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Sound field modelling using SimulUS (May 2008)


Channa Nageswaran, Charles R A Schneider and Colin R Bird

Paper published in Insight, vol.50. no.5. May 2008.


The use of models in the ultrasonic inspection industry is now widespread. Models are used to ensure that the sound fields created by ultrasonic probes are suited to the inspection task at hand, to study the feasibility of targetdetection and for the technical justification of procedures. Several models are available in the market targeting different requirements and it is important to ensure that the models are assessed independently to define the limits totheir validity. SimulUS was developed for the rapid assessment of inspection scenarios and this paper presents the TWI validation evidence. The accuracy of SimulUS in modelling the ultrasonic sound field using the continuous wave inputis sufficient for most applications in industry for single crystal probes and for linear phased array probes within certain limits of validity.

1 Introduction

The motivation behind developing the SimulUS beam model was as a tool for the rapid visualisation of ultrasonic fields generated by phased array transducers (both linear and matrix). However, it is possible to model single crystaltransducers by defining a 1 by 1 matrix. The present work was carried out using the version released in December of 2005 and was part of a larger validation programme in TWI involving other models. The core of the model is based onHuygen's discretisation method of evaluating the free field ahead of the sound radiator (Section 2). SimulUS operates principally in the continuous wave[1] mode but a pulse wave expression isincorporated. The model is primarily designed for rapid visualisations of the sound field but it can be used as a pulse wave modelling package. The computing requirements are less intensive when the input waveform is of a singlefrequency (continuous wave). Hence, in this paper, the evaluation of the performance of SimulUS in its primary continuous wave mode is presented.

SimulUS was envisioned for use in applications such as:

  1. Technique development - such that the operator can rapidly evaluate the suitability of a particular setup for the inspection task.
  2. Probe design - confident design of, in particular, phased array probes such that the designer can ensure that the parameters are correctly specified to achieve optimised field properties for the application.
  3. Training - allowing students to understand the basics of phased array ultrasonics and conventional ultrasonics through an interactive learning approach; where the effects of varying the crystal size, pitch, frequency etc on the ultrasonic field generated by the transducer can be understood and even the effects of failed elements modelled.

2 Principles of ultrasonic field models

Ultrasonic field models have been developed utilising various different theories. In essence, they are all required to evaluate the sound pressure field ahead of a transducer. As well as SimulUS, two other successful models will be described to illustrate the different approaches that have been taken in the past; both models are well established and have been extensively validated within industry and by research organisations. The models developed in the Non-Destructive Testing Applications Centre (NDTAC) of the Central Electricity Generating Board (CEGB) were based on an analytical approximation to the pressure distribution over the sound radiator. The CIVA model, developed by Commissariat à l'Énergie Atomique (CEA), is a semi-analytical model using a theory for electromagnetic wave propagation that was adapted for simulating the elastodynamic propagation of sound waves. [2]

The sound field ahead of the probe face can be described using Huygen's principle, which uses elementary waves of the same frequency which originate at discrete points on the radiating surface.[3] These waves then undergo interference effects in front of the radiating surface and their summation, as derived by Fresnel, gives the strength of the field at any point ahead of the probe. The solution dependson noting that at any given point ahead of the radiator, Huygens' elementary waves from discrete points have taken different paths. Since the sound pressure has an inverse relationship to actual distance travelled from the origin, the path difference dictates actual pressure contributions from each elementary wave at the point of interest. This method of evaluating the resultant pressure contribution is known as zone construction. For each elementary contribution,the path difference is converted to an angular measurement along with the change in pressure at the given distance; vector addition is used to evaluate the pressure and phase at any given point ahead of the radiator. This method can be used effectively for evaluating the pressure values along the beam axis and off axis with modifications to the element configuration. This forms the basis of the SimulUS model where the contribution of wavelets from sources on the transducer is summed on the calculation plane.

