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Der Artikel diskutiert die Herausforderungen bei der Klassifizierung von Zeitfrequenzkarten geführter Wellen aufgrund ihrer dispersiven Beschaffenheit. Es wird eine Methode eingeführt, die auf Vordergrundextraktion und Dimensionalitätsreduzierung basiert und PCA auf transponierten Matrizen verwendet, um die Klassifizierung von geführten Wellensignalen zu verbessern. Die Methode wird durch numerische Simulationen und experimentelle Daten validiert und zeigt ihre Wirksamkeit bei der Identifizierung und Klassifizierung verschiedener struktureller Bedingungen. Der vorgeschlagene Ansatz ist vielversprechend für praktische Anwendungen in der strukturellen Gesundheitsüberwachung.
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Abstract
Guided waves propagating in mechanical structures have proved to be an essential technique for applications, such as structural health monitoring. However, it is a well-known problem that when using non-stationary guided wave signals, dispersion, and high-order vibrational modes are excited, it becomes cumbersome to detect and identify relevant information. A typical method for the characterization of these non-stationary signals is based on time–frequency (TF) mapping techniques. This method produces 2D images, allowing the study of specific vibration modes and their evolution over time. However, this approach has low resolution, increases the size of the data, and introduces redundant information, making it difficult to extract relevant features for their accurate identification and classification. This paper presents a method for identifying discontinuities by analyzing the data in the TF maps of Lamb wave signals. Singular Value Decomposition (SVD) for low-rank optimization and then perform foreground feature extraction on the maps were proposed. These foreground features are then analyzed using Principal Component Analysis (PCA). Unlike traditional PCA, which operates on vectorized images, our approach focuses on the correlation between coordinates within the maps. This modification enhances feature detection and enables the classification of discontinuities within the maps. To evaluate unsupervised clustering of the dimensionally reduced data obtained from PCA, we experimentally tested our method using broadband Lamb waves with various vibrational modes interacting with different types of discontinuity patterns in a thin aluminum plate. A Support Vector Machine (SVM) classifier was then implemented for classification. The results of the experimental data yielded good classification effectiveness within reasonably low computational time despite the large matrixes of the TF maps used.
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1 Introduction
The propagation of guided waves within mechanical structures has shown its important role in applications like structural health monitoring. However, it is well known that ultrasonic signals traveling in mechanical guides such as plates and cylinders can produce highly dispersive transient signals. Extensive research has focused on the interaction of guide waves with discontinuities. The interaction of guided waves with imperfect interfaces, wave–notch interaction, and discontinuity time-dependent scattering have been extensively studied [1‐4]. Additionally, research has been conducted on scattering direction dependence of the wave mode components [5], numerical prediction of lateral scattering for different hole’s diameters [6], the interaction of guided waves with adhesive joints to approximate the geometry and position of defects [7], attenuation of guided waves caused by leakage of a stiffener attached with adhesive [8], and numerical modeling of wave fields around an undamaged and damaged discontinuity [9] have been extensively studied. Model-based damage prediction methods, using numerical, analytical, or semi-analytical approaches have been proposed [10‐12]. Despite the efforts using model-based methods, characterization and correlation of the signals with the propagation properties of the medium can become a computationally complex problem. Alternatively, signal-based methods have been developed, with an emphasis on time-domain signal analysis using thresholding or signal subtraction [13‐15]. The short-time Fourier transform (STFT) has been of particular interest due to its ability to: measure closely spaced vibration modes and increase time–frequency resolution in dispersive signals. [3, 16]. Wavelet transform has also been implemented with similar success for Lamb wave signal characterization as STFT [17]. Moreover, the resulting transformation maps can be analyzed similarly to image processing techniques, where PCA can be used for feature extraction and classification alongside machine learning algorithms [18‐20]. While a time–frequency map has advantages, the amount of information in the matrix of the time–frequency map is several times the size of the initial time domain data vector, making it in some cases computationally expensive to monitor changes directly on the maps, especially when these are small with a low signal-to-noise ratio and not localized in a region of the map.
In addressing the limitations of traditional Lamb wave approaches for signal identification and classification, machine learning has emerged as a powerful tool for uncovering hidden patterns in datasets. Many researchers have proposed methods of pattern recognition in guided waves, combining different machine learning approaches. For instance, utilizing PCA in conjunction with unsupervised neural network self-organizing map (SOM) for damage detection [21].
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To identify an optimal approach for feature extraction of Lamb waves applying artificial neural networks (ANN) and SVM have been reported (i.e. [22, 23]). Deep neural networks (DNNs) in combination with wave damage interaction coefficients have achieved success for damage identification in thin structures, but limitations due to computational demands and dependency on the database size have been found [24]. Recognizing the need for innovation, convolutional neural networks (CNN) emerge as a class of deep learning models capable of extracting features crucial for damage identification. In [25] fully convolutional neural networks (FCN) on scanning laser wave field maps for damage detection in composite materials were used, requiring a substantial quantity of training data. Other recent advancements in artificial intelligence have enabled the application of various machine learning techniques of pattern recognition to deal with the processing and characterization of ultrasonic signals as demonstrated in studies such as [26‐30].
