## 3 Tailoring Stiffness via Topology Optimization
#### Topology Optimization in Compliant Systems as Flexure Elements
After that, [PERSON] et al. [212] designed flexure joints by using stress-constrained Topology Optimization, aiming to reduce the general stress level and remove the sharp corners of the flexure hinges, thereby reducing the stress concentration and improving the range of motion. Various topologies were created for several levels of stress limit. The resulting final topology without stress constraint is similar to the one obtained in [211]. It can be observed that when the stress constraints are added, the length and thickness of the middle segment become longer and thicker, respectively, thus removing sharp corners distributing the stress and reducing stress concentrations. Several designs were created varying the level of stress constraint and changing the mesh refinement of the design domain, only two final topologies are shown in Figure 13f, corresponding to the designs with less stress allowed for a mesh of 50 \(\times\) 50 (see Figure 13(f-i) and for a mesh of 100 \(\times\) 100 (see Figure 13(f-ii). As it can be seen, optimal designs with stress constraints tend to seem blade flexures, therefore losing precision. In this regard, [PERSON] et al. [213] presented a systematic method for designing flexure hinges with distributed stress by using Topology Optimization with the objective of maximizing the bending displacement while minimizing the axial displacement and at the same time to minimize the maximum level of stress. Additionally, a symmetry constraint in both \(x\)- and \(y\)-axes was added, resulting in the non-conventional notch flexure hinge, shown in Figure 13g, that exhibited distributed stress. Finally, this work was extended in [214], where authors designed flexure hinges with a symmetrical distributed configuration that accomplish the desired performances of prescribed compliance and minimal parasitic motion. Several designs were obtained for each desired performance; only three of them are shown in Figure 13h, accounting for general distributed flexure hinges (see Figure 13(h-i), distributed flexure hinges with prescribed compliances (see Figure 13(h-ii), and distributed flexure hinges with minimal parasitic motion (see Figure 13(h-iii). Resulting configurations are similar to the carstwheel flexure hinge which, as was mentioned before, it is well known for being a high precision flexure joint. Finally, [PERSON] et al. [56] used Topology Optimization to design SARFJ to account for the goals of motion-range, compliance, and precision simultaneously. The resulting main topology is shown in Figure 13(i-i). Based on this topology, a second design stage was performed to produce two simplified hinge designs named \"Snowflake\" and \"Half-Snowflake\" shown in Figure 13(i-ii,iii), respectively. After comparing their performance, the new type of flexure hinge showed higher precision and comparable levels of rotational compliance than blade and cross-axis flexures. Additionally, the resulting structures have extra members that could lead to an off-axis stiffness increment, reducing parasitic displacements and making them able to transfer forces. Other optimal designs of SARFJ were created using 2D design domains, based on the geometry of a right circular flexure hinge as shown in Figure 14. For instance, [PERSON] et al. [215] used Topology Optimization to remove material from a right circular flexure hinge in order to increase compliance while retaining high precision, resulting in the multi-notched flexure hinge topology shown in Figure 14(a-i). Several topologies were obtained using different dimensions in the design domain and varying the percentage of removed material; however, they resulted in nearly the same for all examples. Then, a final shape of the multi-notched flexure hinge (see Figure 14(a-ii) was proposed after a postprocessing refinement stage and was used for comparison in its performance with various flexure hinges (right circular [216], corner-filleted [217], elliptical [218], parabolic, and hyperbolic [219]).