## 4 Case Study on the Multiple-Update-Infill Sampling Methods
### Results and Discussion  
All methods were performed with 30 initial samples for the case studies. To generate different initial samples, Mm LHD was performed 10 times. For the validation samples, 20,000 samples were generated using a quasi-random low-discrepancy sequence (that is, a sobol sequence). Based on the initial samples of each repetition, all SS methods were conducted until the number of the final samples was reached. The number of the final samples is tabulated in **Table 5**. Based on the results of the case studies, the performances of all methods were evaluated in terms of convergence, accuracy, sampling efficiency, and computational cost. Firstly, we evaluated the convergence rate by the decreasing rate of the \(RMSE\) values. A rapid decreasing rate of the \(RMSE\) values indicated that the method accelerates the accuracy of global surrogate modeling. Secondly, the accuracy of the surrogate model was evaluated by the \(RMSE\) values at the final samples. The lower \(RMSE\) value indicated the superior global accuracy of the surrogate model. Next, the pairwise correlation and minimum distance were used to evaluate the measure of clustering the infill samples for sampling efficiency. Lastly, the computational cost was compared in terms of normalized time. **Convergence:** The results of the case studies are shown in Figure 11, Figure 12, and Figure 13, respectively. For all case studies, \(MED_{max}+MED_{min}\) and \(MSE+MED_{max}+MED_{min}\) showed the best convergence performance with small variability. In Case Study 2, \(MSE\) and \(EI_{max}+EI_{min}\) were the worst SS methods in convergence (that is, with slow convergence and larger variability). In Case Study 3, \(EI_{max}+EI_{min}\) was the worst performance in terms of convergence (that is, with divergence and larger variability). \(MSE+EI_{max}+EI_{min}\) and \"All\" also provided convergence performances that  
Figure 10: The contour plots for the response surfaces of the five-story frame structure (axial displacement). \begin{table}
\begin{tabular}{l l c c c} \hline \hline  & **Case Study** & **\# Initial Samples** & **\# Final Samples** & **\# Validation Samples** \\ \hline
1 & Friedman Function & & 210 & \\
2 & Borehole Model & 30 & 240 & 20,000 \\
3 & FE Model Based on 3-bay-5- story Frame Structure & & 300 & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The preliminary setups for all SS methods. were superior to those of \(MSE\) and \(EI_{max}+EI_{min}\). However, their convergence performance was not consistent for all case studies. **Accuracy:** Accuracy was evaluated based on \(RMSE\) values. In Figures 11b-13b, the last value of the \(RMSE\) indicates the measure of the accuracy. Stated differently, lower \(RMSE\) values indicate superior accuracy. For simplicity, the \(RMSE\) values of the final surrogate model were used. \(MED_{max}+MED_{min}\) and \(MSE+MED_{max}+MED_{min}\) provided minimal \(RMSE\) values for all case studies. In addition, their \(RMSE\) values were less dispersed than the other SS methods as shown in Figures 11-13. \"All\" also provided minimal \(RMSE\) values but had a larger dispersion of \(RMSE\) values than \(MED_{max}+MED_{min}\) and \(MSE+MED_{max}+MED_{min}\). The other SS methods were not as good as the abovementioned SS methods. In Case Study 3, \(MSE\) showed excellent performance in terms of convergence; however, it showed poor performance in accuracy. **Sampling efficiency**: The sampling efficiency is measured by the uniformity of the samples in the final surrogate model, as shown in Figures 14-16. As expected, \(MSE\) had the smallest values of \(\rho^{2}\) with large \(d_{min}\) values. Similar performances can be observed in \(MED_{max}+MED_{min}\) and \(MSE+MED_{max}+MED_{min}\). This indicates that the final samples are located over the input space as far apart from each other. On the contrary, the other SS methods have larger values of \(\rho^{2}\) and smaller values of \(d_{min}\). This implies that some of the training samples are closely placed.