# On the damage detection of a metallic beam based on the time reversal principle

Mélissandre Huguet, Alireza Ture Savadkoohi, Claude-Henri Lamarque, Manuel Collet

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Mélissandre Huguet, Alireza Ture Savadkoohi, Claude-Henri Lamarque, Manuel Collet. On the damage detection of a metallic beam based on the time reversal principle. Archives of civil and mechanical engineering, 2025, 25, pp.84. 10.1007/s43452-025-01140-7. hal-04931168

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# On the damage detection of a metallic beam based on the time reversal principle

Mlissandre HUGUET¹*, Alireza TURE SAVADKOOHI¹, Claude-Henri LAMARQUE¹, Manuel COLLET²

¹*Univ Lyon, ENTPE, Ecole Centrale de Lyon, CNRS, LTDS, UMR5513, Vaulx-en-Velin, 69518, France.

²Univ Lyon, CNRS, Ecole Centrale de Lyon, ENTPE, LTDS, UMR5513, Ecully, 69130, France.

*Corresponding author(s). E-mail(s): melissandre.huguet@entpe.fr;

## Abstract

The time reversal principle is exploited for damage detection of a homogeneous metallic beam, representative of a structural element, at its different health states. Then, a damage index is used for quantifying detected damages. It is shown that even with a limited number of transducers and restricted ultrasonic frequencies e.g. around 30 kHz, the damages are detectable. This study, is a primary step for monitoring civil engineering structural elements such as bridge cables for detection of their damages at their early developments.

**Keywords:** Time reversal technique, Ultrasonic, Damage detection, Beam

## 1 Introduction

Damage detection and its quantification is an important issue in civil engineering structures such as bridges [1], to ensure their health state and the security of their users. The detection of pathology of structural systems, such as cracks [2] or changes in the dynamic behavior [3], can be carried out via visual inspections for spotting global problems or visible faults [4]. However, for identifying the presence of non visible anomalies in structural elements, other non-destructive techniques [5] should be exploited. A non exhaustive list of such methods are vibration-based [6], acoustic [7], ultrasonic [8], magnetic field [9], radiography and tomography [10] or the ones combined with ultrasonic imaging [11] and visionbased [12] techniques. Most of damage
detection methods are based on observation of changes in physical, mechanical and/or dynamical properties such as eigenfrequencies, eigenmodes or damping [13].

In the domain of civil engineering, one of the challenges is the heterogeneity of materials and their complex geometries. To this end, acousto-ultrasonic methods [14], which can detect small sized damages due to exploitation of properties of short wavelengths, are used. One of the developed techniques in this category is the Time Reversal Process (TRP). The formalized and basic principles of the TRP [15], with first experimental results [16] and the theory of the time reversal cavity [17] were developed in early 1990th. This method has been widely used in aerospace [18], medicine [19] and communication [20] domains for detection of sources, e.g. impact location [21], objects [22], damages [23], or for medical imaging [24]. Meanwhile, the TRP has not been exploited a lot in the domain of civil engineering. However, for some materials such as concrete, the damage should be close to the sensors due to attenuation effect. The TRP also profits from the use of Lamb waves [25], which are easily generated and measured by piezoelectric transducers. These waves are able to interrogate large and complex structures quickly, but they suffer from dispersion effects. Moreover, the TRP improves the sensitivity of classical ultrasonic methods to damage effects as well as signal-to-noise ratios [26]. At its early developments, it was not directly usable for damage detection but the methodology was improved by a pulse-echo mode in order to estimate the location and size of defects in plates [27]. The disadvantage of the pulse-echo mode is that it needs a large number of transducers accompanied by suitable equipments for emissions and acquisitions. In Structural Health Monitoring (SHM), the TRP has been applied numerically and experimentally for different types of elements. For example, Granger et al.[28] have applied this technique for following the state of a cementitious beam via monitoring the propagation of a crack in the beam and following the self healing of the submerged material by placing a sensor near the crack. The task was carried out by measuring the evolution of the energy of reconstructed signals. However, cementitious materials have low attenuation effects with respect to concrete. Olivera et al. [29] and Wei et al [30] endowed an active monitoring system in order to locate and to quantify cracks of an underwater concrete beam. They used the water as a propagation medium to convert the crack into an acoustic source. Sohn et al [31] applied the TRP on a composite plate. They have spotted that the time reversibility can be violated when nonlinearities are generated by a defect along a direct wave path, which can be also an indirect indication of existence of damages. Park et al. [32] investigated on a thin composite plate in order to develop a reference-free diagnostic technique based on TRP for damage identification. Later on, Gangadharan et al. [33] used the TRP to detect closely spaced damages near the edge of a metal plate. They observed that the shape of the reconstructed signals would not be affected if the time reversibility principle is not violated. They have also shown that the use of broadband excitation improves the separation of reflections of defects.

