Sparse Bayesian learning approach for propagation distance recognition and damage localization in plate-like structures using guided waves

Abstract

In ultrasonic guided wave–based damage detection, the propagation distance recognition of wave packets is an essential step. However, it is difficult to perform direct distance extraction from guided wave signals since the multimode, mode conversion, and dispersion effects typically lead to wave packet overlapping and distortion. In addition, the identified damage location may be incorrect due to inevitable uncertainties in the procedure of propagation distance recognition and damage localization. Motivated by these difficulties, a novel two-stage approach for propagation distance recognition and damage localization is proposed based on sparse Bayesian learning framework. In the first stage, prior knowledge of a small number of wave packets contained in a signal is exploited to sparsely represent the guided wave signal and then the corresponding propagation distance and amplitude information of each wave packet can be obtained. In the next stage, only a small number of damages occurring in a structure are exploited and a vector consisting of the propagation distances extracted from the previous stage is used to match the atoms in a pre-defined over-complete distance dictionary matrix, to achieve our goal of localizing structural damage. Both procedures of the two stages are realized by the sparse Bayesian learning algorithm, which obtains the most probable value and the corresponding uncertainty. A sampling strategy is presented to transfer the uncertainty of the propagation distance recognition to the subsequent damage localization. Finally, the effectiveness of the proposed method is validated using numerical simulation and experimental investigation on aluminum plates. The proposed method is only valid for single damage localization in the present form, but it has the potential to be extended for multiple damage localization.

Keywords

Guided wave propagation distance recognition damage localization sparse Bayesian learning plate-like structures

Introduction

Ultrasonic guided wave (UGW) has been used extensively in structural health monitoring and non-destructive testing for plates, tubes, and pipes, owing to their advantages of high sensitivity to small damage and long propagation distance with little attenuation.^1–3 The UGW can detect various types of structural damage, such as corrosion, fatigue crack, delamination, and disbanding.^4–7 Interrogating large areas with a small number of sensors is the common system configuration using UGW, and then damage location and severity, based on the propagation distances⁸ and guided wave (GW) amplitude,⁹ can be obtained from the received signals. Numerous UGW-based damage localization methods have been developed in recent years, such as the triangulation method,¹⁰ focusing array algorithm,¹¹ GW phased array,¹² migration technique,^13–17 and time-reversal imaging method.¹⁸ All of these methods ultimately estimate the damage locations based on the propagation distance. The propagation distance is typically calculated using the time of flight (ToF) extracted from the time domain signal and the group velocity obtained by solving the dispersion equations or from experiment. However, the issue of accurate measurement of the ToF is generally encountered during this process.

Regarding the ToF measurement for propagation distance recognition, various methods can be employed, such as the Hilbert–Huang transform,¹⁹ wavelet transform,²⁰ correlation coefficient,²¹ and Wigner–Ville distribution.²² However, certain natural properties of GW, including dispersion effect, multimode, and mode conversion, may lead to waveform distortion, wave packet overlapping, and other problems. These problems, combined with the noise effect, may lead to the difficulty in accurate ToF measurement and wave packet separation. Extensive research has been conducted to address these problems. Dispersion removal or compensation methods have been developed to recover distorted waves, such as the linear mapping technique,²³ linearly dispersive signal construction method,²⁴ and compressed sensing-based dispersion compensation method.²⁵ Regarding the multimode aspect, Niethammer et al.²⁶ presented the reassigned spectrogram, which has been demonstrated to be capable of distinguishing multimode GWs. Xu et al.²⁷ used the compensation separation technique to transfer the multiple dispersive waveforms into non-dispersive pulses, which can be extracted individually by rectangular time windows. However, the uncertainty in ToF extraction based on the above methods is out of consideration, which leads to the inaccuracy of the results. Actually, there are always substantial modeling uncertainties involved which may arise from many sources. The propagation distance recognition and the subsequent damage localization when treated deterministically are typically ill-conditioned and often ill-posed when using noisy incomplete data.

The presence of substantial modeling uncertainties in UGW-based propagation distance recognition and damage localization has motivated numerous researchers to tackle the problem from a Bayesian probabilistic perspective. Yan²⁸ utilized ToF data to propose a Bayesian inference method in which a Markov chain Monte Carlo (MCMC) algorithm was employed to characterize the posterior distributions of the unknown damage location and wave velocity parameters. To identify multiple cracks, He and Ng²⁹ developed a Bayesian method which first determines the most plausible solution for crack number based on Bayesian model class selection and then identifies the crack locations and the associated uncertainties by Bayesian model updating. Cantero-Chinchilla et al.³⁰ also utilized Bayesian model class selection to choose the most appropriate time–frequency analysis technique for obtaining the ToF information which is most plausible based on the data, before further damage localization and associated uncertainty quantification. Yang et al.³¹ developed a Bayesian quantification method of crack sizes using damage data from in situ Lamb wave test. Two damage features namely phase change and normalized amplitude were extracted.

To alleviate the effects of the ill-conditioning and often ill-posedness in UGW-based propagation distance recognition and damage localization, sparse processing algorithms, including l₀-norm minimization,³²l₁-norm minimization,³³ and l_p-norm (0 < p < 1) minimization,³⁴ have been introduced for regularization purpose in recent years. Sparse processing is generally performed under a linear generative model with the following form: $g = Φ t$ , where the coefficient vector $t \in R^{N_{m}}$ is assumed enable to characterize the data vector g sparsely with the dictionary matrix $Φ = {[φ_{1}^{T}, \dots, φ_{N_{m}}^{T}]}^{T} \in ℝ^{K \times N_{m}}$ , that is, most components in $t$ are equal to zero. Levine and Michaels³⁵ applied the sparse representation techniques of basis pursuit de-noising and orthogonal matching pursuit for signal decomposition to extract damage information. Xu et al.²⁵ presented a sparse UGW processing method by decomposing the signal based on the designed dictionary matrix, composed of multimode and dispersive guided wave packets, but mode conversion was overlooked. Wang et al.³⁶ developed a sparse representation method by designing a dictionary matrix composed of the entire signal of each possible location reflected from the damage and boundaries, so the dictionary atom was excessively complicated owing to the multimode, mode conversion, and boundary reflection. Zhang et al.³⁷ compared the advantages and limitations of various sparse signal representation methods in GW signal processing. However, all of the methods reviewed above have been based on a deterministic framework, which provides only a point estimate, so the uncertainties in the ToF measurement, propagation distance extraction, and damage localization were not explicitly quantified.

