Efficient finite element modelling of guided wave scattering from a defect in three dimensions

ABSTRACT The scattering matrix (S-matrix) encodes the elastodynamic scattering behaviour of a defect. For the case of guided waves in the low-frequency regime, we explore how to minimise the number of incident/scattered directions used to accurately simulate the S-matrix of an arbitrary defect. The general approach is to use three-dimensional finite element (FE) analysis, implemented in a commercial FE package, to simulate the wave-defect interactions. The scattered wave field is measured at monitoring nodes on a circle that surrounds the defect. These scattered waves are decomposed into the multi-modal far-field scattering amplitudes. The angular order of the scattering is found and used to minimise the number of incident and scattered directions that must be computed. The method is then used to simulate the S-matrices of surface-breaking semi-elliptical cracks and circular through-holes with arbitrary sizes. It is found that the required number of incident/scattered directions varies with the scattered mode, and this is due to the differing scattering orders of these scattered fields. These results demonstrate how to achieve more efficient modelling of guided wave scattering, and also contribute to an understanding of experimental defect characterisation.


Introduction
Ultrasonic guided waves are widely used to detect, locate and continuously monitor defects in plate-like structures [1][2][3][4], pipelines [5][6][7][8] and other engineering structures [9][10][11].The scattered waves inherently contain information about the defect and this scattering signature has been proposed as a useful tool for characterisation [12,13].However, defect characterisation using guided waves is challenging due to the complexity of the scattering, which typically includes mode conversion, multi-modal propagation and dispersion.
In bulk-wave based non-destructive evaluation, arrays of transducers have been used to measure the defect scattering response [14][15][16][17].The array of transducers insonifies the defect from different directions thereby enabling the scattering to be measured as a function of incident and scattered angles.Numerical methods, such as finite elements (FE), are commonly used to simulate the elastodynamic scattering of ultrasonic waves when they propagate in solid structures [17,18].Such simulations can then be used to help interpret the experimental scattering measurements for the purposes of defect characterisation.The knowledge of scattering can also be used for parametric studies to optimise measurement configurations.For example, 3D finite element modelling has been used to explore the scattering of guided waves at an arbitrary defect [16].Wave propagation in various wave-guide geometries has also been modelled numerically, including pipes [19][20][21] and plates [22][23][24].These simulations have been used to explore applications such as guided wave tomography for defect characterisation in plates [16,25,26].In 3D elastodynamic simulations, a compromise is typically needed between the accuracy of the numerical results and the computation time due to the need for small elements and hence many degrees of freedom.Therefore, the minimum number of model runs to ensure model accuracy is important to determine before performing a large number of modelling simulations or experiments.
The scattering matrix of a defect encodes the far field behaviour of the scattered waves for each combination of incident and scattered angles.The concept applies equally to bulk and guided waves [18,27,28].Experimentally, the scattering matrix is measured using an array, either fixed in position or synthesised by scanning [14,29,30].
