## 3 Three-Dimensional Finite-Element Simulation
### Simulation Results and Analyses  
#### 3.2.1 Signal Excitation and Processing  
To investigate the nonlinear Lamb wave scattering features for different micro-crack in different directions, the selection and processing of the excitation signal is crucial. The excitation signal is a sinusoidal mixed-frequency signal modulated by the Hanning window:  
\[x(t)=[H(t)-H(t-\frac{N}{f_{c}})](1-\cos(\frac{2\pi f_{c}t}{n}))(\sin(2\pi f_{1} t+\varphi_{1})+\sin(2\pi f_{2}t+\varphi_{2})) \tag{12}\]  
where H(\(t\)) is the Heaviside step function;\(f_{c}\) is the center frequency of the excitation signal, which is equal to the half the sum of the fundamental frequency components (\(f_{1}\) and \(f_{2}\)); \(\varphi_{1}\) and \(\varphi_{2}\) are the phases of the incident frequency-mixing signal, respectively; and \(N\) is the number of wave peaks of the modulated sine signal. The number of cycles for the excitation signals is 22, which ensures that the generated sum and difference frequencies will not overlap with the excitation frequency components in the frequency domain. The time-domain waveform of the excitation signal is shown in Figure 5a. The incident signals with 0\({}^{\circ}\) and 180\({}^{\circ}\) initial phase are used to excite respectively. The corresponding spectrum is shown in Figure 5b.  
The numerical simulation was carried out according to the simulated schematic diagram of the damaged aluminum plate shown in Figure 3. Two actuators P\({}_{\text{t1}}\) and P\({}_{\text{t2}}\) (labeled as P\({}_{\text{t1,2}}\)) were coupled to excite the S\({}_{\text{0}}\) mode Lamb wave, and then eight sensors P\({}_{\text{r1}}\)-P\({}_{\text{r8}}\) received the structural response signals in the form of stress. The received signal collected by P\({}_{\text{r5}}\) (0\({}^{\circ}\)) at the micro-crack (\(l\) = 6 mm, \(w\) = 1 \(\upmu\)m) is shown in Figure 6a. Two out-of-phase damage signals are displayed because of phases of the excitation signal were 0\({}^{\circ}\) and 180\({}^{\circ}\) respectively. In the spectrum of the response signal, a relatively obvious difference frequency (150 kHz) component was produced in the sideband of the basic frequencies (\(f_{1}\) = 450 kHz,\(f_{2}\) = 600 kHz), whereas the sum frequency (1050 kHz) component with small amplitude was barely seen in Figure 6b.  
According to the above analyses, when incident signals with phase 0\({}^{\circ}\) and 180\({}^{\circ}\) were applied to the damaged plate, the corresponding structural response signals were out of phase. On the basis of pulse inversion theory, the collected damage signals at different phases are superposed in the time domain, as shown in Figure 7a and the corresponding frequency spectrum is shown in Figure 7b. In the inset of Figure 7c, the initial damage signal and the superposed signal in the frequency domain are given for comparison; after pulse inversion, the amplitude of the difference frequency increased,  
Figure 5: Excitation signal in the aluminum plate: (**a**) time-domain signal; (**b**) frequency-domain spectrum.  
but the amplitude of the fundamental frequency signals at 450 kHz and 600 kHz were eliminated. Therefore, in the nonlinear ultrasonic detection without filter module, the pulse inversion method can be used to filter the ultrasonic signal to enhance the amplitude of the difference frequency signal and eliminate the fundamental frequency signal.