## 6 Numerical Examples
### Arch Structure Subjected to External Pressure  
First, a topology optimization design for the compliance minimization problem subjected to volume constraints was carried out. The final optimization result is shown in Figure 4. The area was divided into a 100 \(\times\) 50 grid. The objective function value of the optimized structure was 2.44 \(\times\) 10\({}^{-2}\), and the first-order buckling factor was 0.74. The corresponding convergence curve is shown in Figure 5.  
Then, the topology optimization design subjected to buckling constraints was carried out, and the buckling constraints were taken as 0.9, 1.0, 1.2, and 1.3. The results are shown in Figure 6. The corresponding first-order buckling factors and compliance values were 0.91 and 2.30 \(\times\) 10\({}^{-2}\), 1.01 and 2.35 \(\times\) 10\({}^{-2}\), 1.21 and 2.58 \(\times\) 10\({}^{-2}\), and 1.30 and 2.83 \(\times\) 10\({}^{-2}\). In this numerical example, each iteration step lasted 0.8 s. To obtain the relationship between the first-order buckling factor and compliance, Table 1 was established. Figure 6: Topology changes during the iteration procedure for the three algorithms with \(\underline{\lambda}=1.00\). Table 1 shows that when the volume fraction of the structural was constant, given the different buckling restraint values, the compliance of the structure increased with the increase of the buckling restraint value, which indicated that the improvement of the structural stability was at the expense of a small amount of structural stiffness. The above conclusion was the same as that in the literature 8. Figure 6 presents the intermediate topologies when the buckling constraint was 1.0. Figure 7 shows that when the buckling constraint was considered, the structure no longer maintained a complete arch structure, and the left and right supports expanded outward. When the buckling constraint was small, holes were generated inside. As the buckling constraint increased, the left and right supports on both sides changed into a porous structure, and there were more support rods to improve the stability of the structure, which was similar to [PERSON] et al. [10] using a porous filling structure to improve stability. Figure 8 shows the optimization results for the compliance minimization subjected to the volume constraint and the first mode of the optimization results when the buckling constraint was 1.0. Since the low-order mode accounted for a large proportion of the response, and the low-order mode would be of more concern in engineering practice, only the first-order mode is given in this paper. Comparing the first-order modes of the two optimization results, it could be found that the stability was improved by expanding the \begin{table}
\begin{tabular}{c c c} \hline \hline
**The Value of the Buckling Restraint** & \
\begin{tabular}{c} **The First Order Buckling Factor of the** \\ **Optimization Result** \\ \end{tabular} &
\begin{tabular}{c} **The Compliance of the** \\ **Optimization Result** \\ \end{tabular} \\ \hline
0.90 & 0.91 & 2.30 \(\times\) 10\({}^{-2}\) \\
1.00 & 1.01 & 2.35 \(\times\) 10\({}^{-2}\) \\
1.20 & 1.21 & 2.58 \(\times\) 10\({}^{-2}\) \\
1.30 & 1.30 & 2.83 \(\times\) 10\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table for buckling constraints and structural compliance. Figure 7: Optimization results for different buckling constraints. left and right supports outward and adding holes when considering buckling constraints. Although the deformation region of the results considering buckling constraints was similar to the compliance minimization result, the buckling factor values were greatly increased, and the structure was more stable. The final optimization results in this paper are consistent with the results of [PERSON] et al. (2015), which are compared in Figure 9. Figures 10 and 11 show the convergence history of the objective function, the structural volume fraction, and the first-order buckling factor when the buckling constraint was taken as 1.0.