## 3 Tailoring Stiffness via Topology Optimization
After the optimized unit-cell is obtained, a process to obtain the properties of the macromaterial in terms of the effective properties of its unit cell, termed as homogenization [155], is applied to approximate the apparent properties of the Cellular Material [146, 156]. This widely used methodology is termed here as Topology Optimization Applied to Unit Cells (TOAUC), and it is graphically explained in Figure 9. The first implementation of Topology Optimization applied to unit-cell design can be found in the works of [PERSON] [151, 157, 158]. Most recently, [PERSON] and [PERSON] [159] reviewed the key requirements to apply topology optimization process to generate unit cells of periodic structured materials for a variety of customizable properties. Topology Optimization applied to the unit-cells of Cellular Materials has been used to maximize stiffness under general conditions [160, 161, 162], shear forces [163], and predetermined loads [164, 165]. Resulting structures from 3D maximization stiffness problems are commonly closed-cell cellular structures [159, 166]. This type of structure is preferred because their wall elements spans and connects various faces of the unit-cell, carrying loads axially in more than one direction, thus exhibiting stretch behavior in all directions. However, an important disadvantage of closed-cell structures with located enclosed pores is that they are quite difficult to fabricated. Furthermore, TOAUC-resulting cellular materials have been used to match specific stiffness values in bone implants [167, 168, 169, 170]. The maximization of stiffness by combination of the Topology Optimization and Cellular Materials techniques has also been explored. Two main concurrent methodologies were identified: The first begins with the typical Topology Optimization process to generate a conceptual design in the design domain. After that, the generated design is filled with a cellular structure especially selected to improve the optimization of the objective property. This concurrent method is termed here as Multiscale Topology Optimization with Cellular Materials (MTOCM), and it is graphically explained in Figure 10. Examples of relevant works using MTOCM approach can be found in [171, 172, 173, 174, 175]. Figure 9: Process to tailor stiffness using Topology Optimization applied to the unit cells of Cellular Materials. The second concurrent methodology consists in modifying the relative density of the cellular design domain based on a Topology Optimization process, obtaining the so-called functionally graded Cellular Materials. Modification of the unit-cell relative density can be made by altering cell parameters [176] or by using different unit-cell topologies [177]. This methodology is termed as Topology Optimization of Functionally-graded Cellular Materials (TOFCM), and it is graphically explained in Figure 11. Other, examples of relevant works using TOFCM approach can be found in [178; 179; 180; 181; 182]. Topology Optimization for tailoring stiffness with compliance objectives has been used to create compliant mechanisms [53], including micro-grippers [183], bridge-mechanisms [184], positioning stages [185], and path generating mechanisms [186]. However, using the Topology Optimization process to generate complete compliant mechanisms may lead to complex designs with distributed compliance throughout its topology, thus leading to complicated analysis, manufacturability, and further implementation [187]. Therefore, a new approach of Topology Optimization appeared as an alternative to deal with these drawbacks, maintaining the versatility of the process. This new approach is focused on applying the Topology Optimization process only on the flexures of compliant systems, taking the flexure section as the design domain of the Topology Optimization process. This could enhance the analysis of compliant systems through a proper characterization of their force deflection functions while maintaining their original macro-shape and lumped architecture.