In this paper, a simple methodology for detection of damages on a metal beam, representative of a structural element, is presented. Some analytical articulations for showing feasibility of the time reversal method are presented which are followed by some experimental results based on application of the TRP. The paper is organized as it follows: in section 2, a summary of the time reversal method and its working
process are presented. In section 3, an experimental protocol is provided for damage
detection of a metallic beam. The section is followed by presenting several experimen-
tal results and evaluation of damage indices. Finally, some conclusions and discussions
are provided in section 4

2 Time reversal method

In this section, some detailed information about the TRP and its applications are
provided.

## 2.1 The overview of the methodology

The main idea of the TRP as a Non-Destructive Technique (NDT) is to detect a defect.
Here, the input and reconstructed signals collected from transducers are compared for
detecting possible damages in the system [34].

The different steps of TRP are schematically illustrated in Fig.1. First of all, a signal is emitted from an array of transducers at location A. It propagates through the medium and is recorded by another array of transducers at location B. Next, the recorded signal is time reversed, i.e. the end of the signal becomes the beginning, then it is re-emitted from B to A. Finally, the signal recorded in A, also called as *reconstructed signal*, is compared with the original input signal.

<figure><img src="image_2.png" /><figcaption>Fig. 1 Schematic representation of the TRP. The transducers at locations A and B are respectively represented by PZT A and PZT B.</figcaption></figure>

## 2.2 Expression of the reconstructed signal and the damage detection

2.2.1 The response of the structure

Let us set M, C and K as the mass, damping and stiffness matrices of the structures.
They can be written as the sum of each element matrix of the structures:
$$ M = \sum_{i=1}^{n} M^i \qquad C = \sum_{i=1}^{n} C^i \qquad K = \sum_{i=1}^{n} K^i \qquad (1) $$

where n is the number of elements of the structure.

In the case of damaged structure, a variation of mass, damping and stiffness can appear. We express matrices of the damaged structure as it follows:

$$ M_D = \sum_{i=1}^{n} M^i + \delta M^i \qquad C_D = \sum_{i=1}^{n} C^i + \delta C^i \qquad K_D = \sum_{i=1}^{n} K^i + \delta K^i \qquad (2) $$

Where for $i \in [1 : n]$:

$$ \|M^i\| \gg \|\delta M^i\|, \quad \|C^i\| \gg \|\delta C^i\|, \quad \|K^i\| \gg \|\delta K^i\| \qquad (3) $$

And $\|\cdot\|$ stands for the euclidean norm.

For a healthy structure, the equation of motion in frequency domain reads:

$$ (-\omega^2 M + j\omega C + K)X(\omega) = F(\omega) \qquad (4) $$

where $X$ and $F$ are the Fourier transform of displacement (as output) and external excitation (as input), while $\omega$ stands for the angular frequency and $j^2 = -1$.

The transfer function $H(\omega)$ of the healthy structure is expressed as:

$$ X(\omega) = H(\omega)F(\omega) \qquad (5) $$

$$ \text{with: } H(\omega) = (-\omega^2 M + j\omega C + K)^{-1} $$

In a similar way, for a damaged structure the equation becomes:

$$ (-\omega^2 M_D + j\omega C_D + K_D)X_D(\omega) = F(\omega) \qquad (6) $$

Which can also be written as:

$$ \begin{aligned} & \left[-\omega^2\left(\sum_{i=1}^{n} M^i + \delta M^i\right) + j\omega\left(\sum_{i=1}^{n} C^i + \delta C^i\right) + \left(\sum_{i=1}^{n} K^i + \delta K^i\right)\right](X + \delta X)(\omega) = F(\omega) \\ & \left[(-\omega^2 M + j\omega C + K) + (-\omega^2\sum_{i=1}^{n} \delta M^i + j\omega\sum_{i=1}^{n} \delta C^i + \sum_{i=1}^{n} \delta K^i)\right] \\ & \times (X(\omega) + \delta X(\omega)) = F(\omega) \end{aligned} \qquad (7) $$

