## 5 Result and Discussion
### Numerical Simulation of Thermal Shock Resistance  
The thermal shock resistance of a coating is defined by its ability to withstand cyclic thermal impacts. This resistance is determined by the relationship between induced stresses and the coating's strength. To evaluate this, the material damage degree (SDEG) is used as an indirect measure, reflecting the stress-strength correspondence. Each element in the model has an individual SDEG value, and the mean SDEG reflects the overall degradation of the respective components. In the following part, \"mean SDEG\" is consistently used to describe the average damage across the material's components. Graded ceramic coatings enhance thermal shock resistance by optimizing material composition and structure, thus improving stress distribution and reducing stress concentration. However, in extreme thermal shock conditions, such as rapid heating in aerospace applications, even graded ceramics can develop microcracks at weak interfaces or internal defects, which propagate with repeated shocks, leading to failure. Additionally, mismatched thermal expansion coefficients in layered ceramic coatings can cause interfacial debonding, further reducing structural integrity. Therefore, selecting appropriate coatings and transition layers is essential for improving thermal shock resistance [5; 58; 59]. An orthogonal experimental design was utilized to assess the thermal shock resistance of the material system. Orthogonal experimentation is a method used to analyze multi-factor, multi-level problems by selecting orthogonal arrays that ensure balance, orthogonality, and representativeness. By systematically setting factors and levels, this method gathers key insights into the effects of various factors on outcomes while minimizing the number of experimental trials. It is especially effective in research and engineering applications where optimizing design parameters is critical, as it enhances experimental efficiency while reducing costs and complexity. In this study, orthogonal simulation analysis was used to identify the factors most significantly impacting thermal shock resistance and to determine the optimal parameter combinations for practical applications [60]. The analysis employed K-values, representing the sum of sample data for a specific level of a factor, and k-values, which are the averages of the K-values. While the mean SDEG provides an indication of material damage for the nine specific test samples, k-values offer a broader evaluation by assessing the combined effects across the entire sample space. In this \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sample** & **1** & **2** & **3** & **4** & **5** & **Average** & **Simulation** & **Prediction Error** \\ \hline Residual Stress (MPa) before the thermal shock test & 771 & 586 & 843 & 828 & 893 & 784 & 973 & 24.11\% \\ \hline Residual Stress (MPa) after the first thermal shock test & 264 & 86 & 244 & 190 & 426 & 242 & 204 & 15.70\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Residual stress experimental test and simulation results. Figure 8: Residual stress contour map: (**a**) preparation of residual stress and (**b**) residual stress after the first thermal shock. context, lower k-values correspond to reduced material damage under specific coating parameters, preserving more of the material's original performance. Therefore, the parameter with the lowest k-value for each factor is considered the optimal setting. Additionally, the R-value, calculated from the K-values, reflects the magnitude of a factor's influence on the overall experimental outcomes [61]. This method allows for a comprehensive evaluation of parameter interactions, making it a powerful tool for optimizing thermal shock resistance in practical applications. The triple thermal shock analysis results for the coating surface are presented in Tables 3-5. The orthogonal simulation analysis indicates that the optimal parameter configurations for thermal shock resistance remain consistent under multiple thermal shocks.