Potential Uses of Models

By definition, an ultrasonic simulation model uses physical equations and numerical methods to predict the result of an experiment. Because it is a software model, any parameter that effects the results can be changed or varied as needed. One important motivation for using such models is the relatively low cost of software predictions compared to the cost of comparable experiments.

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

Example optimization studies: Where is the best location to place a probe to catch the signal? What is the optimum wedge angle to maximize flaw response? What is the optimum transducer to guarantee coverage? What is the width of the region 3 dB's down from the peak response? What is the best coverage that can be achieved with this transducer configuration, assuming an infinitely small scan spacing? What is the widest scan spacing that meets the scan quality objectives?

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.

What defines a good scan? In the business world, the objective is usually to minimize cost while achieving a specified quality. Quality is defined here as the ability to find all flaws greater than or equal to a predetermined size. Cost is made up of fixed (equipment) costs plus variable (time related) costs. For the purposes of this document, the scope will be limited to high volume inspections where the variable costs dominate the total cost. This defines one objective; to minimize scan time for a specified quality. For most scan plans there is direct relationship between scan time and scan spacing. The example problems in this document will assume that the amount of data being acquired, not the mechanical speed constraints of the scanning device is the limiting factor that determines scan times. This logic leads to the modified objective of minimizing the number of data acquisition points while maintaining a specified sensitivity over the inspected volume.

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:


Industry Objectives:

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.


Modeling Objectives:

1. Maximize scan spacing while achieving a specified sensitivity.

2. Maximize sensitivity given a specified scan spacing.

The scope of this exercise will be limited to automated immersion scans using longitudinal waves at normal incidence in isotropic media. Automated scans were chosen because they are deterministic, repeatable, and capable of very accurate motion. Contact scans were excluded because the variability in coupling in such cases is significant and difficult to model. The material was limited to isotropic media so that it would be easier to validate with experiments. Longitudinal waves at normal incidence were chosen only to limit the scope. All the calculations will be carried out in regions were the interface surface or transition between interface surfaces are smooth and gradual. This requirement on interface surface is required by the particular choice of beam model used in the study. The methods are applicable to a wider range of problems but will be demonstrated with the above mentioned restrictions.

A scan-optimization problem has many significant variables. The transducer frequency, bandwidth, shape, size, and focal lengths, and water-path all affect the incident fields on the part surface. The incident angle, surface curvature, and material properties of both media affect how the sound changes as it couples into the material. The material velocity, density, attenuation properties and metal path will affect the sound as it propagates in the material. The flaw shape, size, orientation, and material properties affect the amount of energy scattered back to the transducer. Changing any one of these variables can have a significant effect on the detectability of a flaw.

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.

Solution Strategy

To solve for a scan plan that minimizes the number of data acquisition points while maintaining a guaranteed level of sensitivity is a very open ended problem. As the title of this paper suggests, the methods presented here will "refine" or improve scans. In only the simplest of cases with well defined constraints can we assert that we have found the scan plan that is the absolute minimum, or optimum design, for the problem. A method will be described that will guarantee the coverage, and provide an optimum solution for chosen family of scan motions. This may be a local minimum, as another family of shapes for the motions may produce a better result. The logic needed to choose a good family of shapes for scan motions will also be given.

Figure 1. Sketch showing scan and index directions

The general solution approach begins with choosing an appropriate scan coordinate system for the transducer motions. In a typical scan over a region, technicians will specify two spacing parameters, the index and the scan spacing (see Figure 1). It is beneficial to choose a coordinate system that minimizes the

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 cross-section 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.

Scanning the volume of an object is inherently a 3D problem but it often can be modeled in a lower dimension. Because it is the dimension of the problem that drives the difficulty level, it is logical to classify the problems using dimension as the discriminator. We found the interface curvatures and the shape of the cross sections of the fields control the required dimension of the problem.

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 C-scan 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 non-unique. Spherical surfaces also have non-unique 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 cross-sections 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 cross-sections 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 non-elliptical cross-sections 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 non-connected 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 cross-sections, 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 bi-cylindrically 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 non-elliptical 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.
Bi-cylindrical, 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.
Bi-cylindrical 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.