This paper concerns guided wave scattering and a simulation of a typical inspection should include the transmitter and receiver array elements as well as the scatterer.The array data associated with a particular defect are composed of the amplitude and phase of scattered signals at all possible incident and scattered angles in the frequency-domain, i.e. over some defined frequency bandwidth.The larger the ratio of defect size to incident mode wavelength, the more complex the scattering behaviour [14,16,18].To fully capture this scattering complexity necessitates the simulation of many incident and scattered directions, with each incident direction requiring a further computation.Guided waves present an additional challenge as they are multi-modal and so the full scattering behaviour includes all possible mode conversions.
In the case of an axisymmetric defect geometry such as a circular hole, the scattering matrix can be simulated using a single angle of incidence, and a number of receiver locations.When the geometry of the defect is non-axisymmetric, further incident angles need to be considered to obtain the full scattering matrix and thereby capture all wavedefect interactions.In some cases, the number of simulations can be reduced, for example planar cracks have two planes of mirror symmetry and so incident angles in a single quadrant can be used to capture the full scattering behaviour efficiently.
In this paper our aim is to develop an efficient S-matrix generation technique for an arbitrary defect in 3D FE models.The same fundamental approach can be used to achieve an optimal scheme for experimental configurations chosen to characterise defects.The paper includes the generation of complete guided wave scattering matrices over both single and multiple frequencies.Although the guided wave propagation is twodimensional, the interactions of the guided waves with the defects necessitate threedimensional modelling.The limitation of the 3D FE models in our study is that at present they are restricted to the low-frequency region and do not simulate irregular defects or include surface roughness.The necessity to use small elements (i.e.fine discretisation) means that these models quickly become very large and time consuming to perform, even on modern high performance computers.A key challenge is therefore to minimise the number of models (effectively computation runs) and that means minimising the number of incident angles.Where possible it is also important to minimise the size of the model, in terms of the number of elements used, as this further reduces computation time.Hence we present a procedure whereby the complete scattering matrix of an arbitrary defect can be simulated using the minimum number of simulations.We note also that the same ideas apply to minimising the number of measurements in an experimental context, for example, applications such as ultrasonic guided wave sensing by mobile robots.The approach presented first quantifies the angular order of the scattered wave field and we explore how this varies with defect size and shape.This understanding of the scattered field enables the number of models required to be minimised.By using available symmetries and also minimising the size of the individual models themselves we create efficient models of the guided wave scattering from arbitrary defects.
This paper is organised as follows.Section 1 outlines the simulation and measurement challenges as well as the numerical approaches used for obtaining scattering matrices.Section 2 describes an efficient and accurate 3D finite element modelling procedure, and generates the benchmark scattering matrices using fine angular increments.Section 2 uses the angular order of the scattering matrices to explore the minimum number of measurements required for obtaining accurate S-matrices of all scattered wave modes.The effect of frequency on the scattering behaviour is also explored.In Section 3 potential strategies for experimental defect characterisation using limited data are proposed and discussed.Finally, Section 4 summarises the findings and draws conclusions.