In recent years, by considering the sparse representation from a Bayesian perspective, sparse Bayesian learning (SBL) has been applied in many areas, such as compressive sensing,^38,39 model-based structural damage detection,^40,41 and feature extraction,⁴² because of its benefits of posterior uncertainty quantification compared to deterministic methods. In addition, the model inversion uncertainty can be reduced by incorporating the sparseness regularization information. SBL has also been exploited in UGW-based damage detection recently. Wu et al.^43,44 introduced the SBL framework to approximate both the non-dispersive and dispersive GW pulses using the Gabor model and parameterized chirp model, respectively. The latter model enables us to recover multimode and dispersive wave packets from noisy signals, which demonstrated the excellent performance of the SBL algorithm on GW signal. Xu et al.⁴⁵ also utilized SBL to decompose the measured GW signal in frequency domain, in order to obtain traveling distance and the corresponding amplitude coefficients, but mode conversion is not explicitly considered. Overall, the exploitation of SBL applications in UGW is still few, and further work is still required to explore the advantages of SBL in the UGW-based damage diagnosis area.

In this article, a new UGW-based damage detection method by using SBL is presented. The method is composed of two stages: estimating the propagation distance and localizing the damage based on the estimated propagation distance. In the first stage, based on prior information whereby only a small number of wave packets exist in the received signal, the received signal is decomposed by designed dictionary matrix and then the propagating distance of each wave packet can be extracted. During the second stage, owing to the small number of damages occurring in plate-like structure, the vector consisting of the damage-related distance from each sensor is matched with a dictionary matrix constituted by distance vectors of all possible points to each sensor. The contribution of this article is mainly in the following aspects. First, an appropriate dictionary matrix is presented, based on GW propagation model considering dispersion effect, multimode, and mode conversion, for representing signal sparsely with the goal to address the wave packet overlapping and distortion problems. Second, prior knowledge of a small number of wave packets being contained in a signal and only a few damages occurring in a structure is exploited in the study. Robust SBL method proposed in Huang et al.³⁸ is employed for the two tasks of decomposing signal and localizing damage, which has the benefit of performing the sparse inversion with higher robustness for propagation distance recognition and damage localization, and quantifying the posterior uncertainty with higher accuracy. Finally, a stochastic sampling strategy is proposed to generate the posterior samples of the damage-related parameters conditional on the received GW signals by incorporating the uncertainty in the propagation distance recognition into the damage localization results, and thereby the results of the two stages are combined effectively. The advantage of our proposed method is that the uncertainties of the propagation distance recognition of mode-converted wave packet, separation of the overlapping wave packets, and the identification of damage location can be reduced significantly by using the SBL method, which may increase the reliability of the identification results.

The remainder of the article is organized as follows. In section “Proposed SBL-based two-stage approach for damage detection”, the new two-stage approach for propagation distance recognition and damage localization is presented. The proposed method is applied to numerical simulations in section “Numerical studies”. Experimental investigations are also performed to verify the effectiveness of the proposed method and discussed in section “Experimental investigations”, followed by conclusions being drawn in section “Conclusion”.

Proposed SBL-based two-stage approach for damage detection

Sparse signal decomposition and propagation distance recognition

This study focuses on plate-like structures, in which only dispersive Lamb waves are considered. Assuming that an incident wave u(t) is excited at the origin, the waveform propagating to position x can be expressed as²⁵

u (x, t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} F (ω) e^{j ω t} e^{- jkx} d ω

(1)

where F(ω) is the Fourier transform of the excitation signal u(t), and the wavenumber k is a function of the frequency ω, which is obtained by solving the Rayleigh–Lamb equations. The relations between k and ω are nonlinear for all Lamb modes, implying that the group velocities vary across the frequency of the traveling waves, which will result in distortion of the wave packets. In equation (1), it is also observed that the received waveform depends on dispersion relation, Lamb mode, and propagation distance.

Figure 1(a) and (b) illustrates two basic diagnosis patterns in UGW-based structure health monitoring: pulse-echo and pitch-catch patterns.⁴⁶ Due to multiple boundaries and damages, many possible propagation paths exist from an actuator to a sensor. Assuming that the excitation frequency is less than the cutoff frequency of higher mode, only S₀ and A₀ modes propagate along the path. Due to mode conversion and the second reflection at the discontinuity in both patterns, multiple wave packets can be received from each path. For example, when only one wave packet is transmitted by the transducer, two wave packets corresponding to S₀ and A₀ modes of Lamb wave will be generated and propagated in the plate subsequently, and then four wave packets caused by the mode conversion will be received by the sensor, shown in Figure 1(c). Assuming that the excitation signal has a unit amplitude, all received wave packets were obtained based on equation (1), without normalization. Accordingly, each wave packet along wave propagation path may be divided into two segments, which correspond to the propagating distance of S₀ and A₀ modes, respectively.

Therefore, the received signal, which contains multiple wave packets from all possible paths, can be expressed as

\begin{matrix} s_{0} (t) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} \frac{C_{mn}}{2 π} \int_{- \infty}^{+ \infty} F (ω) e^{j ω t} e^{- j k_{S_{0}} x_{S_{0}}^{mn} - j k_{A_{0}} x_{A_{0}}^{mn}} d ω \\ = \sum_{m = 1}^{M} \sum_{n = 1}^{N} u_{S_{0} A_{0}}^{mn} C_{mn} \end{matrix}

(2)

where M is the number of propagation paths, N is the number of wave packets propagating along each propagation path, C_mn is the coefficient corresponding to the nth wave packet $u_{S_{0} A_{0}}^{mn}$ on the mth propagation path, and $x_{S_{0}}^{mn}$ and $x_{A_{0}}^{mn}$ are the propagating distance of S₀ and A₀ modes, respectively. Using vector notation, equation (2) can be rewritten as

s_{0} (t) = (u_{S_{0} A_{0}}^{11}, . . ., u_{S_{0} A_{0}}^{mn}, . . ., u_{S_{0} A_{0}}^{MN}) \times (C_{11}, . . ., C_{mn}, . . ., C_{MN})^{T}

(3)

Owing to the prior knowledge of the small number of wave packets, the signal $s_{0} (t)$ can be sparsely represented by an over-complete dictionary matrix. In the waveform dictionary matrix, each column vector (an atom) is a wave packet, which is generated from mode conversion. Moreover, the dispersion effects on the waveform during its propagation have also been considered. The differences of the atoms are caused by the various propagation distances of two basic Lamb modes, where the distance (from 0 to the largest propagation distance) is discretized by a distance interval equal to the product of the sampling interval of the signal and group velocity. Then, each atom can be calculated according to the corresponding propagation distance. The discrete values of S₀ and A₀ modes are expressed as

\begin{matrix} x_{S_{0}}^{p} = c_{g S_{0}} (f_{c}) p / f_{s}, p = 0, 1, 2, \dots, L - L_{0} \\ x_{A_{0}}^{q} = c_{g A_{0}} (f_{c}) q / f_{s}, q = 0, 1, 2, \dots, L - L_{0} \end{matrix}

(4)

where p and q are the sampling points; $f_{c}$ and $f_{s}$ are the center and sampling frequencies, respectively; L₀ and L are the lengths of the input wave packet and received signal, respectively; (L–L₀)/f_s is the longest propagation time in the received signal; $c_{g S_{0}}$ and $c_{g A_{0}}$ are the group speeds of S₀ and A₀ modes, respectively; and $x_{S_{0}}^{p}$ and $x_{A_{0}}^{q}$ are the discrete propagation distances, respectively. The arbitrary combination of the propagating distances of S₀ and A₀ modes (total distance) is used to predict the mode-converted wave packet. The predicted wave packet is given by

u_{S_{0} A_{0}}^{pq} = \frac{1}{2 π} \int_{- \infty}^{+ \infty} F (ω) e^{j ω t} e^{- j k_{S_{0}} x_{S_{0}}^{p} - j k_{A_{0}} x_{A_{0}}^{q}} d ω

(5)

In addition, the propagation time of the predicted wave packet $u_{S_{0} A_{0}}^{pq}$ should be less than the longest propagation time, which means the sum of the propagation times of S₀ and A₀ modes should satisfy the constraint: $p / f_{s} + q / f_{s} \leq (L - L_{0}) / f_{s}$ . And then there are $Z$ combinations.

