## 4 Conclusions  
To increase the computational speed of a high-fidelity model in solving a complex 3-D transient field problem by a numerical approach, a block Arnoldi-based multipoint-model order-reduction scheme is presented. The proposed methodology was used to reduce the size of an NTM high-fidelity model in solving the three-dimensional temperature field of a case study. The computational results have demonstrated:  
1. The model reduced from the proposed methodology could achieve a higher accuracy compared to one reduced from conventional block Arnoldi in a wideband frequency. For point 1, the maximum error of the proposed model was 0.3603, whereas that of the conventional reduced model can reach 3.2275. For point 2, the maximum error of the proposed approach is 0.0986, whereas that of the conventional reduced model can reach 0.9803. For point 3, the maximum error of the proposed model is 0.1459, whereas that of the conventional reduced model can reach 6.3752.
2. The ROM obtained using the proposed scheme can accurately replicate the transient temperature fields of the NTM high-fidelity model through a wide frequency range, while the computational time of the reduced-order model is less than 13% of the original one.  
It should be noted that an MOR method is generally applicable to a linear system. However, for a nonlinear system, a linearization technique must first be implemented in the system before the application of an MOR. Additionally, once the proposed ROM has been learned in the offline stage, it can be used to simulate a transient response of the system for any excitation /(t), as long as its frequency spectrum stays in the learned-frequency interval.  
Conceptualization, N.W. and S.Y.; methodology, N.W., J.C. and H.W.; software, N.W.; validation, N.W. and S.Y.; formal analysis, N.W. and S.Y.; investigation, S.Y.; resources, S.Y.; data curation, N.W.; writing--original draft preparation, N.W. and S.Y.; writing--review and editing, S.Y. and N.W.; visualization, N.W.; supervision, S.Y.; project administration, S.Y.; funding acquisition, S.Y.; All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (NSFC) under Grants No. 51490682.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Analysis Point** & \multicolumn{2}{c}{**Block Arnoldidi (Order = 30)**} & \multicolumn{2}{c}{**Proposed (Order = 28)**} \\ \cline{2-6}  & **max (K)** & **min (K)** & **Average (K)** & **max (K)** & **min (K)** & **Average (K)** \\ \hline Point 1 & 3.2275 & 0 & 0.1869 & 0.3603 & 0 & 0.0689 \\ Point 2 & 0.9803 & 0 & 0.0543 & 0.0986 & 0 & 0.0246 \\ Point 3 & 6.3752 & 0 & 0.1995 & 0.1459 & 0 & 0.0224 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparisons of the absolute errors on different observing points by using the proposed and conventional schemes.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Analysis Point** & **Proposed (Order = 28)** & **Block Arnoldidi (Order = 20)** & **Block Arnoldi (Order = 30)** \\ \hline Point 1 & 0.0066 & 0.0479 & 0.0246 \\ Point 2 & 0.0065 & 0.0298 & 0.0190 \\ Point 3 & 0.0088 & 0.2659 & 0.1627 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparisons of relative errors on different observing points by using the proposed and conventional schemes.