Advanced statistical learning method for multi-physics NDT-NDE

This work presents an innovative multi-physics (MP) Learning-by-Examples (LBE) inversion methodology for real-time non-destructive testing (NDT). Eddy Current Testing (ECT) and Ultrasonic Testing (UT) data are effectively combined to deal with the localization and characterization of a crack inside a conductive structure. An adaptive sampling strategy is applied on ECT-UT data in order to build an optimal (i.e., having minimum cardinality and highly informative) training set. Support vector regression (SVR) is exploited to obtain a computationally-efficient and accurate surrogate model of the inverse operator and, subsequently, to perform real-time inversions on previously-unseen measurements provided by simulations. The robustness of the proposed MP-LBE approach is numerically assessed in presence of synthetic noisy test set and compared to single-physic (i.e., ECT or UT) inversion.


Introduction
Real time accurate inversion solution becomes the main priority in non-destructive testing and evaluation (NDT-NDE) applications. Among different iterative [1,2,3] and non-iterative [4,5,6,7] inversion solutions, Learning by examples (LBE) strategy is getting more attention for having quasi-real time inversion capabilities. In this work, LBE has been adopted for a NDE problem where a narrow crack is occurred around a fastener (e.g., bore hole) within an inspected medium [8]. This is an important problem for the aging aircraft NDE community and Eddy Current Testing (ECT) is widely applied while the structure under test (SUT) thickness is thin. However, the penetration depth of the induced currents is limited by the skin depth. This makes the detection and resolution of defects more difficult as the depth increases. Whereas ultrasound testing (UT) NDT inspection is suitable for high resolution, but the inspection is affected by the surface roughness of the inspected medium [9]. That means, each of these NDT methods has some pros and cons according to their own physics. Moreover, for the mentioned problem at hand, ECT signal is mostly affected for the presence of fastener. Due to the significant probe impedance variation, the area of the fastener is acting as a circular defect within the inspected medium. The impedance variation due to the fastener is much stronger than narrow crack, thus, when the crack is placed deeper inside the SUT, the ECT signals contribution due to the presence of narrow crack becomes weaker. Conversely, UT signals are stronger for subsurface crack compare to the crack placed at the top surface of the SUT. As a consequence, 2 1234567890 ''"" different impacts on the crack characterization and localization performance are expected based on ECT and UT methods. Thus, multi-physics (MP) data fusion (ECT-UT) has been applied to maximize the inversion performance for crack characterization and localization.
In general, LBE is a two phases approach. During the preliminary phase (so called offline phase), a fast and accurate inverse/trained model is built based on a training set made of inputoutput (I/O) pairs by learning algorithm. The developed (trained) model from offline phase is then used to predict the output associated to an unknown test sample during the second phase (online phase). Within the framework of LBE, an adaptive sampling strategy combining Partial Lest Square (PLS) [10] feature extraction and modified version of output space filling (OSF) [11] (i.e., PLS-OSF sampling [4]) has been adopted for obtaining optimal training sets by both ECT and UT methods separately. An updated version of PLS-OSF sampling algorithm has also been illustrated for dealing with ECT-UT data. Support vector regression (SVR) [12] is used to obtain accurate training model and perform real time inversion. Finally, the performance of the MP-LBE inversion schema for crack characterization and localization is compared to single-physic (i.e., ECT and UT) inversion on noisy data.

Mathematical formulation of forward and inverse problem
Let us consider a homogeneous plate made by aluminium 2024 alloy of thickness 6 mm, density 2.77 g.cm −3 , has been investigated by both ECT and UT NDT methods. The plate consists of a fastener (bore hole) of radius 3.75 mm and 6.00 mm height. The plate is affected by a single notch (e.g., narrow crack) of volume Ω having fixed width 0.01 mm and height 2 mm (Fig. 1) which is attached with the fastener. The crack is characterized by total Q = 3 descriptors of length (l c ), ligament (δ c ) and angular distance (φ c ) (i.e., p = (l c , δ c , φ c )).

ECT treatment
The plate is inspected by a single coil working in absolute mode of frequency 1 kHz with lift off 1 mm. The coil impedance variation due to presence of the crack measured at the k−th (k = 1, ..., K) scanning position with respect to the flawless region is given by [13].
I is the current flowing inside the coil while E inc (r|r k ) is the incident field generated at position r in the unflawed plate (r k = (x k , y k ) represents the k-th coil position within the plate). ρ(r|r k ) is the unknown induced current dipole density, which models the presence of the crack and is related to the total field, E tot (r|r k ) that can be expresses by ρ(r|r k ) = [σ(r) − σ]E tot (r|r k ). CIVA simulator [14]

