# Time-Domain Topological Energy Imaging Method of Concrete Cavity Defect by Lamb Wave

Authors: Wen-Fa Zhu; Wei Shao; Le-Le Peng; Guo-Peng Fan; Xing-Jie Chen; Shu-Bin Zheng; Hai-Yan Zhang
Journal: Shock and Vibration
 (2019)

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## Abstract

This paper presents an ultrasonic Lamb wave imaging method based on time-domain topological energy to address artifacts in the results of traditional ultrasound imaging methods. This method is based on topological theory and the calculation of the direct and adjoint sound fields in a defect-free reference medium. It focuses the direct and adjoint sound fields at the cavity defect using time reversal and their time-domain topological energy as the pixel values of the image to reduce the artifacts. The physical mechanism of time-domain topological energy (TDTE) imaging is revealed by finite element simulation and experiment. The feasibility of this method for multilayer concrete cavity defect imaging is verified. Compared with the traditional synthetic aperture focusing technique (SAFT) imaging method, the numerical simulation and experimental results show that the method can overcome the influence of ultrasonic Lamb wave dispersion and locate cavity defects with high accuracy and few artifacts. These features indicate the potential of the method in imaging damage concrete structures.


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## Body

## 1. Introduction

Concrete is the most widely used material in highway, high-speed railway, and other traffic engineering structures. However, its internal damage is substantially increased because of the influence of dynamic load, environmental temperature, and other factors. The internal damages of concrete, such as cavity defect, seriously affect vehicle speed and safety [1–3]. Therefore, the effective characterization of the degree and scope of concrete cavity is an urgent problem for researchers.Ultrasound imaging has widely been used in the detection of concrete structural defects such as bridge deck [4–6], concrete column [7], pavement [8, 9], and tunnel [10]. Existing detection methods often use ultrasonic body waves. Concrete is assumed as a homogeneous medium with a single constant shear wave velocity; thus, synthetic aperture focusing technology (SAFT) is used to image concrete structures (such as MIRA [11], EyeCon [11], and Pundit [12]). However, this assumption only applies to single-layer concrete structures, which limits the application of existing technology in multilayer concrete structures. The difference in acoustic impedance between layers cannot be neglected in cavity defects in highways and high-speed rail structures [13, 14]. In this case, the assumption of a single constant shear wave velocity will inevitably lead to detection errors. Recent advances in SAFT methods (e.g., [15–18]) have theoretically realized the ultrasonic imaging of multilayer concrete structures. However, the detection efficiency of the ultrasonic body wave is low, and meeting the demand of concrete structure in service is difficult [19, 20]. The ultrasonic Lamb wave method, with the advantages of long distance and wide range detection, improves the detection efficiency. However, the echo signal generated by the main lobe pointing to the strong reflective medium on both sides interferes with the incident signal along the main lobe due to the influence of Lamb wave dispersion [21], multimode characteristics, and boundary reflection effect. Moreover, additional artifacts are retrieved in the defective image by the classical imaging method. The details of defects are closely related to scattering. If the scattering and mode conversion characteristics of defects are not considered in the imaging process, then the detailed feature information of defects is difficult to obtain [22–24]. Therefore, the topological imaging method that completely considers the defect scattering characteristics in the defect reconstruction and imaging process is attracting research attention [25].Topological imaging originated from shape optimization in the mathematical field. Defect images can be obtained without assuming the location and shape of defects in advance by using the idea of topological asymptotic, which has been successfully applied in ultrasonic body wave imaging [26–28]. The study of topological imaging methods for Lamb wave detection began in the last 10 years. In 2014, Rodriguez et al. [29] successfully realized ultrasonic Lamb wave imaging of a single through-hole defect using topological imaging. In 2017, Lubeigt et al. [30] studied the application of topographic imaging in nondestructive testing of bounded elastic media. The results of weld seam inspection show that the defect characteristics of the reference samples are obvious and the imaging quality is high. Concrete cavity defect is a common form of damage in civil engineering structures. However, research on cavity defects in multilayer structures is inadequate. This study establishes a finite element model and constructs an experimental system to stimulate the generation of ultrasonic Lamb waves in concrete and obtain ultrasonic scattering signals. The time-domain topological energy method is used to realize the accurate imaging of concrete cavity defects, and the method is compared with the traditional SAFT method. The numerical simulation and experimental results consistently show that the time-domain topology energy has high positioning accuracy and few artifacts.


