A Company Committed to Problem Solving
and Research in the Engineering and Physical Sciences


Harold A. Sabbagh, Elias H. Sabbagh, and R. Kim Murphy
Victor Technologies, LLC
P.O. Box 7706, Bloomington, IN 47407-7706 USA

(Presented at 28th Annual Review of Progress in Quantitative Nondestrucitive Evaluation
Bowdoin College, Brunswick, ME, July 29-August 3, 2001)

(a) Forward Problem: Cracks with Complex, Double-Peaked Shapes

These cracks, such as that shown in Figure 1, are well suited to model damage arising from multiple sites.
Epicyclic crack
Figure 1:A double-peaked, epicyclic crack, that is suited to model multi-site damage.

In [1], Harrison et al have developed a set of four benchmark problems for verification of theoretical calculations of defect size and shape in eddy-current NDE. The benchmark problems are based on careful measurmements of the change in impedance associated with slots of semi-elliptical and double-peaked (epicyclic) profiles. Later, we will give examples of the inversion of such impedance data, but here we will present results of applying VIC-3D® to the computation of the forward problem for slot D2 of [1]. This slot is epicyclic, with two peaks, the larger about 8.94 mm, and the smaller about 6 mm. The length of the slot is 49.78 mm, and the width 0.37 mm.

The coil possesses an air core, and has an inner radius of 2.51 mm, an effective outer radius of 7.38 mm, a height of 4.99 mm, and an effective lift-off of 0.2 mm. It is wound with 4000 turns, and excited at a number of frequencies. In our modeling effort, we considered only excitation at 250 Hz. The effective value of the conductivity of the workpiece is 2.24× 107 S/m.

In modeling the profile of the flaw, we use the approach of [2], wherein we represent the epicyclic profile of the crack in parametric form as

where t is the independent parameter.

Bowler and Harfield [2] have determined that flaw D2 can be approximated using fifth-order epicycles, with coefficients whose values in mm are listed in Table 1

In modeling the flaw with VIC-3D®, it is necessary to use enough cells to capture the shape of the flaw's profile, so we used a mesh of 2× 64×16 cells in, respectively, width, length, and depth. The probe is scanned over a length of 120 mm, in one mm steps, in accordance with the measured data.

Our first task is to use the parametric equations and the expansion coefficients to determine the volume-fractions that define the flaw for VIC-3D®. This is easily done using a simple two-point Gaussian quadrature routine. The volume-fraction data are then quickly read into the .vic file, which VIC-3D® then executes to produce the results.

The results of the model calculations are compared to the measurements in Figure 2. Clearly, the agreement is excellent.

Flaw D2 data
Figure 2:Comparison of forward-model results, calculated by VIC-3D®, with measured data for flaw D2 of [1] at 250 Hz.

(b) Inverse Problem: Cracks with Complex, Double-Peaked Shapes.

In this section we apply VIC-3D® to the problem of reconstructing two flaws, D1 and D2, that are described in [1]. D1 is a semielliptical slot, shown in Figure 3, and D2 is the same flaw that was described in the previous section.
Semielliptical slot
Figure 3:The semielliptical slot, called D1 in [1].

The objective is to take the measured data of [1], and invert it to determine the shape of the scattering body, namely the semiaxes of the flaw. For this exercise, we used data at 250 Hz, comprising the change in resistance and self-inductance (in mH) measured at the five probe positions -22.5 mm, -17.5 mm, -12.5 mm, -7.5 mm, -2.5 mm. These give us 10 data points with which to determine the two real unknowns (the semiaxes). The coil is the same as before, except that the effective lift-off is 0.3 mm, instead of 0.2 mm. The conductivity of the workpiece is 2.25×107 S/m.

As its contribution to the inversion process, VIC-3D® is used to create an a priori table of possible changes of resistance and inductance at the same five points, when the length semiaxis takes on values of 9, 10, 11, 12, and 13 mm, and the depth semiaxis takes on values of 6, 7, 8, 9, 10 mm. This gives us 25 values of resistance and inductance at each position of the probe. In order to generate these data, VIC-3D® is run twenty-five times, with the five scan positions for each run. VIC-3D® includes a semielliptical flaw in its library, so it was not necessary to compute volume-fractions for each of the twenty-five combinations of flaw shape. Each of these twenty-five runs used a grid of 2× 32× 16 cells (width, length, depth).

The inversion process employs a nonlinear least-squares algorithm to fit the solution vector to the data. During the inversion, the function values are determined from the table, with the entire process being completed in a few seconds.

The computed answer for D1 is 8.21 mm and 11.50 mm for the semiaxes, whereas the actual values (as measured from cast rubber replicas) are 8.61 mm and 11.05 mm. This agreement, with an error less than 5%, is considered quite good, given that there was an absolute uncertainty of 0.5 mm in locating the center of the flaw when taking the data.

Now, we go to the reconstruction of D2. By this we mean that we are going to determine the expansion coefficients b(1) to b(5) in the parametric representation of the depth variable, y, while assuming that expansion coefficients, a(1) to a(5), that define x remain fixed. Because these latter coefficients determine the length and center of the flaw (in the x-direction, of course), we are agreeing that we know these parameters of the flaw, and wish to reconstruct the profile in depth.

Of the given (measured) data, which were determined at one-millimeter increments from -60 mm to +60 mm, we chose eleven points, comprising the change in resistance and self-inductance (in mH) from -31 mm to +29 mm in six-millimeter increments. This results in a total of twenty-two equations for the five unknowns.

As in the case of D1, we use VIC-3D® to generate an interpolation table for the five unknowns, with each unknown taking on two values in the table. This means that we must run VIC-3D® 25=32 times in order to generate the table. The thirty-two combinations are generated by letting the b(i) take on the values shown in Table 2:

b(1) = 7 , 8
b(2) = -2 , -1
b(3) = 2 , 3
b(4) = -0.10 , -0.05
b(5) = -1.0 , 0.0
Table 2: Values assigned to the b(i) in order to generate the interpolation table.

We used the same nonlinear least-squares algorithm as in D1 to do the inversion, and got the results shown in Figure 4

Epicyclic flaw reconstruction
Figure 4:Reconstruction of epicyclic flaw D2 of [1].

The computed values of the b(i) are listed in Table 3:


[1] D. J. Harrison, L. D. Jones, and S. K. Burke, ``Benchmark Problems for Defect Size and Shape Determination in Eddy-Current Nondestructive Evaluation,'' J. Nondestructive Evaluation, Vol. 15, No. 1, March 1996, pp. 21-34.

[2] J. R. Bowler and N. Harfield, ``Thin-Skin Eddy-Current Interaction with Semielliptical and Epicyclic Cracks,'' IEEE Trans. Magnetics, Vol. 36, No. 1, January 2000, pp. 281-291.