TY - JOUR
AU - Saenger, Erik, H
AB - Abstract We present in this work an ultrasonic imaging technique based on wavefield cross-correlation that has potential for high-resolution inspecting of plates and plate-like structures. A curved transmit/receive array setup is used for acquiring wide-aperture waveform data beneficial for applying the presented imaging condition. An additional dispersion analysis using waveform data received by a linear array allows one to reveal the excited guided wave modes and possibly determine the shear wave velocity as an input parameter in the 3D wavefield simulation. Using synthetic but realistic data generated by realistic 3D simulations, we demonstrate the invariance of the time reversal process and the capability of the presented imaging approach for ultrasonic testing based on Lamb waves. In addition to the ability to localise and size multiple defects simultaneously, this imaging approach does not require baseline data and involves only minimal waveform data pre-processing. ultrasonic imaging, Lamb waves, reverse-time migration, non-destructive testing 1. Introduction Ultrasonic guided waves are highly efficient among the physics-based nondestructive testing (NDT) techniques for detecting and sizing of damages in engineered structures. The efficiency of ultrasonic guided waves is due to the long-distance propagation of the guided stress waves and their highly sensitive interactions with damages such as cracks, delaminations, and other heterogeneities caused by material degradation. Chimenti (1997), for example, provides a concise summary of guided stress waves in plate-like structures and their applications in NDT. Increasing advancements in ultrasonic probing devices, modelling of the full ultrasonic wavefield and available computing power make it possible to solve challenging problem of imaging defects in condition monitoring of civil and industrial infrastructure. We are concerned in this work with ultrasonic imaging that delivers imaging results in the form of a quantitative 2D or 3D map of the defects in the test object. In respect of the underlying model of wave physics, ultrasonic imaging methods can be grouped into methods that consider simplifications of the wave physics (ray-path wave propagation, acoustic assumption for an elastic solid, and the low-scattering Born approximation, etc.) and methods that take into account the full elastodynamic wave propagation and wave interactions (dispersion, scattering, mode conversion, etc.) with heterogeneities. The commonly used ultrasonic NDT imaging methods have a number of limitations. The delay-and-sum imaging (Michaels & Michaels 2007) and other ray-based imaging methods such as synthetic aperture focusing technique (SAFT, Mayer et al. 1990), and total focusing method (TFM, Holmes et al. 2005) are limited to simple structures and defects in which direct and reflected wave paths can be approximated by straight rays, and resolve low-resolution images (of a wavelength scale). If applied to guided waves, those methods suffer even further from dispersion and multimodal propagation that require careful attention to the configurations for ultrasonic excitation and reception and possibly dispersion compensation in order to avoid inaccurate localisation of the defects. On the other hand, Bayesian and other probabilistic inference based imaging methods (Flynn et al. 2011) promise to deliver a holistic methodology for structural condition monitoring. However, such statistical quantification methods often require an abundance of measurements, sophisticated data pre-processing to extract damage-sensitive features and tedious calibrations of the statistical estimation scheme. One other ultrasonic imaging approach that is enabled by high spatial resolution laser scanning allows the mapping of defect locations by directly processing the full measured wavefield using, for example, the frequency–wavenumber analysis (Ruzzene 2007) or a standing wave filter (Sohn et al. 2011). However, this imaging technique is limited to detecting defects that occur closely underneath the laser-accessible surface only. A versatile and reliable ultrasonic imaging method is preferably based on modelling of the full elastodynamic wave propagation with an imaging algorithm that is free from local convergence issues related to solving an inverse problem. With regard to the use of measurements, the imaging method is desirable to obviate the use of baseline data and involves as little data and data pre-processing as possible. One such powerful method is the so-called reverse-time imaging that is based on cross-correlating the forward source wavefield with the time-reversed wavefield (or the back-propagated wavefield or reverse