Where $\delta X(\omega)$ stands for perturbation of the output in frequency domain due to variations of physical and mechanical properties which are defined in Eq. 2. By considering
Eq.5 we obtain:

$$ \begin{gathered} \left[ \mathbf{H}^{-1}(\omega) + \delta \mathbf{H}^{-1}(\omega) \right] \left( \mathbf{X}(\omega) + \delta \mathbf{X}(\omega) \right) = \mathbf{F}(\omega) \tag{8} \\ \text{with: } \delta \mathbf{H}(\omega) = \left( -\omega^2 \sum_{i=1}^{n} \delta \mathbf{M}^i + j\omega \sum_{i=1}^{n} \delta \mathbf{C}^i + \sum_{i=1}^{n} \delta \mathbf{K}^i \right)^{-1} \end{gathered} $$

Considering the first order terms and neglecting higher order ones, Eq.8 yields to:

$$ \begin{aligned} \delta \mathbf{X}(\omega) &= -\mathbf{H}(\delta \mathbf{H})^{-1} \mathbf{H} \mathbf{F} \\ &= -\mathbf{H} \mathbf{S}_D \mathbf{H} \mathbf{F} \end{aligned} \tag{9} $$

with: $\mathbf{S}_D = \delta \mathbf{H}^{-1}$

Equation 9 shows that the perturbation of the output due to the damage can be evaluated from the transfer function of the healthy structure and the perturbed inverse transfer function $\mathbf{S}_D$ which is generated by the perturbation of physical and mechanical properties of the system due to the damage. Finally, the response of the damaged structure can be written as:

$$ \begin{aligned} \mathbf{X}_D(\omega) &= \mathbf{X}(\omega) + \delta \mathbf{X}(\omega) \\ &= \mathbf{H}(\omega) \mathbf{F}(\omega) - \mathbf{H}(\omega) \mathbf{S}_D(\omega) \mathbf{H}(\omega) \mathbf{F}(\omega) \end{aligned} \tag{10} $$

Equation 10 will be endowed in the next subsection for showing application of the TRP in a considered academic configuration.

### 2.2.2 The application of the TRP for damage detection

Let us apply the explained process in section 2.1 for a particular case. We suppose that we have two sets of transducers placed in A and B as depicted in Fig.2.

An external excitation $\mathbf{u}_A(t)$, defined as a Ricker wavelet, is applied at point A. The response $\mathbf{u}_B(t)$ is measured at point B, see Fig.1. According to Eq.10, the response in the frequency domain can be expressed as:

$$ \mathbf{U}_B = \mathbf{H}_{BA} \mathbf{U}_A - \mathbf{H}_{BA} \mathbf{S}_{D_{BA}} \mathbf{H}_{BA} \mathbf{U}_A \tag{11} $$

where $\mathbf{U}_A$ and $\mathbf{U}_B$ are the Fourier transforms of $\mathbf{u}_A$ and $\mathbf{u}_B$ and $(.)_{\alpha\beta}$ ($\alpha,\beta \to A,B$) stands for evaluated function ($\mathbf{H}$ or $\mathbf{S}_D$) based on input at point $\beta$ and output at point $\alpha$. Note that to facilitate the reading, we have omitted the frequency dependency $\omega$ in all variables. This equation shows that the presence of the damage affects the measured response at B (i.e. $\mathbf{U}_B$) due to the emission at A (i.e. $\mathbf{U}_A$). These effects intervene through $\mathbf{S}_{D_{BA}}$ in Eq.11.
<figure><img src="image_3.png" /><figcaption>Fig. 2 Simplified scheme of influence of damage on propagation.</figcaption></figure>