## 2. Principles of Topological Imaging

The main idea of topological energy imaging in the time domain is to transform the topological progressive process into direct and adjoint problems, that is, direct and adjoint sound fields. The sample containing the scatterer (defect) is called the plate to be testedΩm, as shown in Figure 1(a). The test plate whose attribute is completely the same without the scatterer is called the reference test plate Ω0, as shown in Figure 1(b). If the excitation signal xnt is applied at the nth element of the linear array and received at each point in the imaging region of the reference plate, then the sound field of the entire imaging region can be obtained through calculation, which is called the direct sound field Ux,y,t.Figure 1
Top view of the test plate model: (a) reference test plate; (b) plate to be tested.
(a)
(b)In the process of sound field calculation,tω is set as the cutoff time of the received signal, which should be greater than the time of the received signal at the farthest target point in the imaging region. Tp is the time for the excitation signal to reach the target point px,y in the imaging region. In the Lamb wave propagation model, the point source excitation with xnt as a time function propagates for a certain distance d, and then the signal waveform can be obtained as follows:(1)Dcpxnt;T=F−1Fxntexp−i2πfdcp,where T is the propagation time, F⋅ is the Fourier transform, F−1⋅ is the inverse Fourier transform, and cp is the phase velocity.According to equation (1), the direct sound field in the imaging region of the reference sample is Ux,y,t:(2)Ux,y,t=Dcpxnt;tp.The excitation position is unchanged, and the test plate is changed from the reference test plateΩ0 to the plate to be tested Ωm. The received signal of the block to be tested at the target point is different from the signal received at the same point in the reference sample, as it was scattered by defects. After the scatter signal is windowed and time-reversed, it is used as a secondary excitation source and reexcited in the reference test plate and then received at each point in the imaging area of the reference test plate. The sound field is called the adjoint sound field Vx,y,t. According to the propagation model of the Lamb wave, the adjoint sound field Vx,y,t in the reference plate can also be obtained:(3)Vx,y,t=Dcpz0t;tω−tp,where tω is the time at which the scatter signal reaches the last sensor, tp is the time when the excitation signal reaches the defect, and z0t is the secondary excitation signal.The imaging function of the TDTE method is not about summing the amplitude of the signal in the time domain but the topological energy value. The absolute values of direct sound fieldUx,y,t and of adjoint sound field Vx,y,t are taken first, squared, and finally multiplied in order to obtain, after integrating time, the time-domain topology energy value. The topological energy expression in the time domain is(4)Gx,y=∫0TUX,Y,t2Vx,y,t2dt,where T is the cutoff time of the signal and point x,y is the coordinate of the imaging area.If the topological energy value of each point is mapped individually with the pixel value of that point, then the topological energy imaging result of the whole time domain can be obtained. It should be noted that the topological energy value of the defect’s position is much higher than that of other position.


