## 1. Introduction

Plate-like structures are widely used in a variety of fields, such as material, civil, aerospace, and other industries [1–5]. Confirming the health condition of these plate-like structures is a very meaningful assignment for engineers. One of the various non-destructive techniques, acoustic emission (AE) monitoring is arguably based on the simplest of physical concepts: the release of transient elastic waves in solid materials as a result of a localized source of damage [2–4]. The process of locating the source of these acoustic waves by capturing the related propagating acoustic signals through various sensors or arrays and properly analyzing them is commonly known as the AE source localization technique [3–8]. Common passive monitoring techniques are currently applied for AE source localization in isotropic and anisotropic structures, particularly in two-dimensional plate-like structures [9,10].

The techniques for AE source localization, such as triangulation technique, modal AE, and optimization-based technique, are currently performed by accurately determining the exact time of arrival (TOA) or time difference of arrival (TDOA) [8]. An obvious disadvantage of these methods is that the precision of the time difference measurement has a great effect on the location precision. To avoid this problem, McLaskey et al. [11] introduced a novel beamforming method of AE analysis, which is particularly suited for field applications on large plate-like reinforced concrete structures, such as walls and bridge decks. The advantage of the beamforming technique is that it does not require the TOA or TDOA of a specific wave mode; the technique can therefore handle noisy signals if the noise is a Gaussian white noise. He et al. [12] applied the beamforming technique for source localization in a thin steel plate through finite element (FE) simulation and experiment. Nakatani et al. [13] studied AE source localization on an anisotropic structure and proved that the technique is simple but effective.

Beamforming has been successfully used in source localization by many researchers, but its limitation as a technique is that it is a velocity-dependent method [11]. As such, its localization accuracy and precision are significantly influenced by the propagation characteristics. The propagation characteristics of an AE signal in plate-like structures are complex because of the signal’s repeated reflection in the interface. The wave speed is affected by many factors, such as frequencies, material properties whether isotropic or not, multi-mode, and the dispersion in the propagation process. Xiao et al. [14] proposed a novel beamforming method with two uniform linear arrays in determining the AE source location using an unknown wave speed to reduce the effect of the factors (mentioned earlier) on AE beamforming. However, the accurate two-dimensional position cannot be shown at the same time in one coordinate system, because the localization coordinates are correct along the direction parallel to the array but not along the direction perpendicular to the array.

## 2. Principle of Lamb wave propagation

The Lamb waves are used to interrogate the whole thickness of the plate-like structures. Either the surface defects or the internal damages can be detected.

### 2.1. The dispersion characteristic of Lamb wave

Since Lamb waves are waves of plane strain that occur in a free plate, only displacements through the thickness (x_{3} direction) and in the direction of wave propagation (x_{1} direction) are taken into consideration (**Figure 1**).

Eq. (1) is for the symmetric Lamb wave mode (*S*_{i}, *i* = 0,1,2,3 ⋯.) in which the particle motion of the plate occurs symmetrically about the symmetric axis of the plate. Eq. (2) is for the anti-symmetric Lamb wave mode (*A*_{i}, *i* = 0,1,2,3 ⋯.) in which the particle motion of the plate occurs asymmetrically about the symmetric axis of the plate [15,16].

where the parameters *p* and *q* are defined as follows:

Where *c*_{L} and *c*_{T} are the propagation velocities of longitudinal wave and shear wave, respectively; and wavenumber

The phase velocity dispersion curves can be solved by Eqs. (1) and (2). And, the group velocity dispersion curves can be plotted from the phase velocity dispersion curves using Eq. (3). The group velocity and phase velocity dispersion curves for an aluminum plate are shown in **Figure 2**(**a** and **b**).

