Abstract
Eddy current methods are commonly used in the inspection of riveted multilayer structures. In a practical scenario, steel rivets may be magnetized to varying levels which can distort the magnetic field and pose a major challenge since the distortion can result in false positives and negatives. This paper presents an invariance transformation procedure that renders the signal insensitive to changes in magnetization while retaining the sensitivity to defect information. The approach is first implemented on simulated data, where the signals from magnetized steel rivets can be generated under controlled variation of relative permeability. The algorithm is then validated on experimental data acquired using steel rivets with different residual magnetization levels.
Introduction
Detection of cracks under fasteners in multilayered structures poses one of the major challenges in nondestructive evaluation (NDE) of aging aircraft fleet. Eddy current (EC) coil using magnetoresistance (MR) sensors has recently emerged as a viable modality for rapid inspection of multilayered components with rivets. Ultra-low frequency excitation along with MR sensors have been developed to increase penetration depth, give good signal to noise ratio (SNR) and obtain high sensitivity to deeply embedded defects. A major challenge for EC-MR inspection is that most cracks originate at fastener sites and are hence masked by the dominant response due to the fastener [1,2]. A second issue is that in most riveted structures, the fasteners are made of magnetically permeable materials, e.g. steel, with relative permeability much greater than 1. The presence of fasteners with high relative permeability causes special challenges to MR measurements as the response due to the fastener depends on the magnetization history which in general is unknown.
The EC testing system discussed in this paper uses planar coils for excitation and tunnel magnetoresistance (TMR) sensors to pick up magnetic field signals [3]. Figure 1 illustrates a typical geometry of test specimen with two aluminum layers joined by a fastener with a crack originating at the fastener site in the second layer. A planar coil driven with low frequency current scans over the top layer. TMR sensors placed along the symmetry line of the planar coil pick up the z-component of the magnetic flux density with good sensitivity to defects even in the second layer. Figure 2 shows two images of simulated TMR signals when the fastener is made of aluminum and steel, assuming the relative permeability of steel 𝜇 r is 5. The crack length d is equal to 0.25 inch and the operation frequency is 100 Hz. The response changes dramatically due to different fastener materials and hence, before developing subsequent classification schemes, it is important to obtain a transformation which renders the TMR signal insensitive to permeability variation while retaining sensitivity to the defect.
Schematic of the multilayer structure. Top view (left), lateral view (right).
Simulated TMR images of aluminum fastener (left) and steel fastener with 𝜇 r =5 (right).
In EC testing, the MR signal is a function of defect parameters as well as operation conditions, e.g. frequency, lift-off, fastener permeability, etc. While other conditions are well controlled with a high-precision scanner and signal generator, the permeability of the fastener is highly related to the magnetization history and is usually unknown. This section presents a systematic approach to develop a transformation to render the measured signals insensitive to variations in fastener permeability.
In general, such a transformation falls under the topic of invariant pattern recognition algorithms. In [4], Wood conducted a comprehensive review on techniques for solving the problem of invariant pattern recognition, including classical methods such as integral transforms and algebraic transforms as well as modern techniques such as neural network approaches. In [5], Mandayam pointed out that classical techniques such as Fourier descriptors and moment invariants fail to achieve permeability invariance and simultaneously preserve defect related information since the variations in a defect related parameter and an undesirable operation variable are represented by the same coordinate transformation. This is exactly the case in the variation of the MR signal with fastener permeability and defect length. Mandayam also developed a novel invariant pattern recognition technique based on approximation theory to successfully overcome this difficulty. This paper follows this idea and successfully implements an algorithm for permeability invariance transformation of MR signals.
Invariance transformation algorithm
Given two signals X
A
and X
B
, characterizing the same phenomenon, two distinct initial features, x
A
(d, 𝜇
r
) and x
B
(d, 𝜇
r
), are chosen, where 𝜇
r
represents an operational variable (i.e. relative permeability of the fastener) and d represents a defect related parameter (i.e. defect length). Notice that x
A
and x
B
must be chosen such that they have different variations with respect to 𝜇
r
. The target invariant transformation method is to obtain a feature h, which is a function of x
A
and x
B
and invariant to 𝜇
r
. In other words, we need to find a function, f, such that
Given two functions g
1(x
A
) and g
2(x
B
), a sufficient condition to obtain a signal invariant to 𝜇
r
can be derived as
Hereafter, for an EC-MR system, the two signals X
A
and X
B
that describe the same defect are assumed to be the real (B
r
) and imaginary (B
i
) components of the magnetic flux density along the z direction (B
z
= B
r
+ jB
i
). Changes in the relative permeability 𝜇
r
of the fastener influence peak-peak amplitude of the magnetic field signals. Figure 3 illustrates the definition of the peak-peak amplitudes P
r
and P
i
of the real and imaginary components respectively. Figure 4 shows P
r
and P
i
as functions of relative permeability for three different defect profiles. P
r
and P
i
characterize the same defect signal, but have different variations with respect to 𝜇
r
. We define the initial features as x
A
= [P
r
, P
i
]
T
and x
B
= P
r
for the real component. Applying the idea of Eq. (3), we obtain a permeability-invariant signal for the real component (the homomorphism ° here considered is the usual product):
Magnetic field signal across the center of the fastener. (a) amplitude of the magnetic field in z direction, ∥B z ∥. (b) real component (B r ) of B z , (c) imaginary component (B i ) of B z .