## 3. Finite Element Simulation

A commercial software package called PZFlex is used to create the finite element model, which is shown in Figure2. PZFlex is the time-domain finite element method (FEM). The complex acoustic wave propagation in a three-dimensional model can be simulated by using this method. The upper and bottom surfaces of the model are free, while the remaining four sides are absorbing. The sample is a two-layer concrete bonding structure. The size of the first layer is 1000 × 1000 × 50 mm3, and the physical parameters are as follows: density is 2500 kg/m3, longitudinal wave velocity is 4700 m/s, and shear wave velocity is 3100 m/s. The size of the second layer is 1000 × 1000 × 30 mm3, and the physical parameters are as follows: density is 1800 kg/m3, longitudinal wave velocity is 2484 m/s, and shear wave velocity is 1521 m/s. A cavity defect having a size of 200 × 200 × 15 mm3 is disposed between the first layer and the second layer of the sample to be inspected. Exciting and receiving sensors are disposed on both sides of the cavity defect. The exciting sensor has nine positions. The position numbers are sequentially labeled as S1–S9 with an interval of 60 mm. The S1 coordinates are (300 mm, 260 mm), whereas S9 coordinates are (300 mm, 740 mm), and the sensor diameter is 10 mm. The receiving sensor has nine positions, and the position numbers are sequentially labeled as R1–R9 with an interval of 60 mm. The R1 coordinates are (700 mm, 260 mm), whereas the R9 coordinates are (700 mm, 740 mm), and the sensor diameter is 10 mm. The imaging area is located at the center of the model and is 600 × 600 mm2 in size.Figure 2
Schematic of the finite element model: (a) 3D finite element model; (b) sectional view.
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(b)The five-period sinusoidal pulse with the Hanning window is used as an excitation signal with a central frequency of 30 kHz. The time-domain waveform and spectrum of the excitation signal are shown in Figure3.Figure 3
Excitation signal time-domain waveform (a) and spectrum (b).
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(b)Figure4 shows the phase velocity dispersion curves obtained by a commercial software package called Disperse (version 2.0.20a, Imperial College London, UK). Figure 4(a) shows the phase velocity dispersion curves in the first layer. The phase velocity of the A0 mode is 2410 m/s when the frequency is 30 kHz. Therefore, the wavelength is 78.2 mm. The relationship between the defect size and the wavelength is d/λ ≈ 2.56. Figure 4(b) shows the phase velocity dispersion curves in the second layer. The phase velocity of the A0 mode is 1254 m/s when the frequency is 30 kHz. Therefore, the wavelength is 150.3 mm. The relationship between the defect size and the wavelength is d/λ ≈ 1.33. Figure 4(c) shows the phase velocity dispersion curves in two layers. The phase velocity of the A0 mode is 1476 m/s when the frequency is 30 kHz. Therefore, the wavelength is 49.2 mm. The relationship between the defect size and the wavelength is d/λ ≈ 1.57. Obviously, the phase velocity dispersion curves are strikingly different in the multilayer structure and the single-layer structure. Therefore, we have to take this difference into consideration in order to improve the test results of a cavity defect which is located at the interface between the first layer and the second layer.Figure 4
Phase velocity dispersion curves: (a) the first layer; (b) the second layer; (c) two layers.
(a)
(b)
(c)The signal acquisition process is as follows: the excitation probe is placed at S1, the receiving probe moves from R1 to R9, and nine groups of experimental signals are collected successively [31]. After signal collection at nine positions is completed, the excitation probe is moved to S2, the receiving probe moves from R1 to R9, and nine groups of experimental signals are collected. By analogy, 9 × 9 groups of experimental signals were collected.The finite element simulation data were used to conduct SAFT imaging, and the imaging results are shown in Figure5. The dotted frame in the imaging figure was the cavity area. In the SAFT imaging intensity profile in Figure 6, the intensity peak in the X direction is at X = 432 mm, and the intensity peak in the Y direction is at Y = 471 mm. Therefore, the center position of the cavity defect represented by the SAFT imaging method is (432 mm, 471 mm). The measured position (432 mm, 471 mm) is represented by (xd1, yd1), and the center position of the actual cavity defect (500 mm, 500 mm) is represented by (xd2, yd2). The positioning error of the cavity defect is defined as E, and E = 74 mm is obtained.Figure 5
SAFT imaging results (finite element simulation).Figure 6
SAFT imaging intensity profile in (a) theX direction of the point (432 mm, 471 mm) and (b) the Y direction of the point (432 mm, 471 mm).
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(b)Taking the excitation source S5 whose coordinate position is (300 mm, 500 mm) as an example, the focusing process of the direct and the adjoint sound fields at the defect in topological imaging is described [32]. As shown in Figure 7, V1–V9 are the scattered signals, and U0 is the direct sound field signal. Figure 7(a) is the scattering signal received by all the receiving sensors. Figure 7(b) shows the time reversal of scattered signals received by all the receiving sensors in the time window (0–396.3 μs). Figure 7(c) is the scattered signal reaching the defect. Figure 7(d) is the scattered signal reaching the nondefective point. By comparing Figures 7(c) and 7(d), it is clear that the scattered signal is focused at the defect, whereas the scattered signal reaching the nondefective area is not focused.Figure 7
Focusing of the direct and adjoint sound fields at the defect: (a) scattered signals received by all receiving sensors; (b) scattered signal after time reversal; (c) scattered signals reaching the defect; (d) scattered signals reaching the nondefective point.
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(b)
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(d)On the basis of focusing scattered signals, equation (4) is used to calculate the topological energy value in the time domain, which is taken as the imaging function. The imaging results are shown in Figure 8. In the time-domain topological energy imaging intensity profile in Figure 9, the intensity peak in the X direction is at X = 470 mm, and the intensity peak in the Y direction is at Y = 500 mm. Therefore, the center position of the cavity defect represented by topological energy imaging in the time domain is (470 mm, 500 mm), and the positioning error with the center position of the actual cavity defect (500 mm, 500 mm) is E = 30 mm.Figure 8
TDTE imaging results (finite element simulation).Figure 9
TDTE imaging intensity profile in (a) theX direction of the point (470 mm, 500 mm) and (b) the Y direction of the point (470 mm, 500 mm).
(a)
(b)In terms of positioning accuracy, the TDTE imaging can realize a more accurate positioning of the cavity defect than the SAFT imaging method which can only determine the approximate location of the defect. In terms of representing the region and artifact distribution of the cavity defect, the defect region represented by topological energy imaging in the time domain is relatively concentrated with few artifacts, whereas the defect region represented by SAFT imaging has a large range with additional artifacts.