The NDTAC beam model is an analytical model that was developed to form part of a larger systematic approach to the modelling of ultrasonic inspections. The model was designed to generate the sound fields created by single crystalprobes. It is a modified version of the simple piston oscillator model of a single crystal. The Fresnel integral approach to evaluate pressure values at a general point ahead of the probe face is computationally intensive. The NDTACmodel was envisaged as a module in a systematic approach to evaluating defect interaction; since the beam model needed to be run several times, it needed to be computationally less intensive while providing sufficient accuracy in the prediction of pressure values ahead of the probe. Hence the NDTAC model suggests an approximation for the amplitude profile over the radiating face such that the evaluation of the field ahead can be performed analytically as opposed to resorting to numerical methods.[4] However due to this approach the models are liable to inaccuracy below a range of 0.8 near field lengths.

In the CIVA model the sound radiating area of the transducer is discretised into source points. The influence of these source points on a region of interest ahead of the transducer is calculated using ray theory. Each source point generates a ray (also termed a 'pencil') which is propagated from the source point to the computation point; CIVA is able to account fully for refraction effects at material interfaces. The underlying theoretical basis of CIVA lends itself for modelling quite complex inspection scenarios and TWI makes use of the model for a wide variety of applications.

SimulUS is principally a continuous wave model dedicated to the simulation of the sound pressure field ahead of the transducer with or without a wedge. In the presence of a wedge SimulUS uses Huygen's principle to evaluate the sound field at the wedge/material interface, which then becomes the radiating area for the sound field within the material; refraction is accounted for by using the appropriate velocities. SimulUS was designed to run rapidly so that it can be used in near real time.

The work presented here investigated the use of all three models in their continuous wave mode and the issues relating to the single frequency input were evident. It is important to realise that models are judged according to their fitness for purpose and in the present paper the applicability of SimulUS for its envisioned role is evaluated.

3 Validation methodology

TWI undertook the validation of SimulUS in a staged approach. Single crystal sound pressure fields generated by SimulUS were validated to study the underlying theoretical basis. Subsequently the sound pressure fields of a linear phased array probe generated by SimulUS were validated against experiment and other models. Experimental data for the validation of the single crystals was performed by the Atomic Energy Authority (AEA) for the CEGB (Chapman, private communication), whereas the data to validate the linear probes was collected in TWI. No CEGB model data is available for linear arrays as the models do not currently handle phased array probes. An immersion technique was used to measure the sound field profiles of the linear array, where a spherical target of 1mm diameter was placed at the required range while the probe was raster-scanned in a two-dimensional plane.

Figure 1 shows the parameter definition in SimulUS. It shows the three possible planes on which the model can be evaluated, denoted as the X-Z plane, the perpendicular plane and the focal plane (or any plane parallel to the latter plane). The transducer is generalised to a two dimensional array which becomes a single crystal when the array size is 1 by 1. Note also the definition of the wedge thickness as a key input parameter (referred to as 'Thick' inFigure 1).


Fig. 1. Definition of parameters in SimulUS and the three simulation planes

(Reprinted with permission from Peak NDT, UK)

As an example, Figure 2 shows the pressure field in the X-Z plane for the 4MHz 8mm x 9mm rectangular single crystal.


Fig. 2. Pressure field along the beam axis for the 4MHz 8mm x 9mm rectangular single crystal generated along the X-Z plane by SimulUS.

3.1 Single crystal probes

The parameters of the probes used in the validation programme are given in Table 1. The 70° shear wave probes were chosen to evaluate SimulUS's ability to deal with high refraction angles across the wedge to material interface. The material was plain carbon steel and the experimental sound field measurements were taken using standard calibration blocks. The probes were unfocused and hence the sound fields generated by them are well characterised in past works; the rectangular nature of the crystals of these probes lend themselves for ease of discretisation for the model and so allow evaluation of the underlying theoretical basis without introducing further complexity.

