High-resolution thickness maps of corrosion using SH1 guided wave tomography

Abstract

Quantifying corrosion damage is vital for the petrochemical industry, and guided wave tomography can provide thickness maps of such regions by transmitting guided waves through these areas and capturing the scattering information using arrays. The dispersive nature of the guided waves enables a reconstruction of wave velocity to be converted into thickness. However, existing approaches have been shown to be limited in in-plane resolution, significantly short of that required to accurately image a defect target of three times the wall thickness (i.e. 3 T) in each in-plane direction. This is largely due to the long wavelengths of the fundamental modes commonly used, being around 4 T for both A0 and S0 at the typical operation points. In this work, the suitability of the first-order shear-horizontal guided wave mode, SH1, has been investigated to improve the resolution limit. The wavelength at the desired operating point is significantly shorter, enabling an improvement in resolution of around 2.4 times. This is first verified by realistic finite-element simulations and then validated by experimental results, confirming the improved resolution limit can now allow defects of maximum extent 3T-by-3T to be reliably detected and sized, i.e. a long-pursued goal of guided wave tomography has been achieved.

1. Introduction

The accurate estimation of the remnant wall thickness of pipelines and storage vessels in the presence of corrosion damage is highly important for making service-life prediction in the petrochemical industry. This is especially the case at locations with limited access such as pipe-supports, where corrosion under pipe support (CUPS) is one of the more common causes of pipe failure. Guided wave tomography has been proposed as a solution to this problem by reconstructing thickness maps from guided wave signals that were transmitted through the region of interest using the dispersive nature of certain guided wave modes. It is of great importance to industry to reliably size defects of size 3 T × 3 T, where T is the nominal wall thickness [1]. It is noted that this measure does not make any assessment of defect shape, so the actual in-plane resolution would need to exceed 3 T to capture the necessary complexity present in true corrosion defects to establish maximum depth. In terms of depth resolution, estimates are required to be within 10% of wall thickness, i.e. 0.1 T; while this itself has been achieved through various approaches, the necessary in-plane resolution to achieve this accuracy in practice places a significant requirement on the capabilities of the technology, and one which has not been reliably met up to this point.

The typical approach in guided wave tomography is to use the measured signals to reconstruct a map of velocity. Each velocity value can then be directly mapped to a corresponding thickness value by the guided wave dispersion curves, ultimately producing the required map of thickness. This has been the basis of many approaches in the past, mainly using Lamb guided waves [2–7].

The underlying assumption here is that guided waves scatter from varying thickness in the same manner as acoustic waves in a two-dimensional medium would scatter from the corresponding change in the velocity field. This is a reasonable assumption provided the thickness variation is gradual, such that the dispersion curves, derived assuming constant thickness, can capture the behaviour. As thickness varies faster, Huthwaite [8] showed that the approach cannot capture the guided wave scattering behaviour, which manifests itself as a resolution limit of around two wavelengths for the A0 mode at 50 kHz, which corresponds to 8T in-plane resolution.

There are two options to improve the resolution further. The HARBUTISM [9] approach develops a more sophisticated model which aims to capture the guided wave scattering behaviour more accurately than the dispersion curve and hence enable higher resolution. Francis Rose et al. [10–12] have developed Mindlin plate theory along similar lines for both composite delamination and plate corrosion damage. These approaches are typically more complex and less robust, since they are sensitive to discrepancies in the modelled behaviour of guided wave scattering [13,14].

The other option is to select a guided wave mode and frequency such that the wavelength becomes shorter, which typically would improve resolution. There are practical issues around this, such as the existence of other guided wave modes which can pollute the signal and a typical increase in attenuation (see [15] for a study of various operating frequencies and modes).

In this paper, we consider the SH1 guided wave mode, the first-order shear horizontal mode. This operates at higher frequency than the fundamental modes, and the shear form of the wave avoids much of the attenuation associated with liquid loading which is problematic for most usable Lamb modes. Also, despite being a higher-order mode, it is relatively straightforward to extract pure measurements since there are relatively few SH modes; below the SH2 cut-off, just SH1 and SH0 exist, and SH0 can typically be separated easily because it has a very different wave speed and is non-dispersive.

The SH1 mode should therefore enable guided wave tomography to achieve its potential of imaging defects of 3 T × 3 T reliably and robustly for the first time. This paper aims to demonstrate this. It is recognized that the SH1 cut-off will present issues for deeper defects; this can be addressed by moving to higher frequencies. There will be complications with additional modes which need separating at these operating frequencies, however, solutions to this will be considered in future work and the issue is considered beyond the scope of this paper which aims to demonstrate the underlying potential of SH1.

Section 2 will provide background on guided wave tomography and the algorithm that was chosen for thickness inversion. Section 3 will present the numerical model that was used for pure SH1 excitation and demonstrate the suitability by reconstructing various defects from pure SH1 point source data generated from three-dimensional finite-element simulations. Next, a more realistic, directional excitation of SH guided waves is then explained and the same defects reconstructed from numerical data. In §4, the method is tested and validated experimentally by reconstructing thickness maps from defects that were machined into mild-steel plates. Finally, §5 will present results from a numerical study, comparing the performance of the fundamental Lamb wave modes A0 and S0 with SH1 guided wave tomography of axisymmetric defects of varying width.