wavefield) re-emitted from the receiving locations. The back-propagated wavefield is formed by re-emitting the time-reversed recordings (recordings that are time-flipped) at the receiving locations (usually on the surface). The reverse wavefield propagates backward to where the waves have been emitted, reflected, and/ or scattered. During reverse wavefield back-propagation, if observed at the original wave emanating location, the wavefield is seen to be spatially focused and temporally compressed to reconstruct its past moments (Fink 1992). This time-reversal process (TRP) is invariant in lossless wave propagation media. The spectacular properties of the TRP soon made its way to NDT research and applications. Prada et al. (2002) demonstrated that the TRP of an array-measured ultrasound sequence can be numerically implemented for NDT imaging purposes. Saenger (2011) studied TRP using a rotated staggered grid finite difference method and implemented various imaging conditions to localise ultrasonic sources within a heterogeneous elastic solid. Using simulated data, Mori et al. (2019) showed that modelling the TRP for waveform data emitted by a transmitter and received by a limited number of receivers can locate a point-like notch damage on plates. However, the use of dual transducers that are needed in their procedure to selectively emit and receive either the fundamental S0 mode or A0 mode can be complicated to carry out in practice. This so-called time reverse modelling (TRM) approach seems well suited to detecting localised events such as acoustic sources and point-like scatterers. Resolving finite-size defects by TRM is not purposed by the imaging conditions. The so-called reverse time migration (RTM), which has its roots in seismic imaging using reflection data (Claerbout 1971), applies an imaging condition by cross-correlating the forward wavefield with the back-propagated (reverse) wavefield. Some recent work has demonstrated the superior capability of RTM for NDT applications to bulk ultrasonic waves (Beniwal & Ganguli 2015; Grohmann et al. 2016; Asadollahi & Khazanovich 2019) and guided ultrasonic waves (Wang & Yuan 2005; He & Yuan 2015; Nguyen et al. 2018). When applied to Lamb waves, however, Wang & Yuan (2005) is limited to the assumption that the propagating wavefront of the forward wavefield is non-dispersive. In those demonstrations of RTM for guided waves in plates, the use of Mindlin plate theory for approximating the 3D elastic solid of the plate constrains its applications in the low-frequency limits. Nguyen et al. (2018) applied the zero-lag cross-correlation based on 3D full-wave simulation for enhanced defect mapping in thin-walled pipes. RTM in the frequency domain has also been used for Lamb waves (Rodriguez et al. 2014). Recent studies on the use of RTM (Lu et al. 2019) and least-squares reverse-time migration (LSRTM; He et al. 2019) for ultrasonic defect mapping of thin plates have shown good potential of RTM for practical ultrasonic testing. He et al. (2019) especially have demonstrated successful RTM imaging on laboratory ultrasound measurement. The contribution in this work is twofold. First, we propose to use a curved array instead of a linear array for achieving enhanced imaging quality in thin-walled plates. With the proposed curved array setup of ultrasonic emitters and receivers customised for a wide-aperture acquisition of ultrasonic pulse-echo data, we demonstrate an enhanced imaging quality of a synthetic plate setup having multiple defects. Secondly, the application of RTM for Lamb waves is demonstrated on a full 3D model rather than a 2D or other simplified model. This is advantageous in simulating the realistic dispersive propagation of Lamb waves. In addition, an analysis on the effects of frequency content on the time reversal process of Lamb waves for the presented setup are provided. This paper is structured as follows. Section 2 details the full 3D model for realistic modelling of the guided wavefield in a plate. Also in this section, simulated ultrasonic waveform data are used to reveal dispersion characteristics of the excited Lamb modes. Section 3 studies the time reversal invariance of Lamb waves and details the use of the new imaging condition on Lamb waves. By using synthetic ultrasound data and a simulation model of a full 3D plate, we demonstrate in section 4 the imaging process and its efficiency in imaging defects of scattering and reflecting types in plate-like structures. Section 5 discusses the pros and cons of the imaging procedure in ultrasound NDT and concludes this work. 2. Modelling and dispersion analysis of Lamb waves 2.1. Full 3D simulation The ultrasonic displacement wavefields are simulated by numerically