Figure 6 is smoothed by Laplacian smoothing and remeshed. Figure 7 shows the stress field on the deformed geometry for the 3 rd precision point. The maximum stress of the smoothed geometry increases by 4% compared to the solution in Figure 6f. Again also slight changes in the output displacements are observed. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Precision Point \(j\)** & \(\mathbf{u_{\text{in}}}\) **[mm]** & \(\mathbf{u_{\text{out},x}^{*}}\) **[mm]** & \(\mathbf{u_{\text{out},y}^{*}}\) **[mm]** \\ \hline
1 & 1.5 & 3 & \(-0.18\) \\
2 & 3 & 6 & \(-0.73\) \\
3 & 4.5 & 9 & \(-1.67\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Precision points for the path-generation example. Figure 5: Example definition scheme for an output path generating compliant mechanism (gray: design space, black: non-design space of solid material). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load Case \(i\)** & \(\mathbf{\alpha}\) **[-]** & \(P_{\text{CL},x}\) **[N]** & \(P_{\text{CL},y}\) **[N]** \\ \hline
0 & 1 & 0 & 0 \\
1 & 0.1 & 40 & 40 \\
2 & 0.1 & -40 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Load cases for the path generation example. Figure 6: Comparison of the path generating example mechanism results for the three precision points in the load case \(i=0\) with 150 \(\times\) 150 elements, volume constrained on the left, stress and volume constrained on the right. In Figure 8, results obtained by the multiresolution topology optimization method on a coarser 75 \(\times\) 75 FE mesh are shown and compared to the results of Figure 6. The density meshes are equally sized (150 \(\times\) 150) for both methods used. The coarser FE mesh is chosen in order to demonstrate the ability of the MTOP approach to find high resolution results based on relatively low resolution FE discretization. Results for the volume constrained as well as for the additionally stress constrained problem solutions could be successfully generated by the MTOP method. Comparing the volume constrained solutions in Figure 8a,c it can be seen that for the multiresolution topology optimization lower stress values occur. The reason for this is a less-precise stress modeling due to the larger size of the FE-elements. Due to the different FE discretization, also a different local minimum is found for the multiresolution topology optimization in Figure 8c. In Figure 8b,d, the stress and volume constrained results are compared. For both results the stress constraint successfully reduced the maximum stress to the threshold value. The MTOP result in Figure 8d is influenced by the underestimation of the stress field, hence thicker flexible joint regions are allowed. Computational time could be reduced by the MTOP method on the same desktop PC (Intel Core i7-7500U CPU @ 2.70 GHz, 16 GB DDR4) by approx. 75% for both examples. The results in Figure 8 show that the introduced formulation of the multiresolution topology optimization method can be effectively used for stress constrained problems. Nevertheless, it has to be considered that the minimum size of the FE mesh has to be chosen sufficiently small in order to ensure a correct stress modeling. The output paths and the prescribed precision points for the various results of the path generation mechanism example are plotted in Figure 9. It can be seen that the output path of each result is a clearly nonlinear curve, which approximates the precision points. The impact of the smoothing on the output path can be clearly observed, leading to a higher discrepancy between output path and precision points. This shows that results for path-generation examples are more sensitive to geometrical deviations.