ABSTR ACT
There is a need for innovative methods to provide
accurate sizing of discontinuities in structures,
to guide maintenance actions and better inform
engineering of the structure state. This paper
highlights the development of the capability to size
the length and depth of cracks in multilayer fastener
sites using bolt-hole eddy current (BHEC) techniques.
Technical efforts include improvements to model
calibration, liftoff compensation, and the inversion
process. In addition, an expanded set of surrogate
models was developed that address crack sizing
for titanium, aluminum, and steel structures across
multiple frequencies and for varying hole diameters.
A comprehensive crack sizing evaluation study was
performed under a wide range of test conditions,
demonstrating improved sizing capability over
using peak amplitude. Crack length estimates were
found to have less error than crack depth estimates,
although crack depth is the more critical parameter for
informing maintenance actions.
KEYWORDS: bolt-hole eddy current inspection, BHEC, model-
based inversion, sizing, uncertainty
1. Introduction
There is a need for innovative methods to provide accurate
sizing information on discontinuities, like fatigue cracks in
metallic structures (Lindgren et al. 2007 Lindgren and Knopp
2016 Aldrin and Lindgren 2018). The value of a nondestructive
evaluation (NDE) sizing capability lies in its ability to acceler-
ate subsequent maintenance actions, improve the quality of
material review board evaluations, provide critical data to life
prediction modelers, and support long-term digital twin and
NDE 4.0 initiatives. Practical examples of NDE sizing capa-
bilities include sizing cracks in welds using time-of-flight dif-
fraction (TOFD) methods, using eddy current testing to size
cracks in aircraft propulsion components, and evaluating the
material state of turbine blades using resonance ultrasound
spectroscopy (RUS) (Aldrin and Forsyth 2017). This paper
presents progress on the characterization of discontinuities
such as fatigue cracks in metallic multilayer fastener sites using
bolt-hole eddy current (BHEC) techniques. Specific benefits of
accurately sizing bolt-hole discontinuities include streamlining
repairs (e.g., managing shallow scratches and gouges sepa-
rately from larger fatigue cracks) and delivering quantitative
data to engineering on the structure state.
1.1. Background on Model-Based Inversion for BHEC
NDE of Cracks in Metallic Structures
Given the complexity of many NDE sizing tasks, the devel-
opment of specialized algorithms and procedures has been
pursued to support inspectors as part of the data review
process. While there is a growing desire to leverage emerging
artificial intelligence (AI) and machine learning (ML) capabil-
ities, successful applications have been limited to cases where
the sizing task is relatively straightforward or where very large,
high-quality empirical datasets have been acquired (Aldrin
and Lindgren 2018). Frequently, such large datasets with
well-understood discontinuities are rarely available or are too
costly to practically produce.
As an alternative to AI/ML methods, model-based
inversion—incorporating an iterative, first-principles physics
model—has been proposed to solve complex NDE character-
ization problems. A number of efforts have been embarked
upon over the decades to develop model-based inversion
solutions for eddy current NDE applications (Sabbagh and
UNCERTAINTY EVALUATION OF CRACK
SIZING CAPABILITY INCORPORATING MODEL-
BASED INVERSION APPLIED TO BOLT-HOLE
EDDY CURRENT INSPECTIONS
JOHN C. ALDRIN†*, DOYLE MOTES‡, MALLORI HUGHES‡, DAVID FORSYTH‡, ELIAS SABBAGH§, HAROLD A.
SABBAGH§, R. KIM MURPHY§, GEORGE NUXOLL††, CHRISTINE KNOTT‡‡, AND ERIC A. LINDGREN‡‡
ME
|
TECHPAPER
† Computational Tools, Gurnee, IL 60031
‡ TRI Austin, Austin, TX 78746
§ Victor Technologies LLC, Bloomington, IN 47401
†† UniWest, Kennewick, WA 99336
‡‡ Air Force Research Laboratory (AFRL/RXNW), Wright-Patterson AFB, OH
45433
*Corresponding author: aldrin@computationaltools.com
Materials Evaluation 83 (8): 42–56
https://doi.org/10.32548/2025.me-04536
©2025 American Society for Nondestructive Testing
42
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Sabbagh 1988 Auld and Moulder 1999 Liu et al. 2008 Yusa
2009 Sabbagh et al. 2013). These approaches require accurate
forward models and a process that iteratively compares the
simulated and measurement data to adjust the model parame-
ters until agreement via a desired error metric is reached.
Several key steps were necessary to achieve the vision of
model-based inversion to characterize cracks in metallic mul-
tilayer fastener sites using BHEC techniques. The critical first
step was the development of accurate forward models using
VIC-3D® (Sabbagh et al. 2013) that precisely represented the
split-D differential coil eddy current measurement response
to notches and cracks (Aldrin et al. 2014). We note that other
researchers have developed models pertinent to BHEC inspec-
tions (Mandache et al. 2011 Bao et al. 2025), though those were
applied primarily to probability of detection (POD) evaluations.
