## 3 Tailoring Stiffness via Topology Optimization  
Topology Optimization is a computational design method, which varies the material distribution within a structure or part, considering a predefined performance objective. In contrast to size and shape optimization in the use of Cellular Materials, topology optimization does not require a predefined initial design, and it can generate optimal geometries when intuitive design approaches fail [143]. The general Topology Optimization problem can be described as follows: To find the material distribution that minimizes an objective function \(F\), subjected to a volume constraint \(g_{0}\leq V_{0}\) and possibly \(n\) other constraints \(g_{i}\leq 0\), \(i=1,\ldots,n\). The material distribution is described by the density variable \(\vec{\rho}(x)\) that can take any value among 0 (void) and 1 (solid material) at any point in the discretized design domain \(\Omega\). This optimization problem can be written in a mathematical form as  
\[ \begin{split}\underset{\vec{\rho}}{\text{min}}:& F(\vec{\rho})\\ \text{s.t.}:& g_{0}\leq V_{0}\\ :& g_{i}(\vec{\rho})\leq 0,\ \ \ \ i=1,\ldots,n\\ :&\ \ \vec{\rho}(x)=\text{value among 0 and 1 }\ \forall x\in\Omega.\end{split}  \tag{1}\]  
Considering the density approach, this Topology Optimization problem is typically solved by discretizing the domain \(\Omega\) into \(m\) finite elements and density can take either the value 0 or 1. In this case, the last line of Equation (1) must be rearranged as \(\vec{\rho}_{j}=0\ or\ 1\) for \(j=1,\ldots,m\). On the other hand, the continuous Topology Optimization problem allows the finite elements to take values between 0 (void) and 1 (solid material). In this case, the last line of Equation (1) is reformulated as \(0\leq\vec{\rho}_{j}\leq 1\ \ j=1,\ldots,m\). In general, Topology Optimization approaches are density-based and level set-based. In density-based methods [144], which include the popular Solid Isotropic Material with Penalization (SIMP) method [145], the geometry composed of finite elements is described via a material distribution of two or more phases, of which usually one represents \"no material\" (i.e., the void phase) leading to jagged boundary geometries. On the other hand, level set-based methods [143] define the interfaces between material phases implicitly by iso-contours  
Figure 8: Lattice flexures proposed by [PERSON] [22] to be used in cross-axis flexural joints. (Adapted with permission from ref. [22]. 2021, Elsevier.) of a level set function allowing a crisp description of the boundaries and the accurate enforcement of boundary conditions. For further details on Topology Optimization methods, solution strategies, algorithms, regularization methods, etc. refer to the works in [146, 147, 148, 149]. Thus, by selecting a specific objective function \(F\) in Equation (1), it is possible to create structures with customized properties, e.g., stiffness. When the design domain has a simple geometry and is free of specific boundary conditions, the topology-optimized result is often a cellular pattern [56] (Figure 2b). In addition, structures resulting from Topology Optimization processes need a postprocessing refinement in order to smooth the jagged curves generated from the discretized domain and to achieve manufacturability [150]. As with the use of Cellular Materials, the main efforts of Topology Optimization processes have been focused in reducing mass while maximizing stiffness [151, 152, 153, 154]. Then, the basic process shown in Figure 2b is hereafter termed as Topology Optimization Resulting in Cellular Patterns (TORCP). There are other processes that combine Topology Optimization and Cellular Materials concepts in order to achieve specific stiffness values. For instance, Topology Optimization can be adopted to guide the material distribution and final configuration of unit cells in Cellular Materials.