## 5 Conclusions  
Many engineering applications sometimes require a larger number of time-consuming and computationally intensive models (that is, a high-fidelity model). Surrogate models are solutions that address this computational burden. The SS methods add the infill samples to the currenttraining samples using infill criteria and refine the current surrogate model in a sequential framework. One advantage of SS methods is the generative learning capability on the underlying and unknown response surface. Numerous infill criteria have been developed for SS methods for their exploitation, exploration, and balanced exploitation and exploration. Multiple-update-infill sampling simultaneously uses multiple infill criteria to accelerate global surrogate modeling. This study conducted case studies on several multiple-update-infill sampling methods for the global quality of the surrogate model. Three infill criteria were selected as follows: MSEs (pure exploration), EI, and an MED (balanced exploitation and exploration). Based on these infill criteria, six combinations of these infill criteria were considered. The performances of six different infill sampling methods have been compared through three case studies by increasing the dimensionality of the inputs. Based on the results of the case studies, some observations have been made: Firstly, reducing the sparseness of the input space (that is \(MSE\)) may not result in the improvement on the global quality of the final surrogate model. In Case Study 2, \(MSE\) consistently provides higher \(RMSE\) values than other SS methods, except for \(EI_{max}+EI_{min}\). This indicates that it is more efficient to make infill samples from the particular regions of interest (maximum and minimum) rather than from pure exploration (\(MSE\)). Secondly, some combinations of the infill criteria may result in the numerical instability because of samples that are too closely spaced. Lower values of \(d_{min}\) with higher values of \(\rho^{2}\) were observed in the three combinations, including \(EI_{max}\) and \(EI_{min}\). These infill criteria can increase the risk of making a singularity in the matrix manipulation in the surrogate model construction since they occasionally locate the infill samples near the optima to exploit the response surface. In this context, \(MSE\), \(MED_{max}+MED_{min}\), and \(MSE+MED_{max}+MED_{min}\) are preferred in terms of numerical stability and sampling uniformity. Thirdly, multiple infill criteria can save computational time. As expected, more infill samples at each stage can further reduce computational time. In addition, one further benefit of the multiple infill criteria is a utilization of parallel computing in the high-fidelity model. Considering that the analysis of the high-fidelity model is the most intensive and expensive part in the SS methods, the availability of the parallel implementation can significantly reduce the computational time for the SS methods. Based on the three case studies, the multiple-update-infill sampling methods using MED (\(MED_{max}+MED_{min}\) and \(MSE+MED_{max}+MED_{min}\)) show the best performances in terms of convergence, accuracy, sampling, and computational efficiency for the global quality of the surrogate model. Implementation of both \(MED_{max}\) and \(MED_{min}\) can simultaneously generate infill samples with larger and smaller output values with larger inter-sample distances so that they can generate a dense set in the particular region by avoiding the risk of numerical instability. \(MSE+MED_{max}+MED_{min}\) can save more computational time than \(MED_{max}+MED_{min}\) with the parallel implementation of the high-fidelity model. It is worth noting that the MSE and EI are only available for the kriging model, but the MED can be applied to any surrogate model. In this context, the MED can be good infill criteria for global surrogate modeling to be automatically customized for various response surfaces ranging from smooth to complex ones.