2. Background

(a). Guided wave tomography

The goal of guided wave tomography is to reconstruct a thickness map of a region of interest (ROI) using ultrasonic guided waves which are transmitted through the ROI and received at various receiving positions of an array placed around it (figure 1). This reconstruction is possible since dispersive guided waves modes are dependent on the product of frequency and thickness and by fixing frequency, their group or phase velocity become thickness dependant. Typically, the aim is to generate a velocity map from the measurements and then the dispersion relationship can be used to convert that velocity map to thickness. In figure 2a, the phase velocity dispersion curves for shear-horizontal guided waves (SH) and the two fundamental Lamb wave modes are plotted. As highlighted, we will use the SH1 mode, which has shorter wavelength than the Lamb waves typically considered, meaning it has potential for improved resolution.

Figure 1.

Schematic of the full view guided wave tomography layout employed in this paper: We model a circular array of radius R on a plate. The transducers are used for excitation as well as reception. The arrow pointing towards the defect indicates an excitation event at a transmitting transducer Tx. A defect is placed in the centre of the array. The incident waves are interacting with the defect and the scattered waves (arrows pointing away from the defect) are picked up by the receiving transducers. (Online version in colour.)

Figure 2.

(a) Phase velocity dispersion curves for Lamb and shear-horizontal guided waves in a steel plate. SH guided wave mode shapes for (b) SH0 & (c) SH1 for a single frequency. The grey arrows in (b) and (c) indicate the direction of propagation. (Online version in colour.)

The mode shapes of the fundamental SH mode (SH0) and of the first higher-order mode (SH1) are shown in figure 2b,c. SH guided waves do not have any out-of-plane displacement, which makes them less susceptible to attenuation due to liquid loading, since, neglecting viscosity, no part of the shear motion can be transmitted into the liquid. The fundamental SH0 mode is non-dispersive and cannot be used for thickness inversion. The first anti-symmetric mode, SH1, does not exist below a cut-off frequency-thickness product of f × T ≈ 1.6 MHz mm and in contrast is highly dispersive, making it a promising candidate for thickness inversion. The SH phase velocity c_p and group velocity c_g are defined as

$$c_p=\pm 2\hspace{0.17em}{c}_{\text{sh}}\left(\frac{f\hspace{0.17em}{T}_0}{\sqrt{4{(f\hspace{0.17em}{T}_0)}^2-n^2{c_{sh}}^2}}\right)$$ 2.1

and

$$c_g=c_{\text{sh}}\sqrt{1-\frac{{(n/2)}^2}{{(f\hspace{0.17em}{T}_0/c_{sh})}^2}},$$ 2.2

where c_sh is the shear wave velocity of the specimen, T₀ the nominal thickness and n is the order of the respective SH guided wave mode [16].

Since going to even higher frequencies would lead to a multitude of modes being present, we limit our investigation to this mode, remaining below the SH2 cut-off frequency. Higher-order Lamb modes are also not considered because they are too challenging to excite in a sufficiently pure manner.

In order to generate the velocity map which can then be converted to thickness, a velocity inversion algorithm is needed and, typically, this assumes an acoustic wave behaviour. One of the simpler algorithms is to model the waves as rays and to use their arrival times for estimating the velocity. While this can allow a defect to be located, the resolution of this approach is very poor as it neglects diffraction which occurs when the scattering object is small compared with the wavelength. Many approaches in tomography have used ray theory, including Hinders [17] [18], Volker [4], Simonetti [19], Belanger [20].

Another possibility is the use of the diffraction tomography (DT) algorithm. This is, however, limited by its underlying use of the Born approximation which limits the contrast and size of the object which can be reconstructed [8,21]. The HARBUT algorithm (hybrid algorithm for robust breast ultrasound tomography) has been shown to have good performance for guided wave tomography velocity inversion [8]. In essence, HARBUT is a modification of DT: DT produces a reconstruction by recognizing that the wave scattering process, under the Born approximation and due to the sinusoidal nature of waves at a single frequency, forms a Fourier transform of the scatterer, thus enabling an inverse Fourier transform to be used to obtain an image of the scatterer. HARBUT modifies DT to incorporate an inhomogeneous background from bent ray tomography, accounting for the wave aberrations with an eikonal solver, ultimately increasing the contrast and size of objects which can be imaged. Thus, by combining the complementary strengths of bent ray and DT, the algorithm is capable of achieving high-resolution results at very low computational intensity. HARBUT has been shown to be robust for a huge range of different defects across different modes and with different frequencies [22].

The highly dispersive nature of the SH1 mode would traditionally limit the propagation distance of the wave. However, since HARBUT operates in the frequency domain rather than the time domain where dispersion would be an issue, we bypass this challenge. The maximum propagation distance depends on energy lost from the wave; dispersion does not remove energy but simply redistributes it in time and frequency domain methods are insensitive to this. It is for these reasons that HARBUT has been chosen as the algorithm for velocity reconstruction in this investigation. In the following part, the modelling approach of pure SH1 excitation is explained and results from SH1 point source data presented.

