## 2 Optimization Problem Formulation  
### Problem Statement-Minimum Compliance  
The topological optimization (TO) problem can be defined as a binary problem whose the objective is to provide the best material distribution in the solution domain, according to the specified criteria. The TO problem analyzed herein is the classical formulation for compliance, which minimizes the work done by external forces subject to a desired prescribed volume, \(V^{*}\). The mathematical formulation of this problem may be expressed as  
\[ \begin{array}{ll}\text{\emph{Minimize}}&C(x)=F^{T}u(x)=\sum_{i}^{n}u_{i}^{T }K_{i}(x_{i})u_{i}\\ \text{\emph{subject to}}:&V(x)=\sum_{i}^{n}x_{i}^{T}V_{i}\leq V^{*}\end{array}  \tag{1}\]  
where compliance \(C(x)\) is the objective function ; \(F\) and \(u\) are the global force and generalized displacement vectors, respectively; \(K\) is the global stiffness matrix; \(V(x)\) is the total volume of the structure; \(V_{i}\) is the volume of the element at each iteration; \(V^{*}\) is the prescribed final limit volume ; \(x_{i}\) is the artificial density of the element; and \(n\) is the total number of elements. The evolutionary structural optimization methods are based on a simple and empirical concept that a structure evolves to an optimal, slowly removing the elements with lower desired sensibility. To maximize the structure's stiffness, compliance minimization has replaced the stress criterion, as described in Equation (1). Thus, it can be highlighted that in the ESO, SESO, and SERA methods, the material is added and removed from the design domain until an optimal setting is reached. Therefore, they are bidirectional in nature. The major difference between them is the heuristic of removal and adding elements from the structure domain. The SERA method applies two separate criteria for removing and adding elements from the domain, allowing the status change from \"passive\" to \"active\" and vice versa. In this way, the final topology is constructed with all real materials present in the structure. For more details about the SERA method, see [5; 16]. The SESO method uses only one criterion to perform this procedure, the elements that attend this criterion are removed from the design domain, ordered, grouped, and p% of the groups with lower compliance are discarded and (\(1-p\%\)) are returned to the structure, smoothing the \"hard-kill\" procedure used in ESO, i.e., the total removal of the elements that meet the rejection criterion. In addition, discrete variables ensure that the final topology is free of gray regions, as in continuous methods, such as the SIMP. It's important to point out that SESO and ESO use domain elements as their discrete variables, while SERA [5; 19] and SIMP [1; 2] use element density instead. It is noteworthy that, in the formulation described in Equation (1), the artificial density of the element \(x_{i}\) is the design variable of all methods, which is computed as follows:  
\[ \begin{array}{l}ESO\\ x_{i}=1\ (active\ element)\ x_{i}=x_{min}\ (inactive\ element)\\ SESO:\\ x_{i}=1\ (active\ element)\\ x_{i}=x_{min}\ (inactive\ element)\\ x_{i}=\eta(i)\ (active\ element)\\ SERA:x_{i}=\{\ x_{min},1\}\\ SIMP:x_{min}<x_{i}<1\end{array}  \tag{2}\]  
considering \(x_{min}=1E-9,\ \eta(i)\) being a weighted function, \(0\leq\eta(i)\leq 1\), presented in [13]. The stiffness matrix is updated as presented in the item 2.2 and \(p\) is the penalization factor, with \(p=1\) being considered to ESO, SESO and SERA methods and \(p=1,2,\ldots,p_{max}\) (\(p_{max}>3\)) for SIMP [1; 2]. The way as all the aforementioned methods evaluate the artificial density parameter is defined by the desired objective function by computing the elemental sensibility, which is iteratively applied to remove or keep the element in the design domain. Some methods, such as the SERA [5; 19] or the variant of ESO, BESO [4], can reintroduce elements into the actual design domain, from which the procedure of creating cavities is applied until the criterion is reached.