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Die Publikation befasst sich mit dem innovativen Einsatz von Time Reverse Modeling (TRM), um knackende Schallereignisse in textilverstärktem Beton zu kartieren. Es beginnt mit einem Überblick über bestehende Methoden zur Lokalisierung akustischer Quellen, wobei die Grenzen bei der Kartierung plötzlicher und dynamischer Crackprozesse aufgezeigt werden. Anschließend passt die Studie TRM, die ursprünglich für elastische Körperwellen verwendet wurde, an akustische Wellen an und demonstriert ihre verbesserten Kartierungsfähigkeiten durch vorläufige Tests mit bekannten Randbedingungen. Die Hauptprüfreihe umfasst einen axialen Zugversuch an einem Betonkörper, bei dem TRM eingesetzt wird, um aktive Risse mit hoher Genauigkeit zu überwachen. Die Ergebnisse zeigen, dass die TRM komplexe Rissverläufe und gleichzeitige Schallereignisse effektiv kartieren kann, was einen bedeutenden Fortschritt im Bereich der strukturellen Gesundheitsüberwachung darstellt.
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Abstract
Time reverse modeling (TRM) is successfully applied to acoustic signals from a circular microphone array, for mapping of sudden cracking sound events. Numerical feasibility using synthetic acoustic sources followed by an experimental study with steel pendulum impacts on a steel plate is carried out. The mapping results from the numerical and experimental data are compared and verified using a delay-and-sum beamforming technique. Based on the feasibility and experimental study, a mapping error is estimated. In the main experimental study, cracking sound events obtained during a tensile test on a textile-reinforced concrete specimen are mapped with the TRM. The enhanced capability of the TRM to map simultaneously occurring cracking sound events along crack paths is demonstrated.
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1 Introduction
Mapping of sound events or acoustic sources employing microphone recordings is a vital and exciting research area in various fields of research such as geophysics [1‐3], aeronautics [4], biomechanics [5] and real-time structure monitoring [6‐8]. A review of imaging methods for acoustic source localization using phased microphone arrays can be found in Merino-Martínez et al. [9]. Beamforming techniques are the most commonly applied methods for localizing acoustic sources, to name a few different applications from recent years. Huang et al. [10] developed a differential beamforming algorithm that can form any specified frequency-invariant beam pattern with a microphone array of any planar geometry. Chiariotti et al. [11] reviewed the acoustic beamforming techniques for noise source localization and its applications. Den Ouden et al. [2] proposed a high-resolution beamforming technique in combination with the CLEAN algorithm for the detection of multiple infrasonic sources. Ma et al. [12] investigated several algorithms for the localization of rotating sound sources at free-space conditions in the frequency domain. Shen et al. [13] developed an adaptive, passive acoustic mapping method that combines delay-multiply-and-sum beamforming with a virtual augmented aperture for precise localization of cavitation sources. Karimi and Maxit [14] proposed vibroacoustic beamforming for sound source localization using measured vibration signals.
Alongside conventional signal-based methods, computational intelligence methods became a helpful tool to reduce human interaction and to generalize processes in the localization of acoustic sources. Vera-Diaz et al. [15] used a convolutional deep neural network that transforms the generalized cross correlation between two signals into a Gaussian-shaped signal, which can be used to estimate a three-dimensional (3D) acoustic map. Zeitouni and Cohen [16] introduced a supervised method, using a single-microphone manifold learning approach for estimating both, the location and velocity of a moving acoustic source in reverberant and noisy environments. Chen et al. [17] proposed a hybrid microphone array signal approach for a near-field scenario and combined the beamforming technique with deep neural networks, for a synchronistical localization, separation, and reconstruction of multiple sound sources. Grumiaux et al. [18] provided a broad overview of deep learning methods for single and multiple sound source localization, with a focus on sound source localization in indoor environments, where reverberation and diffuse noise are present.