3. Numerical demonstration of SH1 guided wave tomography

(a). Physical configuration

As curvature can be neglected for the pipe-radii of interest [23], the investigation was simplified by considering corrosion patches in 10 mm thick mild-steel plates (E = 210 GPa, ν = 0.3125, ρ = 8000 kg m⁻³). For excitation, a 5-cycle Hann-shaped pulse at a centre-frequency of f = 250 kHz was used to create the experimental guided wave data which was excited at 120 transducer positions with a circular array of radius R = 200 mm. At this frequency, SH1 has a wavelength of λ = 16.3 mm and with the cut-off frequency thickness product being at about 1.6 MHz mm, a maximum wall thickness-loss of

$$d_{\text{max}}=10\hspace{0.17em}\mathrm{mm}-\frac{1.6\hspace{0.17em}\mathrm{MHz}\hspace{0.17em}\mathrm{mm}}{250\hspace{0.17em}\mathrm{kHz}}=10\hspace{0.17em}\mathrm{mm}-6.4\hspace{0.17em}\mathrm{mm}=3.6\hspace{0.17em}\mathrm{mm}$$ 3.1

was expected to be suitable for imaging. However, for these initial simulations, the maximum wall loss was set to 20%, which, at 2 mm wall loss, is well within the limit posed by the cut-off frequency thickness product. As previously discussed, future work will study the use of SH1 at higher frequencies to enable deeper defects to be imaged.

The choice for 120 transducers was made using the sampling criterion for a circular array by Simonetti et al. [24]

$$N\hspace{0.17em}\approx \hspace{0.17em}\frac{4\pi \hspace{0.17em}{r}_0}{\lambda },$$ 3.2

where r₀ is the radius of a circle which contains the region of interest with a resolution of half the wavelength. If r₀ is to be equal to the array radius R, the number of required transducers for λ = 16.3 mm would have to be N ≈ 155. As the defects considered in this paper will not exceed a diameter of 300 mm, this region is set to 75% of the array radius R or r₀ = 150 mm. This leads to N ≈ 116 but has been increased to N = 120 to be able to move in 3° steps during the experimental stage of the project. The achievable resolution using HARBUT is set by the Born approximation and is limited to half the wavelength. The wavelength and thus resolution in turn is limited by whatever frequency is chosen. Any complexity of a defect, its finer features, will not be captured if their size is smaller than half the wavelength, even if an infinite number of transducers is used. For the case of a full-view circular array configuration, the features that are bigger than half the wavelength will be captured as long as the defect is within the circle of radius r₀, where the sampling criterion holds and a resolution of half the wavelength is possible. Therefore, the number of sensors required has to be adjusted so that the full extent of the defect remains within the circle at the centre of the array in order to capture its complexity to the extent that is possible at the set frequency.

A set of seven thickness maps of two categories were created (figure 3); two synthetic axisymmetric defects and five rough defects based on real laser-scanned corrosion patches provided by industry. The axisymmetric defects were based on the Tukey [25] and Hann [26] window functions. For the Hann-shaped defect (Hann), the thickness function T(x) was defined as

$$T(\mathbf{\text{x}})=\{\begin{array}{ll}{T}_0-\frac{d}{2}[1+\cos (\frac{2\pi |\mathbf{\text{x}}|}{w})]& |\mathbf{\text{x}}|<\frac{w}{2}\\ T_0& |\mathbf{\text{x}}|\ge \frac{w}{2}\hspace{0.17em},\end{array}$$ 3.3

where d is the depth of the defect and w the width. These values were set to d = 2 mm and w = 100 mm for Hann. In order to model an axisymmetric flat-bottomed defect, a Tukey window function was used as a template and adapted to

$$T(\mathbf{\text{x}})=\{\begin{array}{ll}{T}_0-\frac{d}{2}[1+\cos (\pi \frac{(|\mathbf{\text{x}}|-w/2)}{W/2-w/2})]& \frac{w}{2}<|\mathbf{\text{x}}|<\frac{W}{2}\\ T_0-d& |\mathbf{\text{x}}|\le \frac{w}{2}\\ T_0& |\mathbf{\text{x}}|>\frac{W}{2}\hspace{0.17em},\end{array}$$ 3.4

where w is the width of the bottom hole and W the width of the defect at the surface. The window becomes rectangular when W = w and a Hann window when w = 0. In the case of defect Tukey, the width was set to w = 100 mm and an outer width of W ≈ 200 mm. For the five realistic defects, the spatial resolution of the models was limited to 2.5 mm, the maximum diameter was scaled to be less than or equal to 200 mm and the depths were linearly scaled such that the maximum in each case was set to 2 mm. These defects can be grouped into two classes, pitting-corrosion (Alpha and Bravo) and larger corrosion patches (Charlie, Delta, Echo).

Figure 3.

The seven defect models used in the numerical study: Hann-shaped axisymmetric Hann, Tukey-shaped axisymmetric Tukey and laser-scanned defects of corrosion under pipe support (CUPS): Alpha-Echo. (Online version in colour.)

(b). Numerical model

The finite-element package Pogo was employed to capture the full complexity of guided wave scattering as it provides full three-dimensional elastodynamic simulation capabilities [27]. Pogo uses graphics cards to run the explicit time-domain FE model which significantly speeds up the simulation, compared with conventional FE software that runs on CPUs. The plates were modelled as 21 layers of 0.5 mm 8-element hexahedral elements which amounted to a total of 21 million elements (1000 × 1000 × 21), following the approach used in [7]. In order to avoid signal from boundary reflections, 60 mm of stiffness reduction method (SRM) boundaries [28] were applied along the edges of the plate model. At the chosen frequency of 250 kHz, the SH1 wavelength is 16.3 mm which amounts to 32 elements per wavelength in-plane. For the defect region, elements were squashed in the out-of-plane z-direction in order to model the thickness changes inherent within the defects.

