Example Problem 1: Planar Interface with Axially Symmetric Field

In this first example, we simulated a half inch diameter, 2.25 MHz, unfocused probe with a circular crystal being scanned across a 2" thick stainless steel plate at normal incidence. The objective was to use the modeling software to quantitatively show the trade off between scan sensitivity and scan step size. The sensitivity was quantified relative to a the response from a 1/64" flat bottom hole response at normal incidence with a 3" water path and a 2" metal path. This example is shown first because it is the simplest possible combination of parameters (from Table 1). Both the probe and the interface are axially symmetric, so the wave fields will also be axially symmetric. The part geometry is simple and constant as a function of position. Given the simplicity of the geometry and the symmetry of the fields, this problem can be solved without the use of a CAD interface, or a book keeping tool like the voxelizer. Because this is a contrived example that is not modeling a particular existing transducer, a hypothetical efficiency factor, beta, will be synthesized for the probe. A cosine-squared shaped curve with a 60% bandwidth around the center frequency was used as shown in Figure 3. Note that UTSIM can use a measured beta factor just as easily as it can use a hypothetical one. It is logical to use a hypothetical beta factor at an inspection specification stage, before the transducers are purchased. Later, when the actual transducers are in hand, a measured beta can be used to validate the model predictions with experiment results. Because the interface is flat, Cartesian coordinates will work fine as the scan motion coordinate system. Given that the fields from this transducer will be axially symmetric, the beam width will only need to be evaluated on one plane to define the 3D flaw response amplitude isocontour. The one plane will only be evaluated for one transducer location because the interface is constant as a function of position. A small planar reflector at normal incidence was chosen for the flaw model. This particular flaw was chosen because it is computationally efficient. A typical time domain waveform from this flaw is shown as Figure 4. Figure 5 shows the magnitude of the maximum peak to peak voltage response from a transducer. For convenience, the plot has been normalized by dividing the response by the amplitude of the last on axis peak. The single frequency beam amplitude is drawn on the same graph for comparison. Notice that the single frequency amplitude has many peaks and valleys, but in the broad band response most of them are washed out. Although the water path was specified for the reference sensitivity experiment, the water path used for the scan was intentionally left unspecified in the problem statement to show that it can have a significant effect on the scan spacing result. Because of attenuation and diffraction, sensitivity increases as the water path decreases. The diffraction of the beam also causes larger water paths to create wider beam widths in the inspection zone. The maximum allowable scan spacing will be calculated for a broad range of sensitivity levels and water paths. There are several practical constraints on the water path length. It is usually undesirable to have the water path so short that near field structure occurs in the inspection zone. Also if the water path length is too short, a signal will appear in the time gate from a reflection off the probe. To meet this criteria, the water path must be greater than the inspection depth multiplied by the ratio of velocity of the water over velocity of the medium. For this particular inspection, with the plate 2" (5.08 cm) deep, the water path must be at least 1.3 cm. The upper limit on water path may be constrained by the depth of the immersion tank, but a more practical limit is that the focal point or on axis maximum of the beam should occur inside the part. A plot of on-axis response in the steel from a scatterer as a function of water-path is shown in Figure 6. To calculate the shape of the constant sensitivity flaw response amplitude isocontour, the small scatterer was positioned on axis at a given depth to calculate the on axis response. The on axis response value was used to apply a distance amplitude correction as is frequently done in industry. Next, the flaw was moved in a direction perpendicular to the central axis to get evaluate the amplitude relative to the on axis value as a function of width. This cross axis calculation was repeated at many depths through the inspection zone. Figure 7 shows a beam width profile as a function of depth for a water path of 1.3 centimeters. This profile can be thought of as the diameter of the axially symmetric flaw response amplitude isocontour as a function of depth in the material. Now that the flaw response isocontour function is known, we must solve for the overlap between two transducer locations that will maximize scan step spacing while guaranteeing coverage. A diagram of the geometry of the scan from the top view is shown in Figure 8. It shows the tangency condition necessary for the beam at successive transducer locations to maximize scan step. This diagram only applies to completely axially symmetric problems. For a sensitivity 6dB down from the reference measurement figure 7 showed that the minimum beam width was .36 cm.