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With the growing performance of computer clusters, wave-propagation-based localization methods have gained full attention in the field of engineering acoustics. Pan et al. [19] combined iterative time reversal processing with the distributed multiple-input multiple-output processing for the detection of small targets in shallow water environments. Chen and Ma [4] systematically investigated the implementation schemes for multi-source time reversal focusing on airborne sound. Assous and Lin [20] performed feasibility of time reversal methods in a non-homogeneous elastic medium, from data recorded in an acoustic medium, to determine the presence and some physical properties of not-observable solid objects. Li et al. [21] improved the identification accuracy of low-frequency sources in the strong noise environment, by correcting the wavenumber domain, and applying a cross-spectrum time reversal method based on pressure measurements in a near-field plane. Ma et al. [22] developed a time reversal sparse Bayesian learning method to solve the problem of low-frequency sound source localization in enclosed space, and to improve the spatial resolution. Dvořáková et al. [5] presented a time reversal approach along with signal classification in biomedical applications, for localization and statistical classification of ultrasonic non-linearities, such as air bubbles with different sizes and ultrasound contrast agents in a liquid. Godin et al. [3] characterized the seabed with a single-element time reversal mirror of the low-frequency sound propagation. Im et al. [23] applied a time reversal processing procedure using a virtual receiving array, comprised of a single receiving transducer and a chaotic cavity, to image the reverberation inside the cavity. Golightly et al. [24] focused on the use of near-field scatterers placed within a wavelength of time reversal focusing to achieve super-resolution in a one-dimensional pipe system.
A variety of methods with different approaches and applications were introduced that were successfully applied for mapping of single and multiple sound events, with mostly known boundary conditions. However, for mapping a suddenly occurring and dynamic cracking process, where the sound events occur with boundary conditions not known a priori, the literature is limited. Kocur et al. [25, 26] applied time reverse modeling (TRM) to localize active concrete cracks using elastic body waves (= acoustic emissions). The main difference between TRM localization using acoustic waves and acoustic emissions is that sound events are localized on the surface, while body waves can be used to localize events inside the body. Since air is the wave-transmitting medium, cracks that penetrate the sample cannot interfere with communication between microphones—this is the case with body waves. Air is a homogeneous medium, which also facilitates the time reversal process compared to the heterogeneous composite of concrete and fibers.
In this study, we adapt the TRM method presented in [26] and apply it to acoustic waves for enhanced mapping of cracking sound events. We present a feasibility and preliminary test series utilizing near-field sound events with known boundary conditions (i.e. synthetic acoustic sources and pendulum impacts), to define the setup and estimate the accuracy to be expected for mapping of sudden cracking sound events. The TRM results are contrasted against delay-and-sum beamforming results and discussed. In the main test series, an axial tensile test on a textile-reinforced concrete specimen is carried out and the cracking sound events are recorded by a circular microphone array. The time sequence of the nucleating cracks is monitored using a non-contact optical 3D measuring device for surface strains. We apply the TRM method to map the monitored active cracks with confidence. The enhanced mapping capability of the TRM is demonstrated by mapping crack formation along crack paths in their complexity as accumulated and simultaneously occurring sound events.
Fig. 1
TRM principle, a reception and b re-emission of the acoustic pressure wavefield
According to Feynman et al. [27], three features characterize the propagation of acoustic waves in air. Air particles move due to a stimulus and change the density. The change in density corresponds to a change in pressure. Such disturbance in the state of equilibrium manifests in air particles propagating from its source to other positions. The 3D acoustic wave equation is a second-order partial differential equation that describes the evolution of the acoustic pressure p through a material medium (here air), as a function of the position vector \(\textbf{x}\) and time t. The acoustic wave equation can be written in vector notation as
where \(p=p(\textbf{x},t) \in \mathbb {R} ^{3}\) is a space- and time-dependent wavefield, \(\nabla ^{2}p\) is the Laplacian of the pressure field, and c is the speed of sound. The speed of sound is defined by \(c=\sqrt{\kappa _{0}/\rho _{0}}\). The two quantities \(\rho _{0}\) and \(\kappa _{0}\) are the mass density and the bulk modulus of air, respectively. At a defined spatial domain \(\textbf{x} \in \Omega \), the equation of motion can be solved in the time domain to visualize the temporal evolution of \(p(\textbf{x},t)\) and to simulate signals received at sensor positions \(\textbf{x}_{i}\) (here microphones) on the domain’s boundary \(\partial \Omega \) over time.
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2.2 Mapping of Sound Events with Time Reverse Modeling
2.2.1 Principle
According to Fink [28], the Eq. (1) can be considered time reversal invariant because it contains only second-order time derivatives. Assuming acoustic signals \(p(\textbf{x}_{i},t)\) received by the microphones \(\textbf{x}_{i} \subset \partial \Omega \) over the time \(t \in [0;T]\) with end time T, the time-reversed acoustic signals \(p(\textbf{x}_{i},T-t)\) can be re-emitted into the domain \(\Omega \) to focus on the initial acoustic source location. A requirement for such convergence is that, in addition to the time reversal invariance, spatial reciprocity is satisfied by an acoustic phenomenon. This means the acoustic source location and microphone positions must not be changed in order not to alter the wavefield. The TRM procedure consists of two steps, a forward procedure (FW, Fig. 1a) and the inverse simulation (REV, Fig. 1b).