Table 1 The ultrasonic parameters of the rectangular single crystal probes.

ProbeWave typeBeam angle (°)Crystal size (mm)Frequency (MHz) Wells KK MWB 70-4Shear708 x 94 Wells KK WB 70-2Shear7020 x 222

The sound pressure along the main beam axis (on axis) and the pressure profile perpendicular to the beam axis (crossbeam) at several ranges were used for validation. Table 2 summarises the data presented in this paper. The range refers to the distance along the main beam axis from the index point (the geometric point on the interface where the sound is incident on the material from the wedge) along which outputs were generated in SimulUS. The simulation results are presented in section 4.1.

Table 2 Summary of single crystal data presented for validation of SimulUS.

CaseProbeData typeRange (mm) 1Wells KK MWB 70-4On axisUp to 150mm 2Wells KK WB 70-2CrossbeamAt 66mm 3Wells KK MWB 70-4CrossbeamAt 99.5mm

The crossbeam data are evaluated on the focal plane (see Figure 1). The profile along the X' and Y' axes (see Figure 1) are extracted from the two-dimensional output; the sound field is only refracted along the X-Zplane and is un-refracted along the Y-Z plane. Both the on axis and crossbeam data are presented as profiles for comparison with experimental data and the output from the other models. As an example, Figure 3 shows the two-dimensional crossbeam data evaluated on the focal plane by SimulUS for the 2MHz 20mm x 22mm rectangular single crystal at a range of 66mm.


Fig. 3. The crossbeam of the 2MHz 20mm x 22mm rectangular single crystal at a range of 66mm. The sound pressure profiles along the two focal plane axes X' and Y' are traced in white along the red cursors.

3.2 Linear phased array

A 2MHz linear phased array probe (pitch of 1.5mm with 22 elements) focusing at depths of 100mm and 200mm in water was simulated in CIVA and SimulUS. An array with 22 elements was chosen because the focal depths lie within 10% to 90%of the natural near field length of the array aperture, giving optimal focusing performance. Furthermore, the 22-element array was steered at 0°, 2.59°, 4.92° and 6°, which correspond to generating longitudinal wave beams of 32°, 45°, 60° and 70°, respectively, in steel with an immersion wedge of 7.63°. CIVA was operated in its pulse wave mode, more accurately reflecting the actual excitation pulse, while SimulUS was operated in the single frequency (ie 2MHz) continuous wave mode.

Experimental data were collected at TWI at (a) the focal depth position, (b) an upstream position, which is 6dB below peak strength at a range less than the focal point, and (c) a downstream position, which is 6dB below peak strength at a range greater than the focal point. Model outputs of the sound fields at those positions were also generated at TWI.

As in the case of single crystals, pressure profiles along two orthogonal axes on the focal plane are used in the validation. However, the two axes are referred to as the active and passive axes, corresponding to data on there fracted and un-refracted planes, respectively. The differing terminology is used to help differentiate between the direction along which the sound field is electronically manipulated (phased) by the array - the active axis - and the direction along which the radiation of sound is not controlled electronically - the passive axis. The selection of cases presented in this paper are summarised in Table 3 and presented in Figures 10 to 15.

Table 3 Summary of the phased array data presented for validation of SimulUS.

CaseFocal depth (mm)Steering (°)Data positionData type 11000Focal pointActive 21000DownstreamPassive 31006Focal pointActive 41006Focal pointPassive 52004.92UpstreamActive 62002.59Focal pointPassive

4 Results and discussion

The results for the conventional and phased array ultrasonic probes are presented and discussed separately, followed by a general discussion of SimulUS and its role within industry. Note that both SimulUS and CIVA calculate the transmitted (or free field) at the specified range (or region) along the beam axis but the results are validated against received pulse-echo fields from targets. Hence the raw output of the two models is squared to generate the pulse-echo field values; all field profiles presented in this paper are pulse-echo fields.

