# Topology Optimization Based on SA-BESO
## 3 Results and Discussion  
### Example of the Two-Dimensional Simply Supported Beam  
The simply supported beam is shown in Figure 2--its topology optimization design area is the rectangular area. The elastic modulus of the material is \(E=206\) GPa, and Poisson's ratio is \(v=0.3\). The lower left corner of the structure constrains the degrees of freedom in the X and Y directions, and the upper middle corner of the structure is subject to the force of the concentration force \(F=1\) N. According to symmetry, only the design area of the right half can be considered, as shown in Figure 2b. The design area was divided into \(60\times 20\) grids, with a filter radius \(r=3\), penalty factor \(p=3\), evolution rate \(ER=2\%\) and temperature set as \(T=100\)\({}^{\circ}\)C.  
We explored the effect of the values of the minimum crossing rate and the variation rate on the performance of the SA-BESO algorithm. Due to the random nature of the algorithm, each value was run five times and the average value of the results was taken. The relevant result is shown in Table 1. When the minimum crossover rate is \(P_{c.min}\), the value range of the minimum variation rate \(P_{m.min}\) is 40-80%. When \(P_{m.min}\) is 40% or 50%, the algorithm is prone to non-convergence, that is, the minimum variation rate cannot be too low, otherwise the probability of the algorithm generating a new structure is low. As can be seen from Table 1, the minimum rate of variation has a large effect on the efficiency of the algorithm, while the minimum rate of crossing does not have a significant effect on the efficiency. When the minimum variation rate increases, the number of iterations of the algorithm decreases, but if it is overly large, the final structural compliance increases. The efficiency of the algorithm can be tuned by tuning the minimum rate of variation, and the minimum crossing rate can be tuned appropriately to obtain a topological structure with lower compliance. For this example, the preferred value is \(Pm.min_{c.min}\).  
The optimized structure, structural compliance and number of iterations obtained by the SA-BESO algorithm for three consecutive times are shown in Figure 3a-c. Figure 3d is the optimized structure obtained by the standard soft BESO algorithm, and the changes in structural compliance and volume are shown in Figure 4. It can be seen that the optimal  
\begin{table}
\begin{tabular}{c c c} \hline
**The Values** & **Mean Iterative Step** & **Average Compliance of Structure** \\ \hline \(Pm.min_{c.min}\) & 202.6 & 98.0068 \\ \(Pm.min_{c.min}\) & 63.6 & 95.2420 \\ \(Pm.min_{c.min}\) & 56.6 & 96.5007 \\ \(Pm.min_{c.min}\) & 65.0 & 96.1153 \\ \(Pm.min_{c.min}\) & 61.6 & 95.3461 \\ \(Pm.min_{c.min}\) & 68.2 & 95.5784 \\ \(Pm.min_{c.min}\) & 61.6 & 97.0381 \\ \hline \end{tabular}
\end{table}
Table 1: The SA-BESO algorithm for the Michell-type structure—the optimization result data.  
Figure 2: Design domain of a Michell-type structure: (**a**) two-dimensional simply supported beam; (**b**) the design area of the right half of the beam.  
configurations obtained by the two algorithms are similar, with the SA-BESO algorithm obtaining a smaller amount of structural compliance than the BESO algorithm. For structural compliance changes in the iterative process, the BESO algorithm has two large mutations, while the SA-BESO algorithm has no such phenomenon. The structure volume of the SA-BESO algorithm decreases slowly with iteration, while the structure volume of BESO decreases linearly. The variation in the two metrics reflects that the SA-BESO algorithm improves the stability of the iterative process by judging the degree of compliance, and eventually obtains a topological structure with lower compliance.