In Table 1 a concise comparative analysis of various approaches of machine learning for damage detection and identification using guided waves is provided. Each approach, ranging from traditional methods like PCA and SVM to more advanced techniques such as CNNs and FCNs, is evaluated based on the utilized database, advantages, challenges, and the frequency range of operation.
In addition to these advancements, the domain of structural health monitoring (SHM) utilizing multi-sensor data derived from fixed sensors attached to the structure under surveillance has emerged as a machine learning challenge, with a predominant emphasis on dimension reduction techniques for effective damage detection. The purpose is to develop intuitive visualization and implementation of classification methods for structural health monitoring purposes. In [33], enhanced dimensionality reduction techniques with DNN is discussed. However, the use of raw signals can be computationally intensive. To address this, researchers have explored pre-filtering methods to extract essential information from the signals. SHM automatic damage diagnosis using ultrasonic Lamb waves and deep autoencoders for fatigue damage in composite structures is reported in [32].
One challenge these methods face is feature extraction from complex and extensive datasets. For multidimensional data, methods based on probabilistic models, such as SVD have been proposed to reduce data complexity by identifying the direction of maximum variance, the so-called principal components. In recent years, research on SVD applications for image processing and classification has been intensively studied. In image processing, it has been implemented for pedestrian detection and tracking to differentiate between moving objects (foreground) from the static background using low-rank optimization [34, 35].
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In the non-destructive evaluation field, principal component analysis obtained by SVD has been used to detect damage. In [36] vibrations monitoring using SVD was related to damage in gearboxes. Variations of this technique have been studied in applications such as geology, to measure the time in which a seismic event will occur [37]; this analysis was performed on time–frequency representation since identifying a given wave is difficult in the time domain. Modifications of the basic PCA have been studied for seismic signal analysis using Robust Principal Component Analysis, RPCA [38]. Recently, an artificial learning method for the identification of time–frequency maps from non-stationary signals was proposed in [39]. That research was focused on the application of the background subtraction approach without emphasis on the transpose of the foreground matrices to improve map identification and with emphasis on simple patterns.
As discussed, studies on signal classification using machine learning for SHM have made significant advancements. However, they often focus on extracting features from signals in either the time domain or the frequency domain, potentially missing critical information. Here, we present a method based on the analysis of time–frequency (TF) maps. Previous works used PCA on vectorized maps to reduce dimensionality for classification, but this approach can be challenging for maps with low signal-to-noise ratio. To improve detection in these cases, we proposed foreground extraction by removing invariant information (background) and correlation analysis between individual coordinates within the foreground matrices. It is shown that this strategy is simple and computationally efficient, making it well-suited for signal classification in health monitoring of broadband ultrasonic guided waves.
The specific objectives of this study are as follows: first, to develop a method based on low-rank matrix approximation and correlation between coordinates within the resulting vectorized foreground data, to extract and classify relevant information related to small changes in the STFT maps; second, to conduct an ablation study to numerically test the effectiveness of the proposed method; third, design an experimental setup to evaluate the method with dispersive guided wave signals. The experimental setup aims to simulate several scenarios with different boundary conditions on an aluminum plate, to replicate conditions commonly encountered in SHM applications; and, fourth, to evaluate the effectiveness of the proposed method dealing with STFT maps of dispersive signals for accurate identification and classification of different structural conditions.
Fig. 1
Description of the STFT method: a a time domain signal of length p is windowed to perform a number w of FFT having a length v, b construction process of the STFT map by staking all \(FFT_{i}\), and c example of reconstructed STFT map of size \(v\times w\) for a guided wave signal, the propagating modes \(S_{0}\) and \(A_{0}\) located in time are shown
The paper is organized as follows: Sect. 2 provides a comprehensive overview of the developed methodology utilized in our study, encompassing the STFT signal representation, foreground extraction, and dimensionality reduction. It also discusses preliminary simulation results using synthetic maps. In Sect. 3, describes the experimental setup for signal acquisition, and analyzes signal variations across different structural conditions. Section 4 demonstrates the application of the proposed method for identification and classification. It details the process of cluster generation, and implementation of SVM for classification, and presents the results of identifying various discontinuity patterns based on their cluster formations.