4.1 Single crystal probes

The on axis pressure profiles of the 4MHz 8mm x 9mm rectangular single crystal are shown in Figure 4. The experimental data and the model data from the NDTAC are shown along with the modelling results from SimulUS and CIVA.Figure 4 shows that all three continuous wave models are accurate to within 1dB in the far field, ranges greater than 50mm; the experimental evidence is not sufficient to comment on the accuracy of the three models below this range. At ranges shorter than the near field length (N) the sound field is known to fluctuate rapidly with distance along the main beam axis and off axis. It is believed that all models have limited application within this region and find it difficult to predict actual field variation. Continuous wave models exaggerate the fluctuations due to strong interference patterns and for this reason TWI does not recommend the use of any continuous wave models at ranges less than the near field.


Fig. 4. The on axis pressure profiles of the 4MHz 8mm x 9mm rectangular single crystal

Figure 5 shows the sound pressure field evaluated by SimulUS in the perpendicular plane for the 4MHz 8mm x 9mm rectangular crystal. Note the sound pressure field profile along the main beam axis (on axis) outlined in whiteat the bottom.


Fig. 5. The pressure field of the 4MHz 8mm x 9mm rectangular single crystal evaluated in the perpendicular plane by SimulUS

Figures 6 and 7 show the crossbeam profiles along the un-refracted and refracted planes, respectively, at a range of 66mm for the 2MHz 20mm x 22mm rectangular single crystal. The predicted 6dB beam widths are accurate to within 10% along the un-refracted plane (Figure 6), but below the 6dB point from the peak strength, all three models show variation of up to 10dB compared to experiment and they generally overestimate the beam width. Along the refracted plane (Figure 7) the predicted 6dB beam width is again accurate to within 10%. Again, as along the un-refracted plane, the models become inaccurate when more than 6dB below the peak signal strength. The models are accurate, to within 10dB of experimental strength, along the least refracted edge of the beam (left of the peak) and are less accurate (to within 20dB of experiment) along the most refracted edge of the beam (right of the peak). In general, note that CIVA shows closer agreement to experiment along the least refracted edge (left of the peak), while the NDTAC model shows better agreement along the most refracted edge (right of the peak).


Fig. 6. The crossbeam profile along the un-refracted axis for the 2MHz 20mm x 22mm rectangular single crystal at a range of 66mm


Fig. 7. The crossbeam profile along the refracted axis for the 2MHz 20mm x 22mm rectangular single crystal at a range of 66mm

Figures 8 and 9 show the crossbeam profiles along the un-refracted and refracted planes, respectively, at a range of 99.5mm for the 4MHz 8mm x 9mm rectangular single crystal. Along the un-refracted plane (Figure 8)all the models predict the 6dB beam width accurately to within 10% and remain equally accurate up to the 8dB beam width. The continuous wave nature of the models is clearly illustrated at distances away from the main beam (beyond the20dB beam width). Along the refracted plane all the models overestimate the 6dB beam width by up to 60% in the case of SimulUS, 45% in the case of CIVA and only 20% in the case of the NDTAC model. The difference is less along the least refracted edge (left of the peak) than along the highly refracted edge (right of the peak). In this respect the NDTAC model predicts the experimental profile better than both SimulUS and CIVA.


Fig. 8. The crossbeam profile along the un-refracted axis for the 4MHz 8mm x 9mm rectangular single crystal at a range of 99.5mm


Fig. 9. The crossbeam profile along the refracted axis for the 4MHz 8mm x 9mm rectangular single crystal at a range of 99.5mm

For the inspection configurations considered in this study, SimulUS is able to predict the on axis pressure field profile of unfocused probes to within 1dB at ranges greater than 1.2 near field lengths. Similarly, when refraction is involved, the prediction of the 6dB beam widths at ranges greater than N is accurate to within 10% along the un-refracted plane but the accuracy decreases along the refracted plane.

