## 3 Numerical Simulation  
### Thermal Load, Boundary Conditions, and Assumption  
The thermal load applied to the corrugated core sandwich structures was based on the heating rates taken from simulation data at two locations in the X-33 spaceplane vehicle developed by NASA [2; 8], as shown in Figure 6. Two typical heating rates were applied on the outer surface of the corrugated core structures, as shown in Figure 9. We chose the two heating rates of 5.5 W/cm\({}^{2}\) and 3.5-4 W/cm\({}^{2}\) for investigating the thermomechanical performance of the corrugated core structures because it allowed us to use some of the commonly used metallic materials, namely, Inconel 625 superalloy and stainless steel S304. We used \"high heating rate\" to represent a thermal load of 5.5 W/cm\({}^{2}\) and \"low heating rate\" for 3.5-4 W/cm\({}^{2}\). The two heating rates corresponded to two locations that had higher heating than most of the windward and leeward surfaces of the vehicle, except for the locations near the nose and wing edges. The selected heating rates also satisfied the constraints of overheating of the metallic materials. Also note that the region near the nose of the X-33 vehicle was heated to an equivalent temperature of around 1200 K, which is beyond the operating temperature of metallic materials. Therefore, it is preferable to use some typical ceramic composites instead of metallic materials. In our study, the emissivity of the outer surface was 0.6, which is smaller than that of the coated surface of the X-33 spaceplane vehicle; therefore, the equivalent temperature of the two heating rate profiles would be higher than that of the X-33 spaceplane vehicle. Note that the equivalent temperature is the temperature of a surface at which the amount of heat flux transfers to ambient temperature by surface radiation equal to the amount of incident heat flux onto that surface. It depends on the emissivity, and the incident heat flux of the outer surface is given by Equation (2). \[q_{rad}=e\sigma_{s}\left(T_{surf}^{4}-T_{amb}^{4}\right), \tag{2}\]  
where \(q_{rad}\), \(e\), \(\sigma_{s}\), \(T_{Surf}\), and \(T_{amb}\) are the heat flux per unit area of the surface, the emissivity of the surface (0.6 for this study), the Stefan-Boltzmann constant (\(5.67\times 10^{-8}\) W/(m\({}^{2}\)\(\cdot\)K\({}^{4}\))), the absolute surface temperature (K), and the absolute ambient temperature (K), respectively. The thermal boundary conditions consisted of radiation to space at the outer surface and adiabatic conditions at the sides and inner surface. The adiabatic conditions were assumed to be a worst-case perspective as the sandwich structure would not dissipate the heat that was in contact with the thermal insulation in the TPS design. The mechanical boundary conditions depended on how the corrugated core sandwich structures were attached on the TPS panels. A typical TPS panel is described in Figure 7a, consisting of a sandwich structure for the outer surface, insulation material filling the interior of the panel, foil metal enclosing the sides, and corner brackets connecting the sandwich structure to the inner frame. Figure 7b shows the as-fabricated assembled nine TPS panels with overhang areas. The load sharing between panels at the overhang areas was considered as a restriction of three rotational degrees of freedom at the outer edges of the corrugated core sandwich structure [34]. The simply supported boundary conditions were applied for the inner edges of the corrugated core sandwich structure. A schematic of the boundary conditions is shown in Figure 8.  
Figure 5: A picture of the bidirectionally corrugated core sandwich structure. Figure 8: Schematic of the thermal loading and boundary conditions for the heat transfer model. Figure 6: Temperature distribution over the body of the X-33 spaceplane vehicle at Mach 20, a surface emissivity of 0.86, an attitude of 72 km, and an angle of attack of 45 deg [2,8] (unit: Kelvin). Figure 7: Example of the corrugated core sandwich structures attached on the TPS panels.