The finite element model
The scattering matrix of ultrasonic guided waves from a three-dimensional defect describes the multi-modal scattered response as a function of incident and scattering directions.Compared to previous work, e.g [16], the main novelty described in this paper is the exploration of the minimum number of measurements required to obtain the scattering characteristics of a specific incident/scattered wave mode (i.e.S0, A0 or SH0) from a given defect type and size.The model geometry used here is a plate with centrally located defect and is illustrated in Figure 1.All models are built using Abaqus/CAE 6.14 (Simulia UK, Warrington, UK).The explicit time-domain formulation [14,16,31] is used to calculate all possible scattered elastodynamic wave modes.In contrast to a twodimensional FE simulation, the three-dimensional nature of the model naturally captures both out-of-plane and in-plane displacements so can simulate all mode types.Modal decomposition of the scattered field in different directions is performed using an orthogonality relationship and this allows the magnitude of the scattered modes to be separated.The frequency of the incident signal pulse is selected to ensure no higher order modes exist.Therefore, we note that this paper only considers the low frequency regions, i.e. the zeroth-order modes (i.e.symmetric, S0, anti-symmetric, A0 and shear horizontal, SH0 wave modes).
Two models are used to find the scattered field: a model with the defect and a reference model without the defect.The scattered field is calculated by subtracting the response from the model without the defect from that with the defect.This is possible as we assume that the total field is the linear sum of the incident and scattered fields.Although the scattering interactions are inherently three-dimensional, in the far field, the scattered guided wave modes are two-dimensional.Hence, we encode the scattered modes in a number of two-dimensional far-field scattering matrices [14].As shown in Figure 1, each model starts with the excitation of a particular incident guided wave mode.The response is recorded on a monitoring circle that surrounds the central location, in which the defect is positioned.The various scattered modes are then extracted and the scattering matrix, S θ in ; θ sc ; ω ð Þ, for each mode combination found from [14] where θ in and θ sc are the incident and scattered wave angles, U in is the complex displacement amplitude of the incident wave at the defect centre, U sc is the complex displacement amplitude of the scattered wave monitored at a distance d sc from the defect, λ in is the wavelength of the incident signal, ω is the angular frequency and c sc is the phase velocity of scattered waves.Here, the scattering matrix, S, describes the amplitude of the scattered wave and d sc is normalised to one wavelength, λ in .Meanwhile, the displacement magnitude of U sc will decrease with ffi ffi ffi ffi ffiffi d sc p (in the far field).Figure 1(a) shows the 3D FE model used for simulating the far-field scattering matrix of a surface-breaking semi-elliptical crack defect.This model uses a cubic 3D mesh surrounded by an absorbing region.The absorbing region is added to limit reflections from the model boundary enabling the use of a smaller model domain.The resulting model size is 23.6λ c ×23.6λ c (λ c is the wavelength of the incident signal at the centre frequency), with an absorbing region size of 7.35λ c and an element size of λ c /34 (the reflection ratio for its absorbing boundary region is less than 0.9%).This model size is set to ensure that the monitoring circle is in the far field of the scatterer for all scattered modes.The parameters used in this model are shown in Table 1.Note that literature suggests of between 8 and 40 elements per wavelength for elastodynamic finite element modelling, depending on the scenario [9,16,17,31,32], and so the discretisation used in this paper is within these norms.
In this paper we assume the material of the plate waveguide is high-density polyethylene (HDPE).This material has particular relevance to the non-destructive testing and condition monitoring of water pipes, but the process is general and could be broadly applied.This is reasonable due to the high attenuation of guided waves in this material meaning that they only travel relatively short distances, typically less than one circumference.Here, we consider the pipe as an unwrapped isotropic plate and hence the results are only applicable for scenarios where the pipe diameter is much larger than wall thickness [33].A common type of defect in HDPE pipes is a semi-elliptical crack [34].For comparison we also consider a circular hole as an example of an axisymmetric defect.This simpler defect acts as a reference, or calibration, defect.In the FE models, the surface-breaking semi-elliptical crack is given a width of one element and the part depth circular hole is modelled by removing elements to match its shape as shown in Figure 1(b).
One challenge in a 3D FE model is how to achieve the omnidirectional low-frequency generation of a single pure mode.Different approaches are applied for S0 [35] and A0 [36].For S0 generation, an in-plane omnidirectional excitation is required.To decrease the amplitude of A0 wave, an improved method is proposed with two sets of transducers antisymmetrically distributed about a plate [35].To do this, an 8-node source is used to apply radial forces for S0 [14].These forces are applied simultaneously on the top and bottom surfaces of the plate.This symmetrical excitation helps to ensure pure mode generation.Figure 1(a) shows the nature of this excitation for generating the amplitude of S0 wave with a modal purity greater than 99%.This approach has been shown to produce excellent single-mode generation [16].In experiments there are several ways to generate S0 mode waves from a single side with adequate modal purity, for example, using a radially polarised EMAT sensor with a pancake coil pattern [37,38].For antisymmetric modes, i.e.A0, we use two nodes located on the plate surfaces, with the same out-of-plane force applied to each.Although not used in this paper a pure SH0 of the plate model [39][40][41] or a single torsional wave mode T(0,1) generation on pipe-like structures [42] could be generated by multiple sources to provide in-plane tangential forces for suppressing other undesired wave modes.Other pure wave modes of plate-like structures [43] may also be excited successfully in 3D FE models through matching the excitation with their dispersion characteristics and mode shapes.
The second challenge is how to extract the amplitudes of the scattered guided modes from the multi-modal received signals.By adding the signals from the top and bottom we can remove the symmetric wave modes, leaving only A0 with displacements in out-ofplane.Conversely by subtracting the signals from the top and bottom surfaces we can remove the A0 leaving just S0 with displacements primarily in the radial direction.Similarly, by adding signals with displacements in the tangential direction, SH0 can be extracted.