### 2.2. Finite element simulation

Numerical simulation is another main method aside from experimental validation that is used in various fields. Modeling the guided waves conveniently provides modeled data with a high signal-to-noise ratio, which can be used to develop useful advanced signal processing and analysis techniques. A number of numerical methods have been applied to analyze the propagation of elastic waves, such as the finite element method (FEM), finite difference method, boundary element method, and finite strip elements (FSE), all of which have their respective merits and defects. FEM is best understood from its practical application, known as finite element analysis (FEA). FEA is a computational tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, which in turn represent different areas in the physical system; the technique also uses a software program coded with the FEM algorithm. This study simulates the guided wave signal propagation with FEA in the general three-dimensional domain using the commercial package ABAQUS, which has some advantages and is effective. It easily records the solution results of every element or node by time step and extracts results. In addition, it provides sophisticated pre- and post-processing functions and is user-friendly. Many scholars have demonstrated the possibility of using FEM to solve the guided wave formation and guided wave signal propagation in plate specimens.

#### 2.2.1. Explicit dynamic analysis

The explicit dynamics analysis procedure, ABAQUS/Explicit, is based upon the implementation of an explicit integration rule together with the use of diagonal (“lumped”) element mass matrices. The equations of motion for the body are integrated using the explicit central-difference integration rule.

Where *u* is a degree of freedom (a displacement or rotation component) and the subscript *i* refers to the increment number in an explicit dynamics step.

The explicit procedure integrates through time by using many small time increments. The central-difference operator is conditionally stable, in order to avoid numerical instability, there are more than 20 points per cycle of the highest frequency. Therefore, the time step can be expressed as follows [17]:

where *f*_{max} is the highest frequency of interest.

The size of the finite element is depended on the smallest wavelength. In reference [17], it is recommended that 20 nodes per wavelength be used. So that, it can be expressed as

where *l*_{e} is the element size and *λ*_{min} is the shortest wavelength of interest.

In the present FEM analysis,Δ *t* = 0.1 μs and *l*_{e} = 1 mm. The plate was discretized using brick elements C3D8R, and the mesh is shown in **Figure 3**. The physical properties of the aluminum plate areas follows: *E* = 71 GPa, *ν* = 0.33, *ρ* = 2700 kg/m^{3}.

#### 2.2.2. Excitation signal

In the finite element modeling, the excitation signal is a smoothed tone burst obtained by filtering a five-cycle pure tone burst of central frequency *f* = 150 kHz through a Hanning window [**Figure 4**(**a** and **b**)].The Hanning window is described by the following equation [18]:

The number of counts *(N)* in the tone bursts matches the length of the Hanning window:

The smoothed tone burst is governed by the following equation:

#### 2.2.3. Selective excitation of Lamb wave modes

Dual piezoelectric (PZT) actuators are symmetrically mounted on the top and bottom surfaces of a plate, and the frequencies of excitation signals are controlled to a range so as to generate the fundamental symmetric mode S_{0} and anti-symmetric mode A_{0} only. This mode selection technique can be expressed as:

When either of the dual PZTs is energized, both the symmetric mode S

_{0}and the anti-symmetric mode A_{0}will be generated simultaneously.When dual PZTs are in-phase energized, the symmetric mode S

_{0}with the most signal energy is generated, meanwhile, the anti-symmetric mode A_{0}is weaker.When dual PZTs are out-phase energized, the anti-symmetric mode A

_{0}is generated to dominate the majority of wave energy, and the symmetric mode S_{0}is suppressed.

In the FEM, a 1200 mm × 1200 mm aluminum plate with 4 mm thickness was used for the simulation. Because the piezoelectric element C3D8E doesn’t exist in ABAQUS/explicit; so we applied 16 self-equilibrating concentrated forces, as shown in **Figure 5**., to simulate the wave excitation for the round PZT actuators. The mode excitation method is expressed as follows:

When the 16 self-equilibrating concentrated forces are loaded on the top or the bottom of a plate, both S_{0} mode and A_{0} mode are excited simultaneously.

When the 16 self-equilibrating concentrated forces are symmetrically loaded on the top and the bottom of a plate, the mode S_{0} will be excited.

When the 16 self-equilibrating concentrated forces are anti-symmetrically loaded on the top and the bottom of a plate, the mode A_{0} will be excited. The excited Lamb wave modes are shown in **Figure 6**(**a**–**c**).

#### 2.2.4. Lamb wave scattering simulations

### 2.2.4.1. The interaction of the S_{0} mode with a circular hole

It is well known that mode conversion phenomena will occur when the incident S_{0} mode wave interacts with damages; there will be S_{0} mode, A_{0} mode, and SH0 mode waves in the scattering fields.