solving the elastic wave equation: $$\begin{equation} \rho \partial _t^2 \boldsymbol {u} = \nabla \cdot \boldsymbol{\sigma } + \boldsymbol {f}. \end{equation}$$ (1) In equation (1), ρ is the density distribution in the wave propagation medium and |$\boldsymbol {f}$| denotes the source time function. The stress tensor |$\boldsymbol{\sigma }$| is related to the small strain and material bulk and shear wave velocities by Hook’s law. In a waveguide such as for thin-walled plates, the elastic wave equation (1) is subjected to a traction-free boundary condition such that |$\boldsymbol{\sigma } \cdot \hat{\boldsymbol {n}} = 0$|⁠, where |$\hat{\boldsymbol {n}}$| is the unit vector outward normal to the free surface of the wave propagation medium. Among numerical methods for solving the elastodynamic wave equation, various finite difference schemes (Virieux 1986; Saenger et al. 2000), the elastodynamic finite integration technique (Schubert 2004), various commercially available finite element packages (Leckey et al. 2018) and the spectral element method (Komatitsch & Tromp 1999) have become attractive for large 2D and 3D wave simulations of various scales. These solvers can scale on a massive multi-processor computer cluster commonly available today. Some solvers, such as the rotated staggered grid finite different program (Saenger et al. 2000) and the spectral element solver SPECFEM3D (Komatitsch & Tromp 1999), even provide tomographic functionalities for reverse time simulation imaging. We choose to simulate the ultrasonic wave propagation in full three dimensions using SPECFEM3D for the efficiency of the spectral element method in handling traction-free surfaces and its coarser stability conditions on the spatial discretisation and the consequent time stepping. In addition, SPECFEM3D is able to simulate the reverse wavefield together with reconstructing the forward wavefield on-the-fly during the zero-lag cross-correlation. This ability provides nontrivial technical advantages against insufficient computer memory, heavy storage and I/O traffic in performing RTM imaging for large 3D models. The investigated numerical model setup is displayed in figure 1 with detailed material and geometrical properties listed in table 1. Data acquisition by emitters and receivers arranged in a curved array (a semi-elliptical array in particular) is chosen for three reasons. First, this setup allows the forward-propagating wavefield, if simultaneously excited by a number of semi-elliptically arranged emitters, to focus the ultrasonic wave energy automatically in the region of interest (ROI) in the closed semi-elliptical area on the plate. Secondly, this setup widens the angle for receiving waves reflected/ scattered from the defects in ROI which in turn helps in better resolving the defects. Thirdly, the semi-elliptical array also helps in reducing imaging artifacts caused by symmetric propagation of the ultrasonic wavefield about the linear array. Figure 1. Open in new tabDownload slide Spectral element model of a 4-mm thick plate with a semi-elliptical transmit/receive array (unit is in mm). The array elements are evenly distributed on a semi-elliptical curve having the two radii of 250 mm and 200 mm. In this illustration, the array covers an angle from −45° to 135°. Figure 1. Open in new tabDownload slide Spectral element model of a 4-mm thick plate with a semi-elliptical transmit/receive array (unit is in mm). The array elements are evenly distributed on a semi-elliptical curve having the two radii of 250 mm and 200 mm. In this illustration, the array covers an angle from −45° to 135°. Table 1. Properties of the square aluminum plate model. P-velocity S-velocity Density Edge lengths Thickness Vp (m s−1) Vs (m s−1) ρ (kg m−3) lx, ly (mm) lz (mm) 6148.9 3097.3 2700 1200 4 P-velocity S-velocity Density Edge lengths Thickness Vp (m s−1) Vs (m s−1) ρ (kg m−3) lx, ly (mm) lz (mm) 6148.9 3097.3 2700 1200 4 Open in new tab Table 1. Properties of the square aluminum plate model. P-velocity S-velocity Density Edge lengths Thickness Vp (m s−1) Vs (m s−1) ρ (kg m−3) lx, ly (mm) lz (mm) 6148.9 3097.3 2700 1200 4 P-velocity S-velocity Density Edge lengths Thickness Vp (m s−1) Vs (m s−1) ρ (kg m−3) lx, ly (mm) lz (mm) 6148.9 3097.3 2700 1200 4 Open in new tab For high-resolution imaging, the insonification of a high-frequency source is necessary given that the desired guided wave mode(s) is properly excited. This high-frequency excitation applies stability constraints on the spectral element size and time marching step size. In the high-frequency limit, say 400 kHz, we consider a phase velocity of 3000 m s−1; the minimum wavelength is therefore 7.5 mm. In addition, considering a very low phase velocity