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
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
