*3.2. Experimental Investigations*

Experimental tests of the propagation of Lamb waves in the steel plates were performed using a system for generation and registration of ultrasonic waves, the PAQ-16000D system (EC Electronics, Krakow, Poland). The experimental setup is shown in Figure 4. For both transmitting and receiving Lamb wave signals, plate piezoelectric transducers, NAC2024 (Noliac, Kvistgaard, Denmark), were applied. The excitation was a wave packet with a central frequency of 150 kHz, consisting of five sinusoid cycles modulated with the Hann window.

**Figure 4.** Experimental setup (**a**) and view on the steel plate with piezoelectric transducers (**b**).

The inspected region of the plate was a square with dimensions of 30 cm × 30 cm, with a margin of 10 cm from each of the outer edges of the plate. The excitation was applied at 16 points (8 points on two perpendicular edges), and wave propagation signals were also sensed at 16 points. The transmitting points are marked in Figure 5a as *T*1–*T*16, and the receiving points as *R*1–*R*16. During measurements, the transmitter was placed at a specific point, and the output signals were registered at 8 points located on the opposite edge. The transmitter was then moved to the next point, and the measurement was repeated. Altogether, 128 wave propagation time signals were collected.

The image reconstruction process started by dividing the inspected region of the plate into 64 pixels. The number of pixels was derived from the number of measurement points located on each edge of the plate (8 × 8 points). Each pixel had a dimension of 4.3 cm × 4.3 cm. Pixels are marked with dashed lines in Figure 5b. Image reconstruction was made according to the collected signals of the wave transition from transmitters to receivers. The time of flight (TOF) of the ultrasonic wave for each of the *T*–*R* paths was determined. The tomography velocity image was created in MATLAB® (9.3.0.713597, The MathWorks, Inc., Natick, MA, USA), based on the ART method. In the first step, it was assumed that the wave paths propagating from the transmitters to the receivers are straight. In the next step, the actual course of the rays was traced.

**Figure 5.** The plate with the marked inspected area: (**a**) location of transmitters *T*1–*T*<sup>16</sup> and receivers *R*1–*R*16, (**b**) division of the area into pixels.

#### *3.3. Identification of Material Parameters*

Identification of material parameters was carried out for an intact plate #1. The experimentally determined mass density was ρ = 7893 kg/m3. Poisson's ratio was set as 0.3. The dynamic elastic modulus was also determined experimentally based on a comparison of experimental and numerical dispersion curves. At first, dispersion curves of Lamb waves of two basic modes, i.e., symmetric S0 and antisymmetric A0 were determined. For this purpose, guided waves of frequencies ranging from 40 kHz to 400 kHz with a step of 10 kHz were excited and measured in two configurations, shown in Figure 6a,b. Theoretical dispersion curves were then calculated for different values of elastic modulus. Finally, the dynamic elastic modulus was determined by the method of least squares to give the best fit experimental and numerical modes, and its value was found to be 209 GPa (Figure 6c). The obtained values of the material parameters were then used for numerical simulations.

**Figure 6.** The scheme of experimental tests for determination of dispersion curves for (**a**) A0 mode and (**b**) S0 mode and (**c**) obtained dispersion curves.

## *3.4. Numerical Modelling*

Numerical analysis was carried out using the commercial Abaqus/Explicit program (ver 6.14, Dassault Systemes, Vélizy-Villacoublay, France) based on the finite element method (FEM). Four FEM models were prepared corresponding to the tested plates. Plate models were discretized with S4R elements with maximum dimensions of 1 mm × 1 mm. The boundary conditions were implemented free on all edges. The length of the integration step was 10−<sup>7</sup> s. The excitation signal was the same as in the experiment, i.e., the 5-cycle wave packet of 150 kHz frequency, induced perpendicularly to the surface of the plate at points *T*1–*T*16. The output acceleration signals were collected at points *R*1–*R*16.