Thereafter, an over-complete dictionary matrix $U$ can be constructed using $u_{S_{0} A_{0}}^{pq}$ , which contains wave packets at all possible combined distances of S₀ and A₀ modes

U = [u_{S_{0} A_{0}}^{1}, . . ., u_{S_{0} A_{0}}^{i}, . . ., u_{S_{0} A_{0}}^{Z}]

(6)

Each atom in the dictionary matrix is normalized to a unit two-norm using $u_{S_{0} A_{0}}^{i} = u_{S_{0} A_{0}}^{i} / ‖ u_{S_{0} A_{0}}^{i} | |_{2}$ , where $‖ u_{S_{0} A_{0}}^{i} | |_{2}$ denotes two-norm of the vector $u_{S_{0} A_{0}}^{i}$ . The corresponding amplitude weight vector can be written as

c = [C_{1}, . . ., C_{i}, . . ., C_{Z}]^{T}

(7)

Then, the received signal $s_{0} (t)$ can be represented by the following linear regression model, based on the dictionary matrix U and corresponding weight vector c

s_{0} (t) = Uc + ε

(8)

where ε is the measurement noise. In equation (8), most of the weights in vector c are zeros, because the number of wave packets in the dictionary matrix Z is much larger than the real number of wave packets M × N. Any non-zero component in vector c indicates that $s_{0} (t)$ contains a wave packet $u_{S_{0} A_{0}}^{pq}$ . The almost same distances of multiple non-zero components might be induced by two sources: the first is that two damages occur that are close with each other, and the other is mode conversion at a damage. In this work, we define 10% as the threshold value to distinguish the two cases, that is, if the propagation distance differences between two wave packets are larger than 10%, they are considered as reflections from two damages; otherwise, the corresponding wave packets are considered as the results of mode conversion. The schematic diagram of signal sparse representation described above is shown in Figure 2.

For sparse inversion purpose, an unknown prediction error $ε_{1}$ exists due to the model error for specified $\bar{c}$ plus the measurement error $ε$ . Then, the linear regression model can be rewritten as

s_{0} (t) = U \bar{c} + U (c - \bar{c}) + ε = U \bar{c} + ε_{1}

(9)

where $\bar{c}$ is the uncertain weight vector for signal decomposition inferred by the SBL algorithm in Appendix 2. From Appendix 2, the posterior probability density function (PDF) conditional on the received signal $s_{0} (t)$ is computed as a Gaussian distribution

p (\bar{c} | s_{0}) = N (\bar{c} | μ_{\bar{c}}, \sum_{\bar{c}})

(10)

where the posterior mean vector $μ_{\bar{c}}$ and covariance matrix $\sum_{\bar{c}}$ can be formulated based on equation (22) in Appendix 2. As the dictionary atoms are normalized, the posterior mean $μ_{\bar{c}}$ has certain connections with the wave packet amplitudes, which can be used to assess the damage severity. Moreover, the inversion confidence of the weight vector $\bar{c}$ for signal decomposition can be evaluated by their posterior covariance matrix $\sum_{\bar{c}}$ , that is, the confidence intervals are estimated based on the roots of the diagonal elements of the covariance matrix $\sum_{\bar{c}}$ .

Remark 2.1

The over-complete dictionary matrix $U$ designed and used in this section is only applicable for the theoretically predictable wave packet, which assumes that the dispersive relation is known and the plate thickness is constant. When the structure geometry is more complicated, for example, a wind turbine blade, a new dictionary should be designed to consider these effects for effective recognition of the wave packets.

Damage localization based on identified propagation distance

It is supposed that the pitch-catch pattern is used to localize the damage in plate-like structures. The three-point positioning method, as illustrated in Figure 3(a), can be employed, where the basic equation involved is

d_{i} = \sqrt{{(x - x_{i})}^{2} + {(y - y_{i})}^{2}} + \sqrt{{(x - x_{a})}^{2} + {(y - y_{a})}^{2}}, i = 1, 2, 3

(11)

where d_i is the real (computed) distance of the ith actuator-damage-sensor propagation path; (x, y), (x_a, y_a), and (x_i, y_i), i = 1, 2, 3 are the coordinates of the damage, actuator, and the three sensors attached to the plate, respectively.

However, uncertainties always exist in the identified propagation distances, owing to the Heisenberg uncertainty principle, as well as the environment and operational-induced variations in the dispersion curves, and so on. These uncertainties will lead to the damage localized in an area rather than a single point by using equation (11), as indicated in Figure 3(b). Here, $r_{1}, r_{2}, and r_{3}$ , are the recognized propagation distances from the signals received by S1, S2, and S3 based on the method in subsection “Sparse signal decomposition and propagation distance recognition”. To determine the damage location with higher confidence, the prior knowledge of only a small number of defects occurring in the structure will be exploited, and the SBL algorithm in Appendix 2 will be employed for the probabilistic sparse inversion problem. The recognized distances obtained in the previous subsection will be used to match the atoms in a designed over-complete dictionary matrix for damage localization purpose.

First, the over-complete dictionary matrix is constructed. Assuming that the length of the domain of interest is divided into n segments, then there are (n + 1)² discrete points in the plate. Each discrete node is modeled as a damage location, and the distances from the node to the three transducers constitute a vector d_i = [d_i₁, d_i₂, d_i₃]^T, which is formed as an atom of an over-complete dictionary matrix $D$

D = [d_{1}, . . ., d_{i}, . . ., d_{{(n + 1)}^{2}}]

(12)

The identified propagation distances obtained by the procedure in subsection “Sparse signal decomposition and propagation distance recognition” constitute the data vector y

y = [r_{1}, r_{2}, r_{3}]^{T}

(13)

which is matched to the atoms in the dictionary matrix D using the SBL algorithm for damage localization purpose, based on the following linear regression model

y = D \bar{w} + ε_{2}

(14)

where $\bar{w}$ is the weight vector for damage localization inferred by the SBL algorithm, and the non-zero components in the vector indicate possible locations. According to Appendix 2, the posterior PDF can be obtained as a Gaussian form

p (\bar{w} | y) = N (\bar{w} | μ_{\bar{w}}, \sum_{\bar{w}})

(15)

It is noted that the precision of the damage localization using the above procedure is dependent on the grid size. The damage location accuracy becomes higher with the decrease in the grid size. However, a small grid size causes the number of atoms in the dictionary D increase exponentially, and the computation cost is subsequently demanding. Therefore, a multi-scale grid method is employed to address this issue, as shown in Figure 4. First, the plate is divided into a large grid, and an approximate damage location can be determined by $μ_{\bar{w}}$ from running the SBL algorithm. Then, a small region containing this identified location is re-meshed using a smaller grid size to identify a more accurate location. Note that the re-meshing area should be large enough to ensure not to miss the damage location. The procedure can be repeated to get more and more accurate damage location. From our experience, after four levels of meshing, the identified location is usually consistent with the real location with high accuracy.

