## 3 Numerical Simulation
### Thermomechanical Testing and Validation  
It is necessary to point out that the thermal deformation of the outer surface of the sinusoidal corrugated core sandwich structure was too difficult to be measured by the experiment because it was directly exposed to the high heating rate. Moreover, the sinusoidal corrugated core sandwich structure was designed to connect to the inner structure of the TPS panel by support brackets and to operate in various ambient temperatures, so it was difficult to make the experimental setup so that the boundary conditions would be the same as its real condition. [PERSON] et al. [42] also investigated the total deflection of a honeycomb sandwich structure in room conditions and under free-free boundary conditions by measuring the thermal deformation of the backside. In this study, we intended the thermomechanical test to investigate the three-dimensional thermal deformation behavior of the bidirectionally corrugated core sandwich panel. Even though the boundary conditions were different from the real condition and the tests were performed in room conditions for the test structure, we expected that the results from the experiment would provide precise knowledge of the thermomechanical responses of the bidirectionally corrugated core sandwich structures before they would be employed in the TPS panel. In this thermomechanical test, the thermal deformation of the test structure was measured using a three-dimensional digital image correlation (DIC) method, which is a non-contact, full-field deformation measurement method and has been considered to be a highly accurate method in various fields of research, such as for large deformation [43; 44], high-temperature structures [45; 46; 47], and small structures [48; 49; 50; 51; 52]. The inner surface of the test structure was painted with a white background and black dots known as a speckle pattern. The test structure was vertically placed on the stage (two points on the bottom edges of the structure were kept on the stage and the side edge at the inner face sheet of the test structure was learnt up against the baffle plate to guarantee the vertical position). A radiation heater was employed to create a thermal load on the front side of the test structure. The temperature of the front surface was monitored by a thermocouple, while the temperature of the inner surface was recorded by a thermal imaging camera. The images of the inner surface of the test structure were captured by two cameras with the blue bandpass filters as the temperatures of the front surface reached pre-set values. A picture of the experimental setup is shown in Figure 11.  
Figure 12 shows the result of the thermomechanical testing of sandwich structure. The maximum temperature applied on the outer surface of the sandwich structure was 1023 K and the average temperature of the inner surface was 620 K, as shown in Figure 12a. The total deflection curve followed the tendency of the temperature difference curve, as shown in Figure 12b. The total deflection was dependent on the temperature difference between the outer and inner surfaces. The maximum \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Step** & **Time Period (s)** & \
\begin{tabular}{c} **Ambient** \\ **Temperature (K)** \\ \end{tabular} &
\begin{tabular}{c} **Natural Convection** \\ **Coefficient (W/m\({}^{2}\) K)** \\ \end{tabular} & **Emissivity** \\ \hline
1 & 0–500 & 213 & 0 & 0.6 \\
2 & 500–1800 & 243 & 0 & 0.6 \\
3 & 1800–2200 & 273 & 0 & 0.6 \\
4 & 2200–5000 & 295 & 10 & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ambient temperature and convection coefficient in the simulation. deflection was about 3.6 mm at the peaked temperature difference. The out-of-plane deformation field showed a concave shape of the sandwich structure. This maximum deflection was reasonable because the sandwich structure was tested in room conditions where the inner surface could dissipate the heat to the ambient environment.