# A Level-Set-Based Density Method for Buckling Optimization of Structure with Curved Grid Stiffeners
## 5 Numerical Examples
### Example 1  
Within this example, a square plate with dimensions of 10 mm \(\times\) 10 mm, supported on both sides with simply supported boundary conditions, is considered. Uniform compressive loads of magnitude \(F=1\) N/mm are applied simultaneously on both sides. The boundary conditions are illustrated in Figure 7: the bottom-left corner of the plate is restricted in the \(x\) and \(y\)-direction translational degrees of freedom (i.e., \(u=0\) and \(v=0\)), the translational degree of freedom in the y-direction at the bottom-right corner is constrained, and the z-direction degrees of freedom on all four sides of the plate are fixed (i.e., \(w=0\)). The wavelength parameters in Equations (5) and (6) are set to \(\mu_{1}=\mu_{2}=1.4\). The design domain consists of \(200\times 200\) finite element elements, with \(20\times 20\) CS-RBF knots distributed uniformly within the design domain. To maintain uniformity in the width of the stiffeners post-optimization, the parameter \(\xi\) in Equations (19) and (20) is set to 0.3. Several upper bounds of volume are applied in this example, and the stiffeners of the initial design are set in two directions: horizontal (\(0^{\circ}\)) and vertical (\(90^{\circ}\)), as shown in Figure 8a. In Equation (13), the volume constraints for stiffeners are defined as \(\overline{V}=0.8V^{0}\), \(\overline{V}=V^{0}\), and \(\overline{V}=1.2V^{0}\), respectively, where \(V^{0}\) is the initial volume of the stiffeners. The optimization results are illustrated in Figure 8b-d. It can be seen that despite varying volume constraints on the stiffeners, the layout of the stiffeners remains essentially consistent. The effect of volume constraint manifests in the width of the stiffeners, where a larger volume leads to wider stiffeners. Regarding the layout, it is notable that the distance between stiffeners increases in the central region. This enhances the structural strength in the central region. The path of the stiffeners forms outwardly curved arcs, aiming to enhance the load-bearing capacity against axial compression loads on both sides. Figure 9a-c depict the corresponding first buckling modes of the three optimized structures, and it should be noted that the buckling mode in this example, as well as in all subsequent examples, represents the structure's normalized displacement in the \(w\)-direction. Figure 10a-c depict the first two BLFs and volume variation curves under three different volume constraint scenarios. It is apparent that there is no intersection between the curves of the first and second BLFs, indicating the absence of repeated first BLFs. Additionally, the volume changes for each case follow the prescribed volume constraint values, eventually reaching their respective maximum allowable volume. Furthermore, it can be seen that the optimization of the critical BLF converges under all three constraint conditions, demonstrating the validity of the optimization. Table 1 presents the corresponding outcomes of the optimization, indicating a clear trend where increasing volume leads to higher buckling load. Under the maximum allowable volumes of 1 times and 1.2 times the initial volume, the increment in critical buckling load reaches 13.3% and 44.5%, respectively. When \(\overline{V}=0.8V^{0}\), the buckling load decreases on account of the reduced volume of the stiffeners. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\overline{\mathbf{V}}\) & \(\mathbf{\Lambda_{1}^{\#}}\) & \(\mathbf{V^{\#}}\) & \(\mathbf{V^{\#}/V^{0}}\) & \(\mathbf{\Delta\lambda_{1}/\lambda_{1}^{0}}\) \\ \hline \(0.8V^{0}\) & 4916.3 & 40.44 & 80.00\% & \(-12.0\%\) \\ \(V^{0}\) & 6330.8 & 50.54 & 99.98\% & 13.3\% \\ \(1.2V^{0}\) & 8070.1 & 60.64 & 119.96\% & 44.5\% \\ \hline \hline \end{tabular}
\end{table}