Refining automated ultrasonic inspections with simulation models
by Michael Garton *
CHAPTER 1. INTRODUCTION:
SCAN OPTIMIZATION PROBLEM STATEMENT 
CHAPTER 3. FORWARD MODEL TO EVALUATE SCAN QUALITY 
CHAPTER 4. MODELING EXAMPLES 
CHAPTER 5. CONCLUSION 
CHAPTER 1. INTRODUCTION 
The potential uses of the model fall into three basic categories. 1. Can ultrasound reach the region of interest? 2. What are the properties of the field in a region of interest? 3. What is the response from the incident field on a given flaw? Examples of particular questions that the model can answer are given in Table 1. The measurement model, in combination with some intelligent control software can answer even more complex queries. The control software can loop through a range of parameters to find optimum values for the inspection (see Table 2).
Table 1. Examples of questions that can be answered with the models  
1. Can ultrasound reach the region of interest?
 2. What are the properties of the field in the region of interest?
 3. What is the response from the incident field on a given flaw?

Table 2. Examples of optimization studies that can be done with models 
Example optimization studies:

Although UTSIM is capable of answering all of these questions, it is the last three questions related to scan coverage modeling that this document will demonstrate with several examples. Scan coverage modeling makes use of the measurement model, a beam model, a flaw scattering model, and the CAD interface of UTSIM. Note that to predict scan coverage it is essential to use a beam model since the fields of transducers have complex shapes that vary as a function of the interface and position in the part being inspected. The details of these shapes can not be predicted via simple ray theories. An engineer is often interested in the relative beam widths at each location but current industry practice is to simply measure the beam width experimentally. Also, one of the last steps in the design of a typical automated inspection is to measure the spot size and use it as a scan step [1,2], but, if the beam width is measured after the system is assembled, it is apparent that little room for optimization is available.
By using UTSIM as a forward model to evaluate scan
quality, an engineer in industry could see the quantitative effect of
each parameter without performing an experiment and before buying the
equipment. The scans that are refined by this software and then
verified in the lab are likely to be much higher quality than typical
inspections that are designed by rules of thumb and highly simplified
analysis.
CHAPTER 2. SCAN
OPTIMIZATION PROBLEM STATEMENT 
In the event that the specified quality is
unobtainable over the entire volume or that the time required for a
scan of that quality is unacceptable, the objective must be modified.
One alternative is to maximize quality given an allowable cost. Thus,
we could define two general industry objectives as follows:
1. Minimize inspection cost while achieving a specified inspection quality.
2. Maximize inspection quality while staying under a
specified cost.
This paper will be concerned only with the
evaluation of scan quality. It is assumed that the user of this quality
prediction technology can obtain cost estimates. The following
particular modeling objectives can be derived from the industry
objectives.
1. Maximize scan spacing while achieving a specified sensitivity.
2. Maximize sensitivity given a specified scan
spacing.
The examples in this paper assume that the desired flaw response sensitivity level is well above the noise level. With this assumption, the methods and answers are meaningful for relatively clean materials. For example, problem 2 in chapter 4 refines scan spacing in a titanium billet where the calculations include realistic (measured) attenuation as a function of frequency, but the predicted flaw response amplitude contours are evaluated in the absence of noise.
Because there are approximately 20 significant
variables involved in a calculation to predict the response from a
small flaw in isotropic media, look up tables covering all possible
combinations are not practical. Similarly, experiments to make a 20
dimensional data set would be cost prohibitive. A model based approach
using software simulations is the logical approach to evaluating the
response. Here, we classify variables as one of three types: Inputs,
parameters, and optimization variables. In this context, inputs are the
quantities that the user takes as fixed constants. These values were
determined at the time the part was manufactured and can not be
changed. Parameters, in this context, refer to the values that the user
can change between software runs, but will be considered to be constant
during the software run. Optimization variables refer to the values
that the software will solve for during a run. See the complete list of
each variable type in Table 3.
Table 3. Inputs, parameters, and optimization variables  
Inputs:

