## 3 Finite Element Method Simulation and Results  
### Simulation Setup  
The COMSOL Multiphysics software, which is a useful tool to simulate coupled multiphysics using the FEM, is adopted in this paper to simulate laser ultrasounds. Two physical interfaces of thermal stress and solid mechanics and their coupling packages--thermal expansion and temperature coupling--are used to solve the simulation problem that contains the conversion of light energy, thermal energy, and acoustic energy during the laser ultrasound procedure generated in the finite solid medium. Laser-ultrasound-based mathematical models for thermal expansion and acoustic motion are shown in Equations (1) and (2):  
\[k\
abla^{2}T=\rho c\dot{T} \tag{1}\]  
\[\mu\
abla^{2}u+(\lambda+\mu)\
abla(\
abla\cdot u)=\rho\dot{u}+\beta\
abla T \tag{2}\]  
where \(T\) is the temperature distribution at time t inside the sample; \(\rho\), c, and \(k\) are the density, specific heat capacity, and thermal conductivity, respectively; \(u\) is the displacement field; \(\beta\) is the thermoelastic coupling coefficient; and \(\lambda\) and \(\mu\) are Lame constants. Equations (1) and (2) are coupled by the term \(\beta\
abla T\). Finally, with the FEM, Equations (1) and (2) are solved using COMSOL software to obtain reflected and transmitted acoustic fields at oblique surface cracks. Due to the symmetric characteristic of the laser line source vertical to the scanning direction, a simplified 2D model is used by the FEM in this paper. Moreover, aluminum alloy, which is consistent with the following experimental arrangements, is chosen as the simulation material. The geometry of an FEM model, including laser source location, receiver location, crack depth and angle, as well as boundary condition is shown in Figure 2. The physical parameters of the simulated 2D model are presented in Table 1.  
Figure 1: SVM system flow chart. With the COMSOL Multiphysics software, the grid type, grid size, simulation time step, and length are key parameters that seriously affect the simulation results. In the 2D FEM model, the laser line source is described by a product of two Gaussian functions in the spatial and time domains and shown as \(S(x,t)=g(t)^{*}\!f(x)\). The expressions of \(g(t)\) and \(f(x)\) are:  
\[g(t)=\frac{8t^{3}}{v^{4}}e^{-2t^{2}/v^{2}},f(x)=\frac{1}{\sqrt{2\pi}}\frac{2}{ R_{\rm G}}e^{\frac{-2t^{2}}{R_{\rm G}^{2}}} \tag{3}\]  
where \(R_{\rm G}\) is the Gaussian beam radius of 0.1 mm and \(v\) is the laser pulse rise time of 5 ns. Two functions are shown in Figure 3. To satisfy the rule that the grid size is usually less than a quarter of the acoustic wavelength, a free triangle with a minimum element size of 0.01 mm, a maximum element size of 0.1 mm, a maximum element growth rate of 1.1, a curvature rate of 0.2, and a resolution of narrow regions of 10 are the element types in this built model. The simulation time length is 10 us with a time step of 0.01 us to satisfy the expression \(\Delta t=1/(20t_{\rm max})\), where \(t_{\rm max}\) is the central frequency of ultrasound. The 2D simulated model for laser ultrasounds takes 1 h and 16 min on a workstation with a 3.9 GHz CPU, 32 GB of memory, and a 512 GB hard disk. The size of oblique cracks set on the surface of the specimen includes the depth and angle. Ten depths are used, nine of which are 0.2-1 mm per 0.1 mm, and the tenth depth is 1.2 mm. For each depth, 15 angles are considered; 5 angles are 22.5-52.5\({}^{\circ}\) at an interval of 7.5\({}^{\circ}\), 6 angles are 60-120\({}^{\circ}\) at an interval of 10\({}^{\circ}\), 3 angles are 120-135\({}^{\circ}\) at an interval of 7.5\({}^{\circ}\), and the fifteenth angle is 150\({}^{\circ}\). In total, 150 specimens are used to analyze the acoustic field at cracks to extract information on the crack size using the FEM method. Their data are used for SVM training in the following section.