Generating the input signal
As the FE model used is a time-domain simulation (i.e.explicit) excitation of the incident signal is achieved using a pulse with several cycles.The excitation bandwidth can then be controlled by adjusting the number of cycles in the pulse.For example, to minimise dispersion effects the number of cycles can be increased to narrow the frequency bandwidth.Alternatively, if dispersion is less of an issue, fewer cycles can be used to produce broadband results that cover a wider range of frequencies in a single model.
Here the input pulse signal, u t ð Þ, is a Hanning-windowed toneburst input given by [31] where f c is the centre frequency of the incident signal, and n c is the number of cycles.To ensure that the incident and scattered guided waves contain only the lower order modes S0, A0 and SH0, the input signals are limited below 40 kHz, which is the A1 cut-off frequency for the 8.18 mm HDPE plate simulated.This upper simulation frequency can then be adjusted depending on the dispersion curves for the specific plate thickness and material properties of interest.As shown in Figure 2, a 10-cycle, f c = 30 kHz Hanning windowed signal with a frequency bandwidth between 24 kHz and 36 kHz is used here as the incident signal for the excitation of A0, S0 and SH0 guided wave modes.

Size of absorbing regions
Another challenge for the FE guided wave scattering model is to prevent unwanted reflections of the incident or scattered waves from the model edges while keeping the model small.To counter this, absorbing regions are added to the model edges using absorbing as layers with increasing damping (ALID) [44].These layers match the acoustic properties of the HDPE plate but progressively add attenuation.The result is that any waves entering the absorbing regions are attenuated, significantly reducing the amplitude of the edge reflections.This approach leads to the need to model a smaller region of plate and hence reduces the computation time for each FE model.This is implemented in Abaqus by adding Rayleigh damping to the layers with gradually increasing damping coefficient α, which represents the mass proportional damping.Thus, the value of α n at the n-th damping layer can be given by [17] where l n is the distance from the start of the absorbing region to the centre of n-th damping layer, L total represents the total size of the absorbing region, and N is the total number of damping layers.The direction of the increased values of the coefficient α n is set to be normal to the plate edge.The optimal size of the absorbing region and the parameters in Equation ( 3) are set by a process of trial and error.Here, the size of absorbing regions is least six times the incident signal wavelength at the centre of 30 kHz.The length of each individual layer is 0.294λ c .This leads to reflected waves from the model edge having an amplitude of less than 0.9% of the incident waves, which is considered adequate.

Far field determination
The modelling approach described here requires that the scattering behaviour be recorded in the far field where the guided wave propagation is two-dimensional.In this region the scattering matrix extracted using Equation ( 1) at different scattering distances, d sc should be the same.It has been previously shown [14] that this is achieved at monitoring distances greater than 9λ c for defect lengths of less than 2λ c .The near-field length could also be calculated using the Fraunhofer distance, 2l 2 =λ sc which is less than 8λ c .In this paper, defect sizes of less than one incident wavelength are investigated and so the appropriate far-field distance was explored in the model.A monitoring distance greater than 8λ c should therefore be conservative.Figure 3 shows the angular dependency of the scattered wave mode magnitudes from an example planar crack, monitored at three distances.This example is for a 30 kHz S0 mode, incident on a one-wavelength long and half-depth planar crack at 0 degrees.The relative difference in the scattered amplitude is less than 1% when the monitoring distance is greater than 8λ c .This paper focuses on the behaviour of elastodynamic guided waves with far-field amplitudes only.Hence, a plate dimension of 11.8λ c with d in of 11.5λ c and d sc of 10λ c is used to provide some further separation between the excitation and monitoring circles and the start of the absorbing layers from results in Figure 3.