Parameters:

Optimization variables:

Figure 1. Sketch showing scan and index directions 
variation in the beam width as a function of position along a scan line, and maximizes the change in beam width along the index direction. The coordinate system directions should be orthogonal and can change as a function of position. For a scan over the skin of a conventional airplane wing, the scan coordinate direction would be down the length of the wing, and the index coordinate direction would follow along a rib of the wing. Scans over flat plates lend themselves to Cartesian coordinates. Solids of revolution are normally scanned on a turntable with the scan coordinate direction corresponding to the theta direction of the turntable.
The index coordinate direction for the solid of revolution would follow the change in profile of the radial crosssection of the part. When simulating a scan on with a CAD solid model, the "u" and "v" directions of the surface parameterization can be chosen for the scan and index directions because they are usually parallel to the principal curvature directions of the surface.
The second step in the general solution strategy is
to find the flaw response amplitude isocontour at the desired
sensitivity level as a continuous function of the transducer location.
The transducer location will is specified using the above determined
coordinate system. Given this function, we can solve for the discrete
transducer locations that overlap the flaw response amplitude
isocontours from each transducer location in a way that maximizes scan
spacing while guaranteeing coverage. Because of the complexity of the
functions, numerical methods, not analytical expressions, are used to
tabulate them. The calculation of these functions are complicated when
the interface curvatures, and thus the boundary conditions for the beam
model, vary with the transducer position. Often, the general shape of
the narrowest region of the isocontour will be known, so only size
parameters for this shape need to be tabulated as a function of
transducer location. The software system described in this paper is the
enabling tool that allows one to calculate this type of function for
nontrivial cases. This capability is believed to be unique to this
software system.
Most parts to be scanned can be broken into regions, where each region has locally constant curvatures. Even though a volume is being scanned, there are basically two degrees of freedom for the data acquisition. For most part shapes, the two scan motions can be broken down into two related one degree of freedom problems. When using symmetrical probes on symmetrical interfaces, there would be a single one degree of freedom problem. In this case the scan step equals the scan index. An example would be a Cscan over a flat plate with a planar or spherically focused probe at normal incidence.
To clarify the levels of complexity in the scan problem, some terms describing the interface being scanned need to be defined. The principal curvatures of a surface point are the minimum and maximum radii of curvature passing through the point. All points on a smooth surface can be described in this way and the directions of those two radii will always be orthogonal. There is an exception, when the two curvatures are exactly equal, such as a planar surface where both radii of curvature are infinite, and the directions of principle curvature are nonunique. Spherical surfaces also have nonunique directions of principle curvature.
Another term defined in the context of interface surfaces, "constant", means that the curvature of the interface is not changing as one moves in the principal curvature directions. The shape of the flaw response amplitude isocontour will be the same for any two discrete transducer locations if the transducer is moved in the direction of principal curvature.
The shape of the crosssections of the field effects the level of computation required to solve the scan spacing problem. If we know ahead of time that the cross sections through the flaw response amplitude isocontour in the inspection zone will be circular, we only need to calculate the contour on one plane that can be swept 360 degrees to get the 3D isocontour. These circular cross sections will arise when a probe with a round crystal and planar or spherical focal type is transmitting through an axially symmetric interface at normal incidence. A round planar or spherically focused transducer scanning a flat plate at normal incidence is an example.
Similarly, if we can predict ahead of time that the isocontour crosssections will be elliptical in the inspection zone and the direction that the major axis of these ellipses will lie in, We can define the 3D isocontour by calculating the flaw response amplitude on 2 orthogonal planes. The majority of scanning applications on curved surfaces fall into this category.