2 Time–Frequency Maps and PCA
2.1 Short Time Fourier Transform
The STFT has been of interest in the analysis of non-stationary signals, because it facilitates the examination of signals for processing and interpretation, and finds applications in several fields [36, 37, 39]. STFT provides a 2D spectrogram obtained from a 1D time-domain signal showing the signal behavior in both time and frequency domains simultaneously. As shown in Fig. 1, for an input real-valued signal x(t) of length p, the time–frequency representation is obtained by segmenting the signals into overlapping sub-signals (\(h_{i})\) (Fig. 1a), which helps to reduce distortions at the boundaries between them. For each sub-signal \(h_{i}\) a discrete Fast Fourier Transform (FFT) is calculated, then all the \(FFT_{i}\) are appended to create a matrix containing the magnitude and phase of each \(FFT_{i}\) (Fig. 1b). To tune the time and frequency resolution, it is necessary to regulate the following parameters: (1) the sampled window function of length M and (2) the overlap size L, which controls the increment for shifting the window across the signal. The spectrogram representation is generated as a result of the energy of the complex values of STFT, denoted as \(STFT ( v, w) = \mid {\text {STFT}}\mid ^{2}\), resulting in a matrix \(STFT_{ v \times w }\) where v represents frequency bins and w represents time bins (Fig. 1c). Therefore, the resulting spectrogram significantly increases in size depending on the chosen resolution compared to the original time signal, transitioning from a one-dimensional vector to a two-dimensional matrix. A schematic illustration of the STFT method applied to an ultrasonic Lamb wave signal is presented in Fig. 1.
STFT is a valuable tool for time–frequency representation of signals, enabling a detailed analysis of complex phenomena of multimodal dispersive waves. However, it comes with inherent challenges. The uncertainty associated with time and frequency resolution arises from the compensation in choosing parameters like the window length and overlap size. The overlapping frames introduce redundancy in the information, generating concerns about data efficiency. Additionally, the transformation of a 1D signal into a higher-dimensional 2D spectrogram demands increased memory usage, emphasizing the need for a good balance between resolution and resource constraints in practical applications.
The schematic representation in Fig. 1 illustrates the key components of the STFT method applied to an ultrasonic signal. The distribution of energy in the resulting spectrogram contains valuable information about the medium, representing different modes propagating through the structure.
The central idea of this work is to use these energy distribution maps to classify conditions that generate subtle variations in the energy distribution for effective condition monitoring in structural elements.
Statistical techniques such as PCA are useful for classification, as they can extract the most relevant information from a large number of variables. Here, we propose a method based on the PCA analysis of the STFT maps to filter noise, reduce redundancies, and reveal patterns in the signals associated with their classes in lower dimensions.
2.2 Dimensionality Reduction of Vectorized STFT Maps
Exploring patterns and extracting features from short-time Fourier transform maps of multimodal guided waves can be done with PCA. It offers a way to address complexities associated with high-dimensional data. This is, in principle, a dimensionality reduction technique, that is efficient in terms of memory usage while retaining the critical information necessary for tasks such as clustering or classification. It has been used intensively in applications related with artificial image classification [19, 40, 41]. In PCA, a set of measurements for a feature or variable is gathered in a column vector, with as many vectors as variables included in the analysis. The concatenated vectors form a matrix A. Each measurement becomes a row vector in a typically large, tall data matrix (A). In our case, A contains columns with the vectorized STFT maps. Let \(A_{m \times n}\) be a data matrix with m observations (where each row maps the coordinates on the STFT map) and n variables (is the number of vectorized maps). There, the total number of elements (m) in the STFT matrix of dimension \([v\times w]\) is \(m=vw\).
PCA can be performed using SVD which exploits the property that any matrix \(A_{m\times n}\), can be factorized as
$$\begin{aligned} A =U \Sigma V^{T}, \end{aligned}$$
(1)
where U and V are matrices of orthonormal vectors, and \(\Sigma \) is the diagonal matrix of singular values. The principal components (PC) of A are obtained by selecting the singular vectors in \(V^{T}\) corresponding to the largest singular values. Let \(V_{k}\) be the matrix containing the first k eigenvalues, the transformed centered data matrix Y is given by:
where \(PC_{i}\) represents individual principal components. Thus, PCA involves transforming the original data matrix, A, into a new matrix, Y, where each column of Y represents a principal component.
The matrix \(V^{T}\) is commonly determined by using SVD on the matrix \(A^{T}A\) such that using the definition in Eq. (1), we have that
$$\begin{aligned} A^{T}A = V \Lambda V^{T} \end{aligned}$$
(3)
which is the unbiased covariance (C) when divided by \(n-1\). The eigenvectors in V represent the directions of maximum variance in the centered data of A, and the corresponding eigenvalues in \(\Lambda =\Sigma ^{2}\) quantify the amount of variance along each eigenvector. Thus, the quantity \(A^{T}A\) gives the covariance among the columns of the matrix A . The covariance among the rows of A can be found performing the same analysis but on \(A^{T}\), such that
$$\begin{aligned} AA^{T} = U \Lambda U^{T}, \end{aligned}$$
(4)
where U is the direction of maximum variance for the rows of A. A similar approach with vectorized maps of ray path trajectories was implemented in [42]. This work we hypothesized that PCA on \(A^{T}\) of the tall matrix A created with the vectorized STFT maps could increase sensitivity to small changes in the STFT maps by observing the correlation between coordinates of the pixels instead of between the vectorized images. It is, however, not a common practice to construct the matrix \(AA^{T}\) when the number of rows (samples) is higher than the number of columns (features) since SVD has a higher computational complexity [43, 44]. For example, for a tall matrix A (\(m\gg n\)), \(A^{T}A\) has size [\(n \times n\)], while \(AA^{T}\) has a much larger size [\(m \times m\)] affecting the computational complexity of SVD calculation.
