as shown in Figures 2a, 2b, and 2c, respectively. The notches
were modeled as rectangular blocks for the through-thickness
cracks, semi-ellipses for mid-bore cracks, and quarter-ellipses
for corner cracks. Electrical conductivity values were set to
1.897 × 107 S/m for the aluminum alloy, 5.8 × 105 S/m for the
titanium alloy (Ti-6Al-4V), and 1.45 × 106 S/m for the stainless
steel (17-7PH TH1050, with a relative permeability of 75).
To accurately model the observed differences in eddy
current response between wide notches and cracks, the dif-
ference in opening width must be addressed. For cracks
developed under this program, microscopy has shown crack
opening widths as small as 3 µm. To simulate such thin cracks
using the volume element method, the discretization of the
notch in the lateral (length) direction must be greatly increased
to allow the model to converge numerically. Prior convergence
studies of the method of moments volume integral equation
(MOM-VIE) formulation evaluated the convergence of the
numerical method for narrow cracks of varying widths and
increasing notch discretization (Aldrin et al. 2014).
To practically simulate thin cracks, an extrapolation
scheme was developed, leveraging simulated results that
had converged numerically for notch widths of 0.025 mm,
0.076 mm, and 0.153 mm. A second-order polynomial fit to
these data points was used to extrapolate the simulated eddy
current response down to smaller crack opening dimensions.
Very good agreement of less than 1% error was demonstrated
between the extrapolated model-based results and the simu-
lated results for a 0.003 mm notch width, with very high levels
of discretization (Aldrin et al. 2014). This approach was deter-
mined to offer the best compromise between computation
time and accuracy for simulating very thin cracks and notches
using the volume integral method.
Subsequent work by Shell et al. (2015) applied this model
to invert the dimensions of two sets of cracks and one specially
manufactured EDM notch with a fine width of only 0.030 mm
(1.2 mils). The corresponding crack width estimates fell within
a range of 0.003 mm to 0.038 mm—consistent with inversion
results for the fine EDM notch of 0.030 mm, which had esti-
mated widths between 0.012 mm and 0.038 mm.
Simulated studies run for corner, mid-bore, and
through-thickness cracks/notches are summarized in Table 1.
The work involved completing 28 surrogate model builds.
Metadata for each model build includes discontinuity type,
drive frequency, base layer material, and adjacent layer
material. Optimization studies using a set of calibration spec-
imens were performed early in the program, and the mean
liftoff value used for simulations was set to 0.30 mm.
Some special cases of adjacent material, particularly
adjacent aluminum (high conductivity) and adjacent steel
(high permeability), were also built. An approximation
was made using a locally flat aluminum layer for a crack
in a hole, as shown in Figure 2d. The adjacent edge void
TA B L E 1
Details of simulated studies to generate surrogate models for BEHC inversion capability
Base model Base scan parameters
(varied)
Base discontinuity
parameters (varied)
Base material/
frequency (varied)
Adjacent materials
considered Total sets Total solved
points
Mid-bore
crack (no
edges)
Rotation steps (2.54
to –2.54 mm at 0.1
[51] mm steps) × index
steps (4.57 to 0 at
0.25 [19] mm steps
use symmetry) 969
solutions
Length (0.32, 0.64, 0.95,
1.27, 1.91, 2.54, 3.81,
5.08 mm [8 levels]) ×
depth (0.32, 0.64, 0.95,
1.27, 1.91, 2.54 mm [6
levels]) × width (0.51,
0.152, 0.076 mm [3
levels]) +no-discontinuity
case 144 +1 =145
combinations
Aluminum
(200 kHz,
500 kHz), stainless
steel (500 kHz,
1 MHz), titanium
(1 MHz, 2 MHz) 6
combinations
n/a
6 (~simulation
time: 2 days per
set assumes 6
parallel runs)
6 × 144 × 969
=837 216
simulations
(~time: 12
days)
Corner
crack (with
one edge)
Rotation steps (2.54 to
–2.54 mm at 0.1 [51]
mm steps) × index steps
(–1.52 to 4.06 at 0.51
[12] mm steps) 612
solutions
Length (0.51, 1.02, 1.53,
2.54, 3.81, 5.08 mm [6
levels]) × depth (0.32,
0.64, 0.95, 1.27, 1.91,
2.54 mm [6 levels]) ×
width (0.152, 0.076 mm [2
levels]) +no-discontinuity
case 72 +1 =73
combinations
Aluminum
(200 kHz,
500 kHz), stainless
steel (500 kHz,
1 MHz), titanium
(1 MHz, 2 MHz) 6
combinations
Air (low conductivity),
titanium (moderate
conductivity), stainless
steel (moderate
conductivity +
permeability),
aluminum (high
conductivity) 4
combinations
16 (~simulation
time: 7 days per
set assumes 6
parallel runs)
(Note: Not all
combinations of
adjacent material
were needed.)