Fig. 2
3D representation of the normalized inverse pressure wavefield \(\bar{p}_{\max }(\textbf{x})\) due to a synthetic acoustic source located at [0, 20, 0] gp
a Numerical setup of the model \(\textcircled {1}\) for b excitation of acoustic waves at \(n_{z}=0\) mm by an idealized impulse. c Reception of acoustic waves at \(n_{z}=800\) mm
In the forward procedure (numerical or experimental), acoustic signals due to a sound event are received at defined microphone positions and reversed with respect to time. In the inverse simulation, the microphone positions serve as excitation points for the numerical model. The re-emitted pressure wavefields will interfere with each other and focus on the location of the initial acoustic source.
2.2.2 Acoustic Source Mapping
After the inverse pressure wavefield \(p(\textbf{x},T-t)\) has vanished, the initial acoustic source location appears as a dominant interference of the wavefield. As an imaging condition, the maximum absolute value of \(p(\textbf{x},T-t)\) can be calculated by
The maximum pressure magnitude \(p_{\max }\) is stored for every point \(\textbf{x}=[n_{x},n_{y},n_{z}]\) on the spatial domain \(\Omega \), and the entire simulation time T. Equation (2) can be written in a normalized form
where \(\bar{p}_{\max }\) is the global maximum of \(\bar{p}_{\max }(\textbf{x})\). In Fig. 2, \(\bar{p}_{\max }\) depicts the location of the initial acoustic source on the numerical domain, i.e. at [0, 20, 0] grid points (gp).
Table 1
Coordinates of synthetic acoustic sources and sound events with known boundary conditions (\(n_{z}=0\) mm)
#1
#2
#3
#4
#5
#6
#7
#8
\(n_{x}\) (gp)
− 80
− 40
0
40
80
0
0
0
\(n_{y}\) (gp)
40
40
40
40
40
20
0
− 20
Table 2
Coordinates of the microphone positions of model \(\textcircled {1}\) (\(n_{z}=800\) gp, seven microphones\(^{*}\)) and model \(\textcircled {2}\) (\(n_{z}=610\) gp, fourteen microphones)
mics
16\(^{*}\)
32\(^{*}\)
48\(^{*}\)
64\(^{*}\)
80\(^{*}\)
96\(^{*}\)
112\(^{*}\)
29
31
33
47
81
93
95
\(n_{x}\) (gp)
47
6
− 39
− 55
− 30
18
52
50
52
− 15
− 49
− 57
40
18
\(n_{y}\) (gp)
− 29
− 55
− 39
6
46
52
18
− 197
− 104
− 163
− 106
153
199
52
3 Numerical Simulations
3.1 Setup and Synthetic Data Generation
In a numerical feasibility study, the setup and relevant parameters for TRM with numerical (synthetic) and experimental data due to sound events with known boundary conditions were set. For synthetic data generation and the later inverse simulations, the synthetic air model \(\textcircled {1}\) with the dimensions \(n_{x} \times n_{y} \times n_{z} = 200 \times 200 \times 800\) gp\(^{3}\) (gp is equivalent to mm) was built. Assuming the speed of sound \(c=340.5\) m/s, the travel distance of 0.8 m (\(\equiv \,800\) gp) and the signal length of \(t=4.0 \times 10^{-3}\) s, the inverse simulation time was set to \(T=6.5 \times 10^{-3}\) s to ensure wavefield interference of the entire signal. Eight synthetic acoustic sources denoted by #1 through #8 in Fig. 3a were excited. The acoustic sources were idealized by an impulse with the dominant frequency \(f_{\text {dom}}\cong 13\) kHz (see Fig. 3b). This frequency is in line with the frequency content of the pendulum impacts described later. Seven microphone positions located closest to the microphone array center were defined (see Fig. 3c). The coordinates of the source locations and microphone positions are listed in Tables 1 and 2. The simulated acoustic waves were received at the synthetic microphone positions for a duration of 2 s. Two source-to-microphone configurations were investigated: Source within the microphone array (#3, #6, #7 and #8) and source outside the microphone array (#1, #2, #4 and #5) according to Fig. 3a. For the inverse simulations using the cracking sound events, the air model \(\textcircled {2}\) with the dimensions \(n_{x} \times n_{y} \times n_{z} = 140 \times 440 \times 610\) gp\(^{3}\) was built. Assuming the travel distance of 0.61 m (\(\equiv \,610\) gp), the inverse simulation time was set to \(T=6.0 \times 10^{-3}\) s. Fourteen microphone positions located closest to the array center and within the numerical domain \(n_{x} \times n_{y} \times n_{z}\) were defined according to Table 2.