The SVD computational complexity using \(AA^{T}\) can be mitigated by truncating all zero elements in \(\Sigma \). That is, for a matrix \(A_{m\times n}\) with \(m\gg n\), the truncated SVD time complexity of A or \(A^{T}\) is proportional to \({\mathcal {O}}(r^{3})\) [45], where r is the rank given by \(r\le \min (m,n)\), since \(rank(A)=rank(A^{T})\). Here, we have used the MATLAB SVD-economy mode to reduce the time complexity. For example, using non-truncated SVD on a tall matrix \(A_{10,000\times 100}\), the computational time for \(A^{T}\) is 13.203 s using full SVD. However, when the economy mode is used, the time is reduced to only 0.033s.
2.3 Foreground Extraction of STFT Maps
In image processing, background subtraction has received significant attention as an effective method for detecting moving objects from sequences of frames captured by static cameras. For video surveillance, the objective is to detect a moving object by analyzing the difference between a current image and a reference one [46, 47]. A parallel similarity can be seen between fixed cameras and fixed transducers used in structural monitoring: both continuously monitor their surroundings from a fixed position. Therefore, in our experiment to implement background subtraction, fixed sensor positions were used. Considering fixed transducers, only specific regions of the STFT maps are expected to change when small variations (discontinuities) in the medium affect the propagation of guided waves. Echoes from edges and external interference are considered background noise as long as they remain invariant during the monitoring test. This implies the presence of invariant background information in the STFT maps. Subtracting this background could yield sparse foreground matrices containing information solely related to discontinuities or changes. Here, it is assumed that minor variations in STFT maps resulting from different test conditions of the test object can be interpreted as moving objects that could be separate from the background. For a matrix A constructed by vectorization of the STFT maps (\(S_{i}\)), that is
where each \(S_{i}\) is a \([m \times 1]\) vector with m number of elements in the STFT matrix, and n maps. Background subtraction using low-rank approximation is based on the idea that the matrix A\([m\times n]\) (i.e., in our case, a set of vectorized STFT maps) can be decomposed into two matrices
$$\begin{aligned} A=B+F, \end{aligned}$$
(6)
where B is low-rank and F is sparse. The optimization problem can be stated as
$$\begin{aligned} \min _{B,F} {\text {rank}}(B) + \left\| F \right\| _{0} \quad {\text {subject to }} A=B+F. \end{aligned}$$
(7)
Unfortunately, this is a NP-hard non-convex optimization problem. However, this can be solved by convex optimization using Principal Component Pursuit, PCP [47], which gives a solution to the convex optimization problem
$$\begin{aligned} \min \left\| B\right\| _{*} + \lambda \left\| F \right\| _{1} \quad {\text {subject to }} A=B+F , \end{aligned}$$
(8)
where \(\lambda =1/\sqrt{{\text {max}}(m,n)}\) is a relaxation parameter, and \(\left\| \cdot \right\| _{*}\) and \(\left\| \cdot \right\| _{1}\) are the convex nuclear and L1 norms respectively. Despite its potential computational expense, PCP proves robust against noise. Alternatively, a heuristic approach is to implement directly SVD on the matrix A, to find the best rank approximation, r, and then to thresholding columns of \(\Sigma \) such that only the first l singular values with \(l\le r\) are considering, thus truncating further the rank of B. In [48] an interactive SVD method of background removal is discussed. They proposed the slope (\(\sigma _{i}-\sigma _{i+1}\)) of the singular value distribution resulting of SVD as mean to heuristically defining a threshold; a similar approach is used here and compared with the results using PCP.
Thus, the background of A can be found by using a truncated matrix \({\hat{A}}=U_{l}\Sigma _{l}V^{T}_{l}\), where \(U_{l}\) contains the first l columns of U corresponding to the largest singular values in \(\Sigma \). In background removal operation, a vectorized map J is projected in the subspace \(U_{l}\) to obtain the background \( B_{J}\)
The foreground \(F_{J}\) of the map J is then obtained through a simple subtraction
$$\begin{aligned} F_{J}=J-B_{J}. \end{aligned}$$
(10)
In our proposed methodology, background subtraction and PCA are combined to accomplish unsupervised clustering and classification of guided wave signals based on their STFT maps.
