C. Ennaceur* T. H Gan, R. Sanderson, P Mudge and B. Bridge
Paper presented at 4th International Conference of NDT (4th ICNDT) organised by the HSNDT, Chania, Crete, 11-14 October 2007.
A finite element numerical model of time reversal focussing of pulsed guided wave modes in straight pipes is presented. In the model a pure longitudinal or torsional wave mode is launched down a pipe containing a surface-breaking planar defect of constant depth and orthogonal to the longitudinal axis of the pipe. All signals received from the defect and pipe end are computed. Then the pulse received by the defect is time reversed and sent again down the same pipe without the defect present. It is found in this simulation that the wave was focused on the region that previously contained the defect. Quantitatively the model predicts the increase in signal amplitude available for the interrogation of a defect as a result of the time reversal routine. In the final stage of the model the signal received by reflection of the time reversed defect echo is computed. The model is illustrated for an axisymmetricfundamental longitudinal wave mode with a 10 cycle 61 kHz pulse of Gaussian envelope launched in a pipe of 6 inch diameter and 0.28 inch wall thickness. These results were then compared with experimental data on the pipe. The increase in signal to noise ratio of the defect echo produced by time reversal was 4 (12dB) compared with the experimentally observed increase of 2.2 (6.8dB). Theoretically the signal to noise ratio improvement corresponds to a reduction in the area of the minimum detectable size of defect by a factor of 4. Generally, the use of modelling to predict experimental situations in which time reversal focussing will improve defect detection sensitivity, has been confirmed.
Guided wave ultrasonic technique, also known as long range ultrasonics (LRU), is a condition monitoring tool of tremendous importance because of its ability to inspect long runs of pipework from a single access point. This feature generally reduces the time and costs of global monitoring and makes feasible the global monitoring of pipework buried underground, underwater or under various other engineering structures. There are many cases where LRU is the only possible global monitoring solution. As in all comparable inspection methods sensitivity has to be 'traded in' to achieve the range i.e. minimum detectable detect sizes are generally less than what is possible with local techniques(where applicable). Much effort recently has been devoted to methods of increasing sensitivity. A key problem is coherent signal noise caused by mode conversions. Beyond a certain optimum condition which depends on the overall system electronics, further increase in transmitter signal to noise ratio does not improve defect detection sensitivity because the signal to noise ratio of the unwanted mode conversion echoes increases in direct proportion with the wanted defect echoes. Focussing of the incident wave on a defect can be used to overcome this problem.
Time delay (also called phased array) focussing in LRU has been reported recently by Rose and Mudge  and this facility has been incorporated in a new generation of the Teletest System as illustrated in Figure 1. The time reversal method of focussing has recently been reported and compared with the time delay focussing approach in terms of experimental data for LRU in 6 inch pipes by Ennaceur et al  The method was pioneered in the area of biomedical engineering by Fink [3,4] and Fink et al  and in long range underwater communications by Leutinger et al. 
Fig.1. The Teletest LRU system shown with a 24- transducer ring on a 6 inch pipe. This system was use to obtain experimental data for both comparison with and inputting of data to the finite element LRU model
Time reversal focussing has an advantage over the time delay approach in that it does not suffer from aberrations in inhomogeneous media. It also has the ability to focus automatically because the technique does not need to know the exact position of the defect. The time reversed signals will focus automatically on the defect as it is the source of reflector.
In this paper a time reversal focussing experiment was performed on a straight 6 inch steel pipe using a Teletest ® long range ultrasonics instrument ( Figure 1). The experiment was simulated by theoretical modelling and the theoretical results validated by comparison with experiment ( Figures 2 - 8). The potential for the use of modelling to improve the sensitivity of time reversal measurements in long range ultrasonic testing was thus established.
