## 3 Numerical Simulation
### Coupled Displacement-Temperature Analysis  
We conducted a fully coupled displacement-temperature analysis in ABAQUSTM software (ver, 2019, Dassault Systemes, France, 2019) to simulate the thermomechanical phenomena. The geometry of model and heat transfer mechanisms in the sandwich structure are shown in Figure 10. The model consists of an outer face sheet, a corrugated core, and an inner face sheet. The incident heating rates applied to the outer surface of the corrugated core sandwich structure follow the graphs in Figure 10. The heat transfer simulation procedure of the corrugated core sandwich structure is as follows: the heat flux is applied on the outer surface, an amount of heat radiates to the ambient environment, the remaining heat is transferred in the webs of the corrugated core through conduction and in the cavities through radiation, and it finally arrives at the inner surface. The process is described by the 3D transient heat transfer equation as  
\[\frac{\partial}{\partial x}\bigg{(}k\frac{\partial T}{\partial x}\bigg{)}+ \frac{\partial}{\partial y}\bigg{(}k\frac{\partial T}{\partial y}\bigg{)}+ \frac{\partial}{\partial z}\bigg{(}k\frac{\partial T}{\partial z}\bigg{)}- \
abla\cdot\vec{q}_{R}^{+}=\rho c\rho\frac{\partial T}{\partial\tau}, \tag{3}\]  
where \(k\) is the thermal conductivity (W/m\(\cdot\)K), \(T\) is the temperature (K), \(\vec{q}_{R}^{-}\) is the radiative heat flux, \(\rho\) is the density (kg), \(c_{p}\) is the specific heat (J/kg\(\cdot\)K), and \(\tau\) is the time (seconds). The radiation heat flux per unit area into a cavity surface is calculated as Equation (4). \[q_{i}^{c}=\sigma_{s}e_{i}\underset{j=1}{\overset{N}{\underset{i}{\sum}}}e_{j} \underset{k=1}{\overset{N}{\underset{k=1}{\sum}}}F_{ik}C_{kj}^{-1}\big{(}T_{j} ^{4}-T_{i}^{4}\big{)}, \tag{4}\]  
where \(q_{i}^{c}\) is the radiation heat flux per area into a cavity surface \(i\); \(\sigma_{s}\) is the Stefan-Boltzmann constant; \(e_{i}\) and \(e_{j}\) are the emissivity of surfaces \(i\) and \(j\), respectively; \(F_{ik}\) is the geometrical view factor matrix; and \(C_{kj}\) is the reflection matrix given in Equation (5). The approximation of the average-temperature radiation condition is assumed for the cavity as described in Equation (6). \[C_{kj}=\delta_{kj}-\big{(}1-e_{j}\big{)}F_{kj}, \tag{5}\]  
\[T_{av}^{4}=\frac{1}{A_{total}}\underset{j=1}{\overset{N}{\underset{i}{\sum}}} A_{j}T_{j}^{4}, \tag{6}\]  
where \(\delta_{kj}\) is the Kronecker delta; \(A_{j}\) and \(A_{total}\) are the area of \(j\) surface and the total of the cavity surfaces (m\({}^{2}\)), respectively. The total time of the simulation is 5000 s, which represents the flight time in a mission of the vehicle to the earth and after touchdown on the runway. The thermal load is applied for a period of 2200 s to simulate the flight time of the vehicle to the earth. For the standing time on the ground after touchdown (2200 s), the vehicle does not suffer any more heat flux, so natural convection and radiation from vehicle to the ambient environment at the runway is applied for the next 2800 s ([PERSON] et al., 2019). The initial temperature of the sandwich structure is assumed to be 295 K. Owing to the unknown ambient temperature during a flight mission of the vehicle, we also assumed the ambient temperature for the simulation. The natural convection coefficient is determined by considering the Nusselt number  
Figure 10: Heat transfer mechanisms in the corrugated core sandwich structure. (**a**) unidirectional corrugated-core unit cell, (**b**) bidirectional corrugated-core unit cell. and characteristic length of the surface [6; 37; 38]. The ambient temperature and natural convection coefficient are given in detail in Table 4[34]. The temperature-dependent material properties were used to fully consider the material behavior at high temperature. The plastic behavior of the materials was also included in the numerical models.