A time-reversal assisted probabilistic fusion strategy for damage localization in plate-like structures using Lamb waves

Abstract

Precise damage identification and quantification is of great significance to ensure structural performance and provide early-warning for safety maintenance. In this research, a time-reversal assisted probabilistic strategy is developed to accurately localize damage via baseline-free manner in plate-like structures by fusing damage prediction data obtained from the elliptical trajectory location method (ETLM) and reconstruction algorithm for probabilistic inspection of defects (RAPID) algorithm. Damage features, including time difference for ETLM and correlation coefficient index (CCI) for RAPID algorithm, are extracted from the time-reversal focusing signal using the Hilbert transform (HT). To make full use of the damage-related information contained in the time-reversal focusing signals and improve the localizing accuracy, a decision-level data fusion based on Bayesian inference is proposed to fuse damage features and reconstruct damage information. A series of numerical and experimental studies, including different configurations of damage cases and transducer distributions, were conducted to investigate the performances of the proposed time-reversal assisted probabilistic method on damage localization. Results shows that the proposed method could successfully achieve accurate damage localization via baseline-free manner and significantly reduce the damage artifacts compared to traditional methods.

Keywords

Damage localization time reversal elliptical location method RAPID algorithm Bayesian inference probabilistic fusion

Introduction

In recent years, the development of structural health monitoring (SHM) technologies has attracted much attention in both laboratory research and field applications.¹ Guided-waves (GWs), as one of the active damage detection methods, can provide quantitative damage information within seconds.^2,3 Among GWs, Lamb wave detection technology has many advantages, including a wide inspection range, high sensitivity to incipient damage and low cost,^4–6 which makes it a promising approach for online SHM system applications. A large number of theoretical and experimental studies have been conducted to verify the feasibility of Lamb waves on the damage detection and evaluation field.^7–11 The structural conditions can be evaluated by extracting damage-related characteristics of Lamb waves.^12,13 The Lamb waves-based damage detection and quantification could be divided into the baseline method and baseline-free method.¹⁴ Baseline methods, which compare the signals from the intact state and damaged state to obtain damage indexes, have good robustness for structural condition evaluation. However, baseline signals are typically difficult to be obtained in field applications. The baseline-free method, circumventing the difficulties of baseline signal acquisition via various methods,^15–19 has drawn much attention in SHM.

Among them, the time-reversal method, a baseline-free method proposed by Ing and Fink,²⁰ has been extensively adopted in structural damage detection²¹ and condition monitoring²² because the dispersive characteristics of Lamb waves are significantly compensated and the subsequent analysis is simplified. The philosophy of time-reversal method is based on the reciprocity of acoustic waves, which means that the input signal can be well reconstructed at the location of the actuator if the medium is intact. The differences between the reconstructed signal and the original input signal may indicate the occurrence of defects. The simple interpretation of time-reversal focusing signals leads to the extensive application of the time-reversal method on SHM.

The time-reversal focusing signal mainly consists of two parts, namely, the main lobe and the sidebands.²³ The main lobe is produced by the focusing of the direct propagating waves, while the sidebands are generated by the damage-induced scattering waves. It has been proved that the scattered waves contained the rich information about the damage.^24,25 The time differences of the damage induced scattered waves are contained in the sidebands, which could be utilized for damage localization.^26–28 It is interesting to cooperate the elliptical trajectory location method (ETLM) with the time differences obtained from the time reversal method to achieve baseline-free damage localization.

Another widely used damage index of time-reversal focusing signals is the correlation coefficient index (CCI) between the reconstructed signal and original input signal. The CCI has been widely used in damage detection²⁹ and condition monitoring of concrete structures,³⁰ bolt connections,³¹ and plate-like structures.³² A pioneering study is conducted by Wang³³ to combine the CCI of time-reversal signals and a reconstructed algorithm for the probabilistic inspection of damage (RAPID) method to localize the defects in composite structures to circumvent the difficulties of complicated data analysis procedures of Lamb waves induced by anisotropy of composite structures. In RAPID algorithm, the damage index and spatial probability distribution function are used to characterize the damage, and a weighted superposition of the damage indexes of each path gives the amplitude of the imaging points. It needs to be noted that the CCI is sufficient to function as a damage indictor but suffers from poor time-resolution. Then, Wang et al.⁹ used CCI and RAPID to localize defect in aluminum plate and utilized virtual sensing paths to enhance the performance of this approach. After that, Miao et al.³⁴ calculated the CCI via time-reversal reconstructed waveform, and the CCI was utilized as an input damage feature for RAPID algorithm to localize damage. As the damage size changes, the CCI calculated by the traditional main wave packet of the time-reversal focusing signal could not perform sufficient sensitivity to damage. Agrahari and Kapuria et al.²⁹ proposed a new approach to calculate the CCI based on the extended wave packet, and this damage feature showed excellent sensitivity to damage when it is utilized at the best reconstructed frequency. For different forms of damage, the CCI based on the extended wave packet could exhibit good performance.³⁵ Some influential factors (such as: adhesive layer, the thickness of host structure, excitation signal and ambient temperature, etc.) have been considered to investigate the effects on CCI.^22,36 Further, a significant application has been made to fully explore and enhance the combination of CCI based on the reconstructed signal and RAPID algorithm to achieve accurate damage localization.³⁷ Moreover, to eliminate the need of two measurements for each sensing path, the virtual time reversal (VTR) method³⁸ is utilized to improve the efficiency of obtaining CCI, and considerable efforts have been devoted to investigate the combination of CCI based on time-reversal focusing signals and the RAPID algorithm to different application scenarios.^39–41

However, neither of the two aforementioned damage localization methods combined with time-reversal operation can fully exploit the abundant information in the focusing signals. In terms of hardware deployment, both two methods require a sensor network to accurately locate the defects. And it will lead to inadequate utilization of the established sensor network if only one of them is utilized to locate damages. In order to circumvent obstacles, probabilistic method is used to reduce errors and improve accuracy.^42,43 To define the field value of image pixel via probability method, Zhou et al.⁴⁴ proposed a novel probability-based diagnostic method for damage localization with hybrid signal features. Cao et al.⁴⁵ proposed a probability weighted four-point arc algorithm for damage imaging. Guo et al.⁴⁶ used matching pursuit decomposition algorithm to extract damage feature (time-of-flight) and a Gaussian kernel probability density distribution was adopted to characterize the damage.