Ultrasound testing treatment
The plate has been investigated by a ray probe by using water coupling medium (i.e., density 1 g.cm −3 ). The probe is acting both for transmitting and receiving UT signals. More details of the treated problem and probe definition are available in [15]. CIVA uses a hybrid model known as Physical Theory of Diffraction (PTD), based on Kirchhoff approximation and highfrequency Geometrical Theory of Diffraction (GTD) for generating diffraction/scattering waves from planner-like defects. The scattered field, for the PTD model can be expressed by [16] . (2) where, α = L, T V, or T H (Longitudinal, Transverse Vertical or Transverse Horizontal, respectively) incident type wave vector and β = L, T V, or T H is the scatter type wave vector. S β is the distance between the diffraction point r α β and the observation point is the Kirchhoff edge diffraction coefficient. u Kir is the displacement scattered field at the observation r, and the Rayleigh field u Rayleigh (r) comprises the surface waves. The reflections/scatters wave have been collected through C-Scan (e.g., maximum ray amplitude available at each inspection point) and represented by χ U T . CIVA simulator [14] has been utilized as a forward operator Φ U T {.} in order to generate UT data. Unlike, ECT signals, UT signals contain only the real data, hence UT signals are represented by

Data fusion using ECT and UT
In this case, ECT signals and UT signals are generated separately by their own forward solver (i.e., Φ ECT {.} and Φ U T {.}) and both of these data sets are fused by concatenating ECT and UT data. The obtained fused ECT-UT data are represented by

Adaptive sampling through feature extraction
The main goal of the adaptive sampling (i.e., PLS-OSF) is to apply PLS feature extraction for reducing the dimension of the actual features (e.g., F ECT , F U T and F ECT −U T ) by projecting into extracted feature space. After-which, adaptive sampling is performed directly in the extracted feature space to build suitable I/O pairs for building optimal training model by using lowest number of training samples during offline phase. Though, PLS-OSF [4] has been directly applied for ECT and UT data separately, a modified version of PLS-OSF is needed for dealing with ECT-UT data. The following steps describe the updated PLS-OSF sampling strategy.
i Initialization-Generate N 0 number of initial samples by uniform GRID (i.e., full factorial grid) sampling approach. A matrix of defect parameters p = (p (n) ; n = 1, ..., N 0 ) having } generate ECT and UT data respectively, and fill the ii PLS Feature Extraction-In this step, the F ECT −U T = 3K dimensional ECT-UT data are reduced to J number of extracted features where, Among different iterative algorithms, we have used SIMPLS algorithm [17] to obtain the weight matrix W . Y and G contain the for the adaptive step.
iii Adaptive Sampling-Generate V candidate samples by p , which can be described by . The set of extracted features is obtained by iv Stop Criterion-The adaptive sampling step adds new sample iteratively until N iterative = N (N is desired/feasible training size).
At this stage, an ε-SVR [12] has been utilized to train separately q-th set of I/O pairŝ ; p Finally, the q-th crack parameter associated to is estimated (i.e.,p (m) q ) by the corresponding trained model in online phase.

Numerical validation
The ECT probe and the UT probe collect their corresponding NDT data from 81 positions along X directions with a step size of 0.5 mm and from 41 positions along Y directions with a step size of 1 mm, respectively through a raster scan. Therefore, ECT (i.e., impedance variation signal) and UT signals (i.e., reflected rays) are collected from K = 81 × 41 = 3321 number of inspected points. Therefore, for a single crack configuration (i.e., sample), F ECT = 2K = 6642,  (NME) described in [4] has been utilized for evaluating the inversion performance.  ECT signals are mostly corrupted for imposing noise and by combining ECT and UT signals, we can improve the overall inversion performance. Fig. 3 (a) shows that l c estimation by using ECT suffers on noisy data than UT, while on N oiseless test set by both ECT and UT have shown similar prediction accuracy. On the other hand, crack ligament distance δ c estimation is showing lower prediction error for adopting ECT signals than UT signals for both noisy and N oiseless test set ( Fig. 3 (b)). Whereas, UT data shows lower N M E than ECT data for 7 1234567890 ''""  (Fig. 3 (c)). By combining both ECT and UT signals, ECT-UT data fusion contains both information from ECT and UT signals. Whereas, applying PLS-OSF sampling, we can retrieve most significant information from ECT-UT. As a consequence, it improves the learning ability of SVR during training model development. Hence, ECT-UT data fusion has shown higher prediction accuracy than ECT and UT data for all the cases on noisy and N oiseless test sets. Fig. 4 shows the scatter plots of true vs. predicted crack parameters obtained for N = 216 for noisy test set (SN R = 20 [dB]). Qualitatively, ECT-UT data fusion provides better l c , δ c and φ c estimation than ECT and UT signals. Concerning real time solution, it takes 0.03s for testing 1000 samples during online phase.

Conclusions
In this work, we have shown an innovative MP-LBE inversion strategy for crack dimension and position estimation. Within the framework of LBE, PLS-OSF/SVR strategy has been applied for solving a NDE problem by utilizing ECT signals, UT signals and ECT-UT data fusion. By combining two different NDT methods, we first retrieved the variation of actual ECT and UT signals for changing crack parameters. Applying adaptive sampling through PLS feature extraction retrieves most significant information from the actual ECT-UT features that improves the learning ability of SVR. ECT-UT shows better prediction accuracy than ECT and UT methods separately for performing inversion on both noisy and noiseless synthetic test set.