To apply the TRP, the response $\mathbf{u}_B(t)$ should be time reversed. This operation in the frequency domain is equivalent to taking its complex conjugate<fnref n="1" /> $\mathbf{U}_B^*(\omega)$. Then, the signal $\mathbf{U}_B^*$ is emitted from B to A. The measured response in A (i.e. $\mathbf{U}_{A,resp}$) is given by:

$$ 
\begin{aligned}
\mathbf{U}_{A,resp} &= \mathbf{H}_{AB} \mathbf{U}_B^* - \mathbf{H}_{AB} \mathbf{S}_{D_{AB}} \mathbf{H}_{AB} \mathbf{U}_B^* \\
&= \mathbf{H}_{AB} \left[ \mathbf{H}_{BA} \mathbf{U}_A - \mathbf{H}_{BA} \mathbf{S}_{D_{BA}} \mathbf{H}_{BA} \mathbf{U}_A \right]^* \\
&\quad - \mathbf{H}_{AB} \mathbf{S}_{D_{AB}} \mathbf{H}_{AB} \left[ \mathbf{H}_{BA} \mathbf{U}_A - \mathbf{H}_{BA} \mathbf{S}_{D_{BA}} \mathbf{H}_{BA} \mathbf{U}_A \right]^* \\
&= \underbrace{\mathbf{H}_{AB} \mathbf{H}_{BA}^* \mathbf{U}_A^*}_{(I)} \\
&\quad - \underbrace{\mathbf{H}_{AB} \mathbf{H}_{BA}^* \mathbf{S}_{D_{BA}}^* \mathbf{H}_{BA}^* \mathbf{U}_A^*}_{(II)} \\
&\quad - \underbrace{\mathbf{H}_{AB} \mathbf{S}_{D_{AB}} \mathbf{H}_{AB} \mathbf{H}_{BA}^* \mathbf{U}_A^*}_{(III)} \\
&\quad + \underbrace{\mathbf{H}_{AB} \mathbf{S}_{D_{AB}} \mathbf{H}_{AB} \mathbf{H}_{BA}^* \mathbf{S}_{D_{BA}}^* \mathbf{H}_{BA}^* \mathbf{U}_A^*}_{(IV)}
\end{aligned}
\quad (12) $$

Where $(\cdot)^*$ stands for complex conjugate of the matrix.

This equation shows different manners of propagation of the wave from the first input to the last response. Let us clarify different segments of Eq. 12:

* (I) corresponds to a forward and backward propagation from A to B without being affected by the damage.

* (II) corresponds to a forward propagation from A to B with intervention of the damage and a backward propagation from B to A without involvement of effects of the damage.

<footnote n="1">¹e.g. for a time harmonic wave $\psi(x, t) = e^{-j\omega t}h(x)$ it is easy to see that complex conjugation is equivalent to expressing $t \to -t$</footnote>
• (III) corresponds to a forward (from A to B) and a backward (from B to A) propagation respectively without and with intervention of effects of the damage.

• (IV) corresponds to a forward and backward propagation both with influence of the damage.

The four manners of the propagation, detailed in Eq.12, are illustrated schemati-
cally in Fig.3. Moreover, Eq.12 highlights the time delays/advances created by the
presence of the damage and the use of the TRP which can be seen in complex con-
jugates of the transfer (H) and the inverse transfer (S<sub>D</sub>) functions. It states that if
the time reversal operation is well applied, i.e. if an appropriate time window of the
recorded signal is registered and re-emitted, then the reconstructed signal will contain
all necessary information for identifying the damage [25].

<figure><img src="image_4.png" /><figcaption>**Fig. 3** Scheme of the four manners of propagation visible in Eq.12</figcaption></figure>

For undamaged element of structure we will have S<sub>D</sub> = 0, so only the first term
(I) of Eq.12 will intervene on the reconstructed signal. On the contrary if there is a
damage in the system, all terms of Eq.12 are involved. Moreover, Eq. 12 highlights
that during the TRP, the reconstructed signal in a point will be scaled version of
the original input at the same point. In the presence of damage, it will also contain
some side-bands on both sides due to delay and advance time of the three other terms
(II),(III) and (IV). If the TRP is well applied, i.e. if all terms of Eq.12 are recorded,
the two side-bands are symmetric compared to the main one [31]. The schematic representation of reconstructed signals after an input excitation, such as Ricker wavelet, for healthy and damaged systems are presented in Fig.4.

