# Ultrasound Defect Localization in Shell Structures with Lamb Waves Using Spare Sensor Array and Orthogonal Matching Pursuit Decomposition

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Introduction of OMP

^{N}, select K atoms in the dictionary D and make a K-term approximation to y.

_{K}is the subscripts set of the K atoms, corresponding to the first K maximum coefficient of $\left|{y,a}_{i}\right|$.

#### 2.2. Overcomplete Dictionary Construction

_{i}(ω) is the scattered coefficient from the excitation source to the damage of the i-th damage; β

_{i}(ω) is the scattered coefficient from the damage to the receiver of the i-th damage, and d

_{s}

^{i}is the propagation distance of the i-th propagation path. From Equation (2), it can be seen that the signal is obtained by multiplying the atoms in the dictionary with their coefficients, so the value of the scattered coefficient α

_{i}(ω) will be reflected in the coefficient matrix θ. From this, the i-th column of the dictionary D of A or S mode can be expressed as

^{i}is the propagation distance of the i-th atom, F

^{−1}{·} represents the inverse Fourier transform. Thus, the single-mode dictionary can be obtained in the form [12]

#### 2.3. Nondispersive Dictionary

_{i}(t).

_{n}can be built.

_{m}(t-t

_{i}) is the value of the m-th sampling point of the nondispersive signal with time shift t

_{i}.

#### 2.4. OMP-Based Decomposition and Dispersion Removal Algorithm

_{s}contains r wave packets, the received signal y

_{s}can be represented by a linear combination of its atoms [18]

^{N×M}(M ≥ r) is the overcomplete dictionary, θ∈R

^{M}is the coefficient column vector, and n∈R

^{N}is the residual noise term.

_{s}

^{i}(1 ≤ i ≤ r, i∈N

^{+}) of the scattered signal is completely covered by the distance d

^{i}(1 ≤ i≤M, i∈N

^{+}) corresponding to the given atoms, then, the scattered signal can be sparsely decomposed with the dictionary D by the OMP algorithm.

- (1)
- Initialization process. Determine sparsity degree K which means the number of potential wave packets. Build an overcomplete dictionary D as:

- (2)
- Orthogonal matching. Find the column a
_{λ}in the dictionary according to the product value θ_{λ}of acquisition signal y and a_{λ}. Then, record the product value θ_{λ}, known as matching coefficients. ${a}_{\lambda}={\mathrm{argmax}}_{i=1,2\dots N}\left|\langle y,{a}_{i}\rangle \right|$, λ indicates that the atom is the λ-th column in the dictionary, and ${\theta}_{\lambda}={\mathrm{max}}_{i=1,2\dots N}\left|\langle y,{a}_{i}\rangle \right|$. - (3)
- Update and iteration. Update the solution set $\theta =\theta \cup \left\{{\theta}_{\lambda}\right\}$, and update the residual signal by subtracting the selected atoms from the signal of last iteration.

- (4)
- Termination judgment. Determine whether the number of iterations is greater than K. If it is not satisfied, execute the matching and update procedures again.

_{λ}of θ represents the scattering coefficient of the λ-th atom in the dictionary with travel distance L

_{λ}. Each scattering wave packet with a unique travel distance in the collected signal can be recovered using the above equation and the overcomplete dictionary D. The propagation distance of the scattering wave packet can be visually expressed by the travel distance of the λ-th atom. With the same column label, the collected signal can be represented as the combination of nondispersive wave packets using the nondispersive dictionary. Sparsity degree K determines the number of atoms selected from the dictionary. When the value of K is small, some wave packets may not be matched. When the value of K is large, matching performance will be better but the amount of calculation is greatly increased. Therefore, the total number of wave packets m in the signal needs to be predicted, and in general, the value range of K is 2m ≥ K ≥ m. The procedure of the proposed method is shown in Figure 2.

## 3. Methodology Verification

#### Numerical Simulation

_{c}represents the central frequency, and t is time serial ranging from zero to N/f

_{c}.

## 4. Experimental Verification

#### 4.1. Sparse Sensor Array-Based Localization Method

_{A}, y

_{A}) and (x

_{B}, y

_{B}), respectively. Therefore, the propagation distance of the primary scattered wave (as shown in Figure 8) that is reflected only by the defect can be described as

_{c},y

_{c}) for this experiment. In the process of defect localization, at least three different propagation distances need to be determined for a more accurate localization [24]. Traditionally, each propagation distance is confirmed by a pair of PZT wafers. Hence, at least three PZT wafers are needed. In addition to the direct arrival wave and the primary scattered waves, there are also secondary scattered waves reflected from the edges and defects. Figure 9a indicates the signal via path a-b, excited by the exciter A and reflected by the defect. Moreover, sensor B will also receive signal from paths c-d, e-f, and e-d, with two reflections by the defect and the edge. The latter two scattered waves are relatively weak compared with the first one, and are not considered here.

