Passive imaging in composite plates using Green’s function reconstruction from the diffuse field

Aiming at the problem that enables damage detection in real-time without any active excitation sources, we propose a new passive detection way based on diffuse field for structural health monitoring of CFRP structures. In this work, the wave propagation was first simulated with a k-space pseudo-spectra method in CFRPs. A diffuse field can be generated after multiple scattering from fibers and boundary reflections. After that, the reconstruction of Green’s function was investigated through different parameters including the number of noise sources, pulse bandwidth, transducer distance, etc. The delamination inspection process was explored by contrasting the reconstructed Green’s function with the referenced Green’s function on the composite plate. Finally, a probability distribution-based imaging algorithm was further applied for delamination imaging.


Introduction
Carbon Fiber Reinforced Plastics (CFRP) composite materials have merits such as great specific strength, excellent heat resistance, outstanding corrosion resistance, and exceptional thermalmechanical performance.They have become a crucial material in aerospace, automotive, and renewable energy sectors [1] .However, due to fatigue and impact damage, CFRPs are susceptible to delamination, cracking, and other defects in service or manufacturing which significantly affect the material properties.The defect detection is thus crucial for the quality insurance and reliability of CFRPs [2] .
In environments with multiple scattering or reverberation, multiple reflections or scattering are more likely to create diffuse fields.Methods based on reconstructing Green's functions in diffuse fields not only allow real-time monitoring of machinery with noise-like signals but can also make passive structural health monitoring possible for aerospace and marine industries.Such a method finds extensive applications in seismology, underwater acoustics, and ultrasonics.
In the past decade, this method has been widely used for defect detection in isotropic materials.Duroux [3] studied damage detection in complex aluminum plates by cross-correlation of elastic diffuse fields.Chehami [4,5] used numerical simulations and experiments to locate defects based on the differences between the obtained correlation matrices in the presence and absence of defects.In addition to these reference-based methods, Tippmann [6] proposed the reciprocity of the positive and negative Green's functions as a measure of defect.The reciprocity will be broken by the nonlinearity from damage.However, the imperfect diffuse field reconstruction during experiments will bias such methodology.
Short-fiber carbon fiber composite materials are one category of multiple scattering media.Ultrasonic waves within the plate continuously scatter and reflect between the short fibers, facilitating the formation of a diffuse field.This study aims to explore the possibility of using diffuse field crosscorrelation methods for defect localization and imaging in composite material plates.

Theoretical background
Multiple scattering or reverberation occurs when the wavelength of the sound wave is comparable to the size of many microstructures in the CFRPs plate.Multiple scattering or reverberation can help make the distribution of noises more homogeneous to get a more accurate Green's function estimation.Considering the equation in the diffuse field [7] , ( , ) ( ( ) ( , ) ( , )) p t r h D r p t r q t r t By cross-correlating the waveforms received at the two points, we obtain the cross-correlation function.The definition of the cross-correlation of the signals which got at two places is Equation (2) [6] , , , 0 ( ) ( ) ( ) After many averages, we can get a completed cross-correlation function defined as Equation ( 3) where represents cross-correlation functions after many averages.Eventually, the cross-correlation function is linked to the Green's function [7] .

( ) ( ( , , ) ( , , ))* ( )
The feature of the reconstruction of Green's function can be evaluated using the similarity coefficient C between the cross-correlation function and the transient response according to Equation ( 5).The higher the coefficient, the better the quality of the reconstruction.( ) ( ) where G(t) is a modified transient response by convolving the transient response with ( ) ) et is the impulse excitation function [5] .
Introducing a defect can be realized by changing the parameters of the medium.In this paper, an imaging algorithm based on probability density is used to image defects.We can calculate the probability for each path, and then sum up the probabilities for all paths to get an overall probability distribution according to Equation (6).

Simulation Media
The simulation model and transducer distribution are displayed in Figure 1 because the length and width are much bigger than the thickness of the mode and the wavelength is greater than the thickness.This three-dimensional model is clear into a two-dimensional model along the z-axis orientation by averaging.At this time, the sound wavelength is greater than the average free path, which can be considered to have obtained sufficient scattering.The carbon fiber is mixed with resin to form a carbon fiber layer.The propagation process of the complete diffuse of the wave is shown in Figure 2.

Simulation parameter settings
We conducted numerical simulations using the finite difference algorithm based on MATLAB.The setup for the numerical simulation is shown in the following part, with a computation grid of 128 × 128, and each grid size is fixed at 0.0012 × 0.0012 mm 2 .In the simulation, the excitation source we used was a tone burst with a central frequency of 0.5 MHZ.Reducing the time step can ensure the stability of the simulation and reduce numerical errors, but it also extends the computation time and demands more computational resources.To strike a balance between model stability and computation duration, according to the Courant-Friedrichs-Lewy (CFL) condition, we ultimately set the time step at 1.2 × 10 - 8 s.At this point, CFL=0.1.The model boundary is set as a perfectly matched layer (PML).First, a pitch-catch scheme is applied to obtain the transient response as shown in Figure 2, in which one source emits at R1 and receives at R2.Then, by emitting one source at a time, waveforms are collected at the two points and the cross-correlation function can be calculated depending on Equation (2).The completed cross-correlation function can be obtained after ensemble averaging of N crosscorrelation functions.

3.3.1
The number of sources.The number of sources is changed from 1 to 300, and according to Equation (3), the respective cross-correlation functions can be obtained.By calculating the correlation coefficient according to Equation ( 5), we found that as the number of sources increases in Figure 3, the similarity coefficient becomes higher, leading to improved reconstruction quality.

3.3.2
The bandwidth of the excitation signal.It can be observed that in aluminum plates, the larger the bandwidth, the better the reconstruction quality.However, in composite material plates, the wavelength should be in the same order as the length of the scatterer.As the bandwidth reduces to a size around the wavelength, the reconstruction quality improves.When the bandwidth gradually decreases to a range equivalent to the wavelength, the computed C between the transient signal and the cross-correlation function also rises from 0.31 to 0.82 which is displayed in Figure 4 and Figure 5.

Imaging map
In numerical simulations, to detect defects, we set up 6 receivers, so we can get fifteen sensing paths.Then we sum the probability of the fifteen paths.The final defect probability distribution is displayed in Figure 6.

Conclusion
In the work of this paper, the applicability of Green's function reconstruction method in defect detection has been validated for short CFRPs.The effects of the number of noise sources, pulse bandwidth, and the transducer distance on the feature of Green's function reconstruction have been analyzed.Furthermore, the defect can be detected successfully using the imaging algorithm based on probability density.In the future, the features of Green's function reconstruction and the accuracy of various imaging algorithms will be explored experimentally for accurate defect localization.

Figure 1 .
Figure 1.Simulation model and transducer distribution.

Figure 3 .
Figure 3.The effect of source numbers on the feature of Green's function reconstruction.

Figure 4 .
Figure 4. (a) Excitation signal with two periods (b) The spectrogram with two periods (c) The superposition of cross correlation function and transient response with two periods.

Figure 5 .
Figure 5.The effect of bandwidth on the feature of Green's function reconstruction.

Figure 6 .
(a) Imaging map (b) Threshold image (The black circle represents the actual defect location).