## 3 Comparing SESO and SERA with Other Topology Optimization Methods  
Currently, the SIMP method is the most used for TO and has shown its effectiveness in structural engineering applications. The SESO, SERA, and ESO methods implemented in this article use the same eight-node hexahedral finite element proposed by [7]. Consequently, the findings of SESO, SERA, and ESO are compared with Top3d by [7], which make use of SIMP. Issues such as checkerboard, local optimal, and mesh dependence arise with evolutionary optimization models, according to [21]. By using this heuristic filtering scheme to address numerical instabilities in TO, a comparison can be made between two levels: with and without a mesh independence filter. Mathematically, the spatial filter is an additional constraint inserted into the optimization problem as a way of smoothing out the spatial distribution of the design variables in the solution domain, minimizing the mesh dependence, and controlling the topology complexity. Thus, with an increase in the radius value, gradients are restricted to smaller values and the transition between solid (material) and empty (without material) becomes smoother, generating more intermediate compliance elements. Given that the OT procedure is affected by the radius of the filter, particularly when it is larger, the optimization problem would not have a solution using the initial condition proposed, as the gradients of the design variables would be restricted to very low rates of variation. Thus, a simple function for the density filter can be written as follows:  
\[x_{i}=\frac{\sum_{j=1}^{n}H_{ij}v_{j}x_{j}}{\sum_{j=1}^{N}H_{ij}v_{j}} \tag{8}\]  
where the element \(x_{i}\) with volume \(v_{i}\), \(x_{i}\) is the weighted average of the distances from the centers of the neighboring elements \(x_{i}\) within a sphere of radius \(R\) with \(H_{ij}\) its weighting factor defined as follows:  
\[H_{ij}=R-d_{ij} \tag{9}\]  
with R being the radius, see Figure 1, of the sphere centered on the element \(x_{i}\) and \(d_{ij}\) is the distance between the centers of the elements \(x_{i}\) and \(x_{j}\) given by:  
\[d_{ij}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}+\left(z _{i}-z_{j}\right)^{2}} \tag{10}\]where \((x_{i},y_{i},z_{i})\) and \((x_{j},y_{j},z_{j})\) are, respectively, the coordinates of the centers of the elements \(i\) and \(j\). The filtered element is incorporated into the optimization procedure, and the modulus of elasticity of the modified SIMP method is given by:  
\[E_{i}(\overline{x})=E_{min}+\ \ (\overline{x_{i}})^{p}(E_{0}-E_{min}),\ \overline{x_{i}}\in\{0,1\} \tag{11}\]  
Thus, it is possible to determine the sensitivity of the \(x_{i}\) (filtered) elements of the SESO, SERA, and ESO methods which given by the expression:  
\[E_{i}(\overline{x})=E_{min}+\ \ \overline{x_{i}}(E_{0}-E_{min}),\ \overline{x_{i}} \in\{0,1\} \tag{12}\]  
Therefore, SESO, SERA, and ESO have the sensitivity at the level of the filtered elements given by:  
\[\frac{\partial C(\overline{x_{i}})}{\partial x_{i}}=-u_{i}^{T}(\overline{x_{ i}})\Big{[}(E_{0}-E_{min})K_{i}^{0}\Big{]}u_{i}(\overline{x_{i}}) \tag{13}\]  
where \(u\) is the nodal vector of the elements' displacements and \(K_{i}^{0}\) is the stiffness matrix of the element. For the SIMP method:  
\[\frac{\partial C(\overline{x_{i}})}{\partial x_{i}}=-u_{i}^{T}(\overline{x_{ i}})\Big{[}p(\overline{x_{i}})^{p-1}(E_{0}-E_{min})K_{i}^{0}\Big{]}u_{i}( \overline{x_{i}}) \tag{14}\]  
The algorithm of the optimization methods can be described as follows:  
Step 1: Discretize the domain using a refined finite element mesh;  
Step 2: Specific the maximum final volume (\(V^{*}\)) and the parameters for the desired method. ESO and SESO: rejection rate (RR), evolutionary rate (ER) and the weighted function (\(\eta\)). SERA: total number of iterations (\(N_{tot}\)), progression rate (PR) and smoothing ratio (SR).