The array of transducers was modelled by exciting forces and measuring the displacement of nodes around the transducer positions. SH1 was excited by an opposing set of twisting sources on the top and bottom surfaces of the plate; the twisting avoids longitudinal excitation because it has zero divergence, and the opposing sources prevent SH0 and SH2 from being generated. This can be seen in figure 4 by considering a single colour. Note that this is only valid if the desired point source location happens to lie perfectly at the centre of an element. In practice, one solution is to accept a slight shift in the transducer position and move it to be centred on an element, however, in this paper, this excitation is applied to each of the four adjacent elements, shown as different colours in figure 4, with the weighting of each element being determined by a linear interpolation approach, such that the centre of mass of the source lies at the desired location. The case depicted in figure 4a shows the weighting that would be applied for the ideal case of the transducer position being exactly on the centre node (number 5). The principle of reciprocity means that the same weightings that are used during excitation are also used during reception, summing the measured displacement values on each degree of freedom with the same weighting as the excitation term.

Figure 4.

(a) Omnidirectional shear excitation on each of four elements. The source can be positioned arbitrarily by weighting the displacements of the four elements and superposing the signals together. (b) How this shear source can be applied to the top and bottom surfaces; excitation by twisting in opposite directions top to bottom will excite the SH1 mode. (Online version in colour.)

The Pogo model was run on eight NVIDIA Quadro RTX 8000 GPUs which required 29 mins to run for all 120 excitation events.

(i). Data processing

Owing to the modal purity of the data (figure 5), no windowing of the signals was necessary and, for every transmit-receive combination, an FFT of the signals was performed in order to obtain the required frequency domain data for HARBUT. For calibration, a calibration matrix is calculated by assuming an incident field generated by unit point sources [7], given by Green’s function

$$G(r)=-\frac{i}{4}{H}_0^{(1)}(k_{0}r),$$ 3.5

Figure 5.

Point source transducer signal processing: (a) Magnitude of the pure SH1 wavefield at t = 0.13 ms after excitation at Tx(1). (b) Magnitude of the Hilbert envelope of the received SH1 time traces for the excitation at Tx(1) and reception at all 120 transducers. (c) 5-cycle Hann-shaped excitation signal and the received pure SH1 signal. (Online version in colour.)

where r is the distance from the source to the receiver, $H_0^{(1)}$ the Hankel function of the first kind and k₀ the background wavenumber. For every SH1 simulation dataset, a calibration value was generated from a transducer pair that was not consistent with the defect.

Within this paper, the HARBUT algorithm was used starting from a homogeneous background; it was found that with suitable regularization, convergence was still achieved. The regularization steps taken were to (a) apply a Gaussian filter (with the standard deviation of the Gaussian distribution set to σ = 3) to the velocity map of the previous step, which served as background for the next one and (b) to use an additional positivity constraint, that any value lower than the background velocity would imply an increased thickness which is non-physical and therefore any such values were corrected to the background velocity.

Since we reconstruct from through-transmission data only (which corresponds to low-spatial frequency components [21]), we imposed a linear upper limiting gradient of the spatial frequency components to values corresponding to

$$k_{\text{Lim}}=[1.1,\hspace{0.17em}2\times \sin \left(\frac{\theta }{2}\right)]\times k_0\stackrel{\theta ={90}^{\circ }}{\approx }[1.1,\hspace{0.17em}1.4]\times k_0,$$ 3.6

where θ is the scattering angle. Applying a step filter, which removes all frequency content above a specific value, will result in ringing artefacts appearing in the image. Therefore, we use a sine-shaped taper in the filter, from 1 to 0. The two values given correspond to the start and end of the filter. This upper limit was used for every defect type and the absence of any large ring artefacts suggests that the sampling was sufficient and effects from the incident field low [29]. Also, setting the filter boundaries to these values allows for the prediction of an in-plane resolution of ≈ 0.7-1 λ, which, for 250 kHz, corresponds to ≈12-16.3 mm. This is well below the often cited in-plane resolution requirement of $3\hspace{0.17em}\text{T}=30\hspace{0.17em}\mathrm{mm}$.

The convergence criterion Q was based on the average thickness change over the sum of the region of interest relative to the nominal thickness, following the approach in [7], where the parameter Q is defined by

$$Q^{(n)}=\frac{\int |T^{(n)}(\mathbf{\text{x}})-T^{(n-1)}(\mathbf{\text{x}})|\hspace{0.17em}S(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}}{T_0\int S(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}}.$$ 3.7

In this equation, T₀ is the nominal thickness, T⁽ⁿ⁾(x) is the thickness at location x for iteration n and S is a map indicating the extent of the damage area. S is a binary map which was created by setting any reconstructed depth greater than 5% of the background thickness to 1 and everything else to zero

$$S(\mathbf{\text{x}})=\{\begin{array}{ll}1& \frac{T_0-T^{(n)}(\mathbf{\text{x}})}{T_0}\ge 0.05\\ 0& \frac{T_0-T^{(n)}(\mathbf{\text{x}})}{T_0}<\mathrm{0.05.}\end{array}$$ 3.8

We set the convergence value to Q ≤ 4 × 10⁻³ as this led to fewer than 10 iterations required for convergence in most cases.