In this section, the theoretical application of the TRP in damage detection has been presented via articulations and decomposition of different segments of the governing equation of the system in the frequency domain. These developments permit to theoretically understand effects of the damage on the reconstructed data on a point based. In the following, the explained principle in Section 2.1 will be exploited for experimentally application of the TRP for damage identification of a simple beam.

<figure><img src="image_5.png" /><figcaption>Fig. 4 Comparison between input and reconstructed waves for healthy and damaged states.</figcaption></figure>

# 3 Experimental approach

In this section, the experimental test set-up for damage detection of a metallic beam will be presented which will be followed by presenting some results.

## 3.1 The experiment set-up

Two steel beams of dimensions 700 × 50 × 5mm which can be seen in Fig.5 are considered. Both of them contains three equal holes of 7mm diameter on the right side. They are also equipped with four piezoelectric transducers of dimensions 5 × 50mm stuck on the surface of the beam. To reduce the electrical noise an insulating tape under the transducers is included. One of the beams, named as the damaged beam, contains a notch of 4mm width and 2mm depth all along the whole lateral length of the beam, i.e. 50 mm (see Fig.5). This notch represents the damage on the beam.

To perform the experiment, the illustrated set-up in Fig.6 is used. First, we create a Ricker wavelet on LabVIEW via following equation:

$$f(t) = \left(1 - 2\left(\pi f_c \left(t - 1.25/f_c\right)\right)^2\right) \cdot \exp\left(-\left(\pi f_c \left(t - 1.25/f_c\right)\right)^2\right) \quad (13)$$

where $f_c$ is the central frequency of the signal fixed to 30kHz. The signal is then created by a function generator (Agilent 33120A) and is amplified by an amplifier
<figure><img src="image_6.png" /><figcaption>Fig. 5 The test specimens: a healthy and a damaged (with a notch) steel beams.</figcaption></figure>

(TREK PZD350A M/S). Then, the treated signal is sent to the input transducer. The pulse is repeated 12 times during 4 seconds. Representation of the input for emission of 5 pulses is illustrated in Fig.7. The signal on another transducer is recorded by an acquisition system (VR9500 Revolution) set to a sampling frequency of 100kHz. Then, following procedure is applied for treatment of the received signal and its re-emission. The treatment consists of performing an average of the 12 pulses received on the transducer, removing the part of the signal after the first wavefront and then to time reverse it. Finally, the treated signal is emitted from the last transducer to the first one for five times and then all records are again averaged.

## 3.2 Damage detection

The process described in section 3.1, is repeated between several pairs of transducers. We name each transducer by a number which are depicted in Fig.8. The cut average of the first salvo of emission from PZT1 and also the recorded signals from the other transducers are presented in Fig.9. Then, the recorded data are time reversed and are sent for the backward propagation. Figure 10 shows reconstructed signals on PZT1 for the healthy beam based on emission of time-reversed signals from PZT2, PZT3 and PZT4. It can be seen that regardless of the amplitude, the shape of the input signal is almost completely reconstructed (in a symmetric manner due to time reversal principle). The same process is carried out for the damaged beam, where its results are presented in Fig.11. As previous case, the shape of the original input signal is reconstructed. Moreover, from Fig.11,b-d), one can spot the emergence of some “time shifted peaks” in the reconstructed signal. These reveal the presence of a damage or a boundary. Especially by comparing Fig.10,c) and Fig.11,c) one can observe some peaks at 0.65ms and 1.2ms on the obtained data of the damaged specimen which are not present on those of the healthy beam. The same peaks are visible in Fig.11b) at the same instants which is coherent with the fact that PZT2 and PZT3 are close each
<figure><img src="image_7.png" /><figcaption>Fig. 6 The experimental set-up.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7 The schematic representation of the emitted signal (in red), the recorded signal (in blue).</figcaption></figure>

other. As these peaks are present only for the damaged beam, it indicates that they represent the notch.

