2. Approach
This section presents details on the BHEC data acquisition
and key features for classification, as well as the design of the
forward model and surrogate model build for the program.
It also gives details on the entire data evaluation process for
model-based inversion, with a focus on the liftoff compensa-
tion process.
2.1. BHEC Data Acquisition and Features
An example of a BHEC indexed hole scan using an eddy
current inspection system with rotation and depth position
indexing is shown in Figure 1, for a hole with a 1.98 × 2.34 mm
top corner crack in aluminum. The scanner indexes the eddy
current probe in both the circumferential (scan) direction
(Figure 1a) and the vertical (index) direction (Figure 1b), pro-
viding full coverage of the hole surface. The vertical channel
differential probe C-scan response (shown in Figure 1c) is used
to make calls for crack indications.
One notable feature of the eddy current system used is its
ability to acquire absolute coil eddy current response data,
providing indications of where material layers start and end, as
well as the material state (Figure 1d).
For inverting the eddy current data, the C-scan data is
reduced to a pair of characteristic vectors for each channel of
data in the rotational scan (Figure 1e) and hole depth index
(Figure 1f) directions. First, the maximum response in the hole
bore (z) direction is evaluated, providing a correlated response
with the crack length and crack profile (shape). Second, a char-
acteristic eddy current response in the hole circumferential
direction is extracted through the crack/notch peak response
in z. Both vertical (Vy) and horizontal (Vx) components of the
eddy current response are recorded, resulting in four data
vectors being used for inversion.
2.2. BHEC Forward Model and Surrogate Model Build
A model of the BHEC split-D differential probe was imple-
mented in VIC-3D® (Sabbagh et al. 2013). The outer coil
radius, D-coil outer radius, D-core outer radius, and D-core
spacing were 0.86 mm, 0.71 mm, 0.64 mm, and 0.16 mm,
respectively, and the ferrite D-cores were assigned a relative
permeability of 2000. A complete split-D differential probe
model with ferrite cores was created using a 32 × 32 × 4 grid
mesh. Models were required for the three classes of discon-
tinuity types/locations typically found in BHEC inspections:
mid-bore cracks, through-thickness cracks, and corner cracks,
ME
|
CRACKSIZING
b
x (y out of plane)
z
a
b
a
b
a
y
Figure 2. BHEC discontinuity-type categories for the model build: (a) mid-bore, (b) through-thickness, and (c) corner crack/notch (d) 3D diagram of
the through-thickness notch model with edges to be meshed.
0.0
5
0
–5
1.3
2.5
3.8
5.1
6.4
7.6 0 45 90 135 180
Angular position (degree)
225 270 315 360
7
6
5
4
3
2
1
0.0
1.3
2.5
3.8
5.1
6.4
7.6 0 45 90 135 180
Angular position (degree)
225 270 315 360
0
0
5
–5
–10 45 90 135 180
Angular position (degree)
225 270 315 360
0
0.0
1.3
2.5
3.8
5.1
6.4
7.6 2 4 6
Response (V)
8 10
Z Vertical
Horizontal
Vertical
Horizontal
θ
Figure 1. Diagrams of bolt-hole eddy current (BHEC) scans of a near-surface corner crack: (a) circumferential (​ ​ (b) depth (z). C-scan responses
from the eddy current inspection system for a 1.98 × 2.34 mm top corner crack in an aluminum layer: (c) differential (vertical) coil (d) absolute coil
(amplitude). Differential response feature vectors at (e) peak circumferential and (f) depth curves for vertical and horizontal components.
44
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
Depth
in
hole
(mm)
Depth
in
hole
(mm)
Response
(V)
Depth
in
hole
(mm)

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