On the other hand, Data fusion,⁴⁷ a multifaceted process of combining information from different sources, could contribute to a more precise characterization of the inspection items if appropriate data fusion algorithm was employed. Data fusion could be conducted at three levels, including the signal/data level, feature level, and decision level.^48–50 Decision-level data fusion integrates decisions from different sources, which is in an analogous fashion to the decision-making process of human beings. It has been extensively applied in medical image fusion, target recognition in automatic driving and SHM.⁵¹ It integrates different symbols with their uncertainties to generate a combination decision during the decision-making process. Among them, Bayesian inference is the process of fitting the specific probability model to the required data, uses prior information to estimate and update the posterior probability and provides supporting evidence for events.⁵² With the change of evidence, Bayesian inference constantly modifies the brief degree of the hypothetical event by modifying the perceptions of numerous sensors in the sensor network. Probabilistic description provided by Bayes method allows us to consider the various uncertainties in the damage detection process, which is exactly suitable for decision-level data fusion.^53,54 Therefore, the core idea is to use Bayesian inference method to integrate damage features diagnosed by the ETLM and RAPID algorithm for accurate damage localization determination in plate-like structures.

In this paper, a time-reversal assisted probabilistic fusion strategy is established based on Bayesian framework for accurate damage localization using damage features including $Δ t$ and CCI. The flow chart of the proposed approach is shown in Figure 1. The focusing signal is obtained based on the time-reversal operation, and the damage features are extracted using the Hilbert transform (HT). The ETLM combined with $Δ t$ , and RAPID algorithm using CCI are implemented to localize the damage, respectively. To make full use of the time-reversal focusing signals, several sensing paths were selected based on the damage-induced scattered characteristics in time domain, which could reduce the computing cost and validate the robustness of the proposed method. Considering the uncertainties (such as the random errors in measurements and identification process), the posterior distribution is obtained by using a developed Bayesian framework and the probabilistic density function is employed to indicate damage location, which significantly reduces the uncertainty-induced errors from the ETLM and RAPID algorithm. Numerical and experimental studies were performed to validate the effectiveness and accuracy of the proposed approach in a steel plate.

Figure 1.

Flow chart of the proposed time-reversal assisted probabilistic fusion approach.

Methodology

Time-reversal process of Lamb waves in plate structure

The working principle of time-reversal method for damage detection is based on acoustic reciprocity. A probing signal can be well reconstructed at the location of piezoelectric actuator (lead zirconate titanate, denoted as PZT-A) when an output signal recorded at the piezoelectric sensor (denoted as PZT-S) is reversed in time domain and reemitted to the original PZT-A if the medium is intact. The time reversal operation could also realize adaptive focusing and dispersion compensation of Lamb wave signals.

In general, the time reversal process could be divided into two processes, namely the forward process and the backward process, as shown in Figure 2. In order to reduce the complexity of signal processing and calculate the damage index efficiently in the following section, the mainly antisymmetric mode (A mode) is adopted in the forward process. The received signal of the sensor in forward process contains the waves of direct path and scattered path, as shown in Figure 2(a). It needs to be noted that the damage information is mainly incorporated in the scattered wave packets. Then, the received signal including both direct wave packets and scattered wave packets is reversed in time domain and reemitted to the PZT-S in the backward process, as shown in Figure 2(a). Therefore, there are also two possible propagation paths for the reversed signal, the direct wave packets result in the main lobe of the reconstructed signal while the scattered wave packets produce the side lobe in the reconstructed signal. The reconstructed signal $V_{R} (ω)$ at the PZT-A can be expressed as:

\begin{matrix} V_{R} (ω) = V_{DD} + V_{DS} + V_{SD} + V_{SS}, \\ V_{DD} = ζ ζ^{*} G_{1} G_{1}^{*} V, V_{DS} = ζ ζ^{*} G_{1}^{*} G_{2} V, \\ V_{SD} = ζ ζ^{*} G_{1} G_{2}^{*} V, V_{SS} = ζ ζ^{*} G_{2} G_{2}^{*} V \end{matrix}

(1)

where, subscripts D and S represent direct wave path and scattered wave path, respectively. The $V_{DD}$ indicates that the direct wave packets of reconstructed signal received by PZT-A when the direct wave $V_{D}$ of the forward propagating process is reversed in the time domain and used as excitation signal. The subscripts DS, SD and SS are similarly defined. $G_{1}$ and $G_{2}$ are the transfer functions of the structure. $ζ$ is the product of the electromechanical coupling coefficient of the actuator and transducer. V is the excitation signal. The superscript * denotes complex conjugate in the frequency domain.

Figure 2.

Lamb wave propagation in a plate containing a damage. (a) Forward propagating wave from actuator to sensor and(b) backward propagating from sensor after time-reversal to actuator.

