The concept of TR applied to Lamb waves is here briefly recalled with the support of Figure 1. Elastic guided waves are excited into a plate-like structure by means of a tone burst signal U_D(t) at point A (the subscript D is used to distinguish the signal in the direct propagation phase from the reconstructed one, named D,r, in a time reversal experiment). This elastic wave propagates from A to B in the plate where it is recorded as U_acq(t). This signal can be then time inverted U_acq(−t) and used as the input signal in B. The final stage is to acquire the corresponding signal U_D,r(−t) in A again.

If the source is point-like, TR allows to focus back to the source irrespective of the medium complexity (Cassereau and Fink, 1992; Fink, 1992; Wu et al., 1992). Spatial reciprocity is not broken by velocity dispersion, multiple scattering, mode conversion, anisotropy, refraction or attenuation, as long as the latter is linear with respect to the wave amplitude. This remains true even if the propagation medium is inhomogeneous with variations of density and stiffness which reflect, scatter, and refract the acoustic waves. On the contrary, non-linear elastic effects may break spatial reciprocity (and therefore focusing through TR), as do those effects that lead to wave velocity variations along the direct and inverse propagation paths (Park H.W. et al., 2009). Contrary to the case of bulk waves, TR of Lamb waves is complicated by their dispersion and multi-modal nature (Park H.W. et al., 2009).

In what follows the impact localization algorithm originally proposed by Park et al. (2012) is briefly recalled.

First, Lamb guided waves are excited in the specimen under test by means of a surface-mounted piezoelectric (PZT) transducer (red circle in Figure 2A) reproducing an impact-like time-history (a square signal, for instance). A training data set of signals g_i(t), with i = 1, 2, …, m (Figure 2B), is then collected recording the out-of-plane velocity at the desired m points with a Scanning Laser Vibrometer, denoted in Figure 2A with black dots within the target scanning area bounded by the red dashed line.

Now let us suppose that the structure is subjected to an impact within the scanned area (Figure 2C) and that the guided waves response f(t) is recorded by the same piezoelectric used to generate the training dataset (Figure 2D). At this point, the correlations between the actual impact response f(t) and the responses of the training data set g_i(t) are computed. Because of the dispersion of Lamb waves, without any numerical manipulation the correlation is very poor, since the compared signals are completely different. However, it can be mathematically shown that if the correlation is written as a function of inverted signal f(T-t)=f~(t) (being T the duration of the acquired signal), the g_i(t) with maximum correlation to the actual impact response f(t) can be used to identify the impact location. The correlation correlation between f(t) and g(t) is defined as follows (see Park et al., 2012; Miniaci, 2014 for further details):

where ⋆ denotes the correlation operation. On the other hand, the convolution of two functions is defined as:

where ⊗ is the convolution operation. Comparison between Equations (1, 2) reveals that the correlation and convolution are related to each other as follows:

Equation (3) shows that the correlation between two signals is mathematically equivalent to the convolution between one and the time-reversed version of the other one. By applying the Fourier transform, the convolution in the time domain is transformed into a simple multiplication in the frequency domain:

where F denotes the Fourier transform operator. The convolution is reconstructed by taking the inverse Fourier transform of Equation (4):

Since this new expression involves only Fourier and inverse Fourier transforms and point-wise multiplications, the correlation or convolution can be computed effectively:

The maximum correlation value, obtained using Equation (6) is designated as the most likely impact point (Park et al., 2012).

The reviewed impact localization algorithm is tested on a reinforced aluminum plate, schematically shown in Figures 3A,B. The specimen is 1, 000 mm in length and 1, 000 mm in width. It is composed of a flat aluminum 1-mm thick plate reinforced by two unidirectional eccentric stiffeners with L cross-section. The width of both the web and the flange of the stiffeners is 3 mm. The stiffeners are parallel to the specimen edges and are attached to the plate along their full length. Material properties are the following: Young's modulus E = 68 GPa, Poisson's ratio 0.32 and density ρ = 2, 700 kg/m³ (Miniaci, 2014).