Sampling strategy for generating posterior samples of damage-related parameter $\bar{w}$ conditional on GW signals

As mentioned in subsections “Sparse signal decomposition and propagation distance recognition” and “Damage localization based on identified propagation distance”, the GW-based damage detection is achieved by two stages. In the first stage, the posterior PDF of the weight vector ${\bar{c}}_{i}$ conditional on the received signal $s_{i}$ , $p ({\bar{c}}_{i} | s_{i})$ , i = 1, 2, 3, is obtained by decomposing the signal for propagation distance recognition. In the second stage, damage localization is achieved by inferring the posterior PDF of the weight vector $\bar{w}$ conditional on the three propagation distances $r_{1}, r_{2}, and r_{3}$ . Each propagation distance $r_{i}, i = 1, 2, 3$ is obtained by comparing all components in ${\bar{c}}_{i}$ , and selecting the propagation distance corresponding to the component with the largest value as $r_{i}$ , because the corresponding waveform makes the largest contribution to the signal representation. Therefore, we can express the posterior PDF as $p (\bar{w} | {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3}) = p (\bar{w} | r_{1}, r_{2}, r_{3}) = p (\bar{w} | y)$ . Finally, the marginal posterior PDF of the weight vector $\bar{w}$ conditional on the received signals $s_{1}, s_{2}, and s_{3}$ , can be expressed as

\begin{matrix} p (\bar{w} | s_{1}, s_{2}, s_{3}) = \int p (\bar{w}, {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3} | s_{1}, s_{2}, s_{3}) d {\bar{c}}_{1} d {\bar{c}}_{2} d {\bar{c}}_{3} \\ = \int p (\bar{w} | {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3}) p ({\bar{c}}_{1} | s_{1}) p ({\bar{c}}_{2} | s_{2}) p ({\bar{c}}_{3} | s_{3}) d {\bar{c}}_{1} d {\bar{c}}_{2} d {\bar{c}}_{3} \end{matrix}

(16)

In equation (16), the probability product rule is used, that is, $p (\bar{w}, {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3} | s_{1}, s_{2}, s_{3}) = p (\bar{w} | {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3}) p ({\bar{c}}_{1} | s_{1}) p ({\bar{c}}_{2} | s_{2}) p ({\bar{c}}_{3} | s_{3})$ . However, the integral in equation (16) for computing the marginal posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ is analytically intractable. Therefore, a stochastic sampling strategy is presented to generate samples for characterizing this posterior PDF. According to the sampling theory of joint and marginal distributions, the weight vector $\bar{w}$ extracted from the sample pairs ${{\bar{w}}^{(n)}, {\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}}, n = 1, . . ., N$ , where the sample pairs are generated according to the joint posterior PDF $p (\bar{w}, {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3} | s_{1}, s_{2}, s_{3})$ , are distributed as the marginal posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ . Samples ${{\bar{w}}^{(n)}, {\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}}, n = 1, . . ., N$ from the joint posterior PDF $p (\bar{w}, {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3} | s_{1}, s_{2}, s_{3}) = p (\bar{w} | {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3}) p ({\bar{c}}_{1} | s_{1}) p ({\bar{c}}_{2} | s_{2}) p ({\bar{c}}_{3} | s_{3})$ can be obtained by first sampling from the marginal posteriors $p ({\bar{c}}_{1} | s_{1})$ , $p ({\bar{c}}_{2} | s_{2})$ , and $p ({\bar{c}}_{3} | s_{3})$ then using each of these samples to sample from the conditional posterior PDF $p (\bar{w} | {\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3}) = p (\bar{w} | y)$ . In order to characterize the full posterior uncertainty, we take the previously generated samples ${{\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}}, n = 1, . . ., N$ and draw posterior samples ${{\bar{w}}^{(n)}}, n = 1, . . ., N$ from the conditional posterior PDF $p (\bar{w} | {\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}), n = 1, . . ., N$ . Finally, we can simply examine the posterior samples ${{\bar{w}}^{(n)}}, n = 1, . . ., N$ from ${{\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}, {\bar{w}}^{(n)}}, n = 1, . . ., N$ to characterize the marginal posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ . The detailed pseudo-code of the algorithm is summarized in Table 1.

Table 1.

Pseudo-code for generating N samples for characterizing the posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ .

Inputs: the received signal

s_{1}, s_{2}, and s_{3}

Outputs: posterior samples

{{\bar{w}}^{(n)}}, n = 1, . . ., N

For

n = 1, . . ., N

1. Sample the weight vectors for the first sensor signal decomposition

{\bar{c}}_{1}^{(n)} : p ({\bar{c}}_{1} | s_{1})

using equation (10).
2. Sample the weight vectors for the second sensor signal decomposition

{\bar{c}}_{2}^{(n)} : p ({\bar{c}}_{2} | s_{2})

using equation (10).
3. Sample the weight vectors for the third sensor signal decomposition

{\bar{c}}_{3}^{(n)} : p ({\bar{c}}_{3} | s_{3})

using equation (10).
4. Compare all component values in

{\bar{c}}_{i}^{(n)}, i = 1, 2, and 3

and select the propagation distance corresponding to the component with the largest value as

r_{i}^{(n)}

for composing the data vector

y^{(n)} = [r_{1}^{(n)}, r_{2}^{(n)}, r_{3}^{(n)}]^{T}

.
5. Sample the weight vectors for damage localization

{\bar{w}}^{(n)} : p (\bar{w} | {\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}) = p (\bar{w} | y^{(n)})

using equation (15), where we incorporate the

nth

sample

y^{(n)}

.
End
Samples

{{\bar{c}}_{1}^{(n)}, {\bar{c}}_{2}^{(n)}, {\bar{c}}_{3}^{(n)}, {\bar{w}}^{(n)}}, n = 1, . . ., N

are obtained.