Figure 3. Graph of beta for 2.25 MHz probe Figure 4. Typical waveform from small planar reflector Figure 5. On axis response of 2.25 MHz probe in water Figure 6. Effect of water path length on sensitivity for 2.25 MHz probe Figure 7. Beam width as a function of depth for 2.25 MHz probe Figure 8. Geometry required to determine step size for axially symmetric problems Figure 9. Trade off between sensitivity and step size

Using this value and the geometry shown above we can calculate the scan step size as .36 divided by the square root of 2 which equals .25 cm. Because of the symmetry, this value is also used for the index spacing. This calculation was repeated for many water paths and sensitivity levels. The results appear in Figure 9. Note that there is a region where the water path has virtually no effect on the scan spacing. In this region, the beam width near the on axis maximum is the limiting factor that determines

spacing. On the right hand side of the graph, where sensitivity is the highest, the water path has a significant effect. The curves intercept the horizontal axis when the on axis flaw response at the bottom of the zone is no longer above the reference sensitivity. Given this graph the trade off between sensitivity level and scan step size is known for the hypothetical 2.25 MHz transducer. Once the transducer was purchased, the model should be run again using the measured transducer beta factor, effective diameter, and effective focal lengths as more accurate model inputs. Note that this process could be repeated multiple times while varying transducer parameters to possibly specify a better transducer for the inspection.

Constant Cylindrical Interface with Elliptical Field Cross-sections

The titanium used to manufacture jet engine turbine disks is inspected first at the billet stage. Billets are large cylinders of metal the size of telephone poles. The Engine Titanium Consortium (ETC), comprised of General Electric, Pratt & Whitney, Allied Signal, and CNDE have spent many years modeling and improving billet inspections. One of the deliverables of the ETC is an inspection system known as the "Multi-Zone System". The multi-zone system uses large probes with bi-cylindrical lenses to focus sound at different depths (zones) in the billet [24]. This is a difficult problem because the cylindrical surface of the billet causes the beam to diverge or defocus. A lens was affixed to each transducer that produced bi-cylindrical focusing to counteract the effect of the interface. Together, the combination of a bi-cylindrical lens and the cylindrically curved interface form an effective compound lens. Theses lenses can't be designed with simple Snell's law optics because the true focal lengths of real beams are always shorter than what the simple optics equations predict. These large bicylindrically focused transducers are not in general use, so each is custom made. It was cost prohibitive to find the correct lens parameters with trial and error experiments, so beam models were needed. The ETC has used the ultrasonic beam models developed at CNDE to specify lenses for the multi-zone system. Because the lens parameters are proprietary, the actual parameters will not be modeled here. Instead, transducer focal lengths were chosen so that Snell's Law of optics predicts the focal spot to be on the central axis of the billet. This is also similar to what might be specified if the inspection designer did not have access to a beam model. This example will demonstrate calculating the best scan spacing parameters using this non ideal transducer. Two views of the ray modeling to choose the transducer focal lengths are shown in Figure 10. These views were obtained using the capabilities of UTSIM. A 3" water path was specified since this was the water path used in the ETC program. The focal lengths needed to obtain a ray-tracing point focus on the center of the billet were calculated as Fx = 24 cm, and Fy = 76 cm. Where X is in the radial direction of the billet and Y is in the lengthwise direction. The radius of the billet itself was 16.51 cm. The longitudinal wave velocities in the water and titanium were taken as .148 and .607 cm/micro-second respectively. In this example, we will assume that the inspection zone is from 14 cm to 17 cm deep in the billet (Figure 11). This choice of inspection zone provides some coverage overlap of the center point. We will solve for the scan steps in two directions that are consistent with overlapping the 3dB beam widths. From Table 4, we need only calculate the flaw response on 2 planes at one transducer location. The best coordinate system for the inspection is cylindrical with the scan direction being theta and the index direction is along the length of the cylinder. The On axis response in water was again plotted at single frequency with the broad band flaw response superimposed on it as shown in Figure 12. Note the complex nature of the field from this rather exotic transducer. The process to solve for optimum scan step sizes is similar to the previous example with a few added complications. Neither the probe or the interface is axially symmetrical in this example, so the diagram of Figure 8 does not apply. The broad band on axis response in the billet from a small reflector is plotted in Figure 13. One curve was calculated with attenuation off to show the effect of material attenuation on the field. The attenuation was incorporated into the measurement model as a function of frequency using measured values. Note that the attenuation both decreases the amplitude and shifts the location of the peak response. The plot shows that the real focal spot of the beam, as predicted by the beam model is about 2.5 centimeters short of the center of the billet. This is typical of the examples from reference 6 that compare the inadequacy of ray-tracing to predict focal spot locations. The 3 dB beam widths in the x and y directions are plotted in Figure 14. In the X direction, which cuts across the curvature of the billet, the minimum 3dB beam width in the inspection zone is 0.30 cm at 14 cm depth. In the Y direction, the minimum beam width is 0.26 cm at a depth of 15 cm. Because the probe is round and the interface is cylindrical, it is logical that the beam cross sections will be elliptical. The allowable scan spacing in the X and Y directions at a particular depth are proportional to the X and Y points on the ellipse at that depth. To minimize the number of data acquisition points for the inspection, it is logical to choose the overlap point on the ellipse such that it is the corner of a maximum area rectangle inside the ellipse. The process of solving for the maximum allowable scan steps based on the ellipse was repeated many times over the depth of the inspection zone. The limiting case for the X direction is at the top of the zone. The allowable step in the X direction at the top of the zone is 0.22 cm. The allowable step size in the Y direction is .18 cm. Using trigonometry, we can show that .22 cm is approximately a 5 degree chord of a arc at the 14 cm depth. Thus, it would take 72 equally spaced steps around the billet and a scan step of .18 cm along the billet to provide coverage.