Fig.2. A scan display showing echoes from the pipe end and a 9% CSA defect in a 6 inch diameter pipe with 0.28 inch wall thickness using the Teletest Tool configuration described in the text. Signals received by each of4 quadrants, which had been fired simultaneously, were processed in separate channels and then added together to produce these scans. The three traces indicate three reflection modes
Fig.3. As Figure 2, but with defect echoes having a much enhanced signal to noise ratio. These echoes were obtained by time reversing the defect echoes indicated in Figure 2 and using the amplified time reversed signals tore-fire the transducers
2. Simple explanation of the time reversal principle
A simple but comprehensive model of the time reversal focussing technique has been presented by Ennaceur et al in a non mathematical form that can be understood by NDT practitioners.  Firstly a particle mechanics analogy was employed. Secondly, a Huyghens wave reconstruction approach was adopted. The quickest way to describe time reversal focussing is to imagine a wave pulse as a pack of cards. If the ordering of these cards is exactly reversed, we now have a pulse which is the time reversed version of the initial pulse. The unique feature of a time reversed pulse is that if it is re-launched in the opposite direction to the direction of motion of the initial pulse it retraces the path and form of the initial pulse exactly, irrespective of any inhomogeneities in the propagating medium. For example a divergent pulse scattered outwards from a point defect converges back to the defect after time and direction reversal. So if the signal to noise ratio of a time reversed defect echo is improved by some means, sufficiently to overcome absorption and insertion losses, the re-launched echo will give rise to a second defect echo with greater signal to noise ratio than the first.
3. Modelling technique
Finite element modelling was used to simulate wave propagation. The modelling provided a prediction of the three-dimensional distribution of sound wave energy for every point in time that it is solved over. Output from the model can be either a displacement history A-scan style plots or three-dimensional contour plots of the distribution of the propagating stress wave. This technique enables an exact replica of an experiment to be produced and the results can be visualised in a way that is not possible experimentally.
The finite element models were set up, analysed and post processed using the ABAQUS/Explicit version 6.6 modelling software.
A volume model of the experiment test pipe was set up. The pipe was 4.1m long with a 0.28inch wall thickness and a saw cut of constant depth running at a plane at right angles to the pipe axis, 0.6m from one end of the pipe. Thecross sectional area of the saw cut was ~9% of the total cross sectional area. In the model the excitation was initiated at a distance of 1.8m from the defect i.e. 1.7m from the end of the pipe farthest from the defect. This excitation simulated the launching of a real signal from a transducer ring located at that position.
The volume model was meshed so that there were enough elements through the thickness to simulate the internal behaviour of the wave mode. The axial length of the elements was determined by calculating the smallest possible wavelength in the frequency bandwidth excited. The axial length was then chosen so that there were at least 8 elements per minimum wavelength. The overall geometry of the volume model and the finite element mesh around the defect is shown in Figure 4.
Fig.4. Volume model of the 6 inch pipe (not to scale) showing the finite element model geometry and mesh around defect
Material properties of Young's Modulus, Poisson's ratio and Density are required inputs for the propagation analysis. These were assumed to be 207GPa, 0.3 and 7830 kg/m 3 respectively.
4. Experimental set up
The finite element propagation model simulated real transducer displacements. For a 6inch pipe, the Teletest ® tool consists of three rings each with 24 transducers spaced around the circumference. For this work, the transducers were grouped into four quadrants each containing 6 transducers and therefore spanning 90degrees. A 10-cycle 61kHz pulse of Gaussian envelope (Hanning Window) was excited experimentally and echoes from each quadrant received in the 4 channels, one for each quadrant, were processed separately before being summed an presented as a composite A-Scan display. The experimental excitation and reception processes were simulated exactly in the modelling. In the model nodes at the same locations as the transducers were used to input the signal via prescribed loads identical to the experimental inputs.
5. Experimental results
Figure 2 shows the experimental A-scan display of pulses received when a 10-cycle 61kHz pulse of Gaussian envelope is launched into the pipe. The three A scan traces indicate three modes, symmetric, horizontal flexural and vertical flexural, generated by reflections from the defect and pipe end. The strongest of these defect echoes has an amplitude of 43 mV and a signal to noise ratio of 10 where the noise is predominantly coherent i.e. arising from mode conversions.