The benefit of the sampling strategy above is that, by using samples from the posterior PDFs $p ({\bar{c}}_{1} | s_{1})$ , $p ({\bar{c}}_{2} | s_{2})$ , and $p ({\bar{c}}_{3} | s_{3})$ , we can effectively take the uncertainty of ${{\bar{c}}_{1}, {\bar{c}}_{2}, {\bar{c}}_{3}}$ into account and draw samples from the posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ which is conditional on the received GW signals $s_{1}$ , $s_{2}$ , and $s_{3}$ directly. If the group of samples ${{\bar{w}}^{(n)}}, n = 1, . . ., N$ is concentrated in a small area, it can be evaluated that damage occurs at the corresponding location with a high plausibility.

Remark 2.2

In Xu et al.,²⁵ a GW packet extraction method, based on deterministic signal sparse representation, was presented. This method uses the dispersive wave packet propagating at a certain distance as an atom of the dictionary matrix, in which mode conversion is overlooked. The signal is decomposed by the following optimization problem

c = \underset{c}{\arg min} {\frac{1}{2} ‖ s_{0} (t) - Uc ‖_{2}^{2} + γ {‖ c ‖}_{1}}

(17)

where $γ$ is a regularization parameter that balances the one-norm term (sparsity) $‖ c ‖_{1} = \sum_{l = 1}^{K} | c_{l} |$ and two-norm term (accuracy) $‖ s_{0} (t) - Uc ‖_{2}^{2} = \sum_{l = 1}^{K} {(s_{0, l} (t) - U_{l} c)}^{2}$ , and K is the signal length.

Compared with the method in Xu et al.,²⁵ our proposed method offers the following advantages. First, mode conversion is considered in the design of the dispersive wave packet dictionary matrix, which is not the case in the method of Xu et al.²⁵ Second, the SBL algorithm is employed, rather than deterministic l₁-norm minimization, which can quantify the uncertainty in the propagation distance recognition process and obtain the confidence level of the results. Finally, damage localization is investigated by using the identified propagation distance information. Moreover, a sampling strategy is proposed to incorporate the uncertainty in the propagation distance recognition into the damage localization results.

Numerical studies

In this section, we present the numerical studies performed on a 6061 Aluminum plate to validate the effectiveness of the proposed method. The material properties of the 6061 Aluminum plate are as follows: Young’s modulus, E = 69 GPa; Poisson’s ratio, μ = 0.33; and density, ρ = 2700 kg/m³. ANSYS was employed to simulate the propagation of Lamb waves, and their interactions with the damage and boundaries. The plate and piezoelectric wafers were modeled by SOLID185 and SOLID5 elements, respectively. A narrow notch was introduced by deleting corresponding elements to simulate the crack. The size of the elements is supposed to be less than 1/20 of the shortest wavelength among all Lamb modes for effective spatial resolution. Meanwhile, the time step was less than 1/(20f_c), where f_c is the center frequency of the excitation signal.

To excite the S₀ Lamb mode, two piezoelectric wafers were symmetrically attached to the two surfaces of the plate, and the excitation signals were symmetrically applied on the wafers. The other piezoelectric wafers were randomly placed to receive the scattered signal. The excitation signal was a narrowband tone-burst five cycles enclosed in a Hann window, with the center frequency f_c fixed at 110 kHz. Therefore, the time step and the size of element are 0.4 μs and 1 mm × 1 mm × 0.5 mm, respectively. As the center frequency of the excitation signal was less than the cutoff frequency of A₁ and S₁ modes, only A₀ and S₀ modes were received by the sensors. For reference purposes, a finite element (FE) model without notch was also simulated to provide the baseline signal. The difference signal relating to the notch was produced by subtracting the baseline signal from the damage signal.

Propagation distance recognition results of mode-converted wave packets

Figure 5 shows the sketch of the simulated notch and layout of the piezoelectric wafers on the plate, in which the distances between the notch and the wafers are also displayed, and the three sensors are denoted as S1, S2, and S3. The sizes of the plate, notch, and piezoelectric wafer were 1100 mm×1100mm × 1mm, 20 mm×1 mm × 0.5 mm, and 7 mm × 7 mm × 0.1 mm, respectively.

Although the damage location can be determined directly based on the propagation distances of first damage-related wave packets, the information of mode-converted wave packet facilitates successful damage localization. The ToF of mode-converted wave packet can be extracted by conventional methods, but difficulties exist in determining the propagation distance due to mode conversion. In this case, the propagation distance of the mode-converted wave packet can also be recognized by our proposed SBL-based method.

The propagation distances of the multiple wave packets are extracted by the SBL-based method described in subsection “Sparse signal decomposition and propagation distance recognition”. The 95% confidence interval for each component in the weight vector $\bar{c}$ for signal decomposition, which is equal to triple of the posterior standard deviation, is also computed to quantify the posterior uncertainty. When the 95% confidence interval is sufficiently large to include the zero value for a certain non-zero weight, the atom corresponding to the weight is considered to be “irrelevant”, and then pruned out from the model. For illustrative purpose, the results for the difference signal from S2 before and after the pruning scheme are compared in Figure 6(a) to (c). It can be observed that no significant differences exist after pruning away the redundant weights with an excessively large confidence interval, which implies that the proposed post-processing procedure is applicable.

The sparse decomposition results of the difference signals obtained from S1, S2, and S3 are presented in Figures 7 to 9, including the identified propagation distances of S₀ and A₀ modes, the corresponding total distances, and the reconstructed signals. Figure 7(a) to (c) illustrates the posterior mean of weight vector $\bar{c}$ and associated 95% confidence interval, which correspond to the identified propagation distance results. The blue and yellow texts in Figures 7 to 9 refer to the weights corresponding to the first and second wave packets, respectively. It is observed that the weight values of the first wave packet for the three sensors are approximately proportional to the corresponding amplitudes: the amplitudes are 0.000198, 0.000093, and 0.000248, respectively, while the corresponding weight values in $\bar{c}$ are 0.001006, 0.000473, and 0.001293, respectively. Therefore, the weight values of vector $\bar{c}$ can be used to assess the damage severity to some extent, as the wave packet amplitude is an important index for damage severity assessment. Figures 7 to 9(d) compare the reconstructed waveforms with the original difference signals; it is observed that they are reasonably consistent.

Table 2 tabulates the recognized total propagation distances of the two wave packets as well as the relative errors compared to the actual distances, where the relative error is computed by $(r_{i} - d_{i}) / d_{i}$ , where $r_{i}$ and $d_{i}$ are the recognized propagation distance and actual distance for the ith sensor, respectively. It is found that each distance difference between the two wave packets is less than 10%, and then it is considered mode conversion has occurred. It is demonstrated that the relative errors of the identified propagation distances are small. The maximum error is about 6%, which is acceptable for practical applications.

Table 2.

Identified total propagation distances and relative errors compared to actual distances (mm).