4.2 Linear phased array

Figure 10 shows the crossbeam profile of the 2MHz linear phased array probe along the active axis at the focal depth when the beam is steered at 0° and focused to a depth of 100mm. The experimental data was collected at TWI; CIVA was run with a finite broadband pulse input but SimulUS was run in its continuous wave mode. Both models are accurate to within 10% in their estimate of 6dB beam width. The strong side lobes of energy predicted by the SimulUS model are due to the single frequency content in its input pulse (continuous wave), unlike CIVA which has a finite spectrum that effectively averages the energy distribution to eliminate points of maxima and minima. The continuous wave mode appears not to affect the ability of SimulUS to effectively estimate the 6dB beam width.


Fig. 10. The crossbeam profile along the active axis at the focal depth when the beam is steered to 0° and focused at 100mm depth (2MHz 22-element linear array)

Along the passive axis, beyond the focus at the downstream position the 6dB beam width is again estimated to within 10% (similar to CIVA) in the case shown in Figure 11 (beam steered to 0° and focused at 100mm depth),but the general features of the crossbeam profile along the passive axis are exaggerated by SimulUS, whereas CIVA tracks the experimental pressure profile curve better. Similar levels of accuracy in crossbeam profiles at the focal depth are achieved in the case of steering to 6° and focusing to a depth of 100mm, as outlined in Figures 12 and 13, with maximum inaccuracy of 20% for SimulUS (in the active axis case).


Fig. 11. The crossbeam profile along the passive axis at the downstream position when the beam is steered to 0° and focused at 100mm depth (2MHz 22-element linear array).


Fig. 12. The crossbeam profile along the active axis at the focal depth when the beam is steered to 6° and focused at 100mm depth (2MHz 22-element linear array)


Fig. 13. The crossbeam profile along the passive axis at the focal depth when the beam is steered to 6° and focused at 100mm depth (2MHz 22-element linear array)

As in Figure 11, the continuous wave nature of SimulUS is apparent in its prediction of the passive axis profile shown in Figure 13, but the exaggeration of the double hump by SimulUS, apparent in experiment, is less than 2dB.

The 6dB beam width estimate of SimulUS is within 10% for the case shown in Figure 14. The upstream position (as in Figure 14) is within the region of the material where beam forming takes place for a phased array transducer and the sound pressure fluctuates rapidly with position (as in the Fresnel region of single crystal unfocused transducers). In the case of focusing to a depth of 200mm while steering the beam to 4.92° shown in Figure 14, SimulUS performs well with regard to beam width estimates (within 10%) and predicting the double hump feature, but both models overestimate the second hump by ~2dB in comparison to the experiment.

Figure 15 shows the passive axis crossbeam profile where SimulUS underestimates the 6dB beam width by around 20% but is more accurate than CIVA which underestimates the 6dB beam width by up to 30%.


Fig. 14. The crossbeam profile along the active axis at the upstream position when the beam is steered to 4.92° and focused at 200mm depth (2MHz 22-element array)


Fig. 15. The crossbeam profile along the passive axis at the focal depth position when the beam is steered to 2.59° and focused at 200mm depth (2MHz 22-element array).


The 6dB beam width measurements are summarised in Table 4.

Table 4 Summary of beam width measurements for the linear phased array cases (note that the quoted accuracy is rounded up to the nearest multiple of 10 %).

