## 3 Simulation Results and Comparison  
### Simulation of Phused Array Ultrasonic Sound Field and Detection Signal  
In order to study the reliability and efficiency of the fast finite element numerical simulation method, a numerical code based on this method was developed in Fortran Language. This fast FEM code was used to simulate the phased array ultrasonic testing of the internal defects of the type-B sleeve filled weld, as shown in Figure 3. The material of the type-B sleeve and its weld was X70 stainless steel. The parameters of the simulation model are shown in Table 2. To validate the reliability and efficiency of the new fast FEM code, commercial finite element software was also used with the same simulation model for comparison. \begin{table}
\begin{tabular}{c c c c c} \hline
**Element** & **C** & **Mn** & **P** & **S** \\ \hline Proportion of mass fraction (\%) & 0.06 & 1.12 & 0.01 & 0.004 \\ \hline \end{tabular}
\end{table} 
Table 1: Chemical compositions of the X70 pipeline steel and the weld. Figure 3: Specimen and numerical simulation model of the B-type sleeve. Figure 2: Numerical model of PAUT of the type-B sleeve fillet weld defects. Figure 4 shows the simulated PA longitudinal wave focusing field with a focusing angle of 56\({}^{\circ}\) and a focusing depth of 27 mm at a different given time using the fast FEM code and commercial finite element software. It can be seen that the ultrasonic wave fields simulated by the fast FEM code and commercial finite element software were almost the same as each other. In the ultrasonic wave fields, both the longitudinal wave and the transverse wave could be observed. However, because the velocity of the longitudinal wave was used in the focusing rule of Equations (10) and (11), the longitudinal wave had a much larger amplitude and better focus. The actual focusing angle and depth were in good agreement with the set values. The PA longitudinal wave signal of the defect simulated using the fast FEM code and commercial finite element software is shown in Figure 5. It can also be seen that the waveforms of the two signals were very similar to each other. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Material density & 7.85 kg/m\({}^{3}\) \\ Material elastic modulus & \(2.1\times 10^{11}\) N/m\({}^{2}\) \\ Material Poisson's ratio & 0.3 \\ Grid size & 0.1 mm \\ Number of probe array elements & 16 \\ Array element width & 0.6 mm \\ Space between array elements & 0.2 mm \\ Thickness of pipe & 12.8 mm \\ Thickness of sleeve & 20.0 mm \\ Position of defect relative to probe & (22 mm, 15 mm) \\ Diameter of defect & 1.0 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the simulation model. Figure 4: Simulated PA longitudinal wave focusing fields with a focusing angle of 56\({}^{\circ}\)and focusing depth of 27 mm using (**a**) the fast FEM code and (**b**) commercial finite element software. Figure 6 shows the simulated PA transverse wave focusing field with a focusing angle of 56\({}^{\circ}\) and focusing depth of 27 mm at different given times using the fast FEM code and commercial finite element software. The velocity of the transverse wave was used in the focusing rule of Equations (10) and (11). It was found that the longitudinal wave almost disappeared in the ultrasonic field as the focusing angle of the transverse wave was larger than the critical refraction angle. There are only focused transverse waves in the tested specimen during the transverse wave phased array detection. Moreover, when the transverse wave propagated to the location of the defect, it mainly produced an obvious reflected transverse wave rather than an obvious mode converted longitudinal wave signal. These reduced the interference to the identification of the defect transverse wave echo signal. It can also be seen that the ultrasonic wave fields simulated using the fast FEM code and commercial finite element software were almost the same as each other.