## 2 Numerical Model of Laser Ultrasonics for a Solid Cylinder
In the explicit finite element model of laser ultrasonics, the equations of motion for the body are integrated using the explicit central difference integration rule, while the heat transfer equations are integrated using the explicit forward-difference time integration rule. \[\ddot{\textbf{T}}_{n}=\frac{1}{\Delta t}(\textbf{T}_{n+1}-\textbf{T}_{n}) \tag{5}\]  
\[\ddot{\textbf{U}}_{n}=\frac{1}{\Delta t^{2}}(\textbf{U}_{n+1}-2\textbf{U}_{n} \ +\ \textbf{U}_{n-1}) \tag{6}\]  
where \(\Delta t=t_{n+1}-t_{n}\). Equations (5) and (6) are brought into Equations (3) and (4) to obtain the solution of the fully coupled finite element equation. In this research, a laser line source parallel to the cylinder axis in the length direction is used in FEM. A laser source with an approximate Gaussian beam energy distribution is used to thermoelastically generate ultrasonic waves in the solid aluminum cylinder studied. The rise time of the laser beam pulse is 4 ns. The width of the laser line source is 1 mm. The values of the laser energy density is set to \(E=1\) mJ per unit length of the line source. A slot parallel to the cylinder axis is used to simulate the surface crack on the isotropic aluminum cylinder. In this paper, the length of the cylinder, the length of the slot, and the length of the laser line source are assumed to be infinite. Therefore, the ultrasonic field excited by the laser line source in the isotropic cylinder can be simplified to a two-dimensional cross section of the cylinder [45], as shown in Figure 1.  
In procedures of explicit dynamics, the temporal and spatial resolution of finite element simulations are of fundamental importance to the stability and accuracy of the solution. Laser generated ultrasound waves require a very small time-step to accurately resolve their high frequencies components. After several tests, the explicit finite element model converges steadily when the incremental time step \(\Delta t=0.5\) ns is imposed. A reasonable spatial resolution for the propagating waves can be obtained if the size of the finite elements spatial grids is at least 1/20 of the shortest wavelength to be analyzed [50]. \[l_{e}=\frac{\lambda_{\text{min}}}{20} \tag{7}\]where \(l_{e}\) is the element length and \(\lambda_{\text{min}}\) is the shortest wavelength of interest. In this paper, the recommended finite element size is 20 \(\upmu\)m, which is much finer for obtaining the transient ultrasonic field accurately. In order to verify the explicit finite element simulation model, a comparison has been performed using [PERSON]'s study Pan (1980), which contains the results of experimental and calculated normal displacements at an observation angle of 180\({}^{\circ}\) for an aluminum cylinder under thermoelastic generation. In our research, the properties of the aluminum cylinder used for numerical simulations are shown in Table 1. The numerical results are presented in Figure 2a. LL is the first longitudinal wave reflected from the cylindrical surface, and LT are the converted waves of longitudinal and transverse waves converted at the cylindrical surface. The numerical results agree well with the results of [PERSON] (1980). This indicates that the finite element model we established is reasonable and has the potential to be used in a further study of the surface crack identification with laser ultrasonic testing. Furthermore, Figure 2b shows the velocity magnitude map where we can clearly distinguish the spatial distribution of various waveforms. Ultrasonic waves that are excited by a pulsed laser in a cylinder are surface skimming longitudinal wave, surface wave, longitudinal wave, and shear wave [PERSON] (1980). In the ultrasonic wave field generated by a pulsed laser, the surface wave mode traveling along the top surface is most dominant, and the surface skimming longitudinal wave mode having a longitudinal wave velocity propagates just underneath the top surface.