Figure 3 shows the experimental A-scan display obtained when the received defect echo was time reversed and re-launched. The signal to noise ratio had been increased by feeding the time reversed echo into a power amplifier and the output used to refire the transducers. It will be noted that the defect echo amplitude has increased to 73 mV and the signal to noise ratio is enhanced to 22. Time reversal focussing has thus, experimentally, increased signal to noise ratio for the defect echo by a factor of 2.2.
6. Modelling results
Model 1. The received defect echo indicated in Figure 2 was, after time reversal, fed into a finite element model for propagation in an unbounded pipe volume model without the defect present so that only the transmitted signal is present. The purpose of this exercise is to demonstrate theoretically a time reversal event without the presence of extraneous coherent signals and thus to quantity the highest attainable signal to noise ratio for a time reversed signal. The predicted focussing effect is clearly indicated in Figures 5 and 6 which indicate respectively the theoretical three dimensional stress contour plots and axial displacement along the pipe for the time reversed input signal as a function. Figure 5 shows the three dimensional contour plots of the distribution of stress at a point in time when the ultrasound had reached the place where the defect would have been, had it not been 'removed'. A maximum stress level on a 12 point grey scale occurs at the top centre of the pipe exactly coincident with the position of the 'removed' defect. In Figure 6 there was a large pulse arriving at the position of the 'removed' defect at the time expected based on the group velocity of the longitudinal wave mode (5415m/s). There was a 41.6 microsecond delay in the input signal which was factored into this calculation.
Fig.5. Predicted axial stress for time reversed input signal. Original position of defect is shown by black dot
Fig.6. Predicted axial displacement history at the defect location after input of time reversed data
Model 2. Secondly a 6inch pipe model containing the defect was solved with a standard axisymmetric longitudinal input signal, again at 61 kHz and 10-cycles and with a Gaussian window, as in the experimental case. The Ascan presentation of the axial wave displacement as a function of time is shown in Figure 7. The reflection from the defect was then time reversed and re-inputted into a second model. The time dependence of the axial wave displacement then became as in Figure 8. Comparison of Figures 6 and 7 shows that time reversal focussing has increased the defect signal amplitude by a factor of 4. Noise is not visible on the theoretical modelling plots of Figures 7 and 8. However it can be assumed that the noise in the theoretical model is a constant as no coherent noise from mode conversions are present. So the defect signal to noise ratio is taken to have increased by a factor of 4.
Fig.7. Predicted axial displacement history for standard longitudinal test
Fig.8. Predicted axial displacement history for time reversed longitudinal test
7. Comparison of modelling and experimental results
Experimentally an improvement in signal to noise ratio of the defect echo of 2.2 (6.8dB) has been obtained in the circumstances defined above. Theoretically the predicted improvement is a factor of 4 (12dB), implying that minimum detectable defect sizes at any given range can be reduced by a factor of 4, expanding the global monitoring capabilities of LRU. The agreement is fair given the several idealisations present in the theory such as the assumption of a smooth faced defect and the assumption of no energy losses through mode conversions or absorption losses in the finite element model. On the other hand it could well be the case that further improvement in experimental results up to the theoretical limit is possible with refinements in experimental design.
The agreement of theory and experiment is sufficient to confirm the usefulness of modelling for (i) validating the time reversed focussing effect in LRU (ii) improvement in the design of experiments to enhance defect signal to noise ratio in LRU, thus to expand its potential role in large scale condition monitoring. The most important quality of modelling is that it can rapidly predict the range of experimental conditions under which time reversal auto-focussing will provide useful improvement in defect detection sensitivity. Future work will involve a systematic modelling study of time reversal in curved as well as straight pipes as a function of pipe diameter, wall thickness and bend radius.
The authors would like to thank Plant Integrity Ltd for supporting this project. Plant Integrity designs, manufactures and provides services using the Teletest ® system.
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