at the low-frequency limit of the fundamental axis-symmetric Lamb mode, a rather fine mesh is needed to model highly dispersive Lamb wave propagation. With a mesh composed of regular second-order spectral elements (5 × 5 × 5 grid points per element) as shown in figure 1, the mesh consists of 3 elements in the thickness and 768 × 768 elements in the other two planar dimensions. A total of 1769472 elements are distributed on 768 processors to compute the full 3D ultrasonic wavefield. The wavefield is stepped forward with a time step of 1.0E − 5 ms. 2.2. Dispersion image estimation Theoretical dispersion curves provide essential information about the severity of the dispersive behavior of the guided wave modes. Obtaining an image of the dispersion curves from the ultrasonic time-series recordings provides additional information about the actually excited guided wave modes and the corresponding phase velocities. The former can be used to design or adjust the excitation settings (frequency, force polarisation, etc.) and the latter information can help to retrieve a shear wave velocity required in wave propagation modelling. As is often the case in NDT practice, the information about solid bulk and shear wave speeds of the tested structure may not be available, or the material properties have altered due to loading and environmental conditions. The ability to estimate the bulk and shear wave velocities of the measured waveforms makes it possible to perform the entire imaging procedure without prior knowledge about the material’s elastic properties. Park et al. (1998) introduced an efficient phase shift method for imaging the dispersion curves using a limited number of the time-domain waveforms, u(x, t), recorded at discrete positions x and times t by a linear array. The method applies the phase shift operator on the time-frequency Fourier transformed data U(x, f ): $$\begin{equation} U(c,f) = \int e^{i\big(2\pi {\frac{f}{c}}\big)x} \,\, \frac{U(x,f)}{\vert U(x,f)\vert } \,\, dx. \end{equation}$$ (2) In equation (2), for any given frequency f, phase velocity c is spanned over a range of interest. By applying an offset-dependent phase shift |$2 \pi \frac{f}{c} x$|⁠, the propagating phase velocities (corresponding to the excited wave modes) are found at maximal values of U(c, f). Repeating the calculation for all frequencies, an image of the actual dispersive characteristics of the wavefront propagating along the linear array are depicted. In the NDT context, in which the number and spatial coverage of ultrasound receivers is limited, the phase shift method is more advantageous than the frequency–wavenumber analysis by applying a two-dimensional Fourier transform (Alleyne & Cawley 1991) in revealing the multimode characteristics in the array data. We model the same 4-mm thick aluminum plate as shown in figure 1 using the same material properties listed in figure 1. A linear array of receivers pointing to the center of the semi-eliptical excitation array is set up as displayed figure 2a. A broadband Ricker wavelet at 200 kHz center frequency is simultaneously applied to all array elements in the direction perpendicular to the plate. Figure 2b plots the out-of-plane waveform data recorded in the linear array. The imaged dispersion curves plotted in figure 2c reveal that both of the fundamental A0 and S0 Lamb modes are excited. At the high-frequency corner of the A0 dispersion curve, the Rayleigh wave velocity, which is approximately |$92\%$| of the value of the bulk shear wave velocity, can be picked. Figure 2. Open in new tabDownload slide The data acquisition setup with a broadband source centered at 200 kHz (a) and the resulting corresponding waveform data (b) used for constructing the dispersion image (c). Figure 2. Open in new tabDownload slide The data acquisition setup with a broadband source centered at 200 kHz (a) and the resulting corresponding waveform data (b) used for constructing the dispersion image (c). 3. Wavefield cross-correlation imaging condition 3.1. Effects of frequency content on time reversal process Let us experiment using a guided wave mode that is excited at point A and is received at point B. The received time-series waveform at point B, of which the particle displacement is measured in the same direction as of the excitation, is time-flipped and re-emitted there. Then, if we measure the waveform at point A, a wavelet of a shape similar to the original excitation is reconstructed by the wave propagation process itself. Interestingly, the dispersion of Lamb waves is also automatically compensated in TRP (Ing & Fink 1998; Park et al. 2007). In the case of multi-mode excitation, sidebands result around the reconstructed signal (Park et al. 2009). Using the