CaseExperimental 6dB beam width (mm)CIVA 6dB beam width (mm)
(% accuracy)
SimulUS 6dB beam width (mm)
(% accuracy)
1 3 2.96 (10%) 2.9 (10%)
2 17.6 16.5 (10%) 17.1 (10%)
3 3.5 3.25 (10%) 2.9 (20%)
4 20 18 (10%) 20.2 (10%)
5 14.5 15 (10%) 14.2 (10%)
6 18 13.5 (30%) 15.4 (20%)

4.3 General discussion

Ultrasonic modelling is a tool used as an input for inspection design and for assistance with defect analysis; modelling can also be used as evidence in technical justifications. However, it is important to realise that models aretools and their limitations must be established and the users need to be aware of them. TWI recommends a pragmatic approach to the use of sound field (beam) modelling. Hence the models are required to be fit for purpose. In sound fieldmodelling the user is interested in the following parameters:

  • Amplitude with respect to reference amplitude;
  • Beam widths at particular ranges along the beam axis;
  • Beam amplitude with respect to range;
  • Location (and amplitude of) side lobes and diffraction grating lobes.

Practical measurements of the above parameters take place through the measurement of echo amplitudes; however, in practice it is difficult to measure amplitudes to better than 1dB and for most applications an amplitude variation within 2dB is acceptable. Similarly, an experimental inaccuracy of up to ¼ of the beam width in measurement of the 6dB beam width is anticipated. When beam width accuracies below the 6dB point are of concern it is important to recreate the actual input pulse shape of the real probe into the models through sampled inputs, since the frequency content has an important bearing upon the distribution of energy, TWI recommends that models should allow pulse shapes to be sampled from the actual probe to be used in inspection.

Hence from the above assertions it is believed that for practical applications the sound field pressure amplitude does not need to be accurate to better than 2dB with respect to a reference reflector. For very precise or criticalapplications more accuracy may be required. Similarly, an inaccuracy of up to ¼ beam width in the experimentally measured 6dB beam width may account for some of the differences between the models and measured values. This paperhas described the differences between continuous wave and pulse wave models and it is clear that pulse wave models are generally more accurate. But it is also clear from this paper that continuous wave models are pessimistic withrespect to grating lobes and side lobes, but are less intensive computationally. In comparison to CIVA, the time required to set up and run a model in SimulUS is significantly less (depending on required computational accuracy).

In summary, it is believed that SimulUS displays sufficient accuracy, within its limits of validity, for most applications and the roles for which SimulUS was designed. Subsequent to this validation effort, TWI has effectively usedSimulUS for inspection design and for developing advanced phased array probes, testing out many possible scenarios with little time penalties.

5 Conclusion

Within the scope of the validation at TWI, SimulUS displays 6dB beam width accuracy to within ¼ of the measured beam width in the case of single rectangular crystals and linear phased array probes.

The fluctuations evident in the continuous wave model employed by SimulUS would be an issue if the beam profiles were to be used for field-defect interaction modelling, but since the primary role of SimulUS is to allow rapidvisualisation of ultrasonic sound fields, TWI is satisfied with the limits to its validity in the continuous wave mode for the role proposed for the model.

TWI firmly believes that SimulUS can be used effectively as part of inspection design; the relatively small errors and pessimistic predictions will enable it to be used safely by users who appreciate the limitations of themodel.

6 Acknowledgements

The authors are indebted to Bob Chapman of British Energy for providing CEGB validation data and for generally supporting this validation effort. We are also grateful to Industrial Members of TWI for funding the Core ResearchProgramme project.

A special acknowledgement is due for Alison Whittle of Peak NDT for her interest and the insightful support given to TWI, and for illustrating the finer points of modelling.

7 References

  1. A C Whittle, 'Preliminary steps to validate a beam model for ultrasonic phased arrays', Insight, Vol 48, No 4, pp 221-227, April 2006.
  2. G A Deschamps, 'Ray techniques in electromagnetics', Proc IEEE 60, pp 1022-1035, 1972.
  3. J Krautkrämer and H Krautkrämer, 'Ultrasonic testing of materials'. Springer-Verlag, New York, 1990. ISBN 0-387-51231-4.
  4. J M Coffey and R K Chapman, 'Application of elastic scattering theory for smooth flat cracks to the quantitative prediction of ultrasonic defect detection and sizing', Nuclear Energy, Vol 22, No 5, pp 319-333, 1983.

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