## 3.3 The damage index

Let us consider the explained experimental protocol in section 3.2 for a forward propagation from PZT1 and PZT3 on both healthy and damaged beams. Figures 12 and 13 compare reconstructed signals on PZT1 for healthy and damaged beam. It can be seen that for reconstructed signals on PZT1 based on emission of time-reversed signals from certain transducers, some peaks are present, symmetrically to the reconstructed input peak, for the damaged beam. These peaks do not exist in reconstructed data for the
<figure><img src="image_9.png" /><figcaption>Fig. 8 The scheme of the beam and attributed numbers for each transducer.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9 Average of pulses recorded for forward propagation from PZT1 to PZT 2,3 and 4.</figcaption></figure>

healthy beam. These peaks are highlighted on the same figures by arrows. However, some configurations of transducers do not present significant difference, for example the reconstructed signal on PZT1 for a backward propagation from PZT4. This means that for some characteristics of excitation, i.e. its frequency and bandwidth, the defect is undetectable. To quantify the deviation, a damage index (DI) is evaluated between the reconstructed signals on healthy and damaged beams which reads as:

$$DI_{RMSE} = \sqrt{\frac{\sum_{i=1}^{N} (x_i - y_i)^2}{\sum_{i=1}^{N} (x_i)^2}} \quad (14)$$

where x and y stand for the reconstructed signals on the healthy and the damaged beams, respectively. The number of points in the signal is shown by N and RMSE represents the root mean squared error. The DI<sub>RMSE</sub> close to 1, indicates the highly damaged beam while the values near 0, are signatures of less damaged configurations. The values of the DI<sub>RMSE</sub> obtained from reconstructed data from different transducers are presented in Table 1. It can be seen that its values are always higher than 0.5 and it averages 0.7634 which reveals the presence of a damage.
<figure><img src="image_11.png" /><figcaption>**Fig. 10** Original and reconstructed signals on PZT1 for the healthy beam based on emission of time reversed signals from other transducers: a) The original input signal from PZ1 (before the TRP), b) The reconstructed signal from PZT2, c) The reconstructed signal from PZT3, d) The reconstructed signal from PZT4.</figcaption></figure>

<table><caption>**Table 1** Damage index value for different couple of transducers</caption><thead><tr><th>Signal reconstructed on</th><th>Backward propagation from</th><th>DI<sub>RMSE</sub></th></tr></thead><tbody><tr><td rowspan="3">PZT1</td><td>PZT2</td><td>0.8814</td></tr><tr><td>PZT3</td><td>0.8039</td></tr><tr><td>PZT4</td><td>0.6997</td></tr><tr><td rowspan="3">PZT3</td><td>PZT1</td><td>0.9170</td></tr><tr><td>PZT2</td><td>0.7006</td></tr><tr><td>PZT4</td><td>0.5775</td></tr></tbody></table>

# 4 Conclusions and discussions

An experimental study for detection of faults in a homogeneous beam has been presented which endows the principle of time reversibility of the wave equation in a lossless medium. For this study, several pairs of piezoelectric transducers have been exploited. It is shown that via presented simple test set-up and exploited methodology, the damages of the beam can be detected rapidly, even by a single pair of transducers. A simple damage index, based on reconstructed signals on healthy and damaged beams is introduced for representing the severity of the damage on the system. For more precise results, more transducers can be used [35] accompanied by higher frequency excitation. In this paper the damage was supposed to be on the whole length of the surface of the beam. One of perspectives of the current work is to consider partial damages across the beam as numerically was shown by Huguet et al. [35]. Other
<figure><img src="image_12.png" /><figcaption>**Fig. 11** Original and reconstructed signals on PZT1 for the damaged beam based on emission of time reversed signals from other transducers: a) The original input signal from PZ1 (before the TRP), b) The reconstructed signal from PZT2, c) The reconstructed signal from PZT3, d) The reconstructed signal from PZT4.</figcaption></figure>

outlook of this work is to exploit and to adapt the explained methodology, e.g. applying high ultrasonic frequency emissions, for damage detection of other types of civil engineering elements such as cables inside the anchorage.

## Acknowledgement

The authors would like to thank the following organizations for supporting this research: (i) The Ministère de la Transition Écologique and (ii) LABEX CELYA (ANR-10-LABX-0060) of the Université de Lyon within the program "Investissement d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR)

## Declaration

### Compliance with Ethical Standards:

The authors declare that they have no conflict of interest.