2.4 Description of the Proposed Method
The main steps of the proposed methodology are shown in the flow diagram of Fig. 2. First, a number n of time domain signals (\(s_{i}\)) are acquired and transformed into the time–frequency domain and the resulting maps are vectorized and assembled to get a matrix A with dimensions \([n\times m]\) containing a number n of STFT maps each with m elements. Next, background subtraction is implemented on the matrix A to determine the foreground matrix, F which contains the changes among the maps. Third, PCA implementation on \(F^{T}\) to reduce data dimensionality and to promote unsupervised data cluster formation is carried out. It should be stressed that in our proposal, the columns in \(F^{T}\) are the coordinates in the foreground maps, and the rows are the maps. This matrix operation intends to increase sensitivity to changes in the foreground maps. In a final step, data dimensionality reduction can be used in the context of a classifier, for example using a SVM technique. It generates classification hyperplanes between a pair of clusters defined by the principal components, allowing signal classification.
2.5 Preliminary Simulations Using the Proposed Method
In this section, the effectiveness of the proposed methodology is explored using synthetic data. A dataset of synthetic maps were created using the following function as the background
This function involves Gaussian-like distributions with some stationary points. To introduce controlled changes (foreground) in the amplitude distribution, three geometric shapes: triangle, rectangle, and circle (as shown in Fig. 3) were added, creating a three-classes discrimination problem. The location of the geometries was varied randomly about 2 pixels, adding variability among maps. Additionally, noise was added to the function using a random noise generator with normal distribution having mean, and standard deviation parameters set as \(\mu =0\), \(\sigma =0.1\).
Fig. 3
a Surface plot of function in Eq. 11 used as background. Plots with added Gaussian noise and geometric shapes: b triangle, c rectangle, and d circle
The maps were then vectorized and stacked as columns to form the matrix A. Here, our goal is to demonstrate that the maps can be separated into clusters based on their class. As a simple ablation study, first, the results when PCA is applied directly to A instead of \(A^{T}\) as proposed in our method are discussed. In a second numerical test, we examined the results with and without implementing the step of background removal.
Figure 4 illustrates data projected onto the subspace given by the first two principal components. In Fig. 4a, the result obtained with the matrix A shows scatter data without cluster formation or class separation. On the contrary, Fig. 4b depicts three clusters of data when \(A^{T}\) is used, revealing a distinct and efficient separation of the data into clusters. The contrasting results emphasize the significance of the selected transpose matrix \(A^{T}\) in our proposed method. A plot of the cumulative singular values and their slope (difference between two consecutive singular values) is given in Fig. 5a. As discussed in Sect. 2.3, this could be used to establish a threshold for identifying the minimal number of singular components contributing the most information. Here a threshold of \(\tan ^{-1}(slope)=80\), was used. Figure 5b shows the cumulative explained variance, indicating the proportion of total variance in the data captured by each successive principal component. Examining the explained variance plot of PCA(A) and \(PCA(A^{T})\) reveals that the first principal component accounts for nearly \(100\%\) of the variance information, indicating a dominant direction of maximum variance within the dataset. This explains the limited effectiveness of PCA(A) in class separation due to small variations among maps. In contrast, performing PCA (\(A^{T}\)) gives a gradual increment in explained variance. This proves that the covariance structure between rows of A instead of columns plays a crucial role in class separation. Thus, resulting in a larger separation of the clusters associated with the classes.
Fig. 4
Data with an SNR = 5 projected into the subspace of the first two principal components using aA, and b\(A^{T}\) matrix
a Cumulative singular values and its slope given after SVD on A, and b comparison of cumulative explained variance obtained from PCA on A and \(A^{T}\)
Figure 6 illustrates the dimensionality reduction for different values of SNR using the first two principal components obtained from the matrix of \(A^{T}\). It is clear that as the signal-to-noise ratio (SNR) decreases, the formation of clusters exhibits less separability.
Fig. 6
Projection of data on the first two principal components of \(A^{T}\) for different values of SNR a 3, b 2, and c 1
Figure 7 presents the complete implementation of the proposed method, with foreground extraction and their projection onto the first two principal components found after PCA(\({{\textbf {F}}}^{T}\)), using the same SNR values as in Fig. 6. It can be seen that even in the challenging scenario of SNR = 1, the proposed method reaches separation between clusters. Thus, it demonstrates the relevance of using foreground extraction and PCA on \({{\textbf {F}}}^{T}\).
Fig. 7
Projection of foregrounds on the first two principal components of \(F^{T}\) for different values of SNR a 3, b 2, and c 1, and examples of foregrounds used for it respective SNR d 3, e 2 and f 1
A type of ultrasonic-guided wave that travels in plate-like structures with traction-free conditions is known as a Lamb wave. As the ultrasonic energy from the excitation region interacts with the upper and lower surfaces, mode conversion phenomena occur between longitudinal and transverse waves causing the superposition of wave packets and generating guided wave modes. In Lamb waves, these modes can be classified into symmetric (S) and antisymmetric vibration modes (A). For an isotropic plate with traction-free boundary conditions, the resulting characteristic equation can be expressed as
where \(p^{2}=\omega ^{2}/c^{2}_{L}-k^{2}\), \(q^{2}=\omega ^{2}/c^{2}_{T}-k^{2}\), and the spatial frequency or wavenumber \(k=\omega /c_{p}\), \(\omega \) is the angular frequency, \(c_{L}\) and \(c_{T}\) are longitudinal and shear wave velocities. The roots of these equations give the phase velocity as a function of k. Here, a solution was computed by implementing a MATLAB® function using the Aluminum 6061 properties. The calculated group and phase dispersion curves for both cases are shown in Fig. 8. These curves describe multimodes and velocity dependence on frequency.