Mindlin plate theory, Park et al. (2009, 2007) showed that a broadband Gaussian pulse cannot be fully recovered by TRP due to limited recording length. We re-examine here the time reversal process of Lamb waves by using the full 3D simulation tool. The same plate model as previously described is studied except that the edge lengths of the square plate is extended to 1600 mm so that reflections from the plate boundaries do not interfere with the direct waves. We examine here the effects of frequency content on the reconstruction quality of the pure A0 mode. Two numerical experiments with narrowband input and broadband input are respectively carried out. The narrowband and the broadband excitation signals are the 5.5-cycle Hanning windowed toneburst and the Ricker wavelet, shown respectively in figures 3a and 3d. Both input signals are centered at 100 kHz. At a distance of |$600 \sqrt{2}$| mm away, the A0 mode output signal is measured at half the thickness of the plate (figures 3b and 3e). The use of broadband input results in more significant dispersion of the propagating waves (figure 3e). Figure 3. Open in new tabDownload slide TRP of a narrowband source wavelet (5.5-cycle Hanning windowed toneburst) and a broadband source wavelet (Ricker wavelet). Subplots (a), (b), and (c) are, respectively, the input signal at point A, the response at point B, and the reconstructed signal by the TRP of the narrowband excitation. Subplots (d), (e), and (f) are, respectively, the input signal at point A, the response at point B, and the reconstructed signal by the TRP of the broadband excitation. All amplitudes are normalised. Figure 3. Open in new tabDownload slide TRP of a narrowband source wavelet (5.5-cycle Hanning windowed toneburst) and a broadband source wavelet (Ricker wavelet). Subplots (a), (b), and (c) are, respectively, the input signal at point A, the response at point B, and the reconstructed signal by the TRP of the narrowband excitation. Subplots (d), (e), and (f) are, respectively, the input signal at point A, the response at point B, and the reconstructed signal by the TRP of the broadband excitation. All amplitudes are normalised. At this point the received signal is time-flipped and re-emitted at the receiving point. We now observe at the original excitation point (also at half-thickness of the plate). The narrowband signal has been well recovered, as can be seen in figure 3c. The strongly dispersed signal (figure 3e) has also reversed the dispersion behavior, in which low-frequency content of the A0 mode travels more slowly than the high-frequency part on its way back and becomes focused at the original source point (figure 3f). However, compared to the original input, the reconstructed signal has lost part of its content. The mismatch in the reconstructed broadband signal is also seen in the frequency domain figure 4. Because the very low-frequency part of the A0 Lamb mode propagates with very low velocity (see dispersion curves in figure 2c), reconstruction of this part is difficult if the output signal (figure 3e) is recorded just for a short time duration. For the wavefield cross-correlation imaging, the frequency-dependent time-reversal property of Lamb waves has direct effects on the reverse wavefield, which is formed by superimposed wavefields simulated by simultaneously exciting sources at an array of emitters. A narrowband wavelet implies well-constructed reverse guided wavefield. However, this does not necessarily mean better imaging quality compared to the use of a broadband wavelet, due to the occurrence of long-duration wave-packets (Nguyen et al. 2018). Figure 4. Open in new tabDownload slide Frequency spectra of the TRP reconstructed wavelets and their corresponding original wavelet spectra: (a) narrowband case and (b) broadband case. Figure 4. Open in new tabDownload slide Frequency spectra of the TRP reconstructed wavelets and their corresponding original wavelet spectra: (a) narrowband case and (b) broadband case. 3.2. The imaging condition The wavefield cross-correlation imaging condition introduced here for the localisation and sizing of defects in plates is reminiscent of the reverse-time migration technique widely used in seismic imaging Claerbout (1971). In an NDT context, this imaging method has been shown to be successful for concrete testing using the bulk ultrasonic waves Beniwal & Ganguli (2015); Grohmann et al. (2016). Also using ultrasonic bulk waves for NDT, a concise description of RTM and the influence of the used velocity model in the imaging condition of RTM is reported in Nguyen & Modrak (2018). To be complete, we restate the principle of this wavefield cross-correlation here. This model-based imaging procedure begins with an