## 4 Remarks on Tailoring Stiffness via Cellular Materials and Topology Optimization  
Cellular Materials and Topology Optimization can be used effectively for tailoring stiffness. The Cellular Materials approach is based on the proper selection of the unit-cell that constructs the cellular material; note that this approach works on the micro-scale. On the other hand, the Topology Optimization approach generates an optimal topology for specific external load conditions; then, it is said that this approach acts on the macro-scale. Therefore, both approaches can be combined to take advantage of their particular characteristics at the same time. A summary that address advantages and disadvantages of Cellular Materials and Topology Optimization approaches as well as their combinations, is presented in the diagram in Figure 18.  
In addition, the implementation of Cellular Materials and Topology Optimization approaches for tailoring stiffness in compliant systems entails the modification of other mechanical characteristics such as weight reduction, stiffness ratios, stress relief, and motion range. Therefore, a brief discussion on how these characteristics are modified as the stiffness is modified is presented here.  
Stiffness reduction via Cellular Materials and Topology Optimization approaches implies removing material. Therefore, a reduction in stiffness brings with it a weight  
Figure 17: Schematic description of the Topology Optimization process to design the optimal configuration of SATFJ using a 2D design domain based on the geometry of blade flexures. **(a)** Optimal design obtained in [52] i-optimal topology and ii-final shape (reprinted with permission from [52], 2019, Elsevier).  
Figure 18: Summary diagram of the Cellular Materials and Topology Optimization approaches and their combinations.  
reduction. The key aspect relies on a controlled removing of material (or distribution of material), so that this reduction in overall weight can be related to the resulting mechanical properties. A random removal of material will in turn reduce stiffness and weight; however, the control of the resulting mechanics becomes a trail and error process (non-systematic).  
The motion range of a compliant system is related to the elastic displacement. This scales with the elastic modulus of the structure, i.e., its stiffness. Therefore, any attempt to tailor the stiffness of the cellular material or optimize the topology will lead to modifications in the motion range that can be achieved. Another aspect that can come with the incorporation of any of the two approaches is the anisotropy, as the proposed or resulting geometries may be directionally dependent. Structures made of anisotropic materials can have displacements in directions apart from the those aligned to the loading axis.  
The use of Cellular Materials and Topology Optimization approaches for stiffness reduction often leads to geometries with sharp corners, thus incrementing the stress level. However, by controlling the topology of a cellular material, it is possible to tune the localization and magnitude of the stress [100] in a structure. Moreover, certain cellular structures such as the contact-aided patterns have been used to provide stress relief [44; 45; 46], while others such as the auxetic patterns allow a better stress distribution due to their high flexibility [119].  
Stiffness ratios allow one to compare the stiffness levels in different directions. A flexure element commonly needs great flexibility in certain directions while maintaining high levels of stiffness in the others, i.e., its stiffness ratios must be large. The use of Cellular Materials and Topology Optimization approaches for stiffness reduction in certain direction generally produce stiffness reductions also in the others. However, the stiffness reduction in the desired direction is significantly higher than in the non-desired directions, thus incrementing the stiffness ratios [22]. Moreover, the stiffness ratios can be maximized in the Topology Optimization process [52].