## 3 FEM Analysis and Results
### Finite-Element Model of AETs. Frequency Response Correspondence  
The influence of transducer design on its frequency response was investigated by finite element modeling. It is widely known that the damper size and the number of piezoelectric elements can be considered as major design features that caused a significant effect on the transducer bandwidth. In Figure 6 the frequency response of transducers with a damper of various heights was shown.  
The transducer with a piezoelectric element height of 3 mm, a damping protector of 1 mm and an upper damper with a thickness of 0.5 mm had the frequency response presented in Figure 6a. A decrease in the damper thickness resulted in an evident increase in the bandwidth of a receiving transducer up to 350 kHz. As shown in Figure 6b, growth in the thickness of the upper damper caused low sensitivity of the transducer in the frequency range above 200 kHz, as well as a decline in the resonance frequency from 150 to 80 kHz.  
Comparison of the obtained numerical models of transducers included one and five piezoplates was carried out. The studied effect of increasing the number of piezoelectric plates on the transducer bandwidth was inferred in Figure 7.  
From the frequency response of the transducer model with one piezoelectric element, 1 mm damping protector and an upper protector height of 3 mm, the local maximum of the energy density distribution at frequencies of 160 and 340 kHz can be noticed. The greater the number of piezoelectric elements, the more perceptible equalization of the  
Figure 6: The frequency response of transducers with a damper thickness of (**a**) 0.5 mm and (**b**) 5 mm.  
Figure 7: The frequency response of transducers with (**a**) one and (**b**) five piezoelectric elements.  
energy density distribution in the 80-200 kHz passband will be. Based on the simulation results, the optimal design features of the AE source and receiving transducers (AE sensors) were selected for FEM analysis of acoustic signal propagation in a steel plate model. The model of the AE source included four piezoelectric plates 3 mm thick, a damping detector 1 mm thick, and an upper damper 0.5 mm thick, while the model of AE sensors consisted of two piezoelectric plates 3 mm thick, a damping detector 1 mm thick, and an upper damper 4 mm thick. Their frequency response graphs were shown in Figure 8.  
The AE source of broadband acoustic signals had the frequency response given in Figure 8a. Four piezoelectric elements made it possible to expand the passband of the acoustic signal source up to 120-300 kHz. In Figure 8b the frequency response of the receiving transducer model was displayed. The optimal FE model of the AE transducer obtained the same spectral characteristics as the AE sensors from the experimental part had, see Figure 3. To numerically estimate the similarity of the recorded AE signals spectra (Figure 3b) and the frequency response of the receiving transducer FE model, the correlation coefficient \(r\) was computed:  
\[r=\frac{\overline{|H(f)_{1}\cdot H(f)_{2}|}-\overline{|H(f)_{1}|}\cdot\overline {|H(f)_{2}|}}{\sigma_{1}\cdot\sigma_{2}} \tag{4}\]  
where \(H(f)_{1}\) and \(\sigma_{1}\)--the average value and dispersion of the spectral characteristics of the recorded AE signals; \(H(f)_{2}\) and \(\sigma_{2}\)--the average value and dispersion of the spectral characteristics of the receiving transducer model.  
The correlation coefficient value between the results shown in Figures 3b and 8b was \(r=0.82\).