approximate model of the test object. Because we investigate a homogeneous and isotropic elastic plate, the wave propagation model used in the imaging step is an intact homogeneous plate model. The density and Vp and Vs velocities, as well as the excitation time function, are inputs to the elastic wave solver. Those quantities can be directly measured or approximated by an indirect method. The imaging condition is a cross-correlation between the two simulated wavefields following the equation: $$\begin{equation} \boldsymbol {I}(\boldsymbol {x}) = \int \nolimits _{t=0}^{T} \boldsymbol {s}(\boldsymbol {x},t) \boldsymbol {r}(\boldsymbol {x},T-t) {\rm d}t\, , \end{equation}$$ (3) where |$\boldsymbol {s}(\boldsymbol {x}, t)$| is the forward wavefield simulated by applying the source time function at the emitter(s) and |$\boldsymbol {r}(\boldsymbol {x}, T-t)$| is the reverse wavefield simulated by re-emitting the time-flipped recorded ultrasonic waveforms at the receivers. Here, |$\boldsymbol {x}$| denotes discrete coordinates in the 3D ultrasonic wave propagation medium and t is the temporal period of the ultrasonic data acquisition, t ∈ [0, T]. Note that the imaging condition equation (3) is alternatively expressed by |$\boldsymbol {I}(\boldsymbol {x}) = \int \nolimits _{t=0}^{T} \boldsymbol {s}(\boldsymbol {x},T-t) \boldsymbol {r}(\boldsymbol {x},t) {\rm d}t$|⁠, which can provide advantages in the implementation of the wavefield cross-correlation. The obtained image |$\boldsymbol {I}(\boldsymbol {x})$| is a 3D reflectivity map showing spatial intensities of material defects that cause wave reflections and scatterings. We note that, although the simulated wavefields are three-dimensional, in our application of RTM for Lamb waves only the out-of-plane component of the wavefields is subject to the imaging condition equation (3). The use of single component wavefields requires simply a product of the forward and reverse wavefields at every discrete point rather than an inner-product operation as applied for RTM of the vector wavefields (Shi et al. 2019). RTM has long been used and is still the main tool in current seismic imaging (Claerbout 1971). This method has recently shown potential for high-resolution medical imaging (Roy et al. 2016) and rock core sample examination (Shragge et al. 2015). Ultrasonic imaging in an engineering NDT context has also progressed (Grohmann et al. 2016; Nguyen & Modrak 2018). Although there are shared attributes among various applications, the RTM for testing engineered materials and structures has its own characteristics and challenges worth mentioning: The number of emitters and receivers is often limited for inexpensive data acquisition. Data coverage is often limited owing to limitations in access surface for placing ultrasonic measurement devices. Engineered materials and structures are often composed of various components having high contrast in wave velocities (e.g. concrete), making strong wave scattering. Many materials are viscoelastic and anisotropic (e.g. composites). Test structures are often limited in size and therefore suffer from reflections from free surfaces and structural heterogeneities around. Many structures are thin waveguides, leading to strong wave dispersion and multimodal wave propagation. We note that the wavefield cross-correlation is central for ultimately building the velocity maps following the principle of full-waveform inversion (FWI, Virieux & Operto (2009). By defining a least-squares misfit function, the adjoint wavefield is formed by re-emitting the data residual into the model (Tromp et al. 2005). The parameter sensitivities are then calculated by cross-correlating the forward wavefield with the adjoint wavefield. In this case, the only difference between RTM and the adjoint method used to form the sensitivity kernels in FWI is the data to re-emit at the receivers in the time-reversed simulation. FWI has shown promise in challenging NDT applications (Seidl & Rank 2016; Jalinoos et al. 2017; Nguyen & Modrak 2018). However, the computational cost of FWI is at least one or two orders of magnitude more than that of RTM. 4. Synthetic verification In section 3, we have examined the time-reversal process and presented the wavefield cross-correlation imaging condition. It can be derived from there that the ideal conditions for the imaging method to work well are: (1) wide-aperture receivers are used to collect the reflected waves, and (2) ultrasonic excitation is designed to avoid the generation of multiple Lamb modes as much as possible. We have also showed that a narrowband excitation is better than a wideband excitation for dispersion recovery in the reverse wavefield simulation. For the wavefield cross-correlation