Examples of the scattering coefficient matrices for holes and cracks
Next we explore how many FE models are required to capture all the scattering behaviour of typical defects to some prescribed accuracy.Two defect types are modelled: a circular hole and a surface-breaking semi-elliptical crack, both of half-plate-thickness.The hole is representative of axis-symmetric defects such as corrosion.The crack then represents an example of a more realistic defect.Here we focus our attention on small, i.e. � λ c , sized defects as these are particularly challenging to characterise.Also, previous results have suggested an angular interval of 6° as a benchmark for models to extract the scattering matrices if the scatterer size is more than 1.0λ c [16].Therefore, these two defects are used to generate a set of benchmark models with an angular interval of 5° that is conservative due to this scatterer size range.Each incident angle requires a separate simulation, from which all scattering angles can be recorded.The results from fine angular increments are then used here as a benchmark, against which the performance of coarser angular sampling can be assessed.For the hole due to its axisymmetric nature, the results for an incident angle of 0° can be shifted and used to reconstruct the other incident angles which have the same response.The crack has two planes of mirror symmetry and so incident angle ranges from 0° to 90° are needed, from which the results for other incident angles can be reconstructed.The scattering matrices of the scattered wave modes for any defect will have distinct forms dependent on the different incident and scattered wave modes as well as the frequency.Figure 4 shows the S-matrices of all (A0, S0 and SH0) scattered wave modes when S0 is incident.The results are for two scatterer types of half plate thickness and two sizes of l=λ c = 0.12 and l=λ c = 1.0 with f c = 30 kHz.There are distinct differences between the scattering matrices for the semi-elliptical cracks and the circular holes.The axissymmetry of the holes means that their scattering matrices consist of diagonal stripes as the scattering is only dependent on the difference in the incident and scattered angles.
Compared with Figure 4(c,d), Figure 4(a,b) shows that the pattern of the S-matrix for all scattered wave modes with a surface-breaking semi-elliptical crack is more complicated due to its non-axisymmetric shape.Also, in contrast to circular holes, they are not always symmetric about the diagonal line.For a smaller crack in Figure 4(a), it shows a lower maximum scattering magnitude and a wider central region when compared with Figure 4(b).Figure 4(c) shows that the maximum magnitude of S0 lies along the diagonal line θ in À θ sc j j = 0° but on the line θ in À θ sc j j = 180° for the scattered A0.For the larger hole in Figure 4(d), the S-matrices of S0 and A0 contain clear diagonal ridges along the straight line θ in À θ sc j j = 180° but diagonal ridges of the scattered SH0 trend along the straight lines θ in À θ sc j j = 50°.Therefore, we can see that the S-matrix pattern of all scattered wave modes is relevant to the 3D defect shapes and sizes.

The scattering matrices of a scatterer obtained by angular limited data
Once a certain angular resolution is exceeded the scattering behaviour can be reconstructed with very low errors, i.e. −40 dB.The 'conventional approach' is to capture the scattering behaviour in very fine increments, and this is our comparator.In this section, the angular and frequency dependencies of two scatterer types' scattering matrices are investigated to explore the efficient S-matrix generation for a range of defects in 3D FE models.Reconstruction of the scattering matrices of all three low frequency scattered wave modes from limited data is explored in both the angular and frequency domains.The configurations for extracting all defect information could be applied to experimental measurements.

Angular behaviour of S-matrix
The scattering matrices can be represented as 2-dimensional Fourier series because the angular dependence of the scattering matrix is periodic with an interval of 360°.The nature of this function can be used to investigate different angular orders of the scattered fields for these two representative scatterer types of different sizes.Thus, the maximum angular order of information is used to determine the minimum number of incident directions needed for an accurate S-matrix reconstruction.The Root Mean Square Error (RMSE) of S-matrix reconstruction is then used to quantify the difference between the benchmark (i.e.small angular increment) S-matrix and those reconstructed from coarse angular increment data by Fourier interpolation.

