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## The Use of CAD Representations in Ultrasonic Modelling

Andreas Schumm
Electricité de France
1 avenue général de gaulle
92141 Clamart
France

Contact

### Introduction

In ultrasonic testing, one is often faced with the inspection of complex shaped components. While it is often possible to employ simplified geometry models to represent the geometry of the specimen, more complex geometries may require the use of CAD solid models to accurately represent the geometry. Recent developments [1] [2] tend to favor this more general approach.

In this paper, we illustrate the benefits of using CAD solid models in ultrasonic modeling on three real-world examples. From these, we derive requirements on the geometries and the codes employed.

### Model Formulation

The Green's function approach is particularly well suited to treat the problem of refraction through an arbitrary interface, as the interface geometry appears explicitly in the formulation. In this type of model, two integral formulations are combined: A Green's function approach to treat the field of a point source across an interface, and the Rayleigh-integral to account for the finite transducer. Green's integral theorem converts a volume integral into a boundary integral, involving the field values and their normal derivatives on the boundary surface. The Rayleigh-integral then integrates these point source solutions over the transducer surface.

The explicit integration of the entire problem requires two surface integrals over the transducer and the interface surface. To avoid the numerical cost of performing these two integrations, the stationary phase method [3] is used to substitute the integration over the interface with an analytical expression. The stationary phase method is based on the assumption that for sufficiently high frequencies, adjacent half-periods of an oscillation cancel each other, so that the principal contribution to the integral comes from regions for which the first derivative of the phase term is zero. The actual integration is then performed by an expansion around these stationary points and an addition of only these contributions. Applied to the integral equation governing transmission through an interface, the stationary points are those whose corresponding ray paths satisfy Fermat's law of minimal travel time between source and observer coordinates. The stationary phase method is thus closely linked to geometrics optics : The stationary phase condition provides ray paths according to a geometrical optics solution, and the stationary phase solution yields directly the geometrical beam-spread factors known from geometrical optics.

The displacement amplitude at a given observation point is obtained by integrating the transducer point source solution in terms of three components beam-divergence Reff, transmission coefficient T and polarization e (for shear waves). Calculated in the time domain, the contribution of a point source to the impulse response becomes

 (1)

The beam divergence factor Reff involves local curvatures at the stationary point. This requirement has an important impact on the CAD representations discussed in the next paragraph.

The expression for the beam divergence involves curvatures at the stationary phase point, distances before and after the interface, and the local normal. While for short distances before the interface or for large radius of curvature, the exact expression converges towards the solution for planar interfaces and curvatures can thus be neglected, some configurations may require that curvatures be taken into account. The CAD model must be able to provide this information.

### Example I: Focused transducer inspecting conical pipe section

 Fig 1: Tube Inspection with immersed transducer

The first example illustrates the application of the model for an immersed transducer setup. The pipe section as shown in figure 1 is controlled from the inside of the pipe with a 60° longitudinal wave transducer. As the transducer is displaced from the cylindrical section towards the conical section, the focal properties of the sound field change due to a varying interface geometry. Figure 2 shows three sound field calculations on characteristical positions, showing the sound field for a probe entirely in the cylindrical section (left), at the intersection of both sections (middle) and entirely in the conical section. Apparently, the geometry has a significant impact on the sound field : The sound field splits up at the intersection, and progressively the primary sound field vanishes while the second focal spot becomes more and more pronounced.

 Fig 2: Sound field distributions at three characteristical positions (cylinder, intersection, cone)

For this simulation, the geometry was represented by a CAD description using rational B-Splines to represent the geometry. A solid modeler such as ideas or ProEngineer is able to furnish this kind of representation. The B-Spline representation is able to provide derivative information of arbitrary degree, and thus provides curvature information as well.

While the images provide qualitative insight into the physical effects being observed, the calculations were actually carried out at a number of transducer positions to determine quantitatively the loss of amplitude in the primary sound beam at a given focal depth as a function of the position. As such, the model provides valuable information for the limits of the inspection procedure.

### Example II: Focused transducer inspecting conical pipe section

 Fig 3: flange geometry Fig 4: sound field at given position
The main pump thermal barrier flange inspection represents a particular difficulty due to the limited accessibility and a toroid geometry, which has a significant impact on the sound beam properties after transmission.

Figure 3 shows the flange geometry. The flange is accessed from the toroid face. Figure 4 shows the resulting sound field of an immersed transducer with a customized wedge. The model is able to predict the impact of modifications of significant parameters, such as the distance between the transducer and the geometry, or the orientation of the transducer.

For this simulation, a tailored parametrical CAD model was used to represent the geometry. A parametrical CAD model can be parameterized for different flange diameters and sizes. The same applies to the transducer surface: The customized wedge was represented by a parametrical surface definition.

### Example III: Bolt inspection

Bolt inspection is characterized by limited access, which must be used in an optimized way to ensure insonification of the entire profile in order to detect defects located at all possible angular positions. For the problem at hand, simulation was used to compare various concepts before building an actual prototype.

 Fig 5: sound field in incidence plane Fig 6: Sound field in parallel plane

Figure 5 shows the sound field of a 12x5mm 3° contact transducer, displayed within the bolt geometry in the incidence plane of the probe. Due to the asymmetrical size of the transducer, two maxima are observed, corresponding to the focal lengths of the two dimensions. Figure 6 shows the sound field in the parallel plane right under the head of the bolt. Apparently defects oriented 90° around the main axis are less insonified than defects oriented at 0°.

For this simulation, a faceted CAD model was used for the bolt geometry, designed with a commercial solid modeler.

### Conclusion

The use of CAD models in Nondestructive Testing Modeling has become a common functionality in today's codes. CAD models for ultrasonic models must provide precise normal information, and preferably provide curvature information where needed. While generic CAD interfaces are more versatile, the use of parametric models may be appropriate for generic cases.

### References

1. A. Schumm, "Using CAD models in transient field calculations through arbitrarily curved surfaces", in Review of Progress in QNDE, Vol. 15. D.O. Thompson and D.E. Chimenti, Eds., Plenum Press, N.Y., 15A, pp. 1075, 1997.
2. A. Lhemery, P.Calmon, L. Paradis, "Ultrasonic Field Simulations for Scan Coverage Studies" in Review of Progress in QNDE, Vol. 18. D.O. Thompson and D.E. Chimenti, Eds., Plenum Press, N.Y., 1998.

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