_{i}= θ

_{r}, so the scattered signal path c-d can be easily deduced out by the mirror point of sensor B, and its propagation distance is

_{c}denotes the distance of path c that is the distance from the excitation sensor A(x

_{A},y

_{A}) to the defect; ${{d}^{\prime}}_{d}$ denotes the distance from the defect to the mirror point ${B}^{\prime}\left({{x}^{\prime}}_{B},{{y}^{\prime}}_{B}\right)$

_{ab}. The propagation distance of the secondary scattered wave is

_{i}denotes the distance of path i; ${{d}^{\prime}}_{j}$ denotes the distance from the defect to the mirror point ${A}^{\prime}\left({{x}^{\prime}}_{A},{{y}^{\prime}}_{A}\right)$ of sensor A(x

_{A},y

_{A}).

#### 4.2. Experimental Setup

- (1)
- under the intact condition, excite a five-peak sinusoidal wave modulated by Hanning window with a center frequency of 100 kHz from PZT A, collect the signals from PZT B. Then, excite modulated wave from PZT B, and collect the signals from PZT B. These signals are considered to be the reference signals.
- (2)
- in the practical monitoring, repeat the acquisition step above.
- (3)
- subtract the intact signal from the acquisition signal with defects, known as the state-relative signals.
- (4)
- decompose the state-relative signals into wave packets of a single mode by the OMP-based decomposition and dispersion compensation method.
- (5)
- localize the defect by sparse sensor array-based localization method.

#### 4.3. Experimental Results

_{λ}, only the A0 mode waves are extracted, as shown in (b), because the amplitude of S0 mode wave is relatively low. Therefore, the OMP algorithm can remove the system noise well, because the predefined atomic functions are independent of the noise. Moreover, using the columns of the chosen atoms, it is easy to get the travel distance of the corresponding wave packets, which verifies the correctness of the method. Figure 12c shows the recomposed signal using the nondispersive dictionary to remove the dispersion.

## 5. Conclusions

- with the over-completed dictionaries of A0 and S0 mode, the OMP-based algorithm can separate wave packets from collected signals, even if the wave packets are overlapped. Thereafter, with the nondispersive dictionaries, the dispersion part is removed, which transforms the deformed wave packets to the original excitation signal.
- the wave packets reflected by the defect and edge are innovatively used for defect localization, which is the equivalent of mounting a virtual sensor at the mirroring position. The use of these multipath wave packets is beneficial for reducing the use of transducers.
- the dispersion-removed wave packets of multipath can localize the defect position and improve the resolution of defect localization.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Mode separation: (

**a**) Original signal; (

**b**) Separated S0 mode dispersion signal; (

**c**) Separated A0 mode dispersion signal.

**Figure 4.**Dispersion compensation: (

**a**) S0 mode signal; (

**b**) S0 dispersion removal signal; (

**c**) A0 mode signal; (

**d**) A0 dispersion removal signal.

**Figure 12.**Defect-free signal processing: (

**a**) Defect-free signal and recovery signal; (

**b**) Recovery A0 with largest matching coefficients; (

**c**) The result after dispersion removal.

**Figure 13.**Residual signal processing: (

**a**) Original residual signal and reconstructed signal; (

**b**) Reconstructed signal with largest matching coefficients; (

**c**) Reconstructed signal with dispersion removed; (

**d**) Separated wave packet and envelope of path a–b; (

**e**) Separated wave packet and envelope of path c–d; (

**f**) Original residual signal and reconstructed signal; (

**g**) Reconstructed signal with largest matching coefficients; (

**h**)Reconstructed signal with dispersion removed; (

**i**) Separated wave packet and envelope of g–h; (

**j**) Separated wave packet and envelope of i–j;.

**Figure 14.**Damage imaging: (

**a**) The location schematic of PZT; (

**b**) The imaging result with original signal; (

**c**) The imaging result with dispersion signal; (

**d**) The imaging result with non-dispersion signal.

Material | Density (kg/m^{3}) | Elastic Modulus (Pa) | Poisson’s Ratio |
---|---|---|---|

Q235 | 7800 | 2.1 × 10^{11} | 0.33 |

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**MDPI and ACS Style**

Mu, W.; Gao, Y.; Liu, G. Ultrasound Defect Localization in Shell Structures with Lamb Waves Using Spare Sensor Array and Orthogonal Matching Pursuit Decomposition. *Sensors* **2021**, *21*, 8127.
https://doi.org/10.3390/s21238127

**AMA Style**

Mu W, Gao Y, Liu G. Ultrasound Defect Localization in Shell Structures with Lamb Waves Using Spare Sensor Array and Orthogonal Matching Pursuit Decomposition. *Sensors*. 2021; 21(23):8127.
https://doi.org/10.3390/s21238127

**Chicago/Turabian Style**

Mu, Weilei, Yuqing Gao, and Guijie Liu. 2021. "Ultrasound Defect Localization in Shell Structures with Lamb Waves Using Spare Sensor Array and Orthogonal Matching Pursuit Decomposition" *Sensors* 21, no. 23: 8127.
https://doi.org/10.3390/s21238127