By taking inverse Fourier transform of the reconstructed signal $V_{R} (ω)$ , the reconstructed original signal in the time domain can be expressed as^55,56:

\begin{matrix} v_{R} (t) = ζ ζ^{*} (ς_{1}^{2} + ς_{2}^{2}) v (T - t) + ζ ζ^{*} ς_{1} ς_{2} \exp (j ω_{0} Δ τ) \\ [T - (t + Δ t)] \\ + ζ ζ^{*} ς_{1} ς_{2} \exp (- j ω_{0} Δ τ) [T - (t - Δ t)] \\ Δ τ = (R_{S} - R_{D}) / c_{p} (ω_{0}) - Δ t, \\ Δ t = (R_{S} - R_{D}) / c_{g} (ω_{0}) \end{matrix}

(2)

where, $ζ$ is the product of the electromechanical coupling coefficient of the actuator and transducer. $ς_{1}$ and $ς_{2}$ indicate the amplitude component of the structural transfer function $G_{1}$ and $G_{2}$ , respectively. In addition, $ς_{1}$ and $ς_{2}$ could be assumed to be constant at a given central frequency $ω_{0}$ of the input narrow-band signal. $c_{p} (ω_{0})$ and $c_{g} (ω_{0})$ represent the phase velocity and group velocity. $R_{D}$ and $R_{S}$ are the distance of the direct wave and scattered wave propagation path. T is the total duration of the reconstructed signal.

Damage features extraction

Figure 3 shows a typical time-reversal reconstruction signal. The main mode is the time-reversal signal of the excitation signal, which contains only the amplitude term and is independent of the wavenumber. The other two parts appearing symmetrically on both sides of the main mode are side-lobe signals. The time difference $Δ t$ between the side-lobe signal and the focusing main mode in the reconstructed signal is equal to the difference between the propagation times of the scattered wave path and direct wave path, and it could be analytically expressed as (R_S−R_D)/c_g(ω₀), which is consistent with the description of Equation (2). The time difference could be obtained by using the HT of the reconstructed signal.

Figure 3.

Schematic of time-reversal damage information extraction.

According to the acoustic reciprocity, the damage could also be evaluated by comparing the reconstructed signal with the pristine input signal in the time-reversal method. The degree of damage is assessed by the difference between the excitation signal $I (t)$ and reconstructed signal $V (t)$ , which could be expressed as³⁷:

CCI = 1 - \sqrt{{[\int_{t_{i}}^{t_{f}} V (t) I (t) d t]}^{2} / \int_{t_{i}}^{t_{f}} I^{2} (t) d t \int_{t_{i}}^{t_{f}} V^{2} (t) d t},

(3)

The CCI ranges from 0 to 1. A CCI closing to zero, implies that the reconstructed signal coincides well with the pristine input signal, which indicates almost no damage in the wave propagation paths. On the contrary, a high CCI value implies a significant difference between the two signals and the presence of defects in the wave propagation paths.

In the conventional time-reversal method, the CCI is calculated by using the main lobe of the time-reversal reconstructed signal. Nevertheless, with the damage size varying, the CCI calculated by the traditional main wave packet of the time-reversal focusing signal could not perform sufficient sensitivity to damage. Therefore, to increase the sensitivity of the defect indicators, a refined time-reversal method^35,37 is utilized, which uses an extended wave packet ranging between the two side bands to calculate CCI to realize damage localization. In this research, the side bands and main mode contained in the reconstructed signals are utilized to calculate CCI, the start and end times of the extended wave packet are expressed as:

t_{i} = t_{m} - t_{s} - t_{sl} / 2, t_{f} = t_{m} + t_{s} + t_{sl} / 2,

(4)

where $t_{m}$ indicates the time instant of the peak of the main mode, $t_{s}$ denotes the time interval between the peaks of the main mode and side bands, and $t_{sl}$ denotes the time span of the main mode or side band. The envelope of the time-reversal focusing signal is obtained using the HT, as shown in Figure 4.

Figure 4.

Schematic of reconstructed signal for the calculation of CCI.

Time-difference-based ETLM

The ETLM uses a series of ellipse loci with actuator and sensor being two loci, indicating the possible damage locations. Considering a sensor configuration with N sensing paths, assume the velocity $c_{g} (ω_{0})$ being constant before and after interaction with damage, the time difference $Δ t_{i}^{c}$ between the scattered wave path and direct wave path for an arbitrary pitch-catch wave propagation path, can be theoretically defined via a set of nonlinear equations:

\begin{matrix} Δ t_{i}^{c} = (\sqrt{{(x_{i a} - x_{d})}^{2} + {(y_{i a} - y_{d})}^{2}} + \sqrt{{(x_{i s} - x_{d})}^{2} + {(y_{i s} - y_{d})}^{2}} \\ - \sqrt{{(x_{i a} - x_{i s})}^{2} + {(y_{i a} - y_{i s})}^{2}}) / c_{g} (ω_{0}) \end{matrix}

(5)

where $(x_{i a}, y_{i a})$ , $(x_{i s}, y_{i s})$ , and $(x_{d}, y_{d})$ are the coordinates of the central locations of the actuator, sensor, and damage, respectively, in the ith actuator–sensor path. In this research, the time difference between the direct wave packet and the scattered is obtained in the time reversal reconstructed signal. By using the time difference information, a series of complicated ellipse-like curves could be generated and the intersection of a minimum of three ellipses is required to indicate the damage location. In an extraordinary case where damage lies on the sensing path, it will lead to localization error of ETLM because the time difference is hard to be obtained. Therefore, the Bayesian fusion method is proposed in the following section to exploit the complementary merits for accommodating more configurations.

In order to realize probabilistic damage localizing via ETLM, the plate-like structure surface under inspection is meshed into $M \times N$ nodes. Each node corresponds to a damage index value $p_{e} (x, y)$ to indicate the probability of damage presence. A symmetric exponential function⁵⁷ centered at the theoretical $Δ t_{i}^{c}$ was adopted in this study, as shown in Equation (6):

p_{e} (x, y) = \sum_{i = 1}^{N} \exp {- \frac{| Δ t_{i}^{c} - Δ t_{i} |}{τ}}

(6)

where $Δ t_{i}$ is the extracted time difference using time-reversal focusing signal, and $τ$ is a decay factor, which is set as 15 μs. In addition, the point with the maximum value of $p_{e} (x, y)$ is regarded as the estimated damage location.

