## 3 Tailoring Stiffness via Topology Optimization
#### Topology Optimization in Compliant Systems as Flexure Elements
Compared with the constant rectangular cross section flexure hinge of similar mass, the optimized design showed more compliance, capable of undergoing a larger rotation angle when subjected to the same external load, while achieving rotation with better precision. However, the obtained design reaches higher levels of stress. Then, in [208], the same authors extended their previous work by including \"fixing the coordinates of the rotational center \(O\)\" as an additional objective in the Topology Optimization problem. This additional objective was attempted by setting the topology to be symmetric with respect to \(x\)- and \(y\)-axes in the design domain resulting in the topology shown in Figure 13b. However, the resulting design showed lower precision than their previous design, demonstrating that minimizing the displacement of the point \(O\) must be added as a constraint in the Topology Optimization problem instead of an objective. Most recently, [PERSON] et al. [209] improved their work in [207] by maintaining the functional requirements of maximizing the compliance in the desired direction, while minimizing the compliance in the other directions and setting \"minimizing the displacement of the rotational center \(O\)\" and symmetry of the topology in both axes, as constraints of the process. The resulting design was similar to that of the cartwheel flexure joint (see Figure 13c), often regarded as a flexure joint with ultra-high rotation accuracy. Additional designs were created by changing the allowable material usage and the geometry of the design domain. In general, the resulting topologies showed great rotational precision and high levels of compliance in the motion direction. The Topology Optimization of SARFJ, using a 2D rectangular design domain, was addressed also by [PERSON] et al. [210, 211, 212, 213, 214]. In [210], the authors dealt with the Topology Optimization of SARFJ undergoing large displacement by using a geometrically nonlinear elastic analysis in the Topology Optimization process. The optimization problem was formulated to set the stiffness of the flexure joint at the functional direction as low as possible and the stiffness at non-functional direction as high as possible. Additionally, minimizing the displacement of the center of rotation point and symmetry about \(x\)-axis were added as constraints. Two optimal designs were created: one based on linear FEA (see Figure 13(d-i) and other using nonlinear FEA (see Figure 13(d-ii). The resulting designs were compared with a filleded V-shape flexure hinge. Results showed  
Figure 13: Schematic description of the Topology Optimization process to design the optimal configuration of SARFJ using a 2D rectangular design domain. Optimal designs were obtained in (and adapted from) (**a**) [207]; (**b**) [208]; (**c**) [209]; (**d**) [210] i-design based on linear FEA and ii-design based on nonlinear FEA; (**e**) [211]; (**f**) [212] i-design with less stress allowed for a mesh of \(50\times 50\) and ii-design with less stress allowed for a mesh of \(100\times 100\); (**g**) [213]; (**h**) [214] i-design for general distributed flexure hinges, ii-design for distributed flexure hinges with prescribed compliances, and iii-design for distributed flexure hinges with minimal parasitic motion; and (**i**) [56] i-optimal topology, ii-simplified design named “Snowflake”, and iii-simplified design named “Half-Snowflake”. that nonlinear design can reach the highest rotation angle and has the lowest level of stress concentration. On the other hand, the linear design has better precision but the highest level of stress. This was further extended in [211], with the design of a three-segment flexure joint named quasi-V-shaped flexure hinge (QVFH) shown in Figure 13e. The authors derived the in-plane compliance equations for small displacements based on [PERSON]'s second theorem. Compared with the filleted V-shaped flexure hinge, the QVFH showed less compliance, more precision in keeping the center of rotation with the minimal offset, and larger maximum stress level.