Fig. 8
Lamb wave dispersion curves for an Aluminum-6061 plate. a Group velocity, and b phase velocity
The main parameters that limit the use of guided waves for structural health monitoring and wave characterization of propagating elastic wave signals in a mechanical guide are dispersion and mode conversion phenomena as the waves encounter discontinuities. Ideally, generating a guided wave in the transducer’s full frequency bandwidth would increase the amount of information from the propagation medium, but unfortunately, the captured unfiltered signals would be too complex to be analyzed using traditional methods.
3.1 Signal Acquisition and Data Sets
The proposed method was experimentally tested using an aluminum plate with artificially made small discontinuities. The types of discontinuities employed produce a similar acoustic signature as mechanical fasteners or bonded elements found in engineering structures. In all cases, the effect of these discontinuities on the resulting propagating Lamb wave with an emitter-receiver setup is expected to be small.
Fig. 9
a Experimental setup and b description of the experimental configuration
The experimental setup for the ultrasonic tests is depicted in Fig. 9. Lamb waves were excited in an aluminum plate with a thickness of 2 mm. Excitation and tuning of Lamb waves were achieved using angle beam incidence, controlled by a variable angle acoustic wedge, primarily generating \(S_{0}\), \(A_{0}\), and \(A_{1}\) modes. The transducer was coupled to the plate with the aid of glycerin gel. The experiments utilized two commercial 1 MHz longitudinal contact transducers arranged in an emmiter-receiver configuration (refer to Fig. 9a). The excitation pulse was generated by a pulser/receiver OLYMPUS 5058-PR as shown in Fig. 9b. The signals were digitized by an oscilloscope DPO-3012 and were processed in a computer running MATLAB®. Following the theoretical dispersion curves, the ultrasonic tests were tuned to mainly excite the \(S_{0}\) and \(A_{0}\) vibration modes; however, due to the frequency bandwidth, \(A_{1}\) was also observed.
In this study, three sets of data (databases) were obtained. For the first database, two discontinuities were considered as described in Fig. 10. A simple through-hole discontinuity (labeled as CB) with a small diameter (6.3 mm) and a discontinuity created by a solid cylinder (with 6 mm in diameter) bonded to the top surface of the aluminum plate (labeled as LB) were considered. The cylinder was bonded to the surface’s plate using simple epoxy glue. The mechanism of wave interaction with these discontinuities differs, but in both cases, the signals obtained for this experiment exhibited minimal changes between test conditions. In most cases, they proved to be challenging to distinguish in the time domain from the tests carried out on the sample under the free discontinuities (SB) condition.
Thus, three classes were considered: CB, LB, and SB conditions. To introduce variability, the distance between the emitter and receiver transducers was fixed at 40 cm, but the relative position (P) between the discontinuity and the emitter was varied. Three positions were considered: P1 = 15 cm; P2 = 20 cm; and P3 = 25 cm. Five repetition tests were performed for each position (removing and repositioning the transducers). Thus, 15 tests for each test condition to a total of 45 signals were obtained.
For a second database, three small aluminum cylinders with different diameters (6, 18, and 27 mm) were bonded on the surface of the plate to form the patterns shown in Fig. 11. Four repeated measurements (removing and repositioning the cylinders) were taken for each pattern. Additionally, the horizontal direction that connects the emitter and receiver was shifted by 1 cm. By doing this, an additional class condition is added. Thus, a total of five classes, including measurements taken with free surface conditions (no cylinders), were considered.
Fig. 11
Schematic description of experimental pattern configurations using bonded cylinders to test the proposed method: a perspective view of the entire experimental setup; top view of the study cases: b with no bonded elements, c-e scenarios with one, two and three bonded cylinders respectively, and f represent the same pattern case (e) but with a 1 cm shift in the pattern position
Comparison of baseline time signal (SB) with a through hole (CB) and b with bonded area (LB); c signals after baseline (SB) subtraction. On the right column, a comparison of STFT maps for signals taken from SB, CB, and LB test conditions is shown
A final third dataset obtained from a linear array of sensors introduces additional complexity. The use of a linear array is found in applications of structural health monitoring. Unlike previous datasets where signals are captured with a single pair of transducers in a fixed direction, here, parallel rays are projected in two directions around the pattern. This setup captures the interactions of waves with different elements within the pattern. Consequently, the signals obtained for each direction exhibit significant variability. This third database of signals was obtained by projecting four wave trajectories in two directions, \(0^{\circ }\) and \(45^{\circ }\), around the pattern with the three bonded cylinders (as depicted in Fig. 14a). For each scanning direction, 16 sample signals were captured; the complete data set consisted of 32 signals, 16 taken in the \(0^{\circ }\) direction and 16 at \(45^{\circ }\).