Correlation-coefficient-index-based RAPID algorithm

Inspired by computerized tomography, Hay et al.⁵⁸ applied a localizing algorithm (RAPID) to map the damage index to a single pitch-catch path of the inspection region, and used the sensor network to realize the localization. The philosophy of RAPID is based on the simple assumption that the damage on the direct path between the actuator and transducer can produce the most significant signal changes, and the degree of the signal difference will decrease with the increase of distance from the direct paths. To quantify the effects of the defects on the sensing paths, the spatial probability distribution function is introduced as the path weight function. It is assumed that the influence is a linearly decreasing elliptic distribution, where the positions of the corresponding actuator and sensor are the foci (as shown in Figure 5).

Figure 5.

(a) Relative distance of damage location and (b) elliptical distribution function of the RAPID algorithm.

In the present study, the RAPID algorithm integrates the CCI of a single sensing path into all sensing paths covered by the detection area, and calculates the damage probability of a single grid point. Thus, the detection area is meshed into uniformly distributed grids. The probability of the defect being located at position $p_{r} (x, y)$ can be calculated as:

p_{r} (x, y) = \sum_{i = 1}^{N} p_{i} (x, y) = \sum_{i = 1}^{N} CC I_{i} W_{i} [R_{i} (x, y)],

(7)

where N is the total number of paths, $CC I_{i}$ is the signal correlation coefficient of sensing path i, and $W_{i} [R_{i} (x, y)]$ is the weight coefficient of $CC I_{i}$ at point (x, y) in the inspection area of the ith sensing path. The W_i(x, y) is dependent on the parameter, $R_{i} (x, y)$ , which is defined as the relative distance from point (x, y) to the ith sensing path:

R_{i} (x, y) = [L_{AD} (x, y) + L_{DS} (x, y)] / L_{AS} - 1,

(8)

where $L_{AS}$ is the distance from the actuator to the sensor of the ith sensing path. $L_{AD} (x, y)$ and $L_{DS} (x, y)$ are the distances from point (x, y) to the actuator and sensor, respectively. The weight distribution function can be defined as:

W_{i} [R_{i} (x, y)] = {\begin{matrix} 1 - R_{i} (x, y) / β, & R_{i} (x, y) < β \\ 0, & R_{i} (x, y) \geq β, \end{matrix}

(9)

The parameter $β$ is the scale factor that determines the shape of the inspection region and it was set at 0.05 in this study. The imaging algorithm can detect the damage in the test area as long as there are sensing paths with sufficient active sensor network coverage.

Bayesian framework-based probabilistic fusion damage localization

Owing to the measurement errors and environmental uncertainties, the damage localization results may deviate from the actual situations. In the Bayesian inference method, the uncertain parameters can be quantified by probability distributions, which was utilized to fuse damage prediction data from the ETLM and RAPID for accurate damage localization. Specifically speaking, Bayesian framework could consider both direct or indirect measured data as evidences to evaluate these distributions of the parameter space, which makes it attract much attention in structure damage identification.^59–62 Bayesian approach could not only offer statistical explanations for measurement errors and uncertainties, but also quantify these deficiencies of the inference via expectations of the parameter probability distributions.

Based on this concept, the core idea is using Bayesian posterior distribution to quantify the uncertainty in damage localizing. In this research, the prior damage location information is derived from RAPID algorithm. The likelihood function is established by weighing the deviation between the measured time difference and the theoretically calculated time difference based on the ETLM. In order to quantify the uncertainty, the variable describing the deviation of the time difference $Δ t$ between the scattered wave path and direct wave path is introduced and expressed by $ε$ . The probabilistic description of the measured time difference in the ith actuator–sensor path $Δ t_{i}^{m}$ can be expressed as:

Δ t_{i}^{m} = Δ t_{i}^{c} (x_{d}, y_{d}) + ε,

(10)

where $Δ t_{i}^{c}$ is the calculated time difference in the ith actuator–sensor path using Equation (5), with the parameters $(x_{d}, y_{d})$ . For brevity, $ε$ is assumed to be a Gaussian variable with a mean of zero and standard deviation $σ$ . Thus, the likelihood function can be written as:

p (Z | θ) = Π_{i = 1}^{N} \frac{1}{\sqrt{2 π} σ} \times \exp [- \frac{1}{2 σ^{2}} \sum_{i = 1}^{N} {(Δ t_{i}^{m} - Δ t_{i}^{c} (θ))}^{2}],

(11)

where the $θ$ is the parameter vector $θ = [x_{d}, y_{d}, σ]$ , $Z = [Δ t_{1}^{m}, Δ t_{2}^{m}, \dots, Δ t_{N}^{m}]$ is the measured time difference. The likelihood function $p (Z | θ)$ is a probabilistic statement that pertains to the distribution of the time difference between the measured data and theoretical calculated results.