Fourier series representation for scattering matrices of the representative scatterers
For any scattering matrix S θ in ; θ sc ; ω ð Þ, the 2D Fourier coefficients A mn ω ð Þ is given by [45] where m and n are incident and scattered angular orders.
Figure 5 shows the Fourier coefficients of the scattering matrices of all scattered wave modes when the incident is S0 only.The information is extracted from the representative scatterers shown in Figure 4.It can be seen that the different types of scatterers of the same size have similar angular order ranges.We then define a threshold level, i.e. −40 dB (1%) to capture the majority of the information from Figure 4 using Equation (5), to allow us to extract the maximum angular order for each scattered wave mode.For example from Figure 5(a), for the scatterer with a size of l=λ c = 0.12 shown in Figure 4(a), a maximum angular order N = 3 is sufficient to capture Fourier coefficients of the scattered S0 with magnitudes greater than −40 dB.The matrix of scattered wave mode S0 needs to be computed at 2N +1 = 7 intervals between 0° and 360° to capture the information of S0 with 1% accuracy.The maximum angular order then captures all Fourier coefficients of representative scatterers with a precision greater than −40 dB.In this way we choose a specific minimum angular interval for each scattered mode such that we capture the majority of the information in each scattering matrix to 1% accuracy.Comparing the Fourier coefficients from the scatterers of different sizes, it can be seen that a smaller angular sampling interval is needed for a larger scatterer as well as a higher maximum angular order for capturing all information of scattering matrices of other scattered wave modes only compared to S0.Therefore, the highest maximum angular order can be determined by the mode with shortest wavelength.Meanwhile, it can also be seen that scattered SH0 has the most complex shape as in Figure 3 and hence requires the smallest angular interval.

Angular sampling interval determination for different scatterer lengths in angular domain
An equation of minimum angular interval, Δθ as a conservative estimate for the required angular sampling interval, is given by [15]: where N 0 is the maximum angular order for small-sized scatterers,l � λ sc .N 0 = 3 has been estimated numerically for a 2D scatterer (incident S0 input and scattered S0 only) from numerical modelling with the chosen smallest scatterers.For a 3D scatterer, N 0 is a specific value for each scattered wave mode that is related to the complexity of their scattered mode shapes and the geometry of the defect.For these two representative scatterers as in Figure 4, the scattered SH0 is more complex than others when the incident is only S0.In this paper, the N 0 = 3, 5 and 12 have been determined from all chosen smallest scatterers using Equation ( 5) that are chosen to capture all information of the Fourier coefficients of scattered S0, A0 and SH0, respectively.For the 8.18 mm plate example considered in this paper, the wavelength λ sc of scattered S0, A0 and SH0 are 1.0λ c , 0.64λ c and 0.63λ c .
In order to quantify the accuracy of a reconstructed S-matrix the RMSE error function is used: where S R is the reconstructed upsampled S-matrix and S B is the benchmark S-matrix.N in and N sc are incident and scattered angles.Firstly, the scattering matrices with an angular sampling interval of 5° as the benchmark reference for the representative scatterer types are computed (as in Figure 4(a)), and then the matrices at a reduced angular sampling rate are reconstructed using Fourier interpolation in Equation (5).The RMSE of their differences is quantified using Equation (7).The minimum angle increment for a specific scattered wave mode at all chosen scatterer types can be determined with a precision greater than −40 dB.After that, a conservative estimate for the minimum angular interval, Δθ defined that can be calculated with a specific maximum angular order from Equation (6).The results obtained from all scattered wave modes from representative scatterer types with variable sizes and the Equation (7) show that ε � 1% thus validating the proposed angular sampling interval determination.
Figure 6 shows the RMSE of the reconstructed S-matrix compared to the equivalent benchmark S-matrix for the representative scatterer types with specific lengths against the estimated angular sampling interval for each scattered wave mode.The results show that the larger scatterer needs a smaller angular sampling interval that means an increased number of FE models required to cover all information.However, more incident directions are required for scattered wave mode SH0 in order to ensure no information is lost.

Reconstruction of all scattering matrices for the representative scatterers in angular domain
Following the results from Figure 6, the limited data S-matrices are presented with an angular sampling interval chosen depending on the −40 dB threshold based on minimum sampling for that mode in Figure 7. Figure 8 shows the reconstruction of S-matrices calculated from the limited data using Fourier interpolation for comparison with the equivalent benchmark S-matrices.It can be seen that the minimum sampling interval using the threshold of −40 dB determined for each scattered wave mode is sufficient to describe the scattering matrix.Finally, the approach of reduced data sampling in angular domain is validated that can be applied to the S-matrix reconstruction for a specific scattered wave mode at any scatterer.