3.2 Implementation Details
A commercial finite element software (ABAQUS [29]) was used to simulate the acoustic wave propagation. The explicit solver was employed to compute the acoustic pressure wavefield. The finite elements were discretized by 3,920,400 (model \(\textcircled {1}\)) and 4,787,280 (model \(\textcircled {2}\)) eight-node linear acoustic brick elements with reduced integration and hourglass control (AC3D8R). The implemented material properties can be found in Table 3. The numerical stability was ensured by the integration time step \(\Delta t \le 1.0 \times 10^{-6}\) s and element length \(\Delta l=0.001\) m (see Neumann stability criterion in [30]). As the ABAQUS solver performs validation checks before job submission, the Courant-Friedrichs-Lewy condition was not explicitly controlled. The translational and rotational degrees of freedom for the boundary edges were set to zero to prevent rigid-body movement. To minimize the computational effort, but at the same time to prevent reflections from the boundary edges, infinite acoustic elements were implemented surrounding the models \(\textcircled {1}\) and \(\textcircled {2}\). All simulations were carried out at the high-performance computing cluster EULER at ETH Zurich. The computations were executed in parallel with 24 central processing units.
4 Physical Experiments
4.1 Setup and Data Collection
In all physical experiments, the acoustic waves were recorded by a commercial 112 MEMS-microphone array (Bionic M-112, Omnidirectional Microphone ADMP441) in combination with the Noise Inspector (CAE Software & Systems) acquisition system. The microphones were arranged asymmetrically in a circular array configuration (see Fig. 4). The technical specifications of the microphones are listed in Table 4. The microphone array was positioned at a distance of \(z=800\) mm to the steel plate (see Sect. 4.1.1) and of \(z=610\) mm to the concrete specimen (see Sect. 4.1.2), respectively. The z-distance was calibrated by reference laser measurements to ensure parallelism between the array and specimen. The center of the microphone array was centered at the origin of the specimen coordinate system at \([x,y,z] = [0,0,0]\) in millimeters.
Table 3
Material properties of air at \(15\,^{\circ }\)C
\(\rho _{0}\) (kg/m\(^{3}\))
\(\kappa _{0}\) (Pa)
c (m/s)
1.225
142,000
340.5
Table 4
Technical specifications of the ADMP441 microphone
Frequency range
Resolution
Max. sampling rate
Max. noise level
60 Hz–15 kHz
24-bit
48 kHz
0.05 mV
4.1.1 Sound Events with Known Boundary Conditions
For the preliminary experimental series, the locations of the sound events were adopted from the feasibility study. The sound events were excited by pendulum impacts on a steel plate, applied with (mostly) known boundary conditions. The plate with dimensions \(x \times y \times z = 400 \times 400 \times 40\,\text {mm}^{3}\) (see Fig. 5a) was clamped on its lower end at \(y=-\,200\) mm. The pendulum system was realized by a 20 mm steel ball with a cord of 250 mm and an angle to the vertical of 22\(^{\circ }\). The pendulum system was chosen because its impulse can be easily reproduced in experiments and has a defined frequency content. The source strength was governed by the angle to the vertical. The impulse shape and frequency content were controlled by the pendulum mass and impact velocity. The eight impact locations were defined on a rectangular grid on the plate surface (see Fig. 5a). The time of the impact was associated with the time of the pendulum release. The indentations of two impact spots (pendulum impact incl. rebounds) are shown exemplarily in Fig. 5b. A deviation of approx. 5 mm on the horizontal (x-axis) can be observed.
The acoustic waves were recorded at seven microphone positions according to Fig. 3c and Table 2 (gp \(\equiv \) mm) for a period of 6 s, to capture the first impact and several rebounds. For analysis, the first impact and four subsequent rebounds denoted by impact 1 through 5 in Fig. 6 with a duration of 0.5 s were selected and stored.