Sensor	Actual distance	First wave packet	Error (%)	Second wave packet	Error (%)
S1	260	272.2	4.7	252.7	2.8
S2	270	252.7	6.4	259.2	4.0
S3	240	253.8	5.8	253.8	5.8

To compare the performance of the proposed and other methods on the guided wave packet extraction, the analysis results using the proposed method and the l₁-norm minimization algorithm²⁵ are presented in Figure 10. It is observed that both methods can identify the weight corresponding to the correct wave packet; however, several additional small weights are also identified by the method in Xu et al.²⁵ These additional weights correspond to several wave packets with small amplitudes, leading to an irregular reduction in the weight value corresponding to the correct wave packet. This may induce false assessment of the wave packet amplitudes, and then, the damage severity.

Overlapping wave packet recognition

In practical applications, wave packet overlapping frequently occurs owing to multiple damages and boundary reflections. To investigate the capability of the proposed SBL-based method for overlapping wave packet recognition, simulated studies were performed on a square aluminum plate with two through-thickness notches, where the layout of the piezoelectric wafers is illustrated in Figure 11.

As the propagation distances of the direct reflections from two notches are substantially less than that of secondary reflection from boundaries, two wave packets scattered from two notches will be in the beginning part of the difference signal. If two notches along the propagation path are close, the corresponding wave packets will overlap. In order to indicate the overlapping rate, an index is defined as $DO = Δ t / t \times 100 %$ , where Δt and t are the durations of overlapping area and the reconstructed wave packet, respectively. The smaller the difference between the identified propagation distances, the more serious the wave packets overlap.

In Figure 12, the overlapping wave packets are decomposed by the proposed SBL-based method. The reconstructed coefficients and associated 95% confidence interval, as well as the reconstructed waveforms for the overlapping signal, are presented. The corresponding relative errors of the identified propagation distances are tabulated in Table 3. It can be observed that the reconstructed waveforms are consistent with the original difference signals and the relative errors are small. Each reconstructed wave packet based on the identified non-zero weight factor in the posterior mean of the vector $\bar{c}$ from the difference signal of S3 is shown in Figure 12(c)(iii), and the initial time and end time of each packet are also labeled. Note that the overlapping rate of the third difference signal is 87.4%, which means the two wave packets overlap severely and it is difficult to recognize the number of waveforms empirically contained in the overlapping wave packet. However, the SBL-based method is still capable of recovering the two wave packets effectively. In addition, due to severely overlapping wave packets, the confidence intervals of the identified weights are relatively larger than those of independent wave packet recognition in Figures 7 to 9.

Table 3.

Identified propagation distances and relative errors compared to actual distances for overlapping wave packet recognition (mm).

Sensor	Actualdistance		Identifieddistances innoise-free case		Error (%)		Identifieddistances in noise-contaminated case		Error (%)
S1	157	234	170.6	254.9	8.9	8.7	170.6	254.9	8.9	8.7
S2	162	246	181.4	237.6	11.9	3.4	183.6	237.6	13.3	3.4
S3	226	258	224.6	254.9	0.6	1.2	226.8	254.9	0.4	1.2

In Figure 13, the noise effects are also investigated for overlapping wave packet decomposition, where the signal-to-noise ratio of the additional Gaussian white noise added to the difference signal is 10 dB. The relative errors of the identified propagation distances are less than 8%, as indicated in Table 3. Under the combined effects of wave packet overlapping and noise, the propagation distances could still be recognized correctly by using the proposed SBL-based method. The recognized propagation distances are almost the same with those of noise-free case, though the confidence intervals become larger.

Damage localization using propagation distance recognition results

In this subsection, the damage localization performance of the proposed method is investigated based on the previously recognized total propagation distances, using the procedure outlined in subsections “Damage localization based on identified propagation distance” and “Sampling strategy for generating posterior samples of damage-related parameter $\bar{w}$ conditional on GW signals”. When using the sampling strategy to generate N = 2000 samples to characterize the posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ in subsection “Propagation distance recognition results of mode-converted wave packets”, it is found that the indices of the non-zero weights in each sample ${{\bar{w}}^{(n)}}, n = 1, . . ., N$ are always the same. Part of the reason is the utilization of SBL and the posterior uncertainty can be significantly reduced by incorporating the sparseness regularization prior information.

Figure 14 presents the damage localization results at each meshing level, where the posterior means and 95% confidence intervals of the non-zero weights in ${{\bar{w}}^{(n)}}, n = 1, . . ., N$ are indicated. According to the propagation distance results, it is found that only one damage occurs, and the location corresponding to the larger weight value is considered as the more plausible damage location. An area containing this location, indicated by the green area in Figure 14(a), is the re-meshing range, and the new identified damage locations are obtained. By re-meshing the selected area, the damage localization results corresponding to the second to fourth meshing levels g2–g4 are presented in Figure 14(b) to (d). The 95% confidence interval of the non-zero weight in Figure 14(d) is very small, indicating that the damage localization result is highly confident.

Figure 15 presents a comparison study of the results between the triangulation method and proposed SBL-based method. For the triangulation method, the three curves cross over a wide region, which has a large uncertainty. While in the proposed method, the most relevant location to the three identified propagation distances was obtained and the uncertainty is small by promoting the model sparseness. From the results, it is seen that the identified damage locations of all meshing levels are within the intersection area of the ellipses. However, the identified damage location of the meshing level g2 (double cross) is closest to the real notch, which is better than two finer meshing levels. The main cause of this phenomenon is the errors in the identified propagation distances from the first stage of the method, which are employed as the “data” for the damage localization in the second stage. Therefore, the damage localization performance of our SBL-based method is superior to that of the triangulation method. Note that some other triangulation-based methods²⁸ also can produce a peak of amplitude within the wide region, but they did not investigate how to reduce the uncertainty in the damage localization.

It is noted that the multiple damage localization is more complicated, because it is difficult to determine which recognized propagation distance is associated with which damage. In this case, the propagation distances related to the same damage should be pre-clustered and then the damage locations can be identified one by one using the proposed method. In addition, there are many combinations when clustering the propagation distances from multiple sensors, and only one of them is consistent with actual damages, which can be judged by the damage localization results of all possible combinations. Then, the computational efficiency is significantly affected.

Experimental investigations

An experimental test on a rectangle 5083 Aluminum plate was conducted to validate the proposed approach. The plate size was 1600 mm × 1400 mm × 1 mm. The material properties were Young’s modulus, E = 71 GPa; Poisson’s ratio, μ = 0.33; and density, ρ = 2660 kg/m³. The dispersion relation was calculated theoretically based on these material properties. Four circular piezoelectric wafers (diameter = 8 mm and thickness = 1 mm) were attached to the plate surface. A 10 mm × 1.5 mm × 0.8 mm notch was produced by a miniature angle grinder to simulate the crack. The layout of piezoelectric wafers and the artificial notch are illustrated in Figure 16.

The testing devices were the function generator (AFG3022C, Tektronix, Inc.), wideband power amplifier (Tegam 2350), and the digital oscilloscope (DPO 2004B, Tektronix, Inc.), which were used to generate, amplify the excitation signal, and receive the signals from the sensors, respectively. The center frequency of excitation signal was 100 kHz.

