Beam Model

An ultrasonic beam model is a combination of algebraic equations and numerical methods that solve a boundary value problem using the wave equation. The particular beam model used in this study is the Gauss-Hermite Beam Model formulated by Bruce Thompson and Tim Gray [3,4]. The Gauss Hermite Beam Model (GHBM) does calculations in the frequency domain. It uses the product of a Gaussian function with Hermite polynomials as basis functions to represent the amplitude and phase curvature of the beam.

It should be stressed that although the examples were worked with the Gauss-Hermite beam model, this choice is not essential and a different beam model can be substituted into the same process.

The GHBM can model longitudinal or shear waves in isotropic or simple anisotropic media. This particular software implementation can represent elliptical or rectangular transducers. The transducers can have planar, cylindrical, spherical, or bi-cylindrical focal types. It is restricted to bicylindrically curved interfaces whose curvatures do not vary significantly over the footprint of the beam. The transducer is approximated as a piston source with unit displacement across its face. The GHBM makes the paraxial approximation, which involves an expansion of the transducer wave field about a fixed central ray direction. It is known that the paraxial approximation breaks down in the near field and at medium to high incident angles on interfaces when real beams are known to be highly non-symmetric. The effect of this approximation has been explored by Lerch et al [5]. Another approximation in the GHBM is its method of treating mode conversion at interfaces. In the GHBM approach the beam is simply turned to follow the central Snell's law ray, and the amplitude terms are multiplied by the appropriate transmission or reflection coefficient, while the phase curvature is multiplied by a factor that accounts for the effects of the interface on the wave-front curvatures. This transmission method works well for flat or mildly curved surfaces at low to medium incident angles. At high angles and or tightly curved surfaces, however, portions of the incident beam not near the central ray will have an incident angle that is significantly different from the central ray and the Gauss-Hermite will fail. However, such cases are often rather exceptional and the Gauss-Hermite works extremely well in many practical cases.

The original creators of the software algorithms that implement the GHBM were Bruce Thompson, Tim Gray, Byron Newberry, Frank Margetan, and Ali Minachi [3,4]. The algorithms can be likened to the series solution methods of differential equations. At the transducer face, the GHBM solves for coefficients of the basis functions which will be the terms in the series. In a typical analysis, 10 to 25 terms are used. To propagate the field, these coefficients and extensions of the basis functions are all that is needed. Evaluating the field is therefore accomplished by simply summing up the basis functions at the point of interest. Because only a summation needs to be performed at single frequency field point, the GHBM is very fast on a typical workstation. In fact, at a single frequency, the entire field on a particular cutting plane through the beam can be evaluated in real time. The GHBM was originally coded in FORTRAN. This version was rewritten in C++ to increase supportability, allow dynamic memory management, and speed integration with other modern software.

Given the incident field from the beam model, a flaw model is used to simulate the interaction between an incident field and a scatterer. UTSIM can model many types of flaws including the small planar reflector used for the examples in this thesis. This flaw model makes use of the plane wave scattering amplitude concept. A plane wave scattering amplitude is a function that represents the amplitude response from the flaw, assuming that the flaw has been illuminated by a planar incident wave, and the response is measured in the far field (spherically spreading region) of the scatterer. Note that the incident wave does not really need to be planar, all that is required is that the flaw be small relative to the wave-front curvature in the beam. Thus, the incident planar wave restriction is better known as a "small flaw approximation".

The measurement model is a beam model and a flaw model with some additional software to simulate the entire ultrasonic measurement process. It is a combination of an ultrasonic beam model, a flaw model, and a measured "efficiency" factor that represents the electrical to mechanical and mechanical to electrical conversion process that occurs at the sending and receiving transducers. As mentioned previously, the measurement model used here was formulated by R.B. Thompson and T. Gray [[6] and was derived from Auld's reciprocity relation [7]. Since the Thompson-Gray measurement model is a frequency domain model, it is necessary to compute the prediction of the model at many discrete frequencies. The resulting frequency spectrum response is then inverse Fourier transformed to synthesize the voltage versus time (time domain) waveform that is typically measured on an oscilloscope display.

The measurement model also includes a process for calibration. A reference time domain waveform from a known simple scatterer is obtained using the same transducer, pulser setting, and wiring that the user wants to simulate. The reference waveform is transformed into the frequency domain and corrections made for diffraction effects [8]. The resulting frequency spectrum is then an effective system efficiency as a function of frequency. When this measured efficiency factor is used in the measurement model, quantitative predications of waveforms received from flaws can be made.

To set up the boundary conditions for the measurement model, a CAD solid model is used. This CAD models is read into the modeling software through a CAD interface. The particular CAD interface of UTISM reads IGES files. IGES or Initial Graphics Exchange Standard is a standard format that most 3-D solid modeling packages can write. By being able to read the standard files, this software can take input from many different CAD packages. The CAD interface used in this study was initially created for use in an ultrasound ray-tracing package then later integrated with a beam model and flaw models to become the complete UTSIM package.

When combined with some computer graphics output, the CAD solid model has many uses in modeling. First of all, the user can visualize the geometry. He or she can position the transducer relative to the part in 3-D space. Secondly, intersections of vectors and the geometry can be calculated. At these intersections, properties of the surfaces can be sampled. The beam model needs normal vectors, tangents, and principal curvatures. It also needs to know if a given point lies on the exterior, surface or interior of the object. The point test is used to keep track of field locations relative to the probe, part, and flaw.