Fig. 5
a Perspective view of the pendulum system with steel ball exciting a sound event at the steel plate. b Indentations of the pendulum impact incl. rebounds at two exemplary impact spots
a Test stand of the axial tensile test, b geometry of the test specimen No. 19 with crack pattern after failure, c setup of model \(\textcircled {2}\) with the fourteen microphones used for analysis
4.1.2 Sound Events with Unknown Boundary Conditions
The main experimental series was a part of the research project ‘Investigation of sustainable reinforcements made of natural fibers for textile concrete components’ [31]. The project was initiated to study the tensile load bearing and bonding behavior between the cement matrix and textile reinforcement (natural flax fibers). The tensile tests were accompanied by acoustical measurements to monitor the cracking process in detail. In this study, a single test specimen (No. 19) was utilized to generate the cracking sound events due to sudden and randomly occurring concrete crack formation under the axial tensile loading. In contrast to the pendulum impacts, the source strength, excitation and frequency, source location, and time of occurrence were not known a priori. Based on engineering judgment, it could be anticipated that the cracks will form perpendicular to the direction of loading. The tensile test was conducted using a uniaxial testing machine (MTS, Hydraulic Actuator Model 243), displayed in Fig. 7a. To ensure ideal axial load application, cardan joints were used. The force was applied displacement-controlled with a speed of 5 mm/min until failure. The force was continuously recorded by a load cell (MTS, Servohydraulic System Load Cell 661.23F01). The test specimen was of dimensions \(l \times b \times t = 900 \times 100 \times 15(60)\,\text {mm}^3\) (see Fig. 7b). The concrete compressive strength was 47.2 MPa. The flax fibers (1500 tex-yarn) were cast in two layers orthogonally to each other, and exhibited a tensile strength of 412.2 MPa at a failure elongation of 0.018. Investigated were the cracks in the center range (\(400 \times 100 \times 15\,\text {mm}^3\)). For the analysis of the cracking sound events, fourteen microphones according to Fig. 7c (model \(\textcircled {2}\)) and Table 2 were involved. The load-time history with the corresponding acoustical measurements was recorded for a period of 65 s. In Fig. 8, the load-time curve is plotted versus the normalized accumulated absolute amplitudes of the microphone recordings to identify the cracking sound events on the global time axis. Each load drop in the load-time curve due to load redistribution indicates such a cracking sound event. It is interesting to observe that peaks of the cracking sound events match nearly all load drops on the global time axis. An area of \(x \times y = 100 \times 200\) mm\(^{2}\) (restricted by the measuring device) was monitored using a non-contact optical 3D measuring system (ARAMIS 12 M) at a frame rate of 5 fps. This allowed to capture the exact crack locations (see Fig. 9) and the time of nucleation on the load-time curve (see Fig. 8). Sixteen cracking sound events denoted by #1 through #16 were selected for mapping with the TRM. Eight of the cracking sound events were due to monitored cracks. As can be observed in Fig. 9, the cracks do not spread from an initial focal point but ‘jump’ spatially between the upper and lower area of the concrete specimen due to load redistribution.
Fig. 8
Load-time curve of the axial tensile test (specimen No. 19) versus the accumulated microphone recordings of fourteen microphones, and selected sixteen cracking sound events (#1 through #16). Eight monitored cracks are marked
Prior to TRM, the stored experimental microphone data (Sects. 4.1.1 and 4.1.2) was pre-processed. As the primary part of the acoustic wave is of the greatest importance for focusing on the location of the initial source [26], acoustic signals with a duration of \(t=4.0 \times 10^{-3}\) s centered on the primary wave part were used as input for the inverse simulations (see Fig. 10a, b). All input signals were re-sampled to 1.0 MHz to be consistent with the numerical setup in the feasibility study and to refine the temporal resolution of the simulations. The re-sampling does not influence the frequency spectrum of both experimental series. As can be seen in Fig. 10a, the frequency spectrum of the microphone recording due to the pendulum impact covers the entire sensitivity range of the ADMP441 microphones. The acoustic signals were windowed (Blackman) to avoid numerical artifacts due to non-zero amplitudes at the beginning of the inverse simulations. For the pendulum impact, windowing seems to minimize the spectral leakage and improve the signal quality. As the acoustic signals due to the cracking sound events show some noise level (see Fig. 10b), a lowpass filter with a cutoff frequency \(f_{\text {cutoff}}=5\) kHz was applied to all signals. After re-sampling and windowing (and filtering), the acoustic signals were reversed in time \(p_{i}(\textbf{x},t) \mapsto p_{i}(\textbf{x},T-t)\).