Propagation distance recognition results

The received signals from both the undamaged and damaged conditions were collected by the sensors and then the difference signals related to the damages were obtained by subtracting the undamaged signal from the corresponding damaged signals. Because of the difference in the electromagnetic effect between the undamaged and damaged conditions, a wave packet exists at the beginning of these difference signals.

In order to study the effectiveness of extracting information from mode-converted wave packets, the difference signal obtained by S1 was decomposed by the proposed method. Figure 17(a) to (c) illustrates the posterior mean of weight vector $\bar{c}$ and associated 95% confidence interval, which correspond to the identified propagation distance results. The identified total distances of the wave packets are 529.2 and 563.8 mm, and the relative error compared to actual distance are 0.2% and 6%, respectively. Figure 17(d) compares the reconstructed waveforms with the original difference signals; it is observed that they are reasonably consistent with each other. These results validate the effectiveness of extracting information from mode-converted wave packets by using the proposed method.

Based on triangulation method, the first wave packet in the difference signal was analyzed to identify propagation distances, which would be used to localize the damage. Because the first wave packet was known as S₀ mode, the propagation distance of A₀ mode in equation (5) was set as 0 when building the dictionary matrix. The propagation distance recognition results of the difference signals from other two sensors are illustrated in Figure 18. When identifying only one wave packet, the atom in the dictionary matrix U corresponding to maximum absolute weight value is considered the most “relevant”, because it makes the largest contribution to the waveform fitting. The final signal reconstruction results demonstrate that the experimental and reconstructed waveforms are strongly consistent with one another. In Table 4, the recognized propagation distances and relative errors compared to the actual distances are tabulated. It is seen that the maximum relative error is around 1.6%.

Table 4.

Identified propagation distances and relative errors compared to actual distances (mm).

Sensor	Actual distance	First wave packet	Error (%)
S1	530	529	0.2
S2	550	554	0.7
S3	450	443	1.6

Damage localization using propagation distance recognition results

Thereafter, the recognized propagation distances were used to localize the damage based on the procedure presented in subsections “Damage localization based on identified propagation distance” and “Sampling strategy for generating posterior samples of damage-related parameter $\bar{w}$ conditional on GW signals”. Similar to the simulation study, when using the sampling strategy to generate N = 2000 samples ${{\bar{w}}^{(n)}}, n = 1, . . ., N$ from the marginal posterior PDF $p (\bar{w} | s_{1}, s_{2}, s_{3})$ , the index of the non-zero weight in each sample ${{\bar{w}}^{(n)}}$ is always the same, indicating the posterior uncertainty for damage localization is small. In Figure 19, the identified damage locations, and corresponding mean and 95% confidence interval of the non-zero weight in the vector $\bar{w}$ for the four meshing levels are illustrated. The final identified damage location is (881.3 mm, 708.3 mm), which is 8.7 mm away from the real damage location (881 mm, 717 mm). When observing the 95% confidence intervals, the values are small for the identified non-zero weight, particularly for the fourth meshing level g4, which indicates that the damage localization is highly confident.

Figure 20 compares the damage localization results of the proposed SBL-based method and triangulation method. For the triangulation method, the localization result is within an area and away from the actual notch; however, the identified damage location of the proposed SBL-based method in the g4 meshing level (circle) is close to the real notch. These comparison results verify the superior performance of our proposed SBL-based method.

Conclusion

In this article, a new two-stage damage detection approach has been proposed based on UGW testing and SBL. In the first stage, sparse signal decomposition is investigated by using SBL for the purpose of propagation distance recognition. By considering multimode, mode conversion, and dispersion effect, an over-complete dictionary is designed. Based on this dictionary matrix, the propagation distances of noisy wave packets can be extracted effectively. The SBL algorithm also enables us to better decompose overlapping wave packets and obtains the multiple propagation distances with high confidence. In the second stage, the damage localization procedure is studied by matching the propagation distance information in an SBL framework. To account for the uncertainties in the two-stage damage detection procedure, a sampling strategy is proposed to generate samples and to characterize the posterior distribution of the localized damage conditional on the measured GW signals directly. Regarding the comparison between our proposed method and that in Xu et al.,²⁵ several significant features have been discussed and demonstrated for our new method in terms of theoretical and application aspects.

The performance of the proposed method was validated by numerical and experimental studies. When investigated for FE simulation signals, the method not only recognized propagation distances of wave packets (including converted wave packets) with quantified uncertainties, but was also demonstrated as effective for the overlapping wave packet problem. Using this information, the damage location was identified robustly, even under high noise levels. Furthermore, the identified damage location was a position with largest possibility compared to the large area enclosed by the triangulation method, which further validated the effectiveness of the method. For further studies, it would be useful to explore the SBL-based approach in multi-damage detection applications.

Footnotes

Appendix 1

Appendix 2 Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The study was supported by the National Key Research and Development Program of China under Grant Nos 2016YFC0802400 and 2018YFC1505304 and the National Science Foundation of China under Grant Nos. 51978217, 51778192, and 51578191.

ORCID iDs

Wensong Zhou

Yong Huang

Hui Li

References

Yuan

F-G.

Structural health monitoring (SHM) in aerospace structures. New York: Woodhead Publishing, 2016.

Guided Lamb waves for identification of damage in composite structures: a review. J Sound Vib 2006; 295: 753–780.

Yuan

FG.

Lamb wave-based subwavelength damage imaging using the DORT-MUSIC technique in metallic plates. Struct Heal Monit 2016; 15: 65–80.

Sharma

Mukherjee

Monitoring corrosion in oxide and chloride environments using ultrasonic guided waves. J Mater Civ Eng 2011; 23: 207–211.

Cho

Structural health monitoring of fatigue crack growth in aluminum structures using ultrasonic guided waves. Penn State Publ 2009; 11: 393–404.

Sohn

Dutta

Yang

J-Y

, et al. Delamination detection in composites through guided wave field image processing. Compos Sci Technol 2011; 71: 1250–1256.

Huang

Balogun

Yang

, et al. Detection of disbonding in glare composites using lamb wave approach. AIP Conf Proc 2010; 1211: 1198–1205.

Dai

Structure damage localization with ultrasonic guided waves based on a time-frequency method. Signal Process 2014; 96: 21–28.

Staszewski

Mahzan

Traynor

Health monitoring of aerospace composite structures: active and passive approach. Compos Sci Technol 2009; 69: 1678–1685.

10.

Moll

Schulte

Hartmann

, et al. Multi-site damage localization in anisotropic plate-like structures using an active guided wave structural health monitoring system. Smart Mater Struct 2010; 19: 45022.

11.

Leckey

CAC

. Lamb wave-based quantitative crack detection using a focusing array algorithm. J Intell Mater Syst Struct 2013; 24: 1138–1152.

12.

Giurgiutiu

In situ 2-D piezoelectric wafer active sensors arrays for guided wave damage detection. Ultrasonics 2008; 48: 117–134.