Two error metrics were calculated for all defects by comparing them to the values of the true thickness maps. The ‘global error’ e is based on the quality of shape assessment used in [8], which is based on a root-mean-squared (RMS) error

$$e=\sqrt{\frac{\int {[T(\mathbf{\text{x}})-T^{\ast }(\mathbf{\text{x}})]}^{2}w(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}}{T_0^2\int w(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}}},$$ 3.9

where T(x) is the true thickness map, $T^{\ast }(\mathbf{\text{x}})$ is the reconstructed thickness map and w is a windowing function that biases the error metric towards depth changes towards the centre of the array of radius R

$$w(\mathbf{\text{x}})=\{\begin{array}{ll}\frac{1}{2}[1+\cos (\frac{\pi \mathbf{\text{x}}}{R})]& \mathbf{\text{x}}<R\\ 0& \mathbf{\text{x}}\ge R.\end{array}$$ 3.10

Also, the relative error at maximum defect depth was calculated by

$$e_m=\frac{|T_m-T_m^{\ast }|}{T_0},$$ 3.11

where T_m is the true thickness at maximum depth, $T_m^{\ast }$ is the ‘measured’ thickness at maximum depth and T₀ the nominal thickness of the plate.

(c). Thickness maps from SH1 point source data

The reconstruction results for the pure SH1 point source data are shown in figure 6 and all results are listed in table 1. Qualitatively, the geometry and features of the defects are captured extremely well. The two axisymmetric defects stand out for being near-perfect reconstructions. At the location of greatest depth, their respective errors are extremely low at e_m = 0.2% for the Hann-shaped and e_m = 0.1% relative background thickness for the Tukey-shaped defect. Their global error percentage is also below 1% which suggest that not only the maximum depth but also the shape of the defects were captured near perfectly.

Figure 6.

HARBUT reconstruction from point source SH transducer FE simulations. (a–g) True thickness maps, (h–n) display the reconstructed thickness maps and (o–u) show line profile through the deepest points for the true (black) and reconstructed depths (grey). (Online version in colour.)

Table 1.

Error and performance values for point source finite-element results: The ‘global error’ e, the error at maximum defect depth e_m, the number of iterations per case and the defect area.

defect	e (%)	em (%)	iterations	defect area (mm2)
Hann	0.2	0.2	5	7.8
Tukey	0.2	0.1	17	21.8
Alpha	0.8	1.7	5	17.3
Bravo	1.0	1.1	5	11.8
Charlie	0.7	2.1	10	34.8
Delta	0.9	1.3	8	22.6
Echo	0.7	0.6	7	18.2

The five realistic defects also showed very good agreement with the true values. They ranged from 1.5% for Alpha to 2.1% for Delta, which is extremely low. At the maximum depth, the error values are again extremely low, with Echo showing the greatest accuracy with only e_m = 0.6% deviation from the background thickness. Defect Bravo showed the greatest error at 2.1% which corresponds to a deviation of 0.21 mm, which is importantly within 10% desired by industry.

Another interesting result is the number of iterations that were required for convergence. The two axisymmetric defects show the lowest and highest number of iterations at 5 (Hann) and 17 (Tukey). Similar numbers of iterations were observed for the more realistic defects where smaller defects (Alpha & Charlie) required fewer iterations than the larger ones. This is likely to correspond to the phase shift of the guided wave passing beneath the defect.

It is noted that each iteration typically takes around 1 s on a standard desktop computer; meaning that the algorithm run time is not a consideration in the majority of cases.

(d). Directional SH model

In a practical physical set-up, pure omnidirectional SH1 transducers are difficult to implement without simultaneous access to both sides of the plate or the inside of a pipeline which is unfeasible. Having demonstrated the accurate reconstructions produced when using pure SH1 signals, the next step was to investigate more realistic transducers. The excitation of SH waves can be achieved with both EMATs and shear piezoelectric transducers, and in general, the function of these are to excite a force parallel to the surface in a particular region. Following this, the SH transducer was modelled as a simple rectangular shape with a width of 5 mm and a length of 20 mm (figure 7). These parameters were found to be a good compromise between a wide beam spread to ensure the whole of the defect lies inside the transducer wavefield, and directionality to ensure that sufficient energy is transmitted into the medium. The transducer model is made in this way to approximately represent the behaviour and dimensions of the true transducer. In theory, it would be possible to model the resulting forces arising from a full electromagnetic model of the EMAT, but this is seen as unnecessary given the intention is to demonstrate the general performance with a more realistic, directional transducer rather than focusing on the specific example.

Figure 7.

Directional transducer (width w = 5 mm, length l = 20 mm) modelling approach. (Online version in colour.)

A rectangular mask was calculated for every array transducer and the nodes within the rectangular area were selected. In order to generate a homogeneous shearing traction on the surface, all nodes within one transducer area received the same forcing, weighted between the two in-plane degrees of freedom in order to excite the force parallel to the longer dimension of the transducer. The forcing applied in the x- and y-directions for each transducer is given as

$$x_n=\sin ({\theta }_n)$$

and

$$y_n=-\cos ({\theta }_n),$$

where n is the number of the transducer and θ the angular coordinate of the transducer. Due to the use of a structured mesh, transducers that align perfectly with the mesh held more nodes than those at an angle which resulted in uneven signal amplitudes. In order to reduce this effect, an arbitrary bias angle of 2.14° was added to the angular coordinate θ, which helped to even out the number of nodes within all transducers.