5 Results and Discussion
5.1 Feasibility Study with Synthetic Acoustic Sources
The synthetic acoustic signals reversed in time were fed as excitations into the numerical model \(\textcircled {1}\) at \(n_{z}=800\) gp according to Fig. 3c. The normalized inverse pressure wavefield \(\bar{p}_{\max }(\textbf{x})\) was calculated according to Eq. (3). Figure 11a–h depict the two-dimensional (2D) interference patterns of \(\bar{p}_{\max }(\textbf{x})\) due to the synthetic acoustic sources #1 through #8. The synthetic microphone positions are indicated by squares while the reference acoustic source locations by crosses, respectively. The locations of the initial synthetic acoustic sources can be identified. The dominant focii of \(\bar{p}_{\max }(\textbf{x})\) are located within a distance of approx. 10 mm from the reference acoustic source locations (except the sources #1 and #5). The synthetic acoustic sources #1 and #5 outside the microphone array (see Fig. 11a, d) focus less accurate than the remaining sources. The interference pattern of all focii appears symmetrical.
5.2 Mapping of Pendulum Impacts
Analogously to the synthetic acoustic signals, the pre-processed experimental acoustic signals due to the pendulum impacts were reversed in time and fed as excitations into the numerical model \(\textcircled {1}\). The setup, including the microphone positions and reference acoustic source locations, was adapted from the feasibility study. The TRM results of the first pendulum impacts displayed in Fig. 12a–h (top) were computed accordingly.
There, clear focii of \(\bar{p}_{\max }(\textbf{x})\) can be observed near the reference pendulum impact locations. For the pendulum impacts within the microphone array (#3, #6, #7 and #8, see Fig. 12e–h, top), the focii match the reference locations within a tolerance of millimeters, except the impacts #2 and #4 outside the microphone array (see Fig. 12b and c, top) which are in the range of a centimeter. Further pendulum impact locations #1 and #5 outside the microphone array (see Fig. 12a and d, top), focus with a significant spatial deviation in the x-direction compared to the reference locations. This is plausible because the x-coordinate of these impacts lies outside the microphone array. The interference pattern of the focii appears less symmetrical than the pattern of the synthetic acoustic sources displayed in Fig. 11a–h. For a qualitative comparison of the interference patterns of \(\bar{p}_{\max }(\textbf{x})\) depicted in Fig. 12a–h (top), the beamformer responses \(\bar{b}_{f,\max }(\textbf{x}_{j})\) according to Eq. (6) in Sect. A (see Fig. 12a–h, bottom) were calculated using the same input signals as for the TRM; both methods perform with similar mapping accuracy except the pendulum impacts located outside the microphone array (i.e. impacts #1 and #5).
5.3 Mapping Error Estimation for Sound Events with Known Boundary Conditions
For assessment of the mapping accuracy, the TRM results of sound events with known boundary conditions were compared to results calculated using the delay-and-sum beamforming technique according to Sect. A. Although the preliminary experiments were performed in the lab with controlled boundary conditions, the pendulum impact locations might be influenced by the experimental procedure, environment, and measurement equipment. Therefore, the synthetic acoustic source locations computed under fully controlled boundary conditions serve as a benchmark for the ideal performance of TRM. In Fig. 13a, the maximal value of the normalized inverse pressure wavefield \(\bar{p}_{\max }\) and the maximal beamformer response \(\bar{b}_{f,\max }\) are compared in terms of the x- and y-coordinates. The TRM results of pendulum impacts within the microphone array are located close to the reference pendulum impact locations with small error spans, while the pendulum impacts outside the microphone array were mistakenly mapped within the array. The beamforming results are consistently accurate in the range of millimeters. As displayed in Fig. 13b, the mapping error e is quantified for each coordinate axis separately with \(e_{x}=|\Delta \bar{x} |\) and \(e_{y}=|\Delta \bar{y} |\) and in total by \(e=\sqrt{e_{x}^{2}+e_{y}^{2}}\). An error threshold of \(e_{x}=e_{y}\le 5\) mm and \(e\le 7.1\) mm was set to account for the experimental deviations of the pendulum impacts (see Fig. 5b). The inputs \(\bar{x}\) and \(\bar{y}\) are the arithmetic mean of the difference between the predicted first five pendulum impact coordinates (first impact and four rebounds) according to Fig. 6, and the reference pendulum impact location in x- and y-direction, respectively. The pendulum impact locations outside the microphone array (impacts #1 and #5) were mapped with a significant error \(e_{x} > 15\) mm and the impacts #2 and #4 with a moderate error \(e_{x}, e_{y} \le 10\) mm, respectively. The total mapping error of the pendulum impacts within the microphone array is within \(e \le 7.1\) mm. The mapping accuracy between the synthetic acoustic sources and the pendulum impacts is comparable, although the overall performance of TRM applied on synthetic data tends to be better.