13.

Yuan

FG.

Damage identification for composite structures using a cross-correlation reverse-time migration technique. Struct Heal Monit 2015; 14: 558–570.

14.

Leckey

CAC

Leser

, et al. Multi-mode reverse time migration damage imaging using ultrasonic guided waves. Ultrasonics 2019; 94: 319–331.

15.

Rocha

Sava

Guided wave tomography based on least-squares reverse-time migration. Struct Heal Monit. Epub ahead of print 19 October 2019. DOI: 10.1177/1475921719880296.

16.

Rao

Saini

Yang

, et al. Ultrasonic imaging of irregularly shaped notches based on elastic reverse time migration. NDT E Int 2019; 107: 102135.

17.

Rao

Ratassepp

Fan

Guided wave tomography based on full waveform inversion. IEEE Trans Ultrason Ferroelectr Freq Control 2016; 63: 737–745.

18.

Wang

Rose

Chang

FK.

A synthetic time-reversal imaging method for structural health monitoring. Smart Mater Struct 2004; 13: 415–423.

19.

Han

W-Q

Luo

. AE frequency analysis of damage mechanism in CFRP laminates based on Hilbert–Huang transform. In: 2nd annual international conference on advanced material engineering (AME 2016), Wuhan, China, 15-17 April 2016, pp. 1258–1263. Atlantis Press.

20.

Kishimoto

Inoue

Hamada

, et al. Time frequency analysis of dispersive waves by means of wavelet transform. J Appl Mech 1995; 62: 841–846.

21.

Wang

Tse

Mechefske

, et al. Experimental investigation of reflection in guided wave-based inspection for the characterization of pipeline defects. NDT E Int 2010; 43: 365–374.

22.

Qatu

Abdelgawad

Yelamarthi

Structure damage localization using a reliable wave damage detection technique. In: International conference on electrical, electronics, and optimization techniques (ICEEOT), Chennai, India, 3–5 March 2016, pp. 1959–1962. New York: IEEE.

23.

Liu

Yuan

FG.

A linear mapping technique for dispersion removal of Lamb waves. Struct Heal Monit 2010; 9: 75–86.

24.

Yuan

Shi

Qing

, et al. Linearly dispersive signal construction of Lamb waves with measured relative wavenumber curves. Sensors Actuators A Phys 2014; 221: 41–52.

25.

Yang

Chen

, et al. A guided wave dispersion compensation method based on compressed sensing. Mech Syst Signal Process 2018; 103: 89–104.

26.

Niethammer

Jacobs

, et al. Time-frequency representation of Lamb waves using the reassigned spectrogram. J Acoust Soc Am 2002; 107: L19–L24.

27.

Moilanen

, et al. Mode separation of Lamb waves based on dispersion compensation method. J Acoust Soc Am 2012; 131: 2714–2722.

28.

Yan

A Bayesian approach for damage localization in plate-like structures using Lamb waves. Smart Mater Struct 2013; 22: 17.

29.

CT.

Guided wave-based identification of multiple cracks in beams using a Bayesian approach. Mech Syst Signal Process 2017; 84: 324–345.

30.

Cantero-Chinchilla

Chiachío

, et al. A robust Bayesian methodology for damage localization in plate-like structures using ultrasonic guided-waves. Mech Syst Signal Process 2019; 122: 192–205.

31.

Yang

Guan

, et al. A probabilistic crack size quantification method using in-situ Lamb wave test and Bayesian updating. Mech Syst Signal Process 2016; 78: 118–133.

32.

Chen

Donoho

Saunders

MA.

Atomic decomposition by basis pursuit. SIAM Rev 2001; 43: 129–159.

33.

Blumensath

Accelerated iterative hard thresholding. Signal Process 2012; 92: 752–756.

34.

Chartrand

Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process Lett 2007; 14: 707–710.

35.

Levine

Michaels

JE.

Model-based imaging of damage with Lamb waves via sparse reconstruction. J Acoust Soc Am 2013; 133: 1525–1534.

36.

Wang

Bao

Zhou

, et al. Sparse representation for Lamb-wave-based damage detection using a dictionary algorithm. Ultrasonics 2018; 87: 48–58.

37.

Zhang

Harvey

DM.

Sparse signal representation and its applications in ultrasonic NDE. Ultrasonics 2012; 52: 351–363.

38.

Huang

Beck

, et al. Robust Bayesian compressive sensing for signals in structural health monitoring. Comput Civ Infrastruct Eng 2014; 29: 160–179.

39.

Huang

Beck

, et al. Bayesian compressive sensing for approximately sparse signals and application to structural health monitoring signals for data loss recovery. Probabilistic Eng Mech 2016; 46: 62–79.

40.

Huang

Beck

Hierarchical sparse Bayesian learning for structural damage detection: theory, computation and application. Struct Saf 2017; 64: 37–53.

41.

Huang

Beck

Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment. Comput Methods Appl Mech Eng 2017; 318: 382–411.

42.

Zhang

Wang

Jin

, et al. Sparse Bayesian learning for obtaining sparsity of EEG frequency bands based feature vectors in motor imagery classification. Int J Neural Syst 2017; 27: 1650032.

43.

Huang

Chen

, et al. Guided-wave signal processing by the sparse Bayesian learning approach employing Gabor pulse model. Struct Heal Monit 2017; 16: 347–362.

44.

Huang

Krishnaswamy

A Bayesian approach for sparse flaw detection from noisy signals for ultrasonic NDT. NDT E Int 2017; 85: 76–85.

45.

Yang

Qiao

, et al. Traveling distance estimation for dispersive Lamb waves through sparse Bayesian learning strategy. Smart Mater Struct 2019; 28: 085008.

46.

Raghavan

Guided-wave structural health monitoring, 2007, https://deepblue.lib.umich.edu/bitstream/handle/2027.42/77498/Raghavan_PhD_thesis_GWSHM.pdf?sequence=1&isAllowed=y

47.

Beck

Katafygiotis

LS.

Updating models and their uncertainties. I: Bayesian statistical framework. J Eng Mech 2002; 124: 455–461.

Sparse Bayesian learning approach for propagation distance recognition and damage localization in plate-like structures using guided waves

Abstract

Keywords

Introduction

Proposed SBL-based two-stage approach for damage detection

Sparse signal decomposition and propagation distance recognition

Remark 2.1

Damage localization based on identified propagation distance

Sampling strategy for generating posterior samples of damage-related parameter w ¯ conditional on GW signals

Remark 2.2

Numerical studies

Propagation distance recognition results of mode-converted wave packets

Overlapping wave packet recognition

Damage localization using propagation distance recognition results

Experimental investigations

Propagation distance recognition results

Damage localization using propagation distance recognition results

Conclusion

Footnotes

Appendix 1

Appendix 2

Funding

ORCID iDs

References

Sampling strategy for generating posterior samples of damage-related parameter $\bar{w}$ conditional on GW signals