The reliability of the proposed technique is first tested numerically by means of a transient Finite Element (FE) analysis simulating the propagation of guided waves in the aforementioned specimen. The implemented model is shown in Figure 3A. A full 3D propagation field is calculated by using linear hexahedral brick elements of C3D8R type for a total number of 2, 686, 684 nodes. To ensure accuracy to the time-transient FE simulations, the plate domain is discretized with elements of maximum side length L_max = 1 mm and the time integration step kept as t_int ≤ 1e − 8 s (De Marchi et al., 2013). To reproduce the experimental conditions, wave reflection, generated by both plate edges and stiffeners, as well as geometrical attenuation, due to wave radiation, are taken into account. Impact is simulated by imposing an out-of-plane displacement in the form of a sharp square pulse (see Figure 3C), which reproduces the kind of excitation that may occur in an impact event. In the FE analysis only 81 scanning points covering a square scanning area of 900 × 900 mm are considered.

Figure 4 shows three snapshots of the simulated guided wave propagation in terms of normalized Von Mises stress at different time steps t1 = 0.5 ms, t2 = 0.75 ms, and t3 = 1 ms after the simulated impact is applied. It is possible to observe how the stiffeners confine and guide much of the energy, and thus of the available information to determine the impact. This is due to the much higher rigidity of the stiffeners with respect to the plate.

The following impact cases are considered:

Impact location corresponding to a grid point (results are shown in Figures 5A,B);

Corresponding signals are collected and processed as explained in section 2.2. Results are presented in Figure 5, confirming the reliability of the method. Figure 5 reports the normalized correlation values between the signal registered at each grid point and the signal of the actual impact points (1-D plots) as well as the estimated points of impact derived in a 2-D visualization. The gray dots denote the scanning points, the red circle the acquisition point, the green spot the real impact positions and the blue crosses the estimated ones. Specifically, it appears that when the impact point coincides with an acquisition point (Figure 5B), the correlation presents a higher value (Figure 5A) than for a random point (Figures 5C,E) and the impact location can be predicted extremely accurately (Figure 5B). Therefore, numerical simulations show that the farther the impact point is from a scanning point, the smaller the correlation value is. The minimum is achieved when an impact occurs at a location equidistant between scanning points (Figures 5E,F).

In what follows, we verify the effectiveness of the TR-based technique regardless of the material and geometrical properties of the specimen. To do this, additional numerical simulations have been carried out (without loss of generality and for the sake of a reduction in computation time, a 2D plane strain model has been implemented see Figure 6A). First, two additional numerical simulations, in which the material properties of the specimen have been changed by small and large amounts with respect to those considered initially, have been performed (refer to Table 1 for the adopted properties), in the case of an impact occurring inside the area delimited by the stiffeners (in #12) and training data collected in #15 (see Figure 6A). In order to verify that the TR-based procedure is not dependent on the local elastic wave velocity, exactly the same configuration is maintained for the two study cases, but with different material properties (aluminum II in Figure 6B and steel Figure 6C, respectively). Figures 6B,C report the normalized correlation values between the signal calculated at each grid point and the signal of the actual impact points (1-D plots), clearly proving the independence of the procedure from wave velocity. This is in accordance with a fundamental symmetry principle of TR (Fink et al., 2000), if the geometry and excitation/acquisition conditions are left unaltered (Miniaci et al., 2017).

Secondly, the stiffeners have been replaced by tapers running through half of the thickness of the plate, as shown in Figure 7, in order to show the effectiveness of the TR-based technique regardless of the geometrical properties of the specimen. In this case, too, the possibility to correctly locating the impact is fully supported by the numerical simulations.

