## 2 SESO--Subject to Self-Weight Loads  
The SESO method aims to optimize structural efficiency by removing or adding elements to the structure. In the search for the topology of maximum stiffness, it is common to use the minimization of the compliance or the maximum von Mises stress as an objective function, while the constraint is imposed on the structural weight, limiting the maximum volume of material allowed. Compliance represents the work done by the loads applied in the structure's equilibrium state. An alternative approach to maximum stiffness design is to use elastic strain energy as a measure of structural stiffness. Therefore, the compliance minimization problem can be reformulated as a problem of minimizing the total elastic strain energy. Thus, the formulation of the TO problem can be defined as  
\[ \begin{split}\text{Minimize}:&\quad\text{C}( \boldsymbol{x})=U^{T}(\boldsymbol{x})K(\boldsymbol{x})U(\boldsymbol{x})\\ \text{Subject to}:&\quad K(\boldsymbol{x})U( \boldsymbol{x})=F(\boldsymbol{x})\\ &\quad F(\boldsymbol{x})=f^{ext}(\boldsymbol{x})+f^{grav}( \boldsymbol{x})\\ &\quad V(\boldsymbol{x})=\sum_{i=1}^{NE}x_{i}V_{i}-V^{*}\leq 0 \\ &\quad\boldsymbol{x}=\{x_{1}\ x_{2}\ x_{3}\ \dots\ x_{n}\},\ \ x_{i}=1\ or\ \ x_{i}=10^{-9}\end{split}  \tag{1}\]  
where \(C(x)\) is the objective function, \(U(\boldsymbol{x})\) is the nodal displacement vector, \(K(\boldsymbol{x})\) is the global stiffness matrix and \(V_{i}\) is the volume of the element, and V* is the prescribed volume. \(K(\boldsymbol{x})U(\boldsymbol{x})=F(\boldsymbol{x})\) is the equilibrium equation, \(F(\boldsymbol{x})\) is the global force vector, \(f^{ext}(\boldsymbol{x})\) is the external force vector, and \(f^{grav}(\boldsymbol{x})\) is the force vector inertia forces of the structure, \(x_{i}\) is the design variable of the i-th element, and \(\boldsymbol{x}\) is the vector of design variables. The design is binary and \(x_{i}=10^{-9}\) is imposed in order to avoid a singular FEM problem when solving the equilibrium. To start the SESO process, it is necessary to define a problem that specifies a design domain and boundary conditions. Then, a finite element analysis is performed to determine the stiffness distribution. The heuristic of SESO is that the structure evolves into a stationary optimal solution, systematically removing inefficient elements. Through a sensitivity analysis, it is possible to calculate a sensitivity number \(\alpha_{j}\) for each element, which indicates the magnitude of the change in global elastic strain energy resulting from the removal of that element. Elements with a low sensitivity number can be removed without significantly affecting the overall stiffness of the structure. The steps of the SESO method can be summarized according to the flowchart, as seen in Figure 1.  
Figure 1: SESO Flowchart with own weight. Formulation for Self-Weight Structure  
Self-weight loads depend on gravitational acceleration and material properties, in particular, material density. As such, in OT problems formulated with SESO-3D approaches, the self-weight loads also depend on the set of design variables. Thus, it is possible to calculate and apply the self-weight at the nodes of the elements, and the main advantage is that the load is directly proportional to the amount of material used in the structure. For the hexahedral element proposed by [25], a gravitational load is obtained by assigning 1/8 of the element's weight to each node according to Equation (2), whose formulation was proposed in [26]. \[f_{i}^{grav}=\rho_{i}*V_{i}*g*\overline{f}=\frac{1}{8}dx_{i}dy_{i}dz_{i}\rho_ {i}g\overline{f} \tag{2}\]  
where \(\rho_{i}\) is the density of the i-th element, \(g\) is the acceleration of gravity (\(g=9.81\) m/s\({}^{2}\)), and \(\overline{f}\) is given by  
\[\overline{f}= \begin{bmatrix}\overline{f}_{1}&\overline{f}_{2}&\overline{f}_{3 }&\overline{f}_{4}&\overline{f}_{5}&\overline{f}_{6}&\overline{f}_{7}& \overline{f}_{8}\end{bmatrix}