Figure 10a. End view of ray-tracing in the billet. Figure 10b. Side view of ray-tracing in the billet. Figure 11. Sketch showing inspection zone for example 2 Figure 12. On axis response in water for example 2 Figure 13. On axis response in metal for example 2 Figure 14. Beam width in X and Y directions for example 2

To summarize this example, a transducer design was chosen using simple ray tracing as a technician might do in industry. Differences in focal length between the ray tracing and beam model methods were shown. The beam cross section was checked at many depths through the depth of the inspection zone. Because of the nature of the probe and interface, the cross sections were elliptical. The beam widths in the X direction were transformed to a cylindrical coordinate system. The limiting beam widths in each direction were identified. The scan steps in the X and Y directions were chosen so that there would be at most a 3 dB ripple in the response due from a given size flaw.


Example Problem 3:

Bi-cylindrical Varying Interface

For the third example, we simulated a 3/4 inch diameter probe with a 7.5 inch spherical focal length. The part being inspected was an aluminum block with a "wavey" surface on the top side, as shown in Figure 15. The block has complex curvatures that include all permutations of locally concave, convex, and flat interfaces. The object of this example was to show the coverage as a function of position for a predefined normal following scan with a constant step size. The scan was nominally designed to inspect the volume with maximum sensitivity at the back surface of the block. The spacing of the nominal scan was chosen so that the 3dB beam widths on a flat block would overlap.



Figure 15. Block for third example.

The constantly varying curvatures of the interface rule out the semi-analytical solution procedures used for examples 1 and 2. As shown in Table 1, this problem will require the evaluation of the flaw response isocontour at each independent location of the transducer. As will be shown, the major axes of the elliptical beam cross sections rotate to the directions of principal curvature. This effect complicates the book keeping of the calculation. Due to the complex variations in the field as a function of position and the need to show the actual coverage, we chose to use the voxelizer to solve this problem.

The CAD part was read in and automatically discretized with volume elements. The size of the elements was chosen as approximately one tenth of the diameter of the focal spot of the transducer. The scan was executed in the UTSIM simulation software using a number one (1/64" diameter) flat bottom hole as the reference reflector.

A false color (gray scale) image of the scan coverage is shown in figure 16. The shade of gray corresponds to the flaw response amplitude in volts expected from a the reference flaw if it were located in that position. The slice in figure 16 was taken near the top of the volume of the block. At this depth, the equally spaced scan has adequate coverage and the shapes of the beam cross sections are relatively circular.

Figure 16. Scan coverage near top of the block	Figure 17. Scan coverage near bottom of the block.
Figure 18. Scan coverage on vertical slice between scan lines.	Figure 19. Scan coverage vertical slice on a scan line.

Figure 17 shows a second slice through the volumetric scan coverage data, this time near the bottom of the block. At this depth in a flat block, the transducer would be focusing with a circular cross section. The varying curvatures of the block caused many effects that degrade the coverage of the scan. Many of the beam cross sections have become elliptical and rotated with respect to the scan coordinate system. As the transducer followed the surface normals, the concave regions of the surface caused the beam cross sections to tilt away from each other, resulting in coverage gaps at the bottom of the block. Figure 18 shows a vertical slice through the depth of the block. This particular slice was chosen between two scan lines in a relatively flat region of the block. The coverage is adequate at all locations on this particular slice.

Figure 19. Shows a second slice through the depth of the block. This one is centered on a scan line in a region with significant curvatures. The left side of the slice was effected by a locally concave curvature. The concave curvature caused tighter focusing of the beam and also tilted some of the fields away from vertical. The dark regions in the lower left are gaps in coverage at medium to high sensitivity. The right hand side of the slice in figure 19 was effected by a locally convex interface. The convex curvature defocused the beam. Coverage in this region remained adequate.



To summarize this example problem, the coverage of an entire scan over an irregular surface was calculated quantitatively. This volumetric coverage data set can be queried for minimum values. The user can iterate with the scan plan generator and/or probe parameters until the coverage is adequate.