#### **4. Results and Discussion**

This research aimed to investigate the influence of increasing surface damage on the obtained tomographic maps. The analysis of the results of experimental and numerical studies was divided into four parts. The first one included tomographic imaging performed for the non-reference approach, with respect to time of transition between transmitters and receivers for each of the tested plates independently. The next part of the analysis consisted of preparing maps based on the differences in the transition times between the reference model and the three damaged models with surface defects. Then, the influence of the ray-tracing technique, taking into account the possibility of wave refraction, reflection and deflection was analysed. The final part focused on analysing the influence of the pixel mesh density on the possibility of estimating damage size.

#### *4.1. Non-Reference Velocity Reconstruction*

Numerical and experimental signals transmitted and registered at selected points were the basis of tomographic maps. The comparison of selected Lamb wave signals for the experimental and numerical models is presented in Figure 7. The graphs show a high convergence between the results from both models.

**Figure 7.** Comparison of selected Lamb wave signals for experimental and numerical tests in a plate without damage (**a**) and with damage (**b**).

The scheme of determining the time of flight (TOF) is shown in Figure 8. The estimation of this value is based on the first wave packet. The wave propagation time was established through the peak-to-peak method and was the difference between the peak value of the first wave pack of the output signal and the peak value of the input signal. The Hilbert transform was employed to create the signal envelope, which enabled the identification of peaks.

**Figure 8.** Determination of the time of flight (TOF) between transmitters and receivers.

The ultrasonic tomography maps are illustrated in Figure 9. The first column shows the plate scheme; the next two present the tomographic maps based on experimental and numerical signals, respectively. The tomograms are constructed, based on the direct measurement of the wave travel time, which means that the results were not compared with the results obtained for the undamaged plate. Each of the performed tomograms is presented separately, ranging from the minimum to the maximum speed. The maps present results for all considered plates, with the increasing surface damage in the form of holes with a diameter of 2, 5 and 10 cm. The defect with a diameter of 2 cm was properly imagined only for data obtained from numerical tests. On the other hand, the holes with a diameter of 5 and 10 cm were successfully detected in both experimental and numerical results. The locations of this damage are marked as areas with a reduced speed in relation to the surrounding material. However, the size of the holes was difficult to estimate accurately.

The quantitative analysis of the values of wave propagation velocities is presented in Tables 1 and 2 for the experimental and numerical data, respectively. The values of minimum, maximum and the mean of the wave velocity are given for all inspected plates. Tables 1 and 2 also provide measures of variation in the form of standard deviation (SD) and coefficient of variation (CV). The wave velocities were calculated along all 128 paths propagating through the plate. The average velocity values for the experimental results (Table 1) were similar and ranged between 2791.01 m/s and 2835.33 m/s. These values apply to the plate with a 10 cm diameter defect and the undamaged plate, respectively. It is worth noting that the differences between the maximum and minimum velocity increased with increasing damage dimensions. This is because the increasing discontinuity of the material reduced the minimum velocity as the rays had to avoid damaged areas. The standard deviation increased with the growing damage area. A similar relationship was observed for the coefficients of variation, which took values ranging between 0.80% and 1.91%.

**Table 1.** Wave propagation velocities in steel plates for data from experimental tests.



**Table 2.** Wave propagation velocities in steel plates for data from numerical tests.

**Figure 9.** Ultrasonic tomography (UT) velocity maps using direct wave transition in plates with surface damage.

Table 2 gives the results of the quantitative analysis carried out for the numerical simulations. The average value of the wave propagation velocity varied between 2743.33 m/s and 2756.65 m/s. As in the case of results from the experimental studies, the difference between the maximum and minimum speed increased with the increasing damage area. The standard deviation was between 29.22 m/s and 51.36 m/s, while the variation coefficient ranged from 1.06% to 1.87%. These values indicate slight differences in the wave velocity along the path of the examined rays and the high sensitivity of ultrasound tomography to the occurrence of a defect.