Fig. 9
Picture of the monitored area taken at approx. time \(t=55\) s during the axial tensile test highlighting the eight monitored cracks, respectively cracking sound events
Pre-processed acoustic signals due to the sound events with a known boundary conditions and b unknown boundary conditions on the time and frequency domain
TRM results showing the 2D representation of \(\bar{p}_{\max }(\textbf{x})\) due to the synthetic acoustic sources #1 through #8. The synthetic microphone positions are indicated by ‘\(\square \)’ and ‘+’ are the reference acoustic source locations
Comparison of TRM results \(\bar{p}_{\max }(\textbf{x})\) (top) with beamformer responses \(\bar{b}_{f,\max }(\textbf{x}_{j})\) (bottom) due to the first pendulum impacts #1 through #8, respectively. The microphone positions are indicated by ‘\(\square \)’ and ‘+’ are the reference pendulum impact locations
a Comparison of the mapped synthetic acoustic sources and pendulum impacts with their horizontal and vertical error spans, calculated with the TRM, \(\bar{p}_{\max }\), and sum-and-delay beamforming technique, \(\bar{b}_{f,\max }\), in Sect. A versus the reference source locations. b Estimation of the mapping error e between the TRM and beamforming
For the cracking sound events, no statistical error estimation was performed because exact crack patterns are hardly reproducible in practice and repeatability cannot be ensured. Instead, a qualitative error evaluation was performed where the results were verified with the monitored active crack locations and the crack pattern of the presented tensile test (specimen No. 19) after failure (see Sect. 5.4).
5.4 Mapping of Sudden Cracking Sound Events
The sixteen pre-processed cracking sound events (#1 through #16 in Fig. 8) were reversed in time and fed as excitations into the numerical model \(\textcircled {2}\). For verification of the TRM results, the crack pattern of the test specimen No. 19 after failure was extracted (see Fig. 7b, investigated range) and superimposed with \(\bar{p}_{\max }(\textbf{x})\). The entire investigated range displayed in Fig. 7c was covered by the fourteen microphones so that all cracking sound events to be mapped were located within the microphone array. Evaluating \(\bar{p}_{\max }(\textbf{x})\) over the entire inverse simulation time T according to Eq. (3), reveals a very promising feature of the TRM. Kocur et al. [26] demonstrated that multiple cracking events that occur simultaneously at different locations can be accurately mapped with a single set of elastic signals. As can be seen in Fig. 14, single cracking sound events (i.e. events #6, #15 and #16) as well as potential simultaneously occurring cracking sound events were successfully mapped with TRM, using a single set of (fourteen) acoustic signals. For the latter, multiple dominant focii of \(\bar{p}_{\max }(\textbf{x})\) can be observed that formed along the crack path for every cracking sound event. The cracks monitored with the ARAMIS 12 M system were marked in the relevant event in which they occur in accordance to the load-time curve displayed in Fig. 8. In general, the monitored crack locations correspond to the mapped dominant focii of \(\bar{p}_{\max }(\textbf{x})\). In case of the events #4, #8 and #14, the cracking activity is superimposed by other simultaneously occurring cracking sound events located outside the monitored area. This outcome allows an enhanced assessment of the cracking sound events in terms of concluding the structural response. The approaching structural failure can be clearly identified by the dominant single cracking sound events #15 and #16.
Fig. 14
Chronological evolution of \(\bar{p}_{\max }(\textbf{x})\) due to cracking sound events #1 through #16. The colormaps are superimposed with the crack pattern after the failure of the test specimen No. 19. The colorbar is identical to the one in Fig. 2. The monitored cracks are marked by arrows in the relevant events and at y-location in accordance with Fig. 9
The TRM method was applied to acoustic waves for mapping pendulum impacts with known boundary conditions and sudden cracking sound events, where the boundary conditions were not known a priori. Using the TRM method, the pendulum impact locations could be mapped with small spatial deviations in the millimeter range, compared to the reference pendulum impact locations (see Fig. 13a, b). As the impacts were repeated with deviations of approx. 5 mm, the overall mapping accuracy achieved (\(e \le 7.1\) mm) for sound events located within the microphone array (impacts #3, #6, #7 and #8 in Fig. 13a, b) is considered as accurate. For pendulum impacts outside the microphone array (impacts #1, #2, #4, and #5), the TRM method tends to focus within the array, with significant spatial deviations in the x-direction (\(e_{x}>15\) mm, impacts #1 and #5) compared to the reference source positions. Such behavior was already reported by other authors [25, 26, 32]. In addition, the horizontal experimental deviations shown in Fig. 5b contribute unfavorably to the deviation in x-direction. The beamforming results demonstrated a consistently good localization performance with an overall error \(e \le 7.1\) mm with isolated outliers. The beamforming technique performed robustly when applied to pendulum impacts located outside the microphone array.