A robust inversion scheme was then developed, leveraging
fast surrogate models incorporating simulated results (Sabbagh
et al. 2013). A comprehensive approach was created to perform
model-based inversion of cracks and electrical discharge
machining (EDM) notches using eddy current techniques with
an automated BHEC system for various discontinuity scenar-
ios (Aldrin et al. 2016). While prior work demonstrated prom-
ising capability for sizing EDM notches with varying aspect
ratios, some model discrepancy was discovered that resulted
in greater inversion error for smaller corner discontinuities
(Aldrin et al. 2019). Sensitivity to probe liftoff and the calibra-
tion process was also observed during experimental testing,
indicating a need to better compensate for variations in probe
state and adjacent material conditions (Aldrin et al. 2019).
We note that some recent work by other researchers (e.g.,
Hughes and Drinkwater 2021) has investigated model inversion
schemes for sizing the depth of EDM notches. However, only
limited progress has been made to address the sizing of cracks
in complex geometries and under real-world test operations
and conditions (Oneida et al. 2017).
In this paper, progress is presented in addressing out-
standing performance gaps and subsequently validating the
capability to characterize fatigue cracks in metallic multi-
layer fastener sites using BHEC techniques enabled by mod-
el-based inversion. Improvements are presented that enhance
the reliability of model calibration, liftoff compensation, and
inversion steps over a wider range of conditions. While earlier
demonstrations focused solely on the inversion of cracks in
high-electrical-conductivity aluminum, this study extends the
capability to address cracks in titanium (lower conductivity)
and stainless steel (moderate conductivity with permeability
greater than 1). In addition, an approach has been developed
that performs reliably over a wide range of fastener hole diam-
eters and the multiple frequencies used in common aerospace
BHEC inspections.
1.2. Assessing Uncertainty in Sizing Capability
To test the reliability of an NDE sizing capability, validation
procedures are needed to ensure the accuracy of the material
characterization techniques. Comprehensive probability of
detection (POD) evaluation procedures have been developed
to validate the reliability of NDE techniques and are used by
the US Department of the Air Force (DAF) in support of the
Aircraft Structural Integrity Program (ASIP) (US Department
of Defense 2009 ASTM 2021). While standard practices exist
for evaluating measurement system capabilities (ASTM 2014),
there are limits to their direct application for assessing NDE
characterization performance. Prior work has introduced pro-
cedures and performed demonstrations that evaluate NDE
sizing capabilities, specifically for discontinuities in welds (Førli
et al. 1998 Lozev et al. 2005). One frequently cited metric is the
calculation of the 95% safety limit against undersizing (LUS)
bound for quantifying sizing performance. However, care must
be taken with this metric, as key assumptions—such as linear-
ity in the parameters and constant variance with varying crack
size—must be verified.
A recent paper by Aldrin and Forsyth (2017) outlined
specific challenges in sizing studies and presented best prac-
tices for evaluating NDE characterization capabilities. These
approaches are applied in this work.
Another critical point for NDE sizing performance studies
is that the discontinuity state must be very well understood.
Uncertainty in the dimensions of any property or dimension
under test must be far less than the expected performance of
the NDE characterization technique. In an ideal experiment, all
epistemic uncertainty in the NDE response would be explained
by fixed variables (e.g., crack size), leaving only the natural
uncertainty due to aleatory noise. If the aleatory noise is large
compared to the NDE response, the “signal” may be lost in the
noise. However, if the crack size is inaccurately known, then
there will be more epistemic uncertainty in the relationship
between the response and crack size, as this uncertainty stems
from a lack of knowledge (e.g., imprecise measurements of
depth). This can lead to unexpected problems in modeling the
NDE response–crack size relationship, such as spurious trends
(e.g., small cracks being undersized and large cracks oversized)
or added variability (e.g., if sizing error is not dependent on
crack size).
It is important to note that many POD crack sets are inad-
equate for use in sizing demonstrations. A study by Shell et al.
(2015) investigated eddy current model-based inversion predic-
tions for a POD set of cracks in IN-100 alloy. While the length
estimates were found to be fairly accurate, the depth estimates
showed significant scatter compared to the assumed depths of
the crack set. Even when the mean relationship between crack
length and depth was perfectly understood, significant depth
variation was found in real cracks of similar lengths within
the set. Although variability and uncertainty in discontinuity
dimensions do impact POD evaluations to some degree, the
level of observed variability in this set was unacceptable for use
in crack depth sizing studies. This paper highlights the impor-
tance for the design, manufacture, and verification of fatigue
crack specimens where depth uncertainty is well-controlled,
which is necessary for a comprehensive crack sizing perfor-
mance evaluation.
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