#### *4.2. Velocity Reconstruction with the Reference to Undamaged Plate*

Ultrasound tomography often consists of comparing the data obtained for an undamaged structure with a damaged structure [38]. During health monitoring, as the damage size grows, the differences become more significant, and the intensity of changes in tomography images indicate the location of the damage. When comparing ultrasonic signals registered in a structure in the healthy (reference) and damaged (current) states, it is possi-

ble to indicate a difference in the TOF, especially if there is a defect along the inspected path. In the case of damage to the entire thickness of the element, the recorded signals travel along the path around the damage [39] so the paths are curved. Examples of Lamb wave signals propagated in an intact plate and a plate with a 5 cm hole along paths *T*1–*R*<sup>3</sup> and *T*1–*R*<sup>5</sup> are compared in Figure 10. When the wave passes through the hole (path *T*1–*R*5), its amplitude and shape change. Moreover, the wave arrives at the receiver with a delay. On the path propagating beyond the damaged area (path *T*1–*R*3), the first wave packet registered by the receiver is the same for the damaged plate and the plate without damage.

**Figure 10.** Comparison of wave signals propagated in intact plate (**a**) and plate with a 5 cm hole (**b**) along traces *T*1–*R*<sup>3</sup> and *T*1–*R*5.

Figure 11 shows the tomographic velocity maps obtained based on differences in wave propagation TOF between the reference plate and the plate with growing damage. Each map was made on a scale from 0 to 1, where 1 indicates the highest difference between reference and current state, and 0 indicates their full compatibility. The location of the circular hole (Figure 11) was detected on the numerical images as regions with a reduced speed of wave propagation. These maps clearly indicate the location of the holes; however, it is not possible to precisely determine their size. This is due to the number of pixels into which the element was divided, as they define the image resolution. Regardless of the hole diameter, each defect lies on a combination of at least two pixels (see Figure 12). This area is indicated on tomography maps as a place with a reduced propagation velocity of ultrasonic waves. In the case of the UT images reconstructed for experimental signals, it can be noticed that they made it possible to determine the location of defects, regardless of their size. At the same time, it was not possible to assess the damage size, as in the case of the numerical results.

**Figure 11.** Ultrasonic tomography (UT) maps in plates with surface damage with the reference to the undamaged plate using TOF differences.

**Figure 12.** Location of damage with respect to the division of the element into pixels.

#### *4.3. Influence of the Ray Tracing Technique*

At the boundaries between regions with different velocities of wave propagation, an elastic wave can be refracted or reflected. In such a case, the wave rays may bend around a defect or other inclusion with a low wave propagation velocity [40]. The assumption that ultrasound waves propagate along straight paths from the transmitters to the receivers gives good results (cf. [41–50]). However, the quality of the tomography can be improved by using the actual ray path. In this study, ray tracing was performed utilizing the so-called hybrid approach, combining the ray bending methods with the network theory (e.g., [51]). The curved path is determined based on the values in individual pixels. The starting point is the image obtained for straight wave paths. Network theory creates a mesh of nodes on which the wave ray can travel. This mesh is usually denser than the pixel division. There are several ways to move from transmitter to specific receiver. The main purpose of the

method is to find nodes through which the path must pass in order to reach the receiver as quickly as possible. To determine which path it is, Dijkstra's algorithm was used. The solution is based on the velocity of wave propagation between two adjacent nodes and the distance between them. All nodes are divided into two groups. In the first group (group I) there are nodes for which the transit time is known. The second group (group II) contains the remaining elements. The schema of Dijkstra's method includes the following steps (e.g., [11,52,53]):


$$t(S) = \min(t(S); t(N) + t\_{SN}),\tag{3}$$

$$t\_{SN} = \frac{d\_{SN}}{\frac{(v\_S + v\_N)}{2}}\tag{4}$$

where *t*(*S*) denotes the travel time to reach node *S*, *t*(*N*) denotes the travel time to reach node *N*, *tNS* is the travel time between nodes *S* and *N*, *dNS* is the distance between nodes *S* and *N* and *vS* and *vN* are the values of the ultrasonic wave velocity in nodes *S* and *N*, respectively;

5. Repeat steps 2–4 until group II is empty.

Dijkstra's algorithm assumes the checking of each node. When the fastest path was established, its straight sections were divided into an increasing number of straight but not collinear segments. The paths made in this way were naturally curved. The travel time was computed for each iteration, and the process ended when converging.