(i). Directional SH data processing

The most immediate effect of the surface SH excitation at f = 250 kHz compared with the idealized omnidirectional source discussed earlier is the additional excitation and detection of the fundamental SH mode, SH0, which appears within the measured time traces (figure 8). The next higher-order mode, SH2 was not picked up as the excited frequencies were well below SH2 cut-off. Lamb wave modes were excited perpendicular to the directional transducer and therefore did not interact with the defect as absorbing boundaries prevented any reflections. Thus, the primary signal processing problem to solve was the isolation of SH1 from the SH0 + SH1 time traces. This was achieved by applying a simple adaptive windowing technique that relied on knowing the distance between transducers as well as the group velocities (figure 8b,c). The latter were taken from the dispersion curves calculated using Disperse [30,31], which made the calculation of theoretical arrival times straightforward. These matched very well for the non-dispersive SH0 mode but overestimated the arrival time of SH1, which is primarily caused by the wavepacket dispersing and hence part of it arriving earlier. A separation point between the two modes was taken to be the mean of the two arrival times. The length of the SH1 signal component was calculated by multiplying the calculated SH1 arrival time by 1.5 which was found to capture the extent of the useful SH1 data sufficiently, accounting for the dispersion. These two signal SH1 window boundary times are defined as

$$t_1=\frac{l_n}{c_{g,\mathrm{SH}0}}+\frac{1}{2}\times (\frac{l_n}{c_{g,\mathrm{SH}1}}-\frac{l_n}{c_{g,\mathrm{SH}0}})$$

and

$$t_2=1.5\times \frac{l_n}{c_{g,\mathrm{SH}1}},$$

where c_g,SH1/SH0 are the group velocities of the fundamental mode SH0 and the first higher-order mode SH1 and l_n the distances between every transducer pair. Using these two arrival times, the signals were windowed using a Tukey window (figure 8c) defined by the following equation:

$$h(t)=\{\begin{array}{ll}0& 0<t<t_1\\ \frac{1}{2}[1+\cos (\frac{\pi (t-t_1)}{\alpha (t_2-t_1)})]& t_1\le t<t_1+\alpha (t_2-t_1)\\ 1& t_1+\alpha (t_2-t_1)\le t<t_2-\alpha (t_2-t_1)\\ \frac{1}{2}[1+\cos (\frac{\pi (t-t_2)}{\alpha (t_2-t_1)})]& t_2-\alpha (t_2-t_1)\le t<t_2\\ 0& t_2\le t,\end{array}$$ 3.12

where the length of the window was defined by t which is a linear-spaced vector with length t₂ − t₁. Setting α = 0.1 led to an almost rectangular Tukey window with a steep slope, therefore not altering the bulk of the signal as would have been the case with e.g. a Hann window. Further processing was carried out as described in §3b(i).

Figure 8.

Directional transducer signal processing: (a) Magnitude of displacement wavefield at t = 0.11 ms after excitation of SH0 in addition to SH1 due to directional excitation, measured on the top surface of the plate. Modal decoupling has occurred. (b) Magnitude of the Hilbert envelope of the received time traces and the respective arrival times for of SH0 (grey), SH1 (light grey) and the calculated SH1 window boundary times: t₁ (light grey) and t₂ (light grey dashed). (c) Full centre time trace (black) and Tukey window based on t₁ and t₂ (black dashed) and windowed SH1 component (grey). (Online version in colour.)

(e). Thickness maps from SH1 directional source data

The reconstructions from windowed SH1 data using directional transducers are shown in figure 9 and the results are listed in table 2. The same reconstruction parameters were used as in the reconstructions from point source data (k_Lim = [1.1, 1.4], σ = 3, Q_Lim = 0.004). Again, the defect shape and details are reconstructed extremely accurately, with the axisymmetric defects achieving the best values and lowest error values across all metrics. The geometry of the defects is reconstructed very well and the overall error values are well below 2% for both the global error and the error at minimum thickness. The number of iterations for convergence increased as well which can be explained by the added ring artefacts due to the use of the directional surface transducer model. Nevertheless, they are consistent with the amount of iterations required for the ideal point source dataset.

Figure 9.

HARBUT reconstruction from directional SH transducer FE simulations. (a–g) True thickness maps, (h–n) the reconstructed thickness maps and (o–u) show line profile through the deepest points for the true (black) and reconstructed depths (grey). (Online version in colour.)

Table 2.

