## 5 Numerical Example
### Example 4--Simply Supported Beam--Performance Characteristic Curve  
In this problem is considered a simply supported beam under the loading and boundary conditions, as shown in Figure 10, with L = 40, h = 20 and F = 1 kN. The design domain is discretized into 40 \(\times\) 20 \(\times\) 40. The volume constraint is limited to 20% of the design domain during the whole evolutionary process. The material properties are [PERSON]'s modulus E = 100 GPa and [PERSON]'s ratio \(\
u\) = 0.3, and the filtering radius is 1.5 mm. Figure 11 shows the optimal topologies in the solution domain for the models implemented. It is observed that the topologies of Figure 12a,b, respectively, ESO and SESO, whose design variables are \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **SESO** & **SERA** & **ESO** & **SIMP** \\ \hline Objective Function & & & & \\ \hline Surface graphics & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **SESO** & **SERA** & **ESO** & **SIMP** \\ \hline GA & & & & \\ \hline Time (s) & 19.74 & 19.99 & 20.10 & 21.38 \\ \hline Iteration & 60 & 61 & 61 & 60 \\ \hline Contour graphics & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparing Topology Optimization Algorithms for Compliant mechanism (Geometrical Advantage). the domain elements, have similar settings. Figure 12c,d, SERA and SIMP have the element density as design variables and presented different topologies. It is noteworthy again that SIMP converged with greater compliance, approximately 45% higher than other methods. The SESO, SERA, and ESO methods allow the withdrawal of elements with low compliance to improve the performance of the structure. Therefore, a characteristic performance curve for this continuum structure is shown in Figure 13. The weight of a structure is gradually reduced during the optimization procedure while compliance increases. The characteristic curve of the performance of a structure during the optimization procedure can be expressed through the weight of the structure and its strain energy, according to [25]. Structure performance informs the success of the stiffness-optimized design. In addition, it informs the designer of its viability. Figure 11: Design domain, boundary, and loading conditions. Figure 12: Optimal topology—(**a**) ESO, (**b**) SESO, (**c**) SERA and (**d**) SIMP. The designer, when analyzing the curves below, should note that the optimal structure reached with compliance 5% greater than the initial compliance, in the graph \(C_{i}/C_{0}=1.05\), is obtained with an approximate ratio volume \(V_{i}/V_{0}=0.3\). Therefore, a volume less than 30%, for example: \(V_{i}/V_{0}=0.26\) is below the curve and, therefore, violates the compliance constraint, \(C_{i}/C_{0}=1.1\) because of the needed 10% greater compliance than the initial. Additionally, this structural design would not be feasible because it lacks the material to finish it. On the other hand, projecting the compliance ratio, \(C_{i}/C_{0}=1.05\), to a point above the curve, \(V_{i}/V_{0}=0.5\), the amount of volume is more than sufficient for executing the project; that is, the project is feasible, but it is oversized. Therefore, the structural optimization methods SESO, SERA, and ESO can improve the performance of oversized structures while saving material. According to Figure 13, these structures meet the design conditions, are not oversized, and have resulted in considerable material savings, as indicated in [25]. In SIMP, this process is continuous, and the topology evolves by changing the modulus of elasticity continuously. Therefore, the SIMP characteristic curve graph is a parallel line to the compliance axis because the volume has a minimum variation in the order of \(10^{-4}\). Figure 14 shows the surface graphs of these structures showing the differences between the optimal settings. The SERA and SIMP methods, based on density, presented very different topologies from the other two SESO and ESO methods.