Finally, we performed additional numerical simulations to check the reliability of the TR-based technique in the case of an impact occurring outside the area delimited by the stiffeners (for instance in #23 with respect to Figure 8) and training data collected through a transducer still bonded between the two stiffeners (in #15). Here too, without loss of generality and to reduce computation time, a 2D plane strain model has been implemented, as shown in Figure 8A. Black and red arrows have the same meaning as in the previous cases. We found that the accuracy of the technique in detecting impacts outside the area delimited by the stiffeners is strongly correlated to the quantity of energy the geometrical irregularity (i.e., the stiffeners) allow to reach the detection point. Indeed, since the thickness of the stiffeners is three times that of the plate, they confine and guide most of the energy (the amplitude of the wave beyond the stiffeners is more than 3 times smaller than that inside the stiffeners), strongly limiting the information reaching the transducer in #15 (in the case of impact occurring outside the area delimited by the stiffeners and training data collected from a transducer bonded between the stiffeners). Therefore, in this specific case with huge impedance mismatch, another map of training data would be required to correctly identify the impact location (see Figure 8B). This condition corresponds to a small ratio between the maximum of the wave amplitudes calculated outside (A_out) and inside (A_in) the area delimited by the stiffeners (0.283). However, if the rigidity of the stiffeners is decreased, a larger wave amplitude is allowed to pass beyond them and the accuracy of the technique increases, as for instance in the case reported in Figures 8C,D, corresponding to the cases of A_out/A_in = 0.415 and A_out/A_in = 0.511, respectively. Thus, it emerges that the method can still be applied insofar as sufficient wave amplitude is guaranteed beyond the geometrical irregularities (quantitatively a zero-grid point error is reached for a ratio of 0.5 see Figure 8D).

The tested specimen is shown in Figure 9A. The red circle represents the position of both the piezoelectric transducers used as actuator for the data training acquisition and of the SLDV acquisition point in impact tests. The testing region (768 × 803 mm), represented as the yellow rectangular box, is composed of a regular grid with 93 × 91 acquisition points. Impacts are simulated in correspondence of the three points shown in Figure 9A, that were randomly chosen within the area delimited by the stiffeners, and denoted by black stars.

Elastic guided waves are excited in the specimen using a ceramic piezoelectric disk of diameter 10 mm made of Sonox® by CeramTec® glued to the surface of the investigated sample using commercial super-glue. A scanning measurement head (PSV 400 by Polytec®) connected to a data acquisition system and a steering circuit (Figure 9B) is used to perform the out-of-plane measurements of the velocities over the target area. The pulse excitation is fed from a TGA1241 function generator by Thurlby Thandar Instruments and amplified through an EPA-104 amplifier by Piezo Systems® Inc, inducing a 20 V_pp signal. In order to improve measurements accuracy, the investigated specimen is covered with self-adesive retro-reflective film by ORALITE®. This allows to improve the laser vibrometer signal level at each measurement point regardless of the incidence angle of the measurement beam on the surface (Ostachowicz et al., 2011).

The training data collection process is realized using the square pulse shown in Figure 3C applied to the piezoelectric transducer. For each scanning point, 16, 384 samples are collected over 8 ms by the SLDV at a sampling rate of 256 kHz, and signals are averaged 128 times to improve the signal-to-noise ratio. Intervals of 50 ms are provided between two consecutive pulse excitations to allow signals to decay close to the background noise level before a new data collection. Measurement of all time signals from 8, 463 scanning points takes 8 h.

The results are shown in Figure 10, where the normalized correlation values between the signal registered at each grid point and the signal of the actual impact points (Figures 10A,C,E) are presented. The estimated point of impact is then derived and highlighted in Figures 10B,D,F. The gray dots denote all the scanning points, the red circle the acquisition point whereas the green spot the real impact position and the blue cross the estimated one. It clearly emerges that the algorithm is able to precisely identify the impact point for all the three considered cases. A very good accuracy, within 0.5 cm, is obtained regardless of the impact position (Miniaci, 2014).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.