## Data availability statement

The raw measured data presented in this paper can be made available upon request to the corresponding author.
<figure><img src="image_13.png" /><figcaption>Fig. 12 Reconstructed signals on PZT1 for healthy and damaged beams, backward propagation from: a) PZT2 b) PZT3, c) PZT4.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 13** Reconstructed signals on PZT3 for healthy and damaged beams, backward propagation from: a) PZT1 b) PZT2, c) PZT4.</figcaption></figure>
References

[1] Karakostas, C., Quaranta, G., Chatzi, E., Zlfikar, A.C., etindemir, O., De Roeck, G., Dhler, M., Limongelli, M.P., Lombaert, G., Apaydn, N.M., Pakrashi, V., Papadimitriou, C., Yeilyurt, A.: Seismic assessment of bridges through structural health monitoring: a state-of-the-art review. Bulletin of Earthquake Engineering (2023)

[2] Morgese, M., Wang, C., Taylor, T., Etemadi, M., Ansari, F.: Distributed detection and quantification of cracks in operating large bridges. Journal of Bridge Engineering 29(1), 04023101 (2024)

[3] Belmokhtar, M., Schmidt, F., Ture Savadkoohi, A., Chevalier, C.: New approach to monitor bridge piers subjected to scour using rocking vibrations: theoretical and experimental identification of two vibration modes. Journal of Civil Structural Health Monitoring (2024)

[4] Godart, B.: Pathology, assessment and treatment of structures affected by delayed ettringite formation. Structural Engineering International 27(3), 362–369 (2017)

[5] Caussignac, J.-M., Vincent, L., Abraham, O., Dérobert, X., Villain, G.: Évaluation et contrôle non destructifs en génie civil. Techniques de l'Ingénieur, R1410, (2013) (2013)

[6] Avci, O., Abdeljaber, O., Kiranyaz, S., Hussein, M., Gabbouj, M., Inman, D.J.: A review of vibration-based damage detection in civil structures: From traditional methods to machine learning and deep learning applications. Mechanical Systems and Signal Processing 147, 107077 (2021)

[7] Arora, V., Wijnant, Y.H., de Boer, A.: Acoustic-based damage detection method. Applied Acoustics 80, 23–27 (2014)

[8] Nerlikar, V., Mesnil, O., Miorelli, R., DAlmeida, O.: Damage detection with ultra-sonic guided waves using machine learning and aggregated baselines. Structural Health Monitoring 23(1), 443–462 (2024)

[9] Javed, A., Sadeghnejad, A., Rehmat, S.e., Yakel, A., Azizinamini, A.: Magnetic flux leakage (MFL) method for damage detection in internal post-tensioning tendons. Florida International University; Florida Department of Transportation Research Center (2021)

[10] Cloetens, P., Pateyron-Salom, M., Buffire, J.Y., Peix, G., Baruchel, J., Peyrin, F., Schlenker, M.: Observation of microstructure and damage in materials by phase sensitive radiography and tomography. Journal of Applied Physics 81(9), 5878–5886 (1997)

[11] Shoukroun, D., Massimi, L., Iacoviello, F., Endrizzi, M., Bate, D., Olivo, A.,
Fromme, P.: Enhanced composite plate impact damage detection and character-
isation using x-ray refraction and scattering contrast combined with ultrasonic
imaging. Composites Part B: Engineering **181**, 107579 (2020)

[12] Weng, Y., Lu, Z., Lu, X., Spencer Jr., B.F.: Visualinertial structural acceleration measurement. Computer-Aided Civil and Infrastructure Engineering **37**(9), 1146–1159 (2022)

[13] Sinou, J.-J.: A Review of Damage Detection and Health Monitoring of Mechanical Systems from Changes in the Measurement of Linear and Non-linear Vibrations, pp. 643–702 ((2009))

[14] Vary, A.: The acousto-ultrasonic approach. NASA Technical Memorandum **89843** (1987)

[15] Fink, M.: Time reversal of ultrasonic fields - part I. basic principles. IEEE transactions on ultrasonics, ferroelectrics, and frequency control **39**(5), 555–566 (1992)

[16] Wu, F., Thomas, J.-L., Fink, M.: Time reversal of ultrasonic fields - part II. experimental results. IEEE transactions on ultrasonics, ferroelectrics, and frequency control **39**(5), 555–566 (1992)