3.2 Analysis of Signals
After careful observation of the signals, \(S_{0}\), \(A_{0}\), and \(A_{1}\) modes were identified (Fig. 12). A comparison of these signals with and without discontinuities (CB, LB versus SB) shows only very small differences in time and TF maps. That can be explained by the dimension of the discontinuity’s diameter (d) which gives a small \(d/\lambda \) ratio (0.33 and 0.89) at the frequency range of the \(S_{0}\) or \(A_{0}\) modes considered in the experiments. The faster \(S_{0}\) mode (observed in the time range of 80–120 \(\upmu \)s) has a very small time shift for CB and almost no change for the LB condition when compared with SB. On the contrary, \(A_{0}\) exhibits a more noticeable time delay and amplitude variations than \(S_{0}\). In principle, a baseline subtraction method to separate signals with discontinuities from free of discontinuities could be implemented, however, inconsistent results in identifying between types of discontinuities were found. An example is shown in Fig. 12c, after reference subtraction, the resulting signals do not exhibit a clear class separation between CB and LB conditions.
A time–frequency analysis using STFT map was implemented on these signals. From STFT of the SB condition, both \(A_{0}\) and \(S_{0}\) modes, as well as a faint \(A_{1}\) (at about 1 MHz), are identified. They closely follow the theoretical predictions (solid and dashed lines) calculated with Eqs. 12 and 13. Also, one can observe a small patch (100–120 \(\upmu \)s and below 500 kHz) associated with echoes of the waves bouncing inside the angular wedge used to excite the Lamb waves. The STFT maps for SB and CB conditions show a small reduction of amplitude, as well as a subtle redistribution of frequency content observed as a shift in the location of the patches related mainly to \(S_{0}\). These small changes are associated with the effect of the boundary conditions at the discontinuity on the propagation of the waves.
Fig. 13
Comparison of signals of the five conditions showed in Fig. 11, a time domain signals for the different conditions-no cylinders, 1, 2, and 3 cylinders coupled, and the three cylinders shifted 1 cm from the centerline. b STFT map for no cylinder coupled. c STFT map for three cylinders coupled
Signal analysis for the second set of experiments aimed to identify patterns formed by attached cylinders on the surface of the test plate (see Fig. 11). Examples of signals and STFT maps are given in Fig. 13. The time-domain traces correspond to the surface of the test plate without and with 1, 2, and 3 cylinders. Changes related to the patterns are observed. However, in both domains, it is not easy to identify patterns based on their signature.
Fig. 14
Comparison of signals and energy maps for pattern classification using two directions and four shifts: a experimental scheme, b time domain signals for the four shifts at \(0^{\circ }\) and \(45^{\circ }\), c foregrounds of the \(0^{\circ }\) direction, and d foregrounds of the \(45^{\circ }\) direction
The third set of tests considers projections of parallel rays in two directions. The captured signals exhibit amplitude changes, particularly at higher frequencies, due to attenuation of energy by the bonded cylinders pattern (Fig. 14b). It can be observed that signals from rays that do not cross the bonded areas exhibit minimal changes. Figure 14b provides a comparison of signals taken at each projection. Changes in amplitude and phase are observed, particularly in the A\(_{0}\) mode. It correlates with previous observations where this mode is more sensitive to the interaction with discontinuities. This is reflected in the STFT maps, with significant changes in the distribution of energy as a function of the wave paths. Figure 14c, d depict the foregrounds extracted from STFT maps, demonstrating how the maps follow closely the theoretical dispersion curves.
Across all analyzed datasets, the objective is to evaluate the proposed method’s ability to systematically detect and classify changes in the STFT maps according to their predefined classes.
4 Implementation of the Proposed Method
4.1 Results with the First Database
Figure 15 shows several examples of the results using background removal and foreground extraction on the \({{\textbf {A}}}^{T}\) matrix for three repeated tests of SB, CB, and LB conditions. The maps are reconstructed using only the first 2 singular values.The background maps reproduce the spectrogram of the SB condition and align with the theoretical dispersion curves. This is a consequence of the fixed distance between emitter and receiver transducers, which leads to an expectedly invariant background across signals from different structural conditions. The resulting foreground maps have small differences between conditions which can be detected by the subsequent steps of the method, and ultimately allowing for class separation.
To generate the low-rank background matrix, 36 test out of 45 samples randomly selected tests (12 from each condition) were used. The remaining nine samples were set aside for testing the SVM classifier. Figure 16 gives the data projection onto the first two principal components. Even with only the first two principal components (PC1 and PC2), there is a clear separation of the clusters for every pair of classes. Using this data, a classifier based on SVM was implemented. The resulting hyperplanes (Decision boundaries) display a large margin between the conditions SB–CB, SB–LB, and CB–LB. The classification decision boundaries were determined using the foreground data projected onto the subspace PC1–PC2. The classifier was tested with the remaining nine STFT sample maps. Background subtraction was applied to each map before its projection into the subspace. It gave in all cases a successful class identification.