imaging condition, however, the long wave-packets resulting from a narrowband excitation lead to increased imaging noise for a waveguide (Nguyen et al. 2018). Therefore, in verifying the zero-lag wavefield cross-correlation imaging condition for Lamb waves here, we use a broadband Ricker wavelet at 120 kHz center frequency as an excitation function. We examine the quality of the presented ultrasonic imaging condition by using a supershot setup in which all elements in the array can simultaneously fire. This acquisition setup is based on the capability that a dense semi-elliptical array can both transmit and receive ultrasonic signals. Since the number of wavefield cross-correlations required in the imaging step is equal to the number of ultrasound shots, the wavefield cross-correlation for the supershot setup is executed only once. The reader can refer to Nguyen et al. (2018) for detailed discussion on the effects of the supershot setup on the use of the wavefield cross-correlation method on guided waves. The same plate model and array setup as described before (figure 1 and table 1) are used for the verification of the imaging condition here. The two damages in the plate are simulated as having material loss for two thirds of the plate thickness and have openings on the other side from the inspection side. The transmit/receive array is placed on the plate surface and covers a planar angle of 180°. The damage locations are shown in figure 5a in relation to the curved array position. A zoomed-in view on the ROI around the curved array is displayed in figure 5b. The imaging results of this setup hereafter will be shown in this region. Figure 5. Open in new tabDownload slide The reference plate with damages and the curved array for firing and receiving of ultrasound data (a, b). Synthesised waveform data recorded by 97 evenly spaced receivers in the semi-eliptical array covering an angle from −45° to 135° on the reference plate: (c) unprocessed data and (d) data after source muting and time reversing. Figure 5. Open in new tabDownload slide The reference plate with damages and the curved array for firing and receiving of ultrasound data (a, b). Synthesised waveform data recorded by 97 evenly spaced receivers in the semi-eliptical array covering an angle from −45° to 135° on the reference plate: (c) unprocessed data and (d) data after source muting and time reversing. The waveform data are recorded by 97 receiving elements equidistantly placed in the array as shown in figure 5c. The source signature at an early time in the data is simply muted so that all other events in the data are reflections from defects. The time-reversed data that are used to extrapolate the reverse wavefield in the imaging step are plotted in figure 5d. As is examined in the dispersion analysis, for this out-of-plane firing, the anti-symmetric Lamb mode A0 is dominant. Still, the existence of the symmetric Lamb mode S0 is unavoidable due to the firing itself and the mode converted wave scattered from defects. Re-emitting scattered/reflected events in the acquired waveforms in the homogeneous plate model in turn excites both the fundamental Lamb modes in the reverse wavefield. To reduce imaging noise caused by mixed-mode interactions, only the out-of-plane components in the respective forward wavefield and reverse wavefield are subject to a cross-correlation. The imaging result in figure 6b for the current array position shows that both damages are well captured. Rotating the curve array to different angular positions and performing the same imaging procedure provides further images of the defects with other illumination angles (figures 6c and 6d). Stacking the three individual results in a single image greatly emphasises the defects and lessens imaging noise as is seen in figure 7. The full 3D stacked image is shown in figure 8. It can be observed that both the L-shape and the point-like defects are well reconstructed. However, the imaging results also come with large strips of imaging artifacts which are caused by the slow propagating waves of the A0 mode in the low-frequency content. Figure 6. Open in new tabDownload slide The reference damages (a) and wavefield cross-correlation images for various supershots: (b) shot 1, (c) shot 2, and (c) shot 3. Note, only reflectivity values at the middle plane of the 3D image of the plate are shown. Figure 6. Open in new tabDownload slide The reference damages (a) and wavefield cross-correlation images for various supershots: (b) shot 1, (c) shot 2, and (c) shot 3. Note, only reflectivity values at the middle plane of the 3D image of the plate are shown. Figure 7. Open in new tabDownload slide The reference damages (a) and the stacked image (b) resulting