According to Bayes’ theorem, the parameter vector of the posterior probability density function (PDF) is expressed as:

p (θ | Z) = p (Z | θ) p (θ) / p (Z),

(12)

where $p (θ | Z)$ is the posterior distribution of $θ$ , and the marginal distribution $p (Z)$ can be expressed as:

p (Z) = \int p (Z | θ) p (θ) d θ,

(13)

where $p (Z)$ is a normalizing factor, and $p (θ)$ is the prior distribution of $θ$ . The prior PDF of $(x_{d}, y_{d})$ is obtained via RAPID method, as demonstrated in Equation (7). Without any prior information about $σ$ , a probability distribution $p (x) \propto 1 / x$ , can be adopted in this research based on the Jeffrey’s noninformative prior.⁶³ Assuming that the unknown parameters are independent of each other, the prior PDF $p (θ)$ can be expressed as:

p (θ) = \frac{1}{σ^{N}} \sum_{i = 1}^{N} CC I_{i} W_{i} [R_{i} (x_{d}, y_{d})],

(14)

Therefore, substituting Equation (11) and Equation (14) into Equation (12), the posterior distribution of the unknown parameters is obtained as:

\begin{matrix} p (θ | Z) \propto \frac{1}{σ^{N}} \sum_{i = 1}^{N} CC I_{i} W_{i} [R_{i} (x_{d}, y_{d})] \cdot Π_{i = 1}^{N} \frac{1}{\sqrt{2 π} σ} \cdot \\ \exp [- \frac{1}{2 σ^{2}} \sum_{i = 1}^{N} {(T_{i}^{m} - T_{i}^{c} (θ))}^{2}], \end{matrix}

(15)

The posterior distributions used in Bayesian inference are often complex, which makes it difficult for Monte Carlo methods to extract independent samples. In such cases, the Markov chain Monte Carlo method (MCMC) is usually used to conquer this problem. The MCMC process produces a Markov chain whose stationary distribution is equal to the target distribution. The posterior of the unknown parameters can be obtained using the samples, and the corresponding predictions can be made. In this study, an MCMC algorithm was used to sample the posterior distributions of the unknown parameters, and the detailed description about the MCMC algorithm could be found in Nichols et al.⁶⁴ Next, the effectiveness of the proposed damage localization approach was investigated numerically and experimentally.

Numerical study

Finite element model

The finite element method (FEM) can be used to obtain damage-related information so that the accuracy of the proposed and traditional damage localization methods could be compared. Therefore, FEM studies were first conducted to verify the feasibility and demonstrate the effectiveness of the proposed damage localization method.

To generate synthetic scattered Lamb waves, a three-dimensional steel plate model was established in the commercial finite element software ABAQUS. The ABAQUS explicit dynamical analysis scheme was applied to obtain the ultrasonic responses under different conditions. As shown in Figure 6, three-dimensional sizes of the steel plate are $600 mm \times 600 mm \times 2 mm$ . The material properties of the steel plate are density $ρ = 7800 kg m^{- 3}$ , Young’s modulus $E = 206 GPa$ , and Poisson’s ratio $ν = 0.3$ . A hole with the radius of 20 mm is introduced in the central coordinates (340,260) of the plate to simulate the damage.

Figure 6.

Schematic setup for numerical study including sensor array and steel plate with damage.

Due to dispersive characteristic of Lamb waves, the response signals received by transducers are usually notoriously complicated. In addition, fewer mode in the forward process is conducive to reducing the complexity of signal processing and calculating the time difference expediently. Based on dispersion curves, as shown in Figure 7, a narrow-band five-cycle 200 kHz sinusoidal signal modulated by a Hann window was used as the excitation signal. In the simulation analysis, a fine grid featuring at least 10 elements across the wavelength $(λ)$ of A₀ (the antisymmetric mode) is required to ensure good spatial resolution of the propagating waves. Usually, for temporal accuracy, a minimum of 20 points per cycle at the highest frequency $(f_{max})$ have been adopted for time-step $(δ t)$ resolution. In this research, an element size of $1 mm \times 1 mm \times 1 mm$ and a time step of 100 ns, were found to be adequate for the modulated tone burst excitation of the 200 kHz central frequency. The element type is a $C 3 D 8 R$ hexahedral mesh with eight nodes.

Figure 7.

Dispersion curves of a steel plate: (a) group velocity and (b) phase velocity.

Numerical results

Figure 8(a) shows the snapshot of the Lamb wave propagation process in the steel plate specimen when the Sensor1 is loaded with excitation signal. It could be seen that the scattered wave generated by the damage diffused around, and could be received by other sensors in the plate. The excitation signal was applied in the four sensors in sequence, and the responses of other three sensors were extracted. In order to achieve baseline-free damage localization and improve the computing efficiency, the VTR technique was adopted in this research to obtain the reconstructed signal^{38,39,56,65,66} instead of generating and recording ultrasonic wave signals twice. It is also conducive to savings of hardware cost and the transfer function in the VTR process is estimated by using the forward response. As shown in Figure 8(b), the scattered waves in the sensing path 1–2, 1–3, and 2–4 (here, “1–2” means that Sensor1 is the actuator and Sensor2 the receiver, the rest are similarly defined) were obvious. And the ones in other paths, on the contrary, are barely to be distinguished. The possible reasons may be the relative positions of damage, wave source and wave receiver, the superposition of wave packets, and interferences of wave reflection from boundary. It distinctly leads to overlap of wave packets in the time-domain. Only the sensing paths that could perceive the damage information are desired for the damage localization. Otherwise, obstacles may appear for extracting damage features because of the interferences. It is also to be noted that both the direct wave and the damage-induced scattered wave in the selected paths (as shown, the time window in Figure 8(b)) were truncated, and then reemitted to the sensor in the virtual time-reversal process.

Figure 8.

(a) The finite element propagation model with a hole damage and (b) the time-domain wave signals for all sensing paths.

The envelopes of wave signals were obtained by the HT, as shown in Figure 9. The time differences $Δ t$ for the three sensing paths including 1–2, 1–3, and 2–4, are 44.7, 20.7, and 20.6 μs, respectively. The CCIs of the three sensing paths are 0.0998, 0.1245, and 0.1856, respectively.

Figure 9.

The time difference $Δ t$ extracted by Hilbert envelope for all six paths.