Frequency range behaviour of S-matrix for scatterer lengths of representative S-matrices
In this section, the behaviour of S-matrices with respect to the incident frequency range is investigated.This behaviour is explored through the S-matrices for representative scatterer types at a particular incident-scattered angle combination.The magnitude of the S-matrix is now found at particular incident-scattered angles and explored as a function of frequency.
Figure 9 shows examples of a specific incident-scattered angle combination in the S-matrices for the two representative scatterer types.The magnitudes of different scatterer types for some scattered angles θ sc of interest where the magnitudes are much higher than other scattered angles at the scattered S0, A0 and SH0 patterns for different scatterer types in Figure 8 when incident θ in = 0° are strong functions of the ratio of defect length, l to incident wavelength, λ in .We could use this information of that to characterise the defect type and length.Here, at most ten FE models with different scatterer lengths from 0.2l=λ in to 1.0l=λ in were simulated to obtain all information required.For a particularly incident signal and specified incident-scattered directions, the scattered amplitude was extracted from the frequency domain at the −6 dB points.Hence, we could use FE models with a scatterer size to efficiently extract the information of S-matrices for a range of scatterer sizes depending on the incident frequency bandwidth.Meanwhile, a four-dimensional (4D) information database, which contains a set of S-matrices of scatterer sizes for ultrasonic guided wave scattering matrices at representative defects could be achieved integrating Figures 8 with 9 for defect characterisation through inversion approaches.

Discussion of potential applications
The previous section has shown how to develop efficient FE models to generate scattering responses.The reduction in number of angles required to accurately capture the scattering response has important implications for practical measurements as well.For instance if we wish to completely characterise the scattering from a defect, a specific angular pitch of incident and scattered angles is necessary.It may not be possible in reality to use transducers arranged as seen in the numerical simulations on a plate-like, or even large diameter pipes where the experimental apparatus must fit within a limited space and the time to acquire such data may be large.Therefore, it is important to understand the consequences of using limited data based on a relatively small number of measurement locations for characterisation of defects in large structures.In this section, potential inspection strategies and use cases for the arrangement of an array of transducers in angular domain are discussed in three different case studies.
• Case No.1 -full matrix capture In section 2, a full S-matrix obtained using 72 sources and monitoring points and all their combinations has been presented in Figure 4.After that, a 2D S-matrix imaging reconstruction using a minimum angular sampling interval with Fourier interpolation that has been achieved to efficiently reduce overall computational cost (see Figures 7  and 8).For example, we could use 7 transducers for the S0 and A0, and 18 transducers for the SH0 to capture their all information of S-matrices for a 0.12λ c defect length at experiments.In practice, we could use the equivalent number of robots arranged in a circle to obtain a realistic S-matrix for defect characterisation with the number dependent and the mode of interest.

• Case No.2 -pulse-echo
In this case, we consider a single source and receiver combination at the same incident and scattered direction.The experimental apparatus is centred on the incident/ scattering direction and moves around a circle for measurements with a minimum sampling angular interval.An example for these two representative scatterers as 1D results is shown in Figure 10(a).The magnitudes for a surface-breaking semi-elliptical crack have distinct changes in the angular domain but the responses are constant for a circular hole at the diagonal line θ in À θ sc j j = 0° (see Figs. 8(c-d))).In practise, we can use the information to arrange transducers for measurements at these featured locations.

