## 3 FEM (Finite Element Method) Simulation for Smaller PZT Patches  
From the previous section, although the PZT-metal transducer was able to detect debonding damage of composite plates, the impedance signature changed dramatically when debonding occurred below the attached PZT transducer. For this reason, various tests with smaller PZT sizes should be tested to validate the suitability of using smaller PZT transducers, as using smaller PZT could detect small debonding areas. However, using PZT in smaller sizes (compared to the conventional size of 10 mm \(\times\) 10 mm with 0.5 mm thickness) would be very difficult to perform experimentally. For an example, it would be impossible to perform an EMI technique using a 0.1 mm square PZT as soldering of the positive and negative sides of the PZT would be difficult. Thus, to first check the reliability of the simulation, commercial FEM software ANSYS Workbench was used with coupled field analysis. Here, finite element couples the effects of interrelated physics within the element matrices making the electromechanical impedance simulation possible. In addition, the convergence criteria in ANSYS was unaltered as the program itself produced acceptable performance compared to the experimental result. Using ANSYS, a model representing the PZT-metal transducer used in the previous section with 6407 nodes and 1096 elements was created as shown in Figure 4. The average element quality was 0.959. In addition, the properties used for the PZT material can be seen in Table 2. The properties for the metal part of the PZT-metal transducer were selected from the ANSYS Workbench engineering database (stainless steel). Figure 5a shows the actual impedance signature of the PZT-metal transducer in air using the AD5933 evaluation board, and Figure 5b shows the impedance signature from the ANSYS simulation. Comparing both results, although the shapes of the signature are slightly different, the resonance frequency range from the FEM simulation result matches very well with the actual experimental result. Two large resonance peaks exist, where the larger one is located at around 65 kHz for both results (Figure 5a,b), and the smaller resonance peak for both signatures is located near 40 kHz. Furthermore, the amplitude of the large peak is around twice the size of the smaller resonance peak. Now that we had confirmed that the FEM simulation result was reliable, four more FEM models were created to evaluate the impedance signatures. PZT patches sizing from as small as 0.1-5 mm were created to investigate for resonance. Figure 6a shows the FEM model with a 0.1 mm \(\times\) 0.1 mm PZT patch with 0.01 mm thickness attached to a square metallic material on the bottom (0.2 mm \(\times\) 0.2 mm with 0.01 mm thickness), Figure 6b looks identical to Figure 6a but the size is 10 times larger. The PZT size is 1 mm \(\times\) 1 mm with 0.1 mm thickness with the bottom metal plate of 2 mm \(\times\) 2 mm with 0.1 mm thickness. Figure 6c has a 5 mm \(\times\) 5 mm size PZT with 1 mm thickness and a metal size of 6 mm \(\times\) 9 mm with 0.5 mm thickness. The last FEM model, Figure 6d, has the same dimension as the previous figure \begin{table}
\begin{tabular}{c c c} \hline  & **PSI-5A4E** & \\ \hline
**Density** & & 7800 \\
**Damping Ratio** & & 0.0125 \\
**Stiffness Matrix [\(e^{\textbf{c}}\)]** & \(C_{11}=C_{22}\) & 152 \\  & \(C_{12}\) & 102 \\  & \(C_{13}=C_{23}\) & 100 \\  & \(C_{33}\) & 127 \\  & \(C_{44}=C_{55}\) & 21 \\  & \(C_{66}\) & 25 \\
**Piezoelectric Stress Matrix [\(e\)]** & \(e_{31}=e_{32}\) & \(-5.5\) \\  & \(e_{33}\) & 16.4 \\  & \(e_{24}=e_{15}\) & 12.4 \\
**Electric Permittivity Matrix [\(e^{\textbf{s}}\)]** & \(\epsilon_{11}=e_{22}\) & 950 \\  & \(\epsilon_{33}\) & 890 \\ \hline \end{tabular}
\end{table} 
Table 2: PZT properties for ANSYS simulation study. Figure 5: Impedance signature of the PZT–metal transducer before structure attachment: (**a**) By experiment; (**b**) by simulation. but with the thickness of the metal plate increased to 1.5 mm.