Peak-peak amplitude of real (left) and imaginary (right) component of the magnetic field as a function of relative permeability 𝜇 r of the steel fastener.
Various choices of the function g
1 lead to different methods of data interpolations. In this study, g
1 is approximated using a radial basis function (RBF) network. In other words, g
1 is chosen as
In Eq. (6) 𝜙 is a Gaussian function defined as 𝜙(∥x − c∥) = exp(−|x − c|2∕2𝜆2) and 𝜆 is the radius of the Gaussian kernel. Equation (6) is solved at known discrete data points (d i , 𝜇 rk ) and is guaranteed a 𝜇 r -invariant signal at these points. An implementation of Eq. (6) essentially involves training a RBF network to obtain the function g 1.
Numerical validation
A numerical model was built with COMSOL®. For simplicity, the primary field is produced by an infinite current sheet carrying a uniform surface current density. The geometry of the model follows Fig. 1 and details of the parameters are given in Table 1. Ten different values of relative permeability are assumed for the fastener and notch defects of three different lengths are considered. Simulated signals corresponding to 𝜇 r = 1, 3, 5, 7, 9 are used to train the RBF network according to Eq. (6) and signals related to 𝜇 r = 2, 4, 6, 8, 10 are used as a test data set. Figures 5 and 6 show real and imaginary parts as well as amplitude of the magnetic field signal for training and test data sets respectively. Raw data are shown in blue solid lines where overall signal strength increases with 𝜇 r . Processed data are shown in red dashed lines and they collapse all close to the signal corresponding to 𝜇 r = 1, which is used as the target signal during training.
Parameters of the numerical model
Parameters of the numerical model
Training data set and output signals. Blue solid lines: raw data. Red dashed lines: output invariant signal.
Test data set and output signals. Blue solid lines: raw data. Red dashed lines: output invariant signal.
Figure 7 shows the transformed 𝜇 r -invariant signals for three different defect sizes. Signals of varying 𝜇 r collapse to a bunch of curves of very similar shapes though they have some variation at the peak values. The sensitivity to defect length has been retained as the slope at x = 20 mm distinguishes curves of different defect lengths. This implementation shows equally good performance on both training data and test data.
Transformed invariant signals of training data (left) and test data (right).
A linear EC coil was scanned over fasteners with defects in an aluminum sample and the induced magnetic fields were measured using a TMR sensor. Figure 8 shows an overview of the experimental system and a bottom view of the target panel. The primary field was generated by a planar coil with symmetric design. A single TMR sensor, TMR 2301 by MultiDimension Technology®, was mounted on top of the center of the coil. This sensor is capable of measuring magnetic field along 3 axes simultaneously. In this study, only signals in the z direction were used. The probe (sensor and coil) was attached to a 3-axis scanner and the magnetic field images were obtained by conducting a 2D scan. The lift-off between the sensor and the panel was 4 mm and the operation frequency was 100 Hz.
Experiment setup. System overview (left), target panel (right).
The dimensions of the target panel are the same as Table 1. In practice, it is very difficult to precisely measure the relative permeability of fasteners. Instead, we used steel fasteners which have been magnetized during previous use and sorted them by measuring the remanent magnetization using a Gauss meter. The remanent magnetization produces an external magnetic flux density from less than 5 μT to more than 300 μT. The signals of 9 fasteners are used for training purpose and the signals of 6 other fasteners are used for test.
Experimental results. Raw data (left), transformed data (right).
Figure 9 shows raw data as well as the transformed invariant signals. The variation due to different remanent magnetization levels is not as significant as that from varying the relative permeability, as shown in Fig. 9(a). After process, the variation due to magnetization level is further suppressed. Only three curves, each identifying signals related to a specific defect size, are visually observable in Fig. 9(b), though there are in fact the same number of curves as in Fig. 9(a). These curves can easily distinguish defects of different lengths by evaluation of the slope at x = 15 mm.
This paper presents a procedure for permeability invariance transformation related to EC-NDE. Results obtained from numerical simulations and experiments indicate that this method is very robust. The invariance transformed signals can be used in further processing and evaluation for defect classification and quantification.
In order to address issues of misalignment of training and test signals, work is under progress to extend the algorithm to the 2D image data of fasteners, where image features will be used for first aligning the training and test images.