[17] Cassereau, D., Fink, M.: Time reversal of ultrasonic fields - part III. theory of the closed time-reversal cavity. IEEE transactions on ultrasonics, ferroelectrics, and frequency control **39**(5), 555–566 (1992)

[18] Shiloh, K., Bartos, A., Frain, A., Lindgren, E.A.: Ultrasonic detection of corrosion between riveted plates **3994**, 70–79 (2000). SPIE

[19] Marquet, F., Pernot, M., Aubry, J.-F., Montaldo, G., Tanter, M., Fink, M.: Non-invasive transcranial ultrasound therapy guided by ct-scans, 683–687 (2006)

[20] Mokh, A., Khayatzadeh, R., Ourir, A., Kamoun, M., Tourin, A., Fink, M., Rosny, J.: Time-reversal of sub-thz pulses in complex media. Progress In Electromagnetics Research B **95**, 141–162 (2022)

[21] Miniaci, M., Mazzotti, M., Radzieński, M., Kudela, P., Kherraz, N., Bosia, F., Pugno, N.M., Ostachowicz, W.: Application of a laser-based time reversal algorithm for impact localization in a stiffened aluminum plate. Frontiers in Materials **6**, 30 (2019)

[22] Jin, Y., Moura, J.M.F.: Time-reversal detection using antenna arrays. IEEE Transactions on Signal Processing **57**(4), 1396–1414 (2009)

[23] Hong, X., Lin, J., Zhou, J., Yang, D.: Array ultrasonic guided wave spiral focusing detection method for inner damage of messenger cable in covered area. Mechanical
[24] Fink, M.: From ultrafast ultrasound imaging to matrix imaging. The Journal of the Acoustical Society of America **152**(4), 218–218 (2022)

[25] Park, H.W., Kim, S.B., Sohn, H.: Understanding a time reversal process in lamb wave propagation. Wave motion **46**(7), 451–467 (2009)

[26] Sicard, R., Goyette, J., Zellouf, D.: A numerical dispersion compensation technique for time recompression of lamb wave signals. Ultrasonics **40**(1-8), 727–732 (2002)

[27] Ing, R.K., Fink, M.: Time recompression of dispersive lamb waves using a time reversal mirror-application to flaw detection in thin plates **1**, 659–663 (1996).
IEEE

[28] Granger, S., Cabot, G.P., Loukili, A., Marlot, D., Lenain, J.-C.: Monitoring of cracking and healing in an ultra high performance cementitious material using the time reversal technique. Cement and Concrete Research **39**(4), 296–302 (2009)

[29] Olivera, J., González, M., Fuente, J.V., Varga, R., Zhukov, A., Anaya, J.J.: An embedded stress sensor for concrete shm based on amorphous ferromagnetic microwires. Sensors **14**(11), 19963–19978 (2014)

[30] Wei, L., Shen, L., Cao, M., Wang, J., Wang, Z., Ostachowicz, W.: Corrosion damage identification in concrete underwater based on time reversal of stress waves. Mechanical Systems and Signal Processing **194**, 110281 (2023)

[31] Sohn, H., Park, H.W., Law, K.H., Farrar, C.R.: Damage detection in composite plates by using an enhanced time reversal method. Journal of Aerospace Engineering **20**(3), 141–151 (2007)

[32] Park, H.W., Sohn, H., Law, K.H., Farrar, C.R.: Time reversal active sensing for health monitoring of a composite plate. Journal of Sound and Vibration **302**(1-2), 50–66 (2007)

[33] Gangadharan, R., Murthy, C., Gopalakrishnan, S., Bhat, M.: Time reversal technique for health monitoring of metallic structure using lamb waves. Ultrasonics **49**(8), 696–705 (2009)

[34] Furukawa, A., Kojima, K., Kiyono, J., Toki, K., Miura, F.: Fundamental study on structural damage detection method for plate structures based on the time reversal technique. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014, Porto, Portugal, 30 June-2 July 2014

[35] Huguet, M., Ture Savadkoohi, A., Lamarque, C.-H., Collet, M., Rodriguez, J.: Damage detection of structural elements by time reversal principle. Proceeding of
the 30th International Conference on Noise and Vibration engineering, ISMA2022,
Leuven, Belgium, 12-14 September 2022, 1918–1930