Figures 13 and 14 present experimental and numerical tomography maps created using the curved rays determined by the hybrid method. The analysis was carried out for the time of flight measured directly from transmitters to receivers (Figure 13) and by comparing the results between the current and reference state (Figure 14). The first step of the hybrid method is the preparation of a tomographic map for straight rays from transmitters to receivers. These maps are prepared both for the time of flight measured directly and for the differences in signals between the current and reference model. It is the starting point for which further ray bending iterations are carried out [52].

The tomographic velocity maps made on the basis of direct measurements (Figure 13) clearly indicate the location of the defects. They are visible as areas with smaller values of wave propagation velocity. Each of the tomographic images has its own individual scale, with values ranging from minimum to maximum velocity. Maps made for straight paths give satisfactory results. Simultaneously, the use of a hybrid method, combining the network method and the ray bending method, improved the results. In this case, the defect area was more concentrated. However, it was difficult to assess the extent of the damage based on these maps. Analysis of traced rays showed that the paths determined by the hybrid method bypassed the area of the defects.

**Figure 13.** Ultrasonic tomography (UT) maps using direct travel time measurements with curved paths.

**Figure 14.** Ultrasonic tomography (UT) maps using the method of signal differences with curved paths.

Figure 14 shows tomographic velocity maps based on the comparison of the TOF between the undamaged plate and plates with growing defects. Values in the map cover the range from 0 to 1, where 1 indicates the highest measured difference between the map for undamaged and defective pieces, and 0 indicates their full compatibility. The circular

hole in each plate is shown as an increased value, which means a significant difference in wave propagation velocity in this area. Curved rays are concentrated within the defect. In this case, the maps are similar to those prepared with the use of straight rays. However, the damaged area is only a little more concentrated. In this case, it was also impossible to estimate the damage size as in the case of straight rays.

#### *4.4. Influence of the Pixel Grid Size*

The existence and position of the surface growing damage in steel plates were determined based on the time of flight and UT reconstruction for both experimental and numerical data (see Figures 9, 11, 13 and 14). However, the damage size assessment using the indicated methods did not reflect the actual dimensions of the hole. This was due to the size of the pixels into which the tested structure was divided. Such a division resulted from the number of transmitters and receivers used in the study. On two perpendicular edges, eight transmitters and eight receivers on opposite edges were used, which determined the division of the element into 64 pixels. The number of pixels affected the resolution of the obtained tomography image (cf. Figure 12).

In order to improve the identification of the size of damage, the pixel mesh was refined. The split of each edge was increased from 8 to 15 elements, which gave a total number of 225 pixels. The dimension of a single pixel was 2.15 cm × 2.15 cm. The impact of the density grid was assessed for data obtained from numerical simulations. Figures 15 and 16 show the tomography maps prepared for the elements divided into 64 and 225 pixels, respectively, using direct measurement of time of flight and difference in the TOF between the current state and the reference state. Orange colour marks the pixels for which the damage covers more than half of the area. In the case of a 64-cell mesh, the damage was concentrated within two pixels. The exception was damage with a diameter of 10 cm, which occupied 4 pixels. More significant differences in the number of pixels occupied by the damage are visible when the mesh was densified to 225 pixels.

**Figure 15.** Pixel grid for the direct travel time measurement method: (**a**) division into 64 pixels; (**b**) division into 225 pixels.

**Figure 16.** Pixel grid for travel time difference method (**a**) division into 64 pixels; (**b**) division into 225 pixels.

The maps in Figures 15a and 16a concern models divided into 64 pixels. The tomograms are very similar to each other, regardless of the damage size. The images were enhanced significantly by densifying the pixel grid (Figures 15b and 16b). It was possible to estimate the damage size for such prepared tomograms. The size of the area with a lower velocities of wave propagation on these maps increased with the size of the defect.