When the depth of the hole is through thickness, the scattering S_{0} mode and SH0 mode can be captured.

When the depth is 75% of the thickness, the scattering S_{0} mode, A_{0} mode, and SH0 mode can be captured.

The snapshots of S_{0} mode interaction with a circular hole are shown in **Figure 7** and **Figure 8**, respectively.

### 2.2.4.2. The interaction of the S_{0} mode with a crack

When the incident S_{0} mode wave interacts with a crack, the crack-induced scattering wave components are same as a circular hole. The snapshots of S_{0} mode interaction with a crack are shown in **Figure 9**.

## 3. Beamforming algorithm for damage identification

### 3.1. Principle of the beamforming network method

The beamforming technique is a method based on the delay-and-sum algorithm. **Figure 10**
shows that an acoustic event causes an AE source to generate an acoustic wave, which is received by M sensors. Given the difference between the distance of the sensors and that of the sound source, the arrival time must also differ. Although different degrees of attenuation exist on the wave energy, waveform parameters, such as frequency, duration, and the peak of the same sound source remain almost same. The beamforming technique adjusts different sensor waveform signals to the same datum point by time delay to ensure that the entire waveform signal is adjusted to the same wave front. The sensors network output of incident waves in an isotropic plate is expressed by Eq. (11). The variable
**Figure 1**, the

Here the variable *x*_{m} (*t*) represents the measured signal acquired by the *m*_{th} (*m* = 1, 2,…, *M*) sensor *S*_{m}.
*m*_{th} sensor.
*m*_{th} sensor, and *c* is the propagation velocity of the acoustic wave. Different reference points (or focal points) have different time delays
*w _{m} * may be taken as 1.

### 3.2. Hilbert transform-based signal processing

Given the propagation of the elastic wave in materials, the reflection and scattering of signals result in mode conversion, which goes against waveform processing. Research findings have shown that an energy distribution curve with pure signal superposition is not continuous, as shown in **Figure 11(a)**. Beamforming technology needs only the necessary amplitude information and does not require the full waveform information. Thus, to reduce the uncertainty, avoid the boundary reflection effect, ensure the rationality of the superposition, and obtain a smooth curve, the envelopes of the signals obtained using Hilbert transform (HT)-based signal processing *A*_{m} (*t*) (i.e., the first wave packet containing the peak information is intercepted) are used in this study instead of the signal variable *x*_{m} (*t*). The parameter *B* given in Eq. (15) attains the maximum value when the reference point coincides with the AE source point. The HT series given in Eqs. (13) and (14) have the same amplitude and frequency content as the original data and include phase information, which depends on the phase of the original data, as shown in **Figure 11(b)**. The same source results demonstrate that the HT-based signal processing *A*_{m} (*t*) is reasonable and capable of obtaining the energy distribution information of the AE source, as shown in **Figure 11(c)**.

### 3.3. Probability-based diagnostic imaging

Given the existence of errors and uncertainty, the real location of the sound source is likely to be where the energy is larger (near the maximum). Thus, a probability diagnostic imaging technology is introduced to explain the results from the perspective of probability theory. After determining the energy distribution, the structure under detection is meshed using L × K nodes for probability-based diagnostic imaging; this technique is an emerging technique for monitoring the health status of a structure and identifying damages and has attracted much attention in recent years [19, 20]. It uses an easily interpretable image whose pixels exclusively correspond to the spatial points to describe the health status of the structure. The field value of a pixel stands for the probability of the presence of damage at the point of the structure corresponding to the pixel, and the degree of probability is often calibrated through different gray scales. In this study, the normalized energy distribution can be regarded as the probability of the AE source location. When the normalized energy distribution focuses on a peak under the correct circumstances, it is similar to the Gaussian distribution. Thus, a cumulative distribution function, *F* (*z*), is introduced to facilitate this procedure of defining the field value asfollows:

where *f (z)* is the Gaussian distribution function, which represents the probability density of the presence of the AE source at the mesh node (*x*_{i}, *y*_{j})(*I* = 1, 2, …, *L; j* = 1, 2, …, *K*). The variable *z* is the distance of the arbitrary node (*x*_{i}, *y*_{j}) to the node (*x*_{p}, *y*_{p}), whose peak point is in the normalized energy distribution, and *σ* is the standard variance. Thus, the field value at node (*x*_{i}, *y*_{j}), which is the probability of the AE source *I* (*x*_{i}, *y*_{j}) is

The normalized energy and probability distributions of AE localization results are shown in **Figure 11(d)**.

