# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 6 Results and Discussion
Figure 14: Comparison between the wall friction coefficient distribution predicted by the entire model and high-fidelity CFD simulation. Figure 13: Comparison between the wall friction coefficient distribution predicted by the Gappy-POD method and high-fidelity CFD simulation. 2. Transonic flow around the M6 wing  
In this case, the entire high-fidelity sampling method (encompassing both initial sample selection and adaptive refinement) was benchmarked against traditional random sampling. As the training sample size increases, Figure 17 shows the statistical results of prediction errors, including the average error and the 99% confidence interval. With the same number of complete samples, the proposed high-fidelity experimental design method consistently achieves a smaller mean error than random sampling. For example, after 20 adaptive refinement iterations, the proposed method demonstrates a 27% reduction in mean error compared to the random method (0.000344 vs. 0.000471). Additionally, the 0.995 error quantile shows a significant reduction. These results indicate that our experimental design not only enhances overall accuracy, but also mitigates large-error occurrences, aligning with findings from the NACA0012 benchmark case. Based on the input of all training samples and their high-fidelity responses, we employed the POD method and Kriging surrogate model to establish a predictive model for flow fields under different input parameters. For a randomly selected test sample, pressure coefficients predicted by the model and CFD simulation were compared along the six cross-sections spanning from the root to the tip of the wing (cross-section locations illustrated in Figure 5), as shown in Figure 18. The displayed results indicate a high degree of concordance between the model prediction and CFD simulation, even in the complicated shock wave region. Consequently, this prediction model is suitable for large-scale MC sampling to investigate the uncertainty of wall pressures. Figure 16: Mean values and 99% confidence intervals for the wall friction coefficient. Figure 15: Prediction errors of the entire model on all test samples. By performing extensive random sampling on the prediction model, we obtained the uncertainty distribution of the flow field. Figure 19 illustrates the spatial distribution of pressure coefficient standard deviation across the wing surface. Significant uncertainty concentrations are localized within the \(\lambda\)-shaped shock structure on the upper surface,  
Figure 17: Box plot of the prediction error with the entire experimental design algorithm and random sampling method (M6 case). Figure 11:0 Comparison of the wall pressure distribution between model prediction and CFD simulation. contrasting with negligible variability in attached flow regions. This behavior aligns with well-documented challenges in Reynolds-Averaged Navier-Stokes (RANS) simulations, where shock position predictability exhibits strong parametric sensitivity--particularly to turbulence model closure coefficients--due to their direct influence on eddy viscosity modulation near strong pressure gradients.