## 4. Experimental Research

The experimental system is shown in Figure10(a). The data acquisition method of the experimental signals is consistent with that of finite element simulation, as shown in Figure 10(b). JPR-600C, developed by Japan Probe Corporation, is used as the ultrasonic transmitting and receiving instrument. The excitation voltage is 550 V, the carrier frequency is 30 kHz, the number of pulses is 6, the repetition frequency is 100 Hz, and the sampling rate is 1 Mbit/s. Air-coupled ultrasonic probes with a center frequency of 30 kHz are tilted by 8.2°. A sponge is attached to the surface of the plate-like structure between the transmitting and receiving probes to reduce the effect of noise in the air. To improve the SNR, the experimental signals were averaged for 64 times. In addition, the level calibration of the concrete bonded structure was carried out before the experiment.Figure 10
(a) Experimental system and (b) schematic of the signal acquisition mode.
(a)
(b)Taking the excitation source S5 with coordinate position (300 mm, 500 mm) as an example, the scattering signal obtained in the experiment and the preprocessed scattering signal such as filtering and calibration are shown in Figure11.Figure 11
Original and preprocessed scattering signals obtained at S5 position of the excitation source: (a) original scattering signal; (b) filtered scattering signal; (c) scattering signal after calibration.
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(b)
(c)The preprocessed experimental signals were imaged by SAFT imaging, and the imaging results are shown in Figure12. In the SAFT imaging intensity profile in Figure 13, the intensity peak in the X direction is at X = 388 mm, and the intensity peak in the Y direction is at Y = 572 mm. The center position of the cavity defect represented by the SAFT imaging method was (388 mm, 572 mm), and the positioning error with the actual center position of the cavity defect (500 mm, 500 mm) was E = 142 mm.Figure 12
SAFT imaging experimental results.Figure 13
SAFT imaging intensity profile in (a) theX direction of the point (388 mm, 572 mm) and (b) the Y direction of the point (388 mm, 572 mm).
(a)
(b)Taking the excitation source S5 whose coordinate position is (300 mm, 500 mm) as an example, the focusing process of the direct sound field and the adjoint sound field at the defect in topological imaging is described. As shown in Figure14, V1–V9 are the scattered signals and U0 is the direct sound field signal. Figure 14(a) is the scattering signal received by all the receiving sensors. Figure 14(b) shows the time reversal of scattered signals received by all the receiving sensors in the time window (0–424 μs). Figure 14(c) is the scattered signal reaching the defect. Figure 14(d) is the scattered signal reaching the nondefective point. By comparing Figures 14(c) and 14(d), it is clear that the scattered signal is focused at the defect and not at the nondefective area.Figure 14
Focusing of the direct and adjoint sound fields at the defect: (a) scattered signals received by all receiving sensors; (b) scattered signals after time reversal; (c) scattered signal reaching the defect; (d) scattered signal reaching the nondefective point.
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(b)
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(d)Topological energy imaging results in the time domain are shown in Figure15. According to the time-domain topological energy imaging intensity profile in Figure 16, the intensity peak in the X direction is at X = 468 mm, and the intensity peak in the Y direction is at Y = 552 mm. The center position of the cavity defect in the topological energy imaging in the time domain is (468 mm, 552 mm), and the positioning error between the center position of the actual cavity defect (500 mm, 500 mm) is E = 61 mm.Figure 15
TDTE imaging experimental results.Figure 16
TDTE imaging intensity profile in (a) theX direction of the point (468 mm, 552 mm) and (b) Y direction of the point (468 mm, 552 mm).
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(b)The experimental results are consistent with the finite element results. The TDTE method can achieve a more accurate location of the cavity defects, and the defect areas characterized by energy imaging are relatively concentrated, with fewer artifacts, than the SAFT method.


## 5. Conclusion

In this study, an ultrasonic Lamb wave high-precision imaging method for realizing multilayer concrete cavity defects using TDTE is proposed. The method is based on the calculation of direct and adjoint sound fields in the defect-free reference medium, and the excitation signal of the sensor is used as the source calculation. Then, in the sound field, the Lamb wave scattering signal caused by the cavity defect is used as the sound source to calculate the adjoint sound field. Finally, the direct sound field is multiplied by the adjoint sound field to obtain the time-domain topological energy value as the imaged pixel value. The TDTE method is much better for multilayer structure as it is based on the focusing effect of the forward acoustic field and the adjoint acoustic field. More specifically, the forward acoustic field and the adjoint acoustic field will only be focused at the defect’s position. Therefore this method can improve the positioning accuracy of the imaging system and reduces artifacts. On the basis of numerical simulations and experimental results, and compared with the results of the SAFT method, the following conclusions are drawn:(i)
Using the time reversal method, the direct sound field and the accompanying sound field can be focused at the defect instead of the defect-free area. By multiplying the direct sound field and the adjoint sound field, the interference of multimode aliasing at the defect-free point can be eliminated. Therefore the TDTE imaging has the higher positioning accuracy and fewer artifacts than the SAFT method.(ii)
The numerical simulation and experimental results show that the location accuracy of TDTE imaging is higher than that of traditional SAFT.(iii)
The numerical simulation and experimental results show that the artifacts in the TDTE imaging results are significantly less than those in the traditional SAFT method.



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