## 2 Materials and Methods
### PAI-Based Internal Crack Detection Feasibility Using Numerical Simulation  
Finite element method simulations are performed to study the effects of internal damage on the propagating acoustic wave and investigate the information it carries about the internal material structure. COMSOL MultiPhysics(r) (version 5.6) software package is utilized to model the geometry of a solid steel block. The physical properties of the steel material are set as given in Table 4. Internal damage in the form of a void of different sizes is introduced inside the steel block at different locations. The cracks are introduced at one of the external edges only to be able to replicate the geometry for an actual experimentation geometry in a future study. Figure 6 shows the model geometry used for simulations with internal damages. Figure 7 presents the resulting mesh grid showing triangular mesh components while also giving an idea of their approximate size. It is important to note here that the mesh is fine near the defect edges for better analysis but is kept relatively coarse at other areas to save computational resources. As described in the literature [44], the element size should be calculated keeping in consideration the minimum wavelength of the propagating acoustic wave, as expressed in the mathematical relation given inEquation (8). Also, the simulation time step and the maximum frequency are associated with the mathematical relation given in Equation (9).  
\[\Delta I=\frac{\lambda_{min}}{20} \tag{8}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Value** \\ \hline Density & 7850 kg m\({}^{-3}\) \\ Thermal conductivity & 44.5 W/(m\(\cdot\)K) \\ Coefficient of thermal expansion & 12.3 \(\times\) 10\({}^{-6}\) K\({}^{-1}\) \\ Bulk velocity & 6200 m s\({}^{-1}\) \\ Shear velocity & 3200 m s\({}^{-1}\) \\ Young’s modulus & 205 GPa \\ Poisson’s ratio & 0.28 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Properties of steel material used in the numerical simulation setting.  
Figure 6: A representation of all the geometries used for simulation. Origin of the coordinate system is the left bottom corner of the steel block. (**a**) Steel block (dimensions 150 \(\times\) 50 mm\({}^{2}\)) without any internal crack. (**b**) Steel block with a crack (crack width 1 mm height 10 mm) at an offset of (120,0). (**c**) Steel block with a crack (crack width 10 mm height 1 mm) at an offset of (150,40). (**d**) Steel block with a crack (crack width 1 mm height 10 mm) at an offset of (30,0). (**e**) Steel block with a crack (crack width 10 mm height 5 mm) at an offset of (150,40). (**f**) Steel block with a crack (crack width 10 mm height 5 mm) at an offset of (150,40). (**f**) Steel block with a crack (crack width 10 mm height 5 mm) at an offset of (150,20).  
Here, \(\Delta I\) is the element size and \(\lambda_{min}\) is the minimum wavelength of the propagating wave.  
\[\Delta t=\frac{1}{20f_{max}} \tag{9}\]  
Here, \(\Delta t\) is the simulation time step and \(f_{max}\) is the highest frequency of interest. A mechanical force of 10 kN is applied as a load at the face of the block to generate the acoustic waves. The acoustic wave is measured at the designated detection points to mimic the pulse and echo scenario. Figure 8A shows the face of the geometry to which the mechanical actuation is applied using a blue highlighting color. In Figure 8B, the acoustic signal detection point is highlighted as a red-colored dot to pinpoint the sensor location for the measurement of the reflected acoustic wave. All the detection points receive the amplitude intensity of the reflected acoustic wave and store it in data variables. The farthest face of the steel block is configured as a low-reflecting boundary to present it as an infinite geometry to the simulation software and deal with reflections only from inserted internal damages.