## 2 FEM Simulation of Fatigue Crack AE  
In this section, finite element modeling (FEM) was conducted to study the fatigue-crack-generated AE signals. Commercial finite element package ANSYS 17.0 was used to implement and realize the three-dimensional (3D) FEM model. The dimensions of the FE model are 120 mm \(\times\) 60 mm \(\times\) 1 mm. In the simulation, only a half model (Figure 2) was created by applying the symmetric boundary condition to reduce the calculation time. The material properties of aluminum 2024-T3 are given in Table 1.  
Non-reflective boundaries (NRB) developed in Ref. [35] can eliminate boundary reflections, and thus allow for simulation of guided wave propagation in an infinite medium with small-sized models. 3D structural solid elements (SOLID45) were used to mesh the aluminum plate. COMBIN14 spring-damper elements were used to construct the 30-mm NRB around the model. Figure 2 shows the schematic of FEM model for AE simulation. The mesh size is 1/3 mm for the length and thickness of the model. The dipole moment excitation concept [36] was used to simulate the fatigue crack growth source. In this approach, the AE source due to a fatigue crack growth event was considered as self-equilibrating dipole forces (M11 moment tensor) acting at the crack tip. In the FEM simulation, the M\({}_{11}\) dipole excitation was defined using equal and opposite nodal forces. The detailed \begin{table}
\begin{tabular}{c c c} \hline \hline
**Young’s Modulus (E)** & **Poisson’s Ratio (v)** & **Density (\(\rho\))** \\ \hline
73.1 GPa & 0.33 & 2780 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of aluminum 2024-T3. Figure 1: Schematic of the proposed approach for crack length estimation. information can be found in Ref. [36]. The time profile of the excitation was defined as a cosine-bell function with 0.5 \(\upmu\)s as the rise time. The parameter summary of the FEM simulation is given in Table 2.  
First, wavefield analysis was performed to understand the fatigue crack generated AE. The wavefield of surface strain (\(\varepsilon_{\text{xx}}+\varepsilon_{\text{yy}}\)) was extracted from the FEM simulations to analyze the wave propagation pattern, as shown in Figure 3. Figure 3a shows the wavefield pattern in a non-cracked specimen and Figure 3b shows the wavefield pattern of an 8-mm crack specimen. It can be found that an additional wave packet was observed due to wave scattering at the crack tip for the case of an 8-mm crack. The AE wave generated at one crack tip travels to the other tip and generates additional propagating waves. Second, nodal strain responses at the location of piezoelectric wafer active sensors (PWAS) sensor were extracted from the FEM simulation. The PWAS location is 25 mm from the origin as shown in Figure 2. To investigate the effect of various crack lengths on AE signals, both nodal response and PWAS response were extracted and calculated for no crack case, 4-mm, 6-mm, and 8-mm crack length, respectively. In this study, the in-plane strain was extracted because the PWAS senses the in-plane strain of the AE signals. Figure 4 shows the frequency spectra of in-plane strain \(\varepsilon_{\text{xx}}\) and \(\varepsilon_{\text{yy}}\) of the AE signals at the center node where the PWAS is located, obtained from the FEM simulation. It can be noted that no frequency peaks were observed for the no crack case (Figure 4a). As the crack length increases from 4 mm to 8 mm, the number of frequency peaks increases proportionally. The nodal responses of various crack lengths have specific peaks and valleys in frequency spectra. More specifically, the 4-mm crack (Figure 4b) gives 2 peaks, the 6-mm crack (Figure 4c) has 3 peaks, and the 8-mm crack (Figure 4d) gives 4 peaks in the frequency spectrum. Hence, the frequency of spectrum peaks is directly related to crack length. To compare the FEM simulation result with the experiment, the PWAS response was calculated through the area integration of the in-plane nodal strain within the PWAS region.