Error and performance values for directional source finite-element results: The ‘global error’ e, the error at maximum defect depth e_m, the number of iterations per case and the defect area.

defect	e (%)	em (%)	iterations	defect area (mm2)
Hann	0.3	0.2	5	7.8
Tukey	0.5	0.1	19	21.8
Alpha	1.0	3.0	4	17.3
Bravo	1.0	2.5	5	11.8
Charlie	0.9	0.1	9	34.8
Delta	1.2	2.7	7	22.6
Echo	1.2	0.6	5	18.2

4. Experimental demonstration of SH1 tomography

(a). Experimental set-up

For experimental validation, several mild steel plates (1000 × 1000 × 10 mm) were acquired and synthetic defects machined into them. Two types of defects were manufactured with a maximum thickness loss of 20%: an axisymmetric defect (Foxtrot) and the complex corrosion patch Echo considered for the numerical simulations (figure 10). For this, two-dimensional thickness maps were generated in Matlab and imported into plate models in Solidworks which were used as templates during metal milling. Due to metal milling tool tip resolution constraints, the spatial resolution of the Solidworks models was 2.5 mm. In order to guarantee repeatable accurate positioning of the transducers, a plywood rig was designed (figure 11) that consisted of a frame that was attached securely to the steel plates via datum holes in the four corners. A ring with 120 positioning holes and identifier labels was fastened to the frame and served as a rail for the transducers. These were kept in position by being secured to two wooden sleds that fit the curvature of the inner and outer diameter of the ring. Finally, two steel rods were used to fix the positions of the sleds on the ring and therefore the attached transducers.

Figure 10.

Thickness map templates which were machined into mild steel plates used in experiments. (Online version in colour.)

Figure 11.

Experimental set-up: (A) custom plywood rig with 120 array positions, (B,C) QSR1^® transducers [32], (D) PC for data acquisition, (E) QSR1^® base unit, (F) mild-steel plate with machined defects. (Online version in colour.)

For signal excitation and reception, a modified QSR1^® system by Guided Ultrasonics Ltd. was employed [32]. The use of SH1 relies upon EMAT transduction, with a racetrack coil applied to the specimen in conjunction with alternating permanent magnets [33]—this is how the QSR1^® transducers work. This is a key distinction from the A0 mode, which was easily excited with point transducers.

For the axisymmetric defect, data acquisition from one transmission and 51 receiving points was sufficient, since a full dataset can be produced by replicating those measurements and exploiting the symmetry. This was not possible for the rough defect and therefore 120 transmission and 51 receiving positions led to a total of 6120 single measurements. For excitation, a 250 kHz 5-cycle Hann-shaped signal was transmitted, and 16 averages per acquisition were taken to minimize incoherent noise. Data were acquired using two electromagnetic acoustic transducers (EMAT) of the QSR1^® system and recorded on a laptop.

It should be acknowledged that in this paper, we are using a 360° dataset, which may not be available in many scenarios. The intention is to demonstrate the capability of the method using an idealized set-up, and future work will evaluate different set-ups. However, it should be expected that the limitations of other set-ups are aligned with the common issues seen with limited view velocity reconstructions, namely artefacts and resolution loss [34,35].

(b). Data processing

The experimental data were processed in an analogous manner to the directional FE data, except for the application of a frequency filter to eliminate most of the high-frequency noise. For this, a Tukey window was placed in the frequency spectrum of every time trace, filtering out any signals below 150 kHz and above 300 kHz, which helps with determining the correct SH0 arrival time. The excited signals were reflected by the plate boundaries but by applying the arrival time windowing approach from the previous section, these signal components, as well as crosstalk and SH0 components, could be filtered out. Since the exact values for the elastic moduli and density of the experimental plates were unknown, the dispersion curve had to be calculated using the analytical equation for SH phase velocity (2.1). For the case of n = 0, the shear velocity equals the phase velocity of SH0 which is all that is required to calculate the dispersion curves for higher-order modes. Since the distance between the transducers is known, the arrival times of SH0 could readily be determined by calculating the arrival of the maximum of the Hilbert envelope of SH0 for every transducer arriving at its opposite receiver (e.g. 0° and 180°, 3° and 183° etc.). Thus, SH0 phase velocity was determined by taking the mean of all 120 velocity measurements

$${\overline{c}}_{\text{Sh, exp}}=3271.3\hspace{0.17em}\pm \hspace{0.17em}1.6\hspace{0.17em}{\mathrm{m}\hspace{0.17em}\mathrm{s}}^{-1}.$$ 4.1

This provides the mean shear wave speed c_Sh needed for the SH1 dispersion curve calculation.

(c). Thickness maps from experimental data

The experimental results are shown in figure 12. The same reconstruction parameters were used for the reconstructions from numerical data and the reconstruction converged after four iterations for the axisymmetric and seven iterations for the realistic case. Most notably, the maximum depth is captured very well for both cases, with the respective errors being 0.14 and 0.71% of the true thickness. However, the geometry of the axisymmetric defect is not captured well, in comparison with the realistic defect case. This could be due to potential discrepancies between the model that was sent for machining and the real defect. Furthermore, as shown by Belanger [5] and Huthwaite [8], reconstructions from a single set of received data that has been formed into a full array from axisymmetric data is prone to exaggerate any errors present within that dataset. By contrast, using a full dataset will average out more of the incoherent errors. Nevertheless, the global error values are again very low with values of 1.7% for the axisymmetric and 1.1% for the realistic defect.

Figure 12.

HARBUT reconstruction from experimental data: (a,b) True thickness maps, (c,d) the reconstructed thickness maps and (e,f ) show line profiles through the deepest point for the true (black) and reconstructed depths (grey). (Online version in colour.)