18 × 73 × 612
=714 816
simulations
(~time: 112
days)
Through
crack (with
two edges)
Rotation steps (2.54 to
–2.54 mm at 0.1 [51]
mm steps) × index steps
(–1.52 to 16.26 at 0.51
[36] mm steps) 1836
solutions
Length (3.18, 6.35, 9.53,
12.70 mm [4 levels]) ×
depth (0, 0.32, 0.64, 1.27,
1.91, 2.54, 3.18, 3.81, 4.45,
5.08 mm [10 levels]) ×
width (0.152, 0.076 mm [2
levels]) 80 combinations
Aluminum
(200 kHz,
500 kHz), stainless
steel (500 kHz,
1 MHz), titanium
(1 MHz, 2 MHz) 6
combinations
Assumed air (Note:
Effect of adjacent
material assumed
small for through
notches relative
to corner notches.
Verified with
aluminum simulation.)
6 (~simulation
time: 11 days per
set assumes 4
parallel runs)
6 × 80 × 1836
=881 280
simulations
(~time: 66
days)
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 45

regions were modeled as an air block with dimensions of
2.44 × 2.54 × 1.27 mm. The volume element discontinuity
region mesh for the through-thickness notch case included the
combined notch plus two air blocks in a single voxel mesh layer.
The notch-to-crack extrapolation approach for mid-bore
notches was also applied to through-thickness and corner
notches, using notch width solutions of only 0.076 mm and
0.153 mm (due to mesh limitations at the edges and solution
time constraints).
To generate all the model libraries shown in Table 1, a
total of 2 433 312 simulations were run, with a total continu-
ous simulation time of ~190 days on a Dual Xeon PC, with a
minimum of four to six runs executing in parallel. Additional
surrogate model resolution was obtained through the appli-
cation of a cubic spline interpolation of the numerical solu-
tions over the scan (circumferential) and index (depth) direc-
tions to generate simulated responses matching experimental
BHEC scans.
2.3. BHEC Model-Based Inversion Approach
A comprehensive approach is presented in Figure 3 for a model-
based inversion design to ensure the reliability of the inverse
methods. For this application, the discontinuity characteris-
tics of crack/notch length, depth, and width are the key factors
of interest in the forward and inverse models. Although the
MOM-VIE formulation is numerically efficient for the simulation
of eddy current NDE measurements, for inverse problem appli-
cations it is time-consuming to perform such repeated numer-
ical calculations within an inversion scheme in real time. To
address this, surrogate models were created from the numerical
model results to greatly improve the speed and performance of
the inverse methods. A fast N-dimensional cubic spline interpo-
lator was implemented to perform each model call.
While eddy current simulations are designed to represent
the change in impedance of an eddy current probe interacting
with a test specimen, conventional eddy current measurement
systems typically operate with the probe balanced over the
specimen and output voltage data. Thus, a model calibration
step is necessary in practice to equate the measurement data
with the simulated results, following the NDE technique pro-
cedure. A complex “beta” term (gain and phase) was evaluated
based on the response to a known calibration standard. At
this stage, a fast surrogate model calibrated according to the
inspection procedure can be applied for inversion.
A two-step inversion process was designed that first evalu-
ates the material layer type and thickness and classifies the crack
type for the indication. Then, the appropriate surrogate model is
selected for crack sizing. Automated routines were implemented
to extract and align all indications from the scan and identify
each indication’s location in terms of material layer and position
(near corner, far corner, mid-bore, or through-thickness).
The second step in the inversion process evaluates crack/
notch depth, length, and width by iteratively comparing the
model to the eddy current indication. Feature vectors in the
circumferential and depth directions (shown in Figures 1e and
1f, respectively) are used as inputs for inversion. A nonlinear
least-squares estimator (NLSE) is used to perform the inversion
process. An iterative scheme is implemented that varies the initial
conditions of the inverse problem and selects the most repeatable
results—those that minimize the difference between the experi-
mental results and estimated model—to avoid local minima.
Some recent refinements to the inversion scheme include
increased surrogate model resolution, an adaptive calibration
process that compensates for scale differences due to varying hole
diameters, improved signal registration in both depth and angular
directions, and direct inclusion of phase in the inverse fit routine.
Recent work has also addressed some discrepancies in
the surrogate model through a rigorous model transformation
study over a wider discontinuity size range using EDM notch
data. Through this process, it was discovered that high- and
low-pass filters, typically used in rotating BHEC NDE, distort
the circumferential response relative to the base BHEC model.
A process was implemented to use calibration data to optimize
the model fit through both gain and filter parameter adjust-
ments. Also, by controlling the phase of the data in the model
fit to ensure both measurement components are above the
background noise level, the model calibration and inversion
processes were found to be more consistent.
In particular, a large difference in phase was observed for
the titanium and steel cases, leading to increased errors in the
fit with the titanium and steel models. An example of poor
titanium model calibration is shown in Figure 4a, highlighting
the discrepancy within the reactance (red) component, espe-
cially in the depth profile fit (bottom plot). Transformations
ME
|
CRACKSIZING
Model
parameters
Experimental
data
Noise
removal
Data
registration
Feature
extraction
Inverse
method
Estimated
parameters
Forward
model
Model
reduction
(surrogate)
Model
calibration
Calibration
surrogate
model
Calibration
data
Data (signal/image) processing
Figure 3. General inversion process for parameter estimation (Sabbagh et al. 2013).
46
M AT E R I A L S E V A L U AT I O N • A U G U S T 2 0 2 5