## 4. FE model of AE simulation

A sketch of the FE model with the geometric configuration 800 mm × 600 mm × 4 mm is shown in **Figure 12**. The homogeneous isotropic aluminum plate is characterized by the following: Young’s modulus *E* = 71 GPa, Poisson ratio *v* = 0.33, and mass density ρ = 2700 Kg/m^{3}. In this model, damping is not considered, the boundaries are free, and a total of eight sensors (S1–S8) form a rectangular network. The AE source position, datum point, and their location are marked, and the three AE sources are named as cases A, B, and C. All the cases are considered both away from and close to the boundary. All of the sensors are decorated (geometrically) evenly. A Hanning-windowed five-cycle sinusoidal tone burst is used as the transient excitation to simulate the AE wave. The frequency of the excitation signal is 400 KHz. A same off-plane displacement with a peak of 0.1 μ m is loaded on multiple cases, and the time histories of the samples of the sensor nodes are captured. The element size and integration time step in FEA process are designated as 2 mm and 0.1 μs, respectively, to avoid numerical instability [19].

The AE wave is a common form of the Lamb wave propagation in plate-like structures. In most cases, multiple Lamb modes exist simultaneously, and the dispersive properties of such modes are not identical even for the same mode in different frequency ranges [21]. Different mode waves travel with different groups and phase velocities. As a consequence, source localization strategies must be conducted based on a certain wave propagation mode. The fundamental symmetric mode S_{0} wave and the fundamental asymmetric mode A_{0} wave are popularly used in plate-like structures. The localization results are presented using S_{0} mode wave, given that the propagation velocity of the S_{0} mode is faster than that of the A_{0} mode and considering the interception of the first wave packet analyzed in this study. The dispersion curve in **Figure 13**
shows that the propagation speed of the S_{0} mode Lamb wave with a frequency of 400 KHz in 4 mm unconstrained aluminum plate is about 4600 m/s.

### 4.1. Localization results of the numerical simulation

To verify the validity of the positioning, three cases of real sound sources were tested using the delay-and-sum algorithm with the precise wave speed. Their real coordinates were as follows: case A (360 mm, 250 mm), case B (750 mm, 550 mm), and case C (600 mm, 50 mm). The localization results for the rectangular sensor network are exhibited in **Table 1**.

The results show that under the true known wave speed, the beamforming array technique and probability-based diagnostic imaging method can confirm the location of the AE source and produce accurate positioning results, as highlighted in the floor plan image. The case away from boundaries can evidently work much better than the cases close to boundaries, given the same wave speed. The results also prove that HT-based signal processing and probability-based diagnostic imaging are reasonable and effective.

### 4.2. The influence of the wave speed

The localization results in **Table 1**
show that the beamforming technique can effectively confirm the location of the AE source and produce accurate positioning results, as highlighted in the floor plan image. However, it is a wave-velocity-dependent method; a clear limiting factor of this technique is the need to obtain knowledge on its wave propagation speed. For common, uniform, and thick plate materials, we generally find out the dispersion curve to determine the required wave velocity, following the theory of wave propagation. However, given the structure size, material properties, or performance degradation factors, ensuring the dispersive curves of the Lamb wave is not easy, so the sound source localization of the plate structure in the case of unknown wave velocity must be discussed.

According to beamforming theory, the largest impact of wave velocity lies on the length of delay. Under a real wave velocity, signals can be achieved through accurate time delay adjustment to the same wave front, the superposition of energy concentrated, and the maximum output reached. In calculating a wave velocity that is far from the true wave velocity, the energy is unable to concentrate and the peak isn’t the maximum. This finding can be proven by **Figure 14**.