from the three individual supershots. Note, only reflectivity values at the middle plane of the 3D image of the plate are shown. Figure 7. Open in new tabDownload slide The reference damages (a) and the stacked image (b) resulting from the three individual supershots. Note, only reflectivity values at the middle plane of the 3D image of the plate are shown. Figure 8. Open in new tabDownload slide The full 3D stacked image. Note that only the surface of the plate can be seen. Figure 8. Open in new tabDownload slide The full 3D stacked image. Note that only the surface of the plate can be seen. Reflections from the plate boundaries in this case do not hinder the reconstruction of the true defects. However, we note that reflections coming from outside the circular array area (plate boundaries and other heterogeneous features), if coincident with waves reflected from defects within the curved array ROI, may hamper the reconstruction quality of the imaging method. Possible solutions are either to perform various imaging in order to get good knowledge of the surroundings or to develop directional array elements that emit guided waves to the ROI area only. Lastly, to compare the quality of the wavefield cross-correlation imaging resulting from the proposed setup and a linear array setup, we bend the transmit/receive elements for the setup shown in figure 5a (replotted in figure 9b) to a linear array (figure 9a). The wavefield cross-correlation imaging procedure is then applied in the same manner. The imaging result of the linear array is displayed in figure 9c. Compared to the image resulting from the curved array (replotted in figure 9d), the one resulting from the linear array shows significant imaging artifacts. The curved array gets rid of symmetric imaging artifacts caused by bidirectional wave propagation symmetric about a linear array. Using a curved array, imaging artifacts outside of the semi-elliptical area becomes negligible as they are smeared out and much less intense in amplitude. Figure 9. Open in new tabDownload slide Imaging quality comparison between the linear array setup (a) and the curved array setup (b). The resulting wavefield cross-correlation images are shown in panels (c) and (d), respectively. Figure 9. Open in new tabDownload slide Imaging quality comparison between the linear array setup (a) and the curved array setup (b). The resulting wavefield cross-correlation images are shown in panels (c) and (d), respectively. 5. Conclusions As the full wave physics is taken into account, the wavefield cross-correlation imaging method is robust and accurate. Given the relatively known wave velocities and high signal-to-noise ratio in NDT settings, the presented ultrasonic imaging method has been given the good conditions to be widely applied in NDT practice . The ability to resolve high-resolution reflectivity images makes the presented method powerful for challenging NDT tasks. The computation burden associated with simulating the forward and reverse wavefields seems to be the current bottleneck to implement the imaging algorithm on a portable device. Running on conventional computing processors, it is ideal for applications where the imaging can be remotely performed on a computer cluster. Continued progress in accelerated computing devices and wavefield modelling techniques may give rise to quasi-real-time wavefield cross-correlation imaging in the near future. The use of only out-of-plane transmitting and receiving eases the data acquisition process. For this acquisition setup, the use of out-of-plane components of the simulated wavefields in the imaging step results in sharp images of the defects, although there exists mixed-mode interactions. With continued advances in ultrasound actuating and sensing technologies, it may allow to emit and receive a single Lamb mode in a curved array, most preferably the non-dispersive SH0 mode, which helps greatly to enhance the imaging quality of the presented method further. Acknowledgements This work is supported by the “Förderprogramm FH ZEIT für FORSCHUNG” by the Ministry for Innovation, Science and Research of the North Rhine-Westphalia State Government of Germany. In addition, the authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Forschungszentrum Jülich. Conflict of interest statement. 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TI - Guided ultrasonic wavefield cross-correlation with a curved array for high-resolution plate inspection
JF - Journal of Geophysics and Engineering
DO - 10.1093/jge/gxaa004
DA - 2020-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/guided-ultrasonic-wavefield-cross-correlation-with-a-curved-array-for-w8v3077IcB
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -