## 4 Conclusions  
1. In this paper, through combining an improved explicit integration algorithm with the non-zero element one-dimensional compressed storage method, a fast finite element numerical simulation method for PAUT was established. Based on this new numerical method, a fast FEM simulation program for PAUT in Fortran Language was developed.  
Figure 8: Comparison of the numerical simulation calculation and storage between the fast finite element program and commercial finite element software.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \begin{tabular}{c} **Theoretical** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Compmercial Finite** \\ **Element Software** \\ **Value** \\ \end{tabular} & \multicolumn{2}{c}{**Fast FEM Code**} \\  & \begin{tabular}{c} **Value** \\ \end{tabular} & \begin{tabular}{c} **Error** \\ \end{tabular} & \begin{tabular}{c} **Value** \\ \end{tabular} & \
\begin{tabular}{c} **Error** \\ \end{tabular} \\ \hline Longitudinal & 6020.18 m/s & 5964.17 m/s & 0.93\% & 5875.34 m/s & 2.4\% \\
\begin{tabular}{c} Transverse wave \\ \end{tabular} & 3217.92 m/s & 3196.88 m/s & 0.65\% & 3166.47 m/s & 1.6\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the results of the ultrasonic wave velocity using the different methods.  
Figure 9: Phased array fan scan image in (**a**) the real experiment and (**b**) the simulation.  
2. To verify the reliability and efficiency of this new fast simulation method and program, a simulation of PAUT of the type-B sleeve weld was realized and compared with the commercial finite element software and experiment measurements. Through the simulation results, it was found that both the ultrasonic fields and measured defect signals from the fast FEM simulation method and the commercial finite element software were in good agreement with each other.
3. Furthermore, the computation time and memory consumption of the new method was only about a 15 th and a 40 th of the commercial finite element software, respectively. Because of its high efficiency and low storage space, this method could not only greatly shorten the calculation time, but could also be used for the simulation of phased array ultrasonic testing of large structures, which is difficult to carry out through conventional numerical simulations.
4. Finally, the PA fan scan imaging of the defect in the type-B sleeve fillet weld was simulated using this fast FEM simulation method and compared with the experiment results. The reliability of this fast numerical simulation method was verified again. At present, this method has only been compiled into a preliminary program and the operability of this program can be further optimized in the future. In brief, this method has broad application prospects in the field of phased array ultrasonic testing simulations.  
Conceptualization, C.P.; Data curation, H.Z. and P.J.; Formal analysis, C.P. and Y.L.; Funding acquisition, Y.W.; Investigation, Y.W., Y.L. and P.J.; Resources, H.Z.; Writing--original draft, Y.W.; Writing--review & editing, Y.W. and C.P. All authors have read and agreed to the published version of the manuscript. This research was funded by [the National Key R & D Projects of China] grant number [No. 2020 YFB1901504]. Not applicable. Not applicable. Not applicable. Data Availability Statement: Not applicable. The authors would like to thank the National Key R & D Projects of China (No. 2020 YFB1901504) for funding. The authors declare no conflict of interest.