## 3 Tailoring Stiffness via Topology Optimization
#### Topology Optimization in Compliant Systems as Flexure Elements
Results showed that the multi-notched flexure hinges can have larger rotational compliance, higher rotational precision, and lower stress levels than any other type of flexure hinge when certain dimensional conditions are satisfied. Additionally, [PERSON] et al. [209] obtaineda similar topology, shown in Figure 14b, using Topology Optimization with a multi-criteria objective function and a symmetry constraint. The resulting design showed great rotational precision and high levels of compliance in the motion direction. Most recently, 3D design domains were also used in Topology Optimization to generate SARFJ as shown in Figure 15. [PERSON] et al. [220] used 3D continuum Topology Optimization to find the optimal topology of a SARFJ in a 3D design domain. The resulting topology is the single-axis multi-cavity flexure hinge (MCFH) shown in Figure 15(a-i). After that, a postprocessing stage was used to simplify the configuration of the hinge resulting in the final design shown in Figure 15(a-ii). Dimensionless empirical equations of the stiffness performance were obtained based on FEA for this simplified MCFH. Then, the new flexure hinge was compared with other types of hinges demonstrating that the MCFH have greater compliance. Then, [PERSON] et al. [221] presented a dual-objective Topology Optimization process for maximization of compliance and precision performance of flexure hinges based on 3D continuum Topology Optimization. The resulting topology, shown in Figure 15(b-i), was further postprocessed and parameterized generating the single-axis quasi-leaf porous flexure hinge (QLPFH) shown in Figure 15(b-ii). Dimensionless empirical equations of the stiffness and rotational center performance were derived with the aid of FEA, and they were used for comparison with a leaf flexure hinge with the same envelope space dimensions. The results showed that the rotational compliance of QLPFH is much larger than of the leaf flexure. In addition, the precision performance of the QLPFH was also improved significantly. Topology Optimization has also been used to find the optimal material distribution in a less common type of flexure joint called \"compliant revolute joint\" (Figure 16). The compliant revolute joint consists of a thin cylinder of compliant material with a inner hollow where  
Figure 14: Schematic description of the Topology Optimization process to design the optimal configuration of SARFJ using a 2D design domain based on the geometry of a right circular flexure hinge. Optimal designs obtained: (**a**) i-optimal topology and ii-final shape (Adapted with permission from ref. [215]. 2021, Elsevier) and (**b**) optimal topology obtained in [209] (Adapted from ref. [209], 2019, Springer). Figure 15: Schematic description of the Topology Optimization process to design the optimal configuration of SARFJ using a 3D cubic design domain. Optimal designs obtained in (**a**) i-optimal topology and ii-final shape (Adapted with permission from ref. [220]. 2021, Elsevier), (**b**) i-optimal topology and ii-final shape (Adapted from ref. [221]. 2020, Springer). its external face is fixed and its internal face is subject to a rotational displacement produced by a tangential force applied at the same face. [PERSON] et al. [222; 223; 224] presented a series of works dealing with the Topology Optimization of this type of flexure joint. In [222], the authors designed new compliant revolute joints reconstructed after removing material with a Topology Optimization process. The obtained designs are shown in Figure 16(i-iv). FEA was used to compare the performance of the optimized designs with the non-optimized compliant revolute joint. Results showed that optimized designs can achieve more rotational displacement. However, their stress levels were also increased. Further experimental validation in the rotational angle and stress levels of the optimal designs was performed in [223], obtaining a maximum error between experimental and FEA results of 8.7%.