Fig. 16
Results for second database. SVM was implemented between pairs of conditions a SB–CB, b SB–LB, and c CB–LB where green markers represent the test data projected on the PC subspace set as (Color figure online)
For the group of signals taken from various patterns of the cylinders attached to the surface of the plate, four repeated measurements were taken for each condition (pattern) to total 16, plus 4 additional for the free of discontinuities condition. The clusters were identified as classes: 0 (no cylinders), 1, 2, and 3 cylinders of diameter 6 mm, and a fifth class (shifted sensors) was obtained by shifting the direction of wave propagagation by 1 cm as described in Fig. 11f.
Fig. 17
Results for the database obtained with the bonded cylinders attached to the surface of the test plate, a using \(PCA(A^{T})\), and b with \(PCA(F^{T})\)
The outcomes of the method implementation on the data are given in Fig. 17. The results are compared with those obtained using PCA on \({{\textbf {A}}}^{T}\) without background subtraction. As observed, it is not possible to separate some of the projected data on the subspace PC1–PC2. On the other hand, the separation of classes through cluster formation improves considerably when using PCA on \({{\textbf {F}}}^{T}\) as described in the proposed method.
4.3 Results Using Data from Projected Signals at Two Directions
As a final experiment, we examined the method’s capability to classify information based on signals collected from two directions about the pattern location having three cylinders bonded to the test plate, as depicted in the diagram in Fig. 14a. Signals were taken at \(0^{\circ }\) and \(45^{\circ }\) with four signals at each projection obtained with parallel shifts of the transducers by \(\pm 2\) cm from the central line of the pattern. Four repetitions for each shifted position were taken. Thus, the complete dataset consisted of 32 signals, 16 from the \(0^{\circ }\) direction and 16 from the \(45^{\circ }\) direction. Out of the total 32 signals, 24 were used for training and 8 for testing.
Fig. 18
Dimensional reduction and SVM classification: PCA for signals propagation in 0\(^{\circ }\) and 45\(^{\circ }\) directions
In Fig. 18, the results depict the projection of foreground data onto the subspace of the first two principal components, displaying clear cluster separation for the two directions. An SVM classifier was successfully implemented to establish the decision boundary for the two-class problem, with all newly labeled test foreground samples correctly classified upon projection onto the subspace.
The analysis highlights the method’s sensitivity to variations in foregrounds between the two projected directions (\(0^{\circ }\) and \(45^{\circ }\)), primarily arising from the interaction of Lamb waves with the pattern formed by bonded cylinders. Remarkably, variations between projection directions are more pronounced than those resulting from shifted positions, further enhancing cluster separation.
The method demonstrated its ability to discern and classify complex patterns generated by the interaction of Lamb waves with bonded cylinders. Unlike previous experiments where signals within a class were similar due to the receiver-transmitter configuration, here, signals within a class exhibit notable differences due to their interaction with the cylinder pattern for the various ray path directions. This underscores the method’s capacity to detect correlations between different signals representing a projection condition, a property that could be utilized with an array of sensors.
In all experiments, results indicated that the proposed method for cluster formation and classification has promising applications for structural health monitoring. The ability of the method to handle Lamb wave signals interacting with single and multiple discontinuities was successfully demonstrated. Using fixed positions of the transmitter and receiver transducers, the background is invariant despite the presence of echoes from edges or other fixed sources, thus allowing identification and classification of the complex patterns considered in the experimental tests.
5 Conclusion
In this work, a method for identifying and classifying STFT maps of non-stationary dispersive guided wave signals was presented. The method was tested using Lamb wave signals propagating in a thin plate with various types of discontinuities. It employs low-rank matrix optimization by SVD to extract the sparse matrix of map’s foregrounds \({{\textbf {F}}}\) and dimensionality reduction by PCA on (\({{\textbf {F}}}^{T}\)) to facilitate identification and classification of the maps. Numerically studied with synthetic data demonstrated that this approach using \({{\textbf {F}}}^{T}\) enhances sensitivity to spatially localized changes in the maps within the considered classes. All numerical tests with synthetic maps and different levels of noise resulted in clear cluster formation according to their class.
Experimental validation involved propagating Lamb waves in a thin aluminum plate with embedded discontinuities. We employed a through-transmission setup for data acquisition. However, under the chosen conditions, the impact of discontinuities on the signals proved minimal, posing a significant challenge for time-domain methods to achieve reliable identification. The proposed method was implemented for three data sets taken from complex discontinuity patterns. The implementation of the method on the STFT maps has shown the capability for class separation in clusters. These clusters then served as the data for developing a SVM classifier, which consistently achieved excellent classification results with minimal computational expense across all datasets. The proposed method successfully identified and classified maps generated from relatively wide bandwidth signals, even with a limited number of samples. This highlights its potential for real-world structural health monitoring with practical sensor arrays and complex discontinuity patterns.
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Conflict of interest
All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
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