Preliminary case

The propagation velocity of the A₀ mode Lamb wave measured from the undamaged model was 2662 m/s at 200 kHz in the numerical study. The PDF of unknown parameters could be obtained from the Bayesian posterior distribution using the prepared $Δ t$ , CCI, and calculated velocity, as shown in Equation (15). The built-in function mhsample of the general computing software MATLAB 2021a was used to calculate the sampling process of MCMC for solving Equation (15). To reduce the effect of initial value selection, 100,000 samples were conducted for each parameter and the first 40,000 were set as the burn-in period. Gaussian distribution with standard deviation 30, 30, 10 was chosen for unknown parameter distribution for each parameter in vector $θ$ . 20 repeated running of the calculation process were conducted with an Intel i9-10900K desktop 3.70 GHz in Windows 10 64-bit and the average time for each calculation was 2.556 s. After 20 repetitions with the same settings, the average computation time for the following cases was within 3 s.

The histograms of the x –y coordinates of the defect are shown in Figure 10. And the damage localization results for the proposed method and the two traditional methods (ETLM and RAPID) are compared, as shown in Figure 11. The localization results are achieved by obtaining the maximum value of the pixel in both ETLM and RAPID, while represented by the mean value in the probabilistic fusion method. The ellipse-like curves could be obtained via solving the equations using the damage feature $Δ t$ and wave velocity, as shown in Equation (5). The ETLM identifies the damage by using the intersection of multiple paths. Unfortunately, it can be seen from Figure 11(a) that many artifacts were also produced by this method, which may lead to false judgement. In terms of RAPID algorithm, the damage index at each point in the inspection area is defined by Equation (7) with the help of damage feature CCI. The RAPID algorithm does not show good accuracy of damage localization which makes the prediction deviate significantly from the actual damage location. The potential reason for the deviation is that the chosen frequency of the excitation signal may result in the poor performance of CCI-based RAPID. It has been shown earlier³⁷ such sweet spot frequencies corresponding to a single mode will yield poor performance for damage localization. On the contrary, the joint PDF possesses only one peak and its projection in xoy dimension meet well with the actual damage location, as shown in Figure 11(c).

Figure 10.

Histograms of MCMC samples for the parameters: (a) the x-coordinate (x_d) and (b) the y-coordinate (y_d).

Figure 11.

Damage localization results for preliminary study: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method. (the mark “○” shows the actual damage position, and the mark “+” shows the result of the damage identification).

Parameter analysis

In order to explore the performance of the proposed approach under various configurations including different sensor arrangement, various damage types and locations, a series of parameter studies are carried out, as shown in Table 1.

Table 1.

The cases of parameter studies.

Case	Case I	Case II	Case III	Case IV
Scenarios	Square array with a hole damage	Square array with a crack damage	Circle array with a crack damage	Circle array with a crack damage at a different location and orientation

Case I: square array with a hole damage

In this scenario, the locations of transducers were altered, as shown in Figure 12. Four sensing paths, including $1 - 2$ , $1 - 3$ , $2 - 4$ , and $3 - 4$ , were selected for damage localization based on the time-domain analysis. The diameter of the damage is 40 mm. The identification results for the three methods (ETLM, RAPID, and probabilistic fusion method) are displayed in Figure 13. It can be seen that the probabilistic fusion method could well integrate the advantages of the two traditional damage location method and achieve more precise localization under different sensor configurations.

Figure 12.

Schematic setup in the numerical studies for the case of square array with a hole damage.

Figure 13.

Damage localization results for the case I: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method.

Case II: square array with a crack damage

In order to further investigate the damage localization performances of the proposed method on different types of damages, a crack damage was explored in this case. The three-dimensional sizes of the steel plate are $1000 mm \times 1000 mm \times 2 mm$ , and the crack with a size of $35 mm \times 4 mm \times 2 mm$ locates at (520,500). Details of transducer configuration are displayed in Figure 14(a).

Figure 14.

Schematic setup in the numerical studies: (a) model dimensions and transducer configuration diagram and (b) the finite element propagation model with a crack damage.

As shown in Figure 14(b), the most significant difference between crack and hole damage is that the scattered wave induced by the crack mainly propagate along one specific direction, which brings much difficulties in damage localization. To investigate the interaction between ultrasonic waves with different propagation direction and the crack damage, the incident Lamb wave with different orientations (0°, 45°, and 90°) to crack were applied to the specimen, and the probing signal is introduced to the finite element model through out-of-plane forces, as shown in Figure 15. It can be seen that the amplitudes of the scattered wave in the directions of 180°, 135°, and 90° are maximum with the incident waves in the directions of 0°, 45°, and 90° with the crack. Therefore, a dense sensor network with eight PZT transducers was adopted to enhance the location accuracy and robustness for the crack with different orientations. Six sensing paths, including 1–2, 1–8, 8–2, 8–3, 3–2 and 3–7, were selected to localize the crack damage base on the scattered characteristic in time domain. Figure 16 shows the damage localizing results for the two traditional methods and proposed method, respectively. The localization result of the ETLM deviates little with the real crack while the damage location obtained from the RAPID algorithm significantly diverge from the real position. On the other hand, the identified possible damage location from the joint PDF meets well with the central coordinate of the crack, and the proposed method significantly reduces the damage artifact compared to traditional methods.

Figure 15.

Three local finite element models for 0°, 45°, and 90° crack orientation with the incident Lamb wave.

Figure 16.

Damage localization results for the case II: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method (the rectangular mark shows the real damage position. The mark “+” shows the result of the prediction).

Case III: circle array with a crack damage

To investigate the influence of sensor arrangement on the damage localization, a circle array of transducer configuration was used to explore the effect on the damage localization, as shown in Figure 17(a). Eight sensors were attached on a circle area with a radius of 150 mm. Six sensing paths, including $1 - 2$ , $1 - 8$ , $3 - 2$ , $3 - 7$ , $8 - 2$ , and $8 - 3$ , were chosen for further analysis. The identification results of three different methods are presented in Figure 18. The locating results from two individual methods differ from the real damage to a different degree while the proposed probabilistic fusion method exhibits better localization accuracy.