The larger the deviation from the real wave velocity, the higher the divergence of the energy; the smaller the deviation from the real wave velocity, the more concentrated the stack power. The normalized energy peak curves of the FE model under different wave speeds are shown in **Figure 15**. When the wave speed is 4700 m/s, the peak of the energy is at its maximum, so that the real wave velocity is around 4700 m/s. At this rate, the localization can also be confirmed by comparing the energy distribution or peak value under the different wave speeds, with no need for the known real wave velocity.

## 5. Experimental verification

To further verify the effectiveness of AE source localization using beamforming network technology for plate-like structures, a pencil-lead-broken experiment on an aluminum plate and a steel cylinder experiment are implemented.

### 5.1. Aluminum plate experiment

The size of the aluminum plate adopted in the experiment was 1000 mm × 1000 mm × 4 mm. It was a homogeneous isotropic aluminum plate with the following characteristics: Young’s modulus *E* = 71 GPa, Poisson ratio *v* = 0.33, and mass density *ρ* = 2700 Kg/m^{3}. A rectangular network with eight AE sensors was built, with each of the sensors placed 100 mm away from the closed boundary. An AE acquisition system from Physical Acoustics Corporation was applied in the experiment, and the sampling frequency used was 1 M/s. The pencil-lead-broken experiment was used to simulate the AE source (**Figure 16**). Multiple cases of the source location were set in Cases A (500 mm, 500 mm), B (200 mm, 500 mm), and C (700 mm, 300 mm).

According to the chapter 3.１ mentioned method, the normalized energy peak curves of experimental aluminum plate under different wave speeds were shown in **Figure 17**. The AE propagation speed was about 5200 m/s. The localization results for the rectangular sensor network are exhibited in **Table 2**.

### 5.2. Thin-walled steel cylinder experiment

For a thin-walled cylinder with a large radius, the propagation characteristics of the Lamb wave are the same as those of plate-like structure [22]. Thus, this study constructed a sensor network in conformity to a plate on a steel thin-walled cylinder to verify the effectiveness of the beamforming method for AE source localization. The cylinder had an inner radius, external radius, and height of 102, 110, and 700 mm, respectively, and was characterized by the following: Young’s modulus *E* = 209 GPa, Poisson ratio *v* = 0.3, and mass density *ρ* = 7800 Kg/m^{3}. The sensors were arranged, such that the circles with the heights of 100 and 600 mm were stepped by 90° (**Figure 18**). The chosen generatrix was 0°, and the X-axis was transformed into a degree measure. The AE source position was designed at (90°, 350 mm); according to the normalized energy peak curves of the 8 mm steel cylinder in **Figure 19**, the wave velocity was 4900 m/s. Given the strong symmetry of the cylinder, some of the sensors could be used more than once, so that beamforming network algorithm was achieved at a step of 90°. The localization result and energy and probability distribution are exhibited in **Table 3**. It was verified that the algorithm can be applied to AE source localization for thin-walled steel cylinders.

## 6. Conclusion

This study investigates AE source localization in plate-like structures using the beamforming network method and probability-based diagnostic imaging. Some conclusions based on the analysis of the localization results are drawn below.

(1) The beamforming technique is effective for AE source localization without the TOA or TDOA. Unlike other localization techniques, it is simple because it avoids the acquirement of detailed information on signals. Through the beamforming network algorithm, the AE source location information is highlighted in one coordinate system as energy distribution.

(2) An HT-based signal processing method was developed. The testing results demonstrate that the HT-based signal processing method is reasonable and capable of obtaining the energy distribution information of the AE source. Moreover, a probability diagnostic imaging technology is introduced to improve localization accuracy. The field value of a pixel stands for the probability of the presence of damage at the point of the structure corresponding to the pixel, and the degree of probability is often calibrated through different gray scales.

(3) When the wave speed is known, the beamforming technique can accurately determine the source localization, no matter how far away from or close it is to the boundaries of the structure. Even if the wave velocity is unknown, the localization can also be confirmed by comparing the energy distribution or peak values under different wave speeds, as proven accurately in this study.

(4) The algorithm can also be applied to AE source localization in thin-walled steel cylinders pipe-like structure. However, its applicability in long axial length structures is yet to be verified.