# Topology Optimization Based on SA-BESO
## 3 Results and Discussion
### An Example of the L-Shaped Bracket  
The example solves an optimization problem with an L-shaped support. Its shape and parameters are shown in Figure 5. The upper end of the support is fixed and the right top of the support is subject to a vertical downward concentrated force \(F=4\) N. The elastic modulus of the material is \(E=206\) Pa, Poisson's ratio is \(v=0.3\), and its target volume fraction is \(V*=0.4\). By comparing the results of the SA-BESO algorithm and the soft kill BESO algorithm, the design area is uniformly divided into \(40\times 40\) structural elements. The filtering radius of the algorithm is \(r=1.5\), the penalty factor is \(P=3\), the deletion  
Figure 4: Evolution histories of structural compliance and volume fraction: (**a**) the changes in the structural compliance; (**b**) the changes in the volume. Figure 3: Optimal topology for the Michell-type structure. (**a**) \(C=94.4720\), \(t=63\). (**b**) \(C=95.6902\), \(t=60\). (**c**) \(C=96.0235\), \(t=57\). (**d**) \(C=96.6927\), \(t=56\). rate \(er\) ranges from 2 to 4%, the minimum and maximum cross probability are \(Pc.max_{c,min}\), respectively, and the maximum mutation probability is \(P_{m,max}\). When the deletion rate takes different values, the variation curve of the L-shaped structure compliance value calculated by the BESO algorithm is shown in Figure 6a. As can be seen from the figure, the efficiency of the algorithm keeps improving as the deletion rate \(er\) increases, but its structural compliance value may oscillate during the iterations, and the structural compliance value of the topology optimization will eventually increase. When \(er=2.5\%\), the final structure compliance value is the lowest, which is 37.7607, and the number of iterations is 41. When \(er=3.5\%\), its efficiency is the highest, the number of iterations is 27, and the final structure compliance value is 37.983. As shown in Figure 6b, when the minimum variation rate P gradually increases from 50% to 70%, the structural compliance value increases steadily during the iteration and does not increase suddenly. When \(er=2.5\%\), the minimum mutation probability of the SA-BESO algorithm is set as \(P_{m,min}\), and the algorithm performs five random operations. As shown in Figure 7, the optimized topology configurations (a) and (b) obtained by the SA-BESO algorithm are similar to the topology configurations (c) obtained by the BESO algorithm. The structural compliance and iteration times obtained by the SA-BESO algorithm are shown in Table 2. The first four times of compliance values are all lower than those obtained by the BESO  
Figure 5: A loaded knee structure. Figure 6: Evolution histories of structural mean compliance at different rates: (**a**) the changes in the structural compliance using different deletion rate; (**b**) the changes in the structural compliance using different minimum variation rate. algorithm, and the iteration times are all lower than that of the BESO algorithm, indicating that the SA-BESO algorithm has higher robustness and can obtain a topology structure with lower compliance while having higher computational efficiency. When \(er=3.5\%\), the minimum mutation probability of the SA-BESO algorithm is set as \(\text{P}=0.7\), and the algorithm performs five random operations. As shown in Figure 8, the optimized topology configurations (a) and (b) obtained by the SA-BESO algorithm are similar to the topology configurations (c) obtained by the BESO algorithm. The structural compliance and iteration times obtained by the SA-BESO algorithm are shown in Table 3. The five times of structural compliance values are all lower than those obtained by the BESO algorithm, and the average number of iterations of the five operations is 26.6, which is slightly lower than that of the BESO algorithm. This shows that the SA-BESO algorithm has high robustness and can obtain a topology structure with lower structural compliance, and there is little difference in computational efficiency between the two algorithms.