The most complex and computationally expensive case occurs when we can make no assumptions about the shape of the field in the inspection zone. For instance, if a large paint brush probe is scanned with a small water path over aluminum sheet stock. The fields radiated from rectangular probes have nonelliptical crosssections with a lot of peaks and valleys in the amplitude across the beam in the near field. The isocontour from such a probe may have multiple nonconnected regions if part of the inspection zone is in the near field. When no assumptions can be made about the shape of the field, we must evaluate the entire inspection volume for each change in interface to guarantee the scan coverage.
To classify the problem based on the interface curvature, consider first a region with a cylindrical interface. This interface is planar in one of the principal directions and curved in the other. It is constant in both principal curvature directions, but not axially symmetric. The cylindrical interface problem can, therefore, be treated as two related one degree of freedom problems. The problems are related because the optimum scan step distance in one direction is dependent on the width of the isocontours in both directions (this will be explained in detail in example 2).
Consider next a region where the principal curvature is constant in one principal direction but varies as the transducer moves in the other principal direction. A cone geometry is an example. The shape of the flaw response amplitude isocontour will be constant along a scan line (around the cone) but it will change with the index direction (along the length of the cone). To calculate the widths of the isocontour as a function of position, we chose to evaluate the isocontours at "m" sample locations along the varying direction. If the field is has elliptical crosssections, the widths of the isocontours on the two principal curvature planes at each of the m locations will define the 3D shape of the isocontour. For well behaved geometries like the cone, small values of m such as 3 or 4 might define the function. In near worst cases, an m value equal to the length of the scan line divided by the typical beam widths should be adequate to sample the function.
Finally, consider an arbitrarily curved region which has radii of curvature that are not constant along a scan line. This case can not be reduced to one dimension and requires some additional software tools to solve. Such an region can be described as bicylindrically curved at each surface point with the values of the curvatures changing as a function of position. The inner region of a propeller blade is a good example of an arbitrarily curved region. Because the interface changes as a function of position, the scan increment or scan step or both are not constant. The surface will need to be sampled at a grid of m by n points to define the isocontours. Table 4 shows the problem dimension broken down by interface and field type.<
Table 4. Problem classification based on interface and field symmetry  
Field Interface Type (example)  Field with circular cross sections in inspection zone  Field with elliptical cross sections in inspection zone  Field with nonelliptical cross sections in inspectionzone 
Planar Constant (flat plate)  Calculate beam width profile in 1 cross plane at 1 probe location to get 2 constant (equal) step sizes.  Calculate beam width profiles in 2 cross planes at 1 probe location to get 2 constant step sizes.  Calculate beam width isocontour in 3D volume at 1 probe location to get 2 constant step sizes. 
Constant Spherical (spherical pressure vessel)  Calculate beam width profile in 1 cross plane at 1 probe location to get 2 constant (equal) step sizes.  Calculate beam width profiles in 2 cross planes at 1 probe location to get 2 constant step sizes.  Calculate beam width isocontour in 3D volume at 1 probe location to get 2 constant step sizes. 
Cylindrical Constant (cylindrical fuselage)  Not Applicable  Calculate beam width profiles in 2 cross planes at 1 probe location to get 2 constant step sizes.  Calculate beam width isocontour in 3D volume at 1 probe location to get 2 constant step sizes. 
Cylindrical varying with index direction (cone or extrusion of a spline curve)  Not Applicable  Calculate beam width profiles in 2 cross planes at M probe locations along the varying direction to get a constant scan step and a variable index step.  Calculate beam width isocontour in 3D volume at M probe locations along the varying direction to get a constant scan step and a variable index step. 
Bicylindrical, one curvature varying with index direction (fillet on solid of revolution)  Not Applicable  Calculate beam width profiles in 2 cross planes at M probe locations along the varying direction to get a constant scan step and a variable index step.  Calculate beam width isocontour in 3D volume at M probe locations along the varying direction to get a constant scan step and a variable index step. 
Bicylindrical varying with two directions (inner region of a propeller)  Not Applicable  Calculate beam width profiles in 2 cross planes at each of M*N locations to get variable scan and index steps.  Calculate beam width isocontour in 3D volume at M*N locations to get variable scan and index steps. 
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