Figure 17.

Schematic setup in the numerical studies: (a) the circle array with a crack damage and (b) the circle array with a crack damage at a different location.

Figure 18.

Damage localization results for the case III: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method.

Case IV: circle array with a crack damage at a different location and orientation

To investigate the performances of the three methods in damage location with different crack location and orientation, a slant crack with angle of 45° was introduced in the numerical model, as shown in Figure 17(b). Three-dimensional sizes of the crack are also $35 mm \times 4 mm \times 2 mm$ , and the central coordinates of the damage are (500,500). The transducer configuration is the same as case III, and six sensing paths, $1 - 2$ , $1 - 3$ , $3 - 2$ , $3 - 4$ , $8 - 3$ , and $8 - 4$ , were selected to realize damage localization. The results are shown in Figure 19. It can be seen that the localization result of RAPID method largely deviates from the crack and both the ETLM and the proposed probabilistic fusion method exhibit little location error. However, the fusion results eliminate the artifacts by integrating damage features to further exploit, which is beneficial for structural diagnosis than using only either of them.

Figure 19.

Damage localization results for the case IV: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method (the slanted rectangular mark shows the real damage position, and the notation “+” shows the result of the prediction).

To further quantify the accuracy in term of the estimation error, the following error term $η_{l}$ is defined as:

η_{l} = \sqrt{{(x_{e} - x_{d})}^{2} + {(y_{e} - y_{d})}^{2}},

(16)

where $(x_{e}, y_{e})$ and $(x_{d}, y_{d})$ are the estimated and actual damage locations, respectively. The estimated damage locations of the three methods for all cases are compared with the real location, and the results are presented in Table 2. It is obvious that the proposed probabilistic fusion method yields more accurate results than two traditional methods.

Table 2.

Summary of error analysis of all simulation studies.

Case number/central damage location	The ETLM		The RAPID algorithm		The proposed method
Case number/central damage location	Estimated value (mm)	Error (mm)	Estimated value (mm)	Error (mm)	Estimated value (mm)	Error (mm)
Preliminary case /(340,260)	(316,269)	25.63	(180,300)	164.92	(340, 262)	2.00
Case I /(340,260)	(334,263)	6.71	(320,180)	82.46	(337,261)	3.16
Case II /(520,500)	(525,511)	27.32	(499,653)	154.43	(520,508)	8.00
Case III /(520,500)	(524,505)	6.40	(500,649)	150.34	(520,504)	4.00
Case IV /(500,500)	(495,511)	12.08	(397,399)	144.26	(498,507)	7.28

ETLM: elliptical trajectory location method; RAPID: reconstruction algorithm for probabilistic inspection of defect.

Experimental validation

Experimental setup

To further demonstrate the effectiveness of the proposed method on damage localization, experimental studies were conducted on a steel plate specimen with dimensions of $1000 mm \times 1000 mm \times 2 mm$ . A sensor network containing eight PZT transducers with a diameter of 20 mm and thickness of 1.5 mm was deployed on the specimen. The PZT transducers are pasted with epoxy. The size of the crack was $35 mm \times 4 mm \times 2 mm$ which followed the same damage dimensions investigated in numerical simulations case II and case III. The origin of the coordinate system is set at the bottom left of the plate. The center of the introduced crack was located at (520,500) mm in the coordinate system. The experimental setup is shown in Figure 20. The experimental system consisted of a multifunction data-acquisition system (SC-HY-PZT-2.0, Sanchuan Inc., Jiangsu, China), a power amplifier (Trek 2100HF, Trek Inc., New York, America) for the signal enhancement, and a laptop for controlling the whole system and saving measurement data. The excitation signal is generated by the multifunction data acquisition system and then amplified by the power amplifier. The amplified signal is sent to PZT actuator for generating Lamb waves. The response signal is received by the PZT sensors and then obtained by the multifunction data acquisition system at a sampling rate of 2 MS/s. Based on dispersion curves, as shown in Figure 7, a Hann window modulated five-cycle sinusoidal tone burst signal with a peak amplitude of 5 V and central frequency of 160 kHz was applied as the excitation signal. It needs to be noted that the main mode of this experiment is A₀ mode, and the propagation velocity of the A₀ mode Lamb wave was obtained from calibration test, which was 2598 m/s.

Figure 20.

Experimental setup.

Experimental results

In this experimental research, two different sensor configurations were investigated to localize the crack damage, as shown in Figure 20. In case I, six sensing paths, including $1 - 3$ , $1 - 8$ , $3 - 5$ , $3 - 6$ , $8 - 2$ , and $8 - 6$ were selected for the damage localization, as shown in Figure 21(a). Analogous to the numerical study, the HT was used to analyze the wave signals to obtain the damage features including $Δ t$ and CCI. The envelops of the corresponding sensing paths are shown in Figure 21(b). The measured time differences and CCIs are displayed in Table 3 and Figure 22, respectively.

Figure 21.

The signal processing for the case I: (a) time-domain wave signals for all sensing paths and (b) the time difference $Δ t$ extracted by Hilbert envelope for all sensing paths.

Table 3.

Measured time difference $Δ t$ of scattered waves for the case I.

Paths	1–3	1–8	3–5	3–6	8–2	8–6
Δt (μs)	44.50	45.01	51.50	55.50	49.50	47.00

Figure 22.

CCIs for the selected sensing paths for the case I.