Figure 2. Coordinate system description for scan path generation

To generate scan motions relative to the CAD part a software module was created that can generate 6 degree of freedom scan motion paths. It is very similar to algorithms used for computer numerically controlled (CNC) machining. Instead of creating a motion path for a machining tool, this software creates a motion path for the transducer. The first five degrees of freedom correspond to the x, y, z, theta, and phi used in CNC (Figure 2). The sixth degree of freedom is psi which controls the direction of the transducer. Psi is not needed for CNC because the cutting tool is symmetrical and spinning about its axis.

The scan path generator uses the two dimensional (u, v) parameter space of the scan motion coordinates to drive the three dimensional locations of the transducer. In most cases, this coordinate system was coincident with the

parametric directions from the CAD surfaces. To create a nominal scan, a Cartesian grid is dropped onto parameter space. The two dimensional parameter space locations of the grid are evaluated to create three dimensional global space points on the surface. Also the first two derivatives of the surface are calculated at each of the grid points. The derivatives are used to calculate tangents and then normal vectors at the surface points. The transducer can be placed with its axis on the surface normal, tilted by a specified incident angle, and translated upwards by a specified length of the water path. This method will define a grid of transducer locations and orientations in global space. The second derivatives of the surface are used to calculate surface curvatures for the beam model. If the Cartesian grid that is dropped on to parameter space had constant increments of u and v, then the resulting array of global space positions, in general, will not have a constant increment of global space between them. An intelligent control algorithm was created to change the parameter space grid as necessary to achieve the desired real space step size.

There are many cases were the total scan coverage can not be determined without a volumetric book-keeping tool. For instance, if the interface curvature varies continuously as a function of position, the beam width will not be constant, and a method of keeping track of coverage is needed. Also, if the inspection volume is interrogated by more than one direction, coordinate transformations will be needed between beam and global coordinates at each location, making the application of semi-analytical methods more difficult. A piece of software called the "voxelizer" was created to fill the need. A voxel is simply square volume element. It is the three dimensional analogy of a two dimensional pixel. The design intent of the voxelizer was to provide an efficient means to store field quantities as a function of position throughout the volume of an arbitrarily shaped part. Voxels are commonly used in the medical industry to visualize volumetric data sets. The basic elements of the voxelizer are listed in Table 5. The voxelizer reads in a CAD solid model to mathematically define the geometry. The voxelizer has no restrictions on geometric complexity.


Table 5. Elements of the voxelizer

Mathematical description of the part (uses CAD part) Intersection engine to intersect a field of regularly spaced vectors with the part description 3. A data structure to keep track of the intersection points Functions to quickly generate a subset of the evenly spaced points inside the part based on the intersection locations Functions that allow random access storage of a quantity at any location Display routines that draw a false color image of any slice through the data based on the stored values

The solid model can be made up of any number of trimmed non-uniform-rational-b-spline surfaces. The software user then defines the desired voxel resolution of the volumetric elements inside the voxelizer. The software automatically creates a bounding box that just encloses the solid model with its sides parallel to the global Cartesian axes. The voxelizer creates a working plane parallel to one of the box sides. A grid is created on the plane with the resolution previously specified by the user. From each point in the grid, a ray is created normal to the plane and intersected with the solid model. The intersection points are stored in a memory efficient data structure. Field quantities will be calculated along each ray at increments equal to the voxel resolution. The voxelizer was designed with a high priority on memory efficiency, because the required data sets are proportional to the volume of the object. The number of voxels used in a typical run is 10,000 to 1,000,000. The Cartesian location of each of the voxel points along the ray is not stored to save memory. The voxelizer can quickly calculate the subset of points along a ray that are inside the geometry. By always calculating the points in the same order, it can recalculate the position for a given field value whenever required. This allows random access storage and retrieval of field quantities at any point in the part without storing the location itself.

To use the software system, a user first reads in the CAD part and the software automatically creates the voxel data structure. Second, all of the probe, flaw, and material parameters are set by the user. Third, the scan region and initial spacing parameters are set based on the parametric CAD surfaces. The user then presses the button on the graphic user interface to start the scan. The software moves the transducer to the first location in the scan plan. The transducer shoots out a central ray to intersect the geometry. At the point of intersection, the surface properties are calculated. The surface properties and material properties are used to automatically set up the boundary conditions for the beam model. The voxelizer sends all of the points near the central ray to the measurement model. For each voxel point, the measurement model positions the flaw at the point and calculates a time domain waveform. The peak to peak amplitudes of the waveforms are stored back in the voxelizer data structure. The software then moves the transducer to the next scan plan location and repeats the measurement model calculations. After one value has been stored at a point location in the voxelizer, it can only be replaced by a higher amplitude value. Another way of saying this is that the voxelizer keeps track of the maximum flaw response amplitude of the current scan location or the past scan locations. At each point, P, it is storing magnitude of the largest signal that the flaw in that location produced. This effectively makes a scan coverage map as a function of position. The voxelizer can then be queried for information such as the lowest flaw response amplitude at a particular depth. Scan coverage can then be viewed as false color images on slices through the part.