The acoustic signals due to the pendulum impacts were less noisy than those due to concrete cracking (compare Fig. 10a, b). A lowpass filter with a cutoff frequency \(f_{\text {cutoff}}=5\) kHz was applied to all signals due to concrete cracking. Applying the TRM to the cracking sound events without filtering would lead to less sharp foci (not explicitly shown here). Prior to the reverse simulations, the acoustic signals were windowed (Blackman) to emphasize the primary wave part and to improve the TRM focusing as well. If an acoustic event occurs on a lower energy level, then all corresponding signals will be recorded on similarly low amplitudes. The TRM might not be problematic in this case.
The greatest feature of TRM is revealed when applied to cracking sound events that occur randomly inside a specimen. As concrete cracks form as a combination of single and accumulated sound events, the evaluation of \(\bar{p}_{\max }\) as shown in Fig. 13a (for pendulum impacts) would be incomplete and misleading in terms of the structural response. The TRM method computes and evaluates the interference of inverse pressure wavefields which can be due to a single or multiple sound events. The entire wavefield \(\bar{p}_{\max }(\textbf{x})\) can be utilized to visualize the cracking sound events that form along the crack paths, regardless whether it is a single sound event or multiple sound events occurring simultaneously at different locations. Although applications of the beamforming technique exist to localize multiple acoustic sources [2, 17, 18], applications on the localization of active cracks are not documented. The TRM method presented here fills this gap and encourages further investigations.
Acknowledgements
The authors would like to thank Mr. Uwe Navrath from the laboratory of the Institute of General Mechanics (IAM) at RWTH Aachen University, and Prof. Marcus Ricker, Ms. Katrin Zecherle, Mr. Theo Baumann and Mr. Lars Neuschl from the Biberach University of Applied Sciences (HBC) in Germany for their support with the experimental work. The numerical simulations with ABAQUS were performed in cooperation with Prof. Eleni Chatzi from the Institute of Structural Engineering (IBK) of the ETH Zurich using the EULER cluster at the scientific computing facilities of the ETH Zurich in Switzerland.
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A robust approach to map acoustic sources is the delay-and-sum beamformer in the time domain. The delay-and-sum technique, here applied in the near field, seeks a phase delay by steering the array of microphones toward a particular point in space. A delay and an input weight are applied to the microphone recordings \(p(\textbf{x}_{i},t)\). The spatial domain of interest \(\partial \Omega \), is discretized by a mesh of control points. The delayed signals are summed to a beam response associated with each of the control points. If the acoustic source and control point coexist spatially, the delayed signals add constructively leading to constructive interference with the maximal beamformer response. According to Chiariotti et al. [11], the beamformer response \(b_{f}\) with respect to any control point \(\textbf{x}_{j}=\left[ x_{j},y_{j},z_{j}\right] \), \(j=1 \ldots N\) at spatial domain \(\textbf{x}_{j}\in \Omega \), and time t can be calculated by
where p is the acoustic signal amplitude, \(w_{i}\) is an amplitude weight and \(A_{i}\left( \textbf{x}_{j},\textbf{x}_{i}\right) =4 \pi \left\| \textbf{x}_{j}-\textbf{x}_{i}\right\| \) is a scaling factor to account for an amplitude reduction at the i-th microphone. The microphone recordings at each array location \(\textbf{x}_{i} \subset \partial \Omega \) are weighted, scaled, and delayed with respect to their relative distance to the control point \(\textbf{x}_{j}\), and then summed to yield the maximal value of the beamformer response \(b_{f}\left( \textbf{x}_{j},t\right) \) according to Eq. (4). Here, acoustic beamforming was applied to the model \(\textcircled {1}\) at \(z=0\) mm. The control points were discretized on a rectangular grid \(1 \times 1\) gp\(^{2}\) (1 gp \(\equiv \) 1 mm). The amplitude weight \(w_{i}\) was set to =1.0 for simplicity. The seven microphones located closest to the array center were used for analysis (same setup and input as for the TRM). Evaluated was the root mean square of the beamformer response
for visualization purposes of the maximum beamformer magnitude \(\bar{b}_{f,\max }\) displayed in Fig. 12a–h (bottom).
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