Then, the MCMC procedure with the same parameter settings in numerical studies was performed using MATLAB built-in function mhsample. Figure 23 shows the histograms of the x –y coordinates of the central location of the damage, which are generated by 60,000 samples. The localizing results of the three methods were presented in Figure 24. There exist many artifacts in the localization results of the ETLM, which was the intrinsic problem of the ETLM. The localization results of the RAPID algorithm largely deviate from the actual location. On the contrary, the proposed probabilistic fusion method yields exclusive and accurate results, which demonstrated that the proposed probabilistic fusion method could efficiently integrate damage features to realize more accurate damage localization.

Figure 23.

Histograms of MCMC samples for the parameters for the case I: (a) the x-coordinate (x_d), (b) the y-coordinate (y_d).

Figure 24.

Damage localization results for the case I: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method (the rectangular mark shows the real damage position, and the notation “+” shows the result of the prediction).

To further experimentally explore the effect of the sensor distribution manner on the damage localization accuracy, case II with square sensor configuration was investigated. Six sensing paths including $1 - 7$ , $1 - 8$ , $2 - 1$ , $2 - 3$ , $3 - 4$ , and $3 - 5$ , were chosen for further analysis. Figure 25 presents the localization results of the three methods. It can be seen that the ETLM suffers from various artifacts while RAPID method is bothered by large error. The probabilistic fusion method generates more accurate damage localization and significantly reduces the damage artifacts, which indicated that the proposed method is robust for different sensor arrangements.

Figure 25.

Damage localization results for the case of square array with a crack damage in the experiment: (a) ETLM, (b) RAPID algorithm, and (c) probabilistic fusion method.

To further evaluate the localizing accuracy, the error term defined by Equation (16) was used to compute the estimated errors. All the estimated results are calculated and compared to investigate the effectiveness of each method, as shown in Figure 26. Similar to the numerical findings, the proposed probabilistic fusion method could not only realize more accurate and reliable damage localization than the ELTM and RAPID, but also eliminate the artifacts compared to ELTM and RAPID by integrating damage features to fully exploit, as shown in Figures 24 and 25.

Figure 26.

The estimated errors for the three methods in the experiment (case I: the circle array with a crack damage and case II: the square array with a crack damage).

Discussion

According to the numerical and experimental studies, including different configurations of damage cases and transducer distributions, the localization results of the RAPID algorithm largely deviate from the actual location, and the ETLM exhibits relatively better accuracy than RAPID but has more visible artifacts. On the contrary, the proposed probabilistic fusion method yields exclusive and accurate results compared to two existing methods, which demonstrated that the proposed probabilistic fusion method could efficiently integrate damage features to realize more accurate damage localization and eliminate the artifacts. In order to further investigate the damage localization performances of the proposed method on different sensor network coverage areas, three coverage areas were considered by using numerical simulations. Figure 27 presents the localization results of the proposed probabilistic fusion method. With the increase of sensor network coverage areas, the estimated errors of three cases were 2, 4.6, and 5.7 mm, respectively. It is reasonable that the localization errors of the proposed method increase to some extent. The proposed probabilistic fusion method shows reliable localizing results, again demonstrating the effectiveness of the proposed method.

Figure 27.

Compared studies with three different coverage areas: (a) 28,800 mm², (b) 45,000 mm², and (c) 80,000 mm² (the mark “○”shows the actual damage position, the mark “+” shows the result of the damage identification and the yellow circle indicates the position of sensor).

Nevertheless, there are still some shortcomings that need to be further explored. Some influential factors (such as various damage sizes and pristine damage location, different probing frequencies) need to be considered and analyzed to investigate the performance of the proposed method in damage localization. In addition, intelligent algorithms (such as the deep learning) can be incorporated into distinguishing the wave packets induced by damage to improve the effectiveness of the proposed method on automated monitoring, and these will be considered in the authors’ future research.

Conclusions

In this paper, the Lamb wave-based time reversal method was first utilized to achieve baseline-free manner for obtaining damage features $Δ t$ and CCI. Then, a Bayesian framework was established to take uncertainties into consideration and improve damage localization accuracy by integrating two independent damage features. The performances of the two traditional methods (ETLM and the RAPID algorithm) and the proposed probabilistic fusion localization approach were investigated via numerical and experimental studies including different configurations of damage cases and transducer distributions. The results show both the ETLM and the RAPID algorithm cannot locate the damage accurately because of the intrinsic deficiency. On the contrary, the proposed probabilistic fusion method could fully exploit the merits of damage features and locate damage more precisely. Moreover, the proposed Bayes framework provides a probabilistic description for the damage location by using the probabilistic density function of the unknown parameters and quantifies the uncertainties in the locating process, which is more in accordance with the nature of estimating or predicting defects rather than giving deterministic results.

Specifically, the proposed probabilistic fusion method could not only realize more accurate damage localization than the ELTM and RAPID algorithm, but also eliminate imaging artifacts. The efficient practical transducer configuration is determined by investigating the interaction between the Lamb wave and damage. The effect of different configurations of damage and transducers on the proposed probabilistic fusion method is less essential than that on the ELTM and RAPID algorithm, and it needs more transducer configurations and sensing paths for the ELTM and RAPID algorithm to achieve accurate damage localization. In addition, with the increase of sensor network coverage areas, the localization errors of the proposed method increase slightly, but it is still relatively small and acceptable. It should be noted that there are some interferences for the time-reversal method to extract damage index because of the superposition of wave packets and the influence of the boundary reflected waves. In the future, a more efficient and effective method for determining the sensing paths and extracting damage index for damage localization needs to be explored under a variety of operational and environmental conditions, and the feasibility of the proposed damage localization method based on Bayes approach on structures with complex geometries (rail, H-beam and steel truss, etc.) will be explored.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the National Natural Science Foundation of China (Grant number 52020105005). The authors greatly appreciate financial supports.

ORCID iD

Qingzhao Kong

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