## 2 Numerical Model of Laser Ultrasonics for a Solid Cylinder  
The principle of generation of laser ultrasonics can be expressed as follows: When laser pulses illuminate the surface of a metallic material, one part of the total energy is absorbed by the surface region of the material, causing temperature rise by heat absorption. Thermal energy then diffuses into the specimen as thermal waves. The heated region undergoes thermal expansion, and thermoelastic stresses generate elastic waves (ultrasound) that propagate deep within the sample [44]. Various modes of ultrasound are formed under the constraints of the material surface and internal conditions. In this paper, the discussion of laser power is restricted to a safe level so that ultrasonic waves can be generated and used for non-destructive testing (NDT) applications without harming target specimens. The thermoelastic fields are governed by the coupled equations of thermoelasticity. The governing equations for an isotropic solid are represented as follows [45]:  
\[k\
abla^{2}t_{i}-\rho c_{v}t_{i}-t_{i0}\beta\
abla\cdot\dot{u}=-q \tag{1}\]  
\[\mu\
abla^{2}u+(\lambda+\mu)\
abla(\
abla\cdot u)=\rho\ddot{u}+\beta\
abla t \tag{2}\]  
where \(t_{i}\) is the temperature field inside the sample; \(t_{i0}\) is the ambient temperature; \(u\) is the displacement vector field; \(k\) is thermal conductivity; \(\rho\) is the density of the sample; \(c_{v}\) is the specific heat; \(q\) is the energy of a pulsed laser; \(\beta\) is the thermoacoustic coupling constant: \(\beta=(3\lambda+2\mu)\alpha_{0}\); \(\alpha_{0}\) is linear expansion coefficient; \(\lambda\) and \(\mu\) are the Lame constants. Without considering viscous damping, the finite-element formulation of governing equations, Equations (1) and (2), can be expressed as [46]:  
\[\mathrm{C_{TU}\dot{U}+C_{T}\dot{T}+K_{T}T=Q} \tag{3}\]  
\[\mathrm{M\ddot{U}+K_{UT}T+K_{U}=F} \tag{4}\]where T is the temperature matrix; U is the displacement matrix; M is the mass matrix; K\({}_{\text{UT}}\) is the thermo-mechanical coupling matrix; K\({}_{\text{U}}\) is the stiffness matrix; C\({}_{\text{TU}}\) is the mechanical-thermal coupling matrix; C\({}_{\text{T}}\) is the heat capacity matrix; K\({}_{\text{T}}\) is the heat transfer matrix; F is the node load vector; Q is the temperature load matrix. A fully coupled thermal-stress method is achieved by the setting of the parameters K\({}_{\text{UT}}\) and C\({}_{\text{TU}}\). The explicit finite element method is used in this paper to solve the laser-ultrasonic coupling (Equations (3) and (4)). Solutions of the thermal and mechanical equations are simultaneously obtained by an explicit coupling matrix. The explicit approach does not require iterations or solving tangent stiffness matrices and therefore occupies fewer computing resources than an implicit solution; it has a high computational efficiency. However, the explicit finite element method is only conditionally stable. The time increment used in an explicit approach must be smaller than the stability limit of the operators for the center difference method and forward difference methods [47; 48; 49]. The implicit finite element method is unconditionally stable, and there is no inherent limit to the size of the time increment. The number of incremental steps used in a typical implicit simulation is several orders of magnitude smaller than that in an explicit simulation. However, the laser-induced stress wave has a high frequency and a wide bandwidth. In order to obtain a high accuracy of stress wave propagation, the incremental step must be very small, so the advantages of an implicit mode are not prominent. In summary, the explicit finite element method has a higher computational efficiency and requires fewer computer resources than the implicit method when dealing with thermoelastic problems [48; 49].