• Case No.3 -pitch-catch
In experiments operating in pitch-catch, the scattering response from a defect and the incident signal received may overlap when the time of arrival is too close (i.e.θ in À θ sc j j = 180°).Therefore, another interesting case is the single source and receiver combination at different incident and scattered direction to avoid this problem.One is that the magnitudes obtained from the scattered angles between 0° to 360° when incident direction is fixed at any incident angle, such as θ in = 0°, see results in Figs. 10  (b-c)).There is, in addition, an angle between the source and receiver.The results for a gap between sensors at θ in À θ sc j j = 15° (see Figure 10(d)) and θ in À θ sc j j = 55° (see Figure 10(e)) that show different scattered wave modes have specific magnitude levels at a particular angular gap.It could be a simplest method to provide scattering data for defect characterisation.Thus, several ultrasonic guided wave sensing robots can be employed for determining a defect type and size.Meanwhile, the measured information could potentially be inverted to obtain the source location.Here, we could use the 4D FE modelling information database for S-matrices at representative 3D defects that predict how many robots or source locations at least and how far the distance between the robots and the defect of interest.Moreover, the number of measurements could also be estimated to acquire sufficient information for a specific scattered wave mode (i.e.S0, A0 or SH0) using the earlier angular order information.Various additional issues such as time-triggering, and transducer positioning are expected to be encountered during experiments that have not been explored in our simulations.In terms of triggering, guided waves arrays have been used to construct images that rely on phase coherence [37,38,46].From this it is known that timing accuracy should be better than 1/10 th of the period of the wave.At the operating frequency of 30 kHz proposed in our paper, that equates to a timing precision of 3.3 µs, which is well within the capabilities of modern electronics.Similarly, in terms of the positioning of the transducers, it is known that if this is better than 1/10 th of a wavelength then amplitude errors due to lack of phase coherence are small.For S0 at 30 kHz the wavelength is 34 mm and so, positioning accuracies better than 3.4 mm would be required.Hence, whilst this experiment would require good positioning, it is well within the capabilities of modern equipment.

Conclusions
In this paper, we proposed an efficient 3D finite element modelling procedure with a reduced calculation time to obtain the scattering matrix of two representative defects of interest.An efficient finite element model has been developed to simulate an arbitrary defect size in the low-frequency regime as well as sampling approaches for the minimum number of model runs to ensure good model accuracy.For modelling irregular and rough defects in FE modelling, it is expected that smaller element sizes will be needed to capture subtle scattering behaviour.The reconstruction of S-matrices of guided waves are efficiently achieved using a sampling interval for each wave mode with Fourier interpolation method.The method can accurately reconstruct the full S-matrix at all angle directions from the S-matrices over a limited number of angles.Here, the simulation procedure described could enable the selection of the minimum number of measurements needed to obtain a specific scattered wave mode (i.e.S0, A0 or SH0) with a given defect type and size.This knowledge relates to strategies in the number of measurements for guided acoustic wave inspection using a swarm of autonomous robots.adaptive methods to sampling the output for an iterative scheme could be adopted, for example, the angular interval is progressively reduced until no further changes in the scattering behaviour is found.The limited modelling output for magnitudes of the far-field scattering coefficient of all scattered waves at a scatterer can be obtained for determining the defect geometry within a reliable incident frequency range.The finding from the practical case studies will help guide potential applications using limited transducer numbers, such as robotic NDT or SHM on pipeline networks.

Figure 1 .
Figure 1.Schematic of 3D FE model to obtain scattering matrices of guided Lamb waves at defect: (a) FE modelling in detail, (b) two typical defect shapes for representative scatterers.

Figure 2 .
Figure 2. The incident signal pulse for the guided wave excitation, a 10-cycle 30 kHz Hanning windowed signal:(a) Time domain, (b) Frequency spectrum.

Figure 3 .
Figure 3. Scattered wave-field patterns for the S-matrix of the crack defect size.

Figure 6 .
Figure 6.The RMSE of the reconstructed S-matrix using Fourier interpolation with an increased angle increment compared to the equivalent benchmark S-matrix of the scatterers with different lengths: crack for (a) 0.12λ c ; (b) 1.0λ c ; and hole for (c) 0.12λ c ; (d) 1.0λ c .Noted that the dashed lines are the −40 db threshold: black for S0; red for A0 and blue for SH0.

Table 1 .
Parameters used for finite element modelling.