5. Performance analysis

Having demonstrated the concept with complex defects from both numerical and experimental data, we now confirm the limitations of the method, as well as compare performance with the two main modes considered previously, A0 and S0. For these purposes, we will focus on simple axisymmetric defects of the same type as the Tukey defect used previously. A0 was used at 50 kHz on the 10 mm thick plate, S0 was at 175 kHz and SH1 was at 250 kHz. These operation points are the ones commonly identified for guided wave tomography, recognizing practical constraints [15]. In all cases, point sources were used and simulations performed using FE, as discussed in Sect. b. The data were post-processed in the same way and images produced via the HARBUT algorithm. The filter was set to [1.1, 1.4] for all reconstructions. At these operating points, the wavelength was 3.8 T for A0, 2.8 T for S0 and 1.6 T for SH1.

Firstly, the performance of SH1 compared with the A0 and S0 modes is evaluated. Tukey-type defects of width 10 T down to 1 T and 0.1 T depth are used for this. Figure 13 shows the line plots of the true (grey) thickness profiles and of the reconstructions of A0, S0 and SH1. From these, it is clear that the A0 mode performs the worst through all the widths, indicating that it has the lowest resolution. SH1 appears to have the best resolution, particularly capturing the shape for width 1 T and maintaining a good accuracy as the width increases. There is a clear ‘overshoot’ visible for both S0 and SH1 at 1.5–2 T for SH1 and 2.5–3 T for S0. This is very likely to be associated with the effective change in density of the plate causing additional scattering to the guided wave, which is not accounted for in this paper, and is most significant for small defects of the scales considered here. As explained in [22], the effective density is assumed to be proportional to the mass per unit area and therefore proportional to thickness if there are no material density changes and only play a role whenever there are sharp boundaries. This is demonstrated for both A0 and S0 modes, and it is reasonable to expect that other modes, including SH1, exhibit this too. A full investigation of this is beyond the scope of the paper, however.

Figure 13.

Comparison of HARBUT reconstructions using Lamb wave modes A0, S0 and shear-horizontal modes SH0 and SH1 for Tukey-type defects with widths ranging from 1 T to 10 T, where T is the nominal plate thickness. (Online version in colour.)

For the smallest defects, the S0 reconstruction is resolution-limited so it underestimates the wall loss, an effect noted in [36]. As the defect becomes larger, resolution is less of an issue so the wall loss estimate increases, but at one point the mode overestimates the value because of the density effect discussed above. As the defect width increases further, the density effect becomes less significant compared with the velocity contribution, and the overshoot reduces.

This same pattern is visible with SH1, although the transitions occur at smaller defect widths. The overshoot of SH1 is small at widths of 2.5 T and above, indicating that SH1 is reliable for reconstructing all defects in this range and hence demonstrating that the 3 T requirement can be met. It is noted that the performance below 2.5 T is still good. By contrast, S0 requires defects of width 4 T and above to minimize the overshoot. While these images give a good qualitative overview, they are not quantitative. Figure 14 compares the global error (see equation (3.9)) across all defects for the three modes; this error indicates the overall behaviour of the modes and avoids sensitivity to overshoots. The results show, that the wavelength of SH1 is significantly below the other modes and, based on this, would lead to a resolution improvement of $\frac{3.8\hspace{0.17em}\text{T}}{1.6\hspace{0.17em}\text{T}}\approx 2.4$ times when compared to A0.

Figure 14.

Global error plot of the Tukey defect diameter sweep for A0 (dashed), S0 (dot-dashed) and SH1 (solid). (Online version in colour.)

6. Conclusion

For the first time, the first higher-order shear-horizontal guided wave, SH1, has been applied to guided wave tomography enabling a significant improvement in resolution (≈2.4 times), based on the wavelength reduction) and hence accuracy of the depth estimate. Realistic three-dimensional FE simulations of a 10 mm thick steel plate were performed for various cases of defects and validated experimentally. We have shown that it is possible to achieve highly accurate thickness maps, confirming that guided wave tomography can now reliably reconstruct defects with sizes smaller than 3 T × 3 T, where T is the nominal wall thickness, conquering a long-standing threshold. The best results were achieved by using pure SH1 point sources in excitation as well as reception which is evident by the extremely low error at the deepest points of the various cases that were investigated. Nevertheless, the more realistic simulation with a directional surface excitation only led to a small drop-off in quality, as again, the most important error at deepest points was well below 3% for all cases. Finally, experimental results validated the FE predictions and show that even in the presence of noise, similar accuracy thickness maps can be created. The ultrasonic beams generated by the EMATs were highly directional, which is undesirable for tomography. However, this paper has shown that even with transducers not optimized to reduce directionality, it is still possible to generate accurate images. Future work will be to take this forward for industrial use.

Data accessibility

Readers interested in accessing data associated with this paper can find information and models at http://dx.doi.org/10.5281/zenodo.3825136.

Authors' contributions

All authors contributed to the writing of the article. B.P. provided access to and support for experimental equipment and data capturing. P.H. provided substantial contributions to conception and design of the model and interpretation of the results. A.A.E.Z. performed simulations, experiments, method development and implementation as well as data analysis. All authors approved the final version and agree to be accountable for all aspects of the work.

Competing interests

We declare we have no competing interests.

Funding

Dr P. Huthwaite is funded by EPSRCunder grant no. EP/M020207/1. A.A.E.Z. is funded by EPSRC under grant no. EP/L015587/1.