## 4 Results and Discussion
### Three-Dimensional Thermal Buckling Optimization  
Thermal buckling analysis was conducted under two critical conditions separately in this study. The smallest eigenvalues and corresponding buckling modes of the initial design and proposed optimized structures are displayed in Figure 12. Local buckling is the major design driver for the ITPS. Web buckles at both Time 1 and Time 2, which could affect the deflection of the TFS to a large extent. Focusing on the lower webs that buckled in OPT1 and OPT2, while the angle between the web and face sheet declines, the load-bearing capacity in \(z\)-direction declines. Furthermore, compared to the original  
Figure 11: Temperature distribution comparison of (**a**) origin design and OPT-T and (**b**) origin design and OPT-P. Maximum deformation comparison of (**c**) origin design and OPT-T and (**d**) origin design and OPT-P.  
design, the thickness of the web in the optimized design was reduced. Consequently, the critical load factor values decreased.  
It is still worth noticing that, as shown in Figure 12, critical load of the ITPS increased when parts of the web material were removed. It is possible to argue that by appropriately distributing materials on the webs, the stress distribution could be optimized, and then the buckling characteristics could be improved. Therefore, two possible approaches were provided in this study to modify the ITPS structural performance. Geometric parameters for the optimized 3D configurations are listed in Table 6, and the corresponding geometry is as shown in Figure 7. Results obtained from size optimization and shape optimization are denoted as OPT-SIZE and OPT-SHAPE, respectively.  
The critical load factors for OPT-SIZE at Time 1 and Time 2 are 1.018 and 1.164, increased by 2.2% and 31.1%, respectively. For OPT-SHAPE, the critical load factors, \(L_{cr1}\) and \(L_{cr2}\), are improved by 53.1% and 74.2%, respectively. This result indicates that shape optimization might be a better approach for strengthening the buckling capacity of the ITPS. On the other hand, the optimized webs may change the buckling mode of the members as depicted in Figure 12. The buckling mode corresponding to the smallest eigenvalue for OPT-SHAPE at Time 2 shows that local buckling occurred at the top surface of the ITPS. Compared to the webs, the top face sheet of OPT-SHAPE is more sensitive to the buckling load, which is distinct from other structures. Thus, special attention was paid to the buckling analysis at Time 3 for two optimized structures to validate the reliability under that condition. Leaving out the buckling that occurred on the bottom surface, the smallest local buckling eigenvalue of OPT-SIZE and OPT-SHAPE is 1.490 and 1.348, respectively, as shown in Figure 13. This value is slightly decreased compared to the initial design; nonetheless, it is still possible to suggest that the optimized structures fulfilled all design requirements as noted before.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**ITPS Designs** & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{t_{w1}/mm}\) & \(\mathbf{t_{w2}/mm}\) & \(\mathbf{t_{w3}/mm}\) & \(\mathbf{L_{cr1}}\) & \(\mathbf{L_{cr2}}\) & \(\mathbf{L_{cr3}}\) & **m/kg** \\ \hline INI & - & - & - & - & - & - & 0.996 & 0.888 & - & 10.101 \\ OPT3 & 0.00 & 0.75 & 0.00 & 5.0 & 5.0 & 5.0 & 1.351 & 1.537 & - & 11.131 \\ OPT-SIZE & 0.00 & 0.75 & 0.00 & 2.0 & 3.5 & 4.3 & 1.018 & 1.164 & 1.490 & 10.104 \\ OPT-SHAPE & 0.75 & 0.75 & 0.33 & 5.0 & 5.0 & 5.0 & 1.525 & 1.547 & 1.348 & 10.006 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Geometric parameters and critical load factors for 3D optimized designs.  
Figure 12: Buckling modes of initial structure and optimized models at Times 1 and 2.