IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 4, APRIL 2017
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Precise Inversion for the Reconstruction of Arbitrary Defect Profiles Considering Velocity Effect in Magnetic Flux Leakage Testing Senxiang Lu, Jian Feng, Fangming Li, and Jinhai Liu School of Information Science and Engineering, Northeastern University, Shenyang 110819, China In magnetic flux leakage type nondestructive testing (NDT), there exists velocity effect, which may cause the distortion of the defect signals and reduce the estimated accuracy of defect profile. In this paper, the distortions are analyzed by using finite-element method (FEM), and the influence of velocity effect on reconstruction of defect profiles is discussed under the condition of the simulation and experiment. This paper proposes an effective method for reconstructing arbitrary defect profiles in different velocity conditions. In the proposed method, the FEM considering velocity effect is employed as the forward model. A weighting conjugate gradient algorithm is applied to update the defect profile iteratively in two gradient directions. The algorithms effectiveness is tested on a series of artificial defects under various velocity conditions. The results demonstrate that the proposed model can achieve the better reconstruction accuracy than the ignoring velocity effect models. The properties of the presented method are so stable and robust that they make our approach a promising technique for the practical application of profile reconstruction for NDT. Index Terms— Finite-element method (FEM), inverse problem, magnetic flux leakage (MFL), velocity effect, weighting conjugate gradient.
I. I NTRODUCTION
T
HE use of nondestructive testing (NDT) evaluation to detect and analyze defects in ferromagnetic test objects is widespread in the energy industry. A particularly popular nondestructive evaluation method for ferromagnetic pipelines is the magnetic flux leakage (MFL) technique, which has been used for a long time [1]. MFL tools use permanent magnets to magnetize the pipe wall near to saturation flux density. If there exists a metal loss (defect), the flux lines will leak out from the pipe wall around the defect. An array of Hall sensors installed around the circumference of the detector between the two poles of the magnetizer is used to measure the leakage flux [2]. The estimation of defect profiles from the measurement MFL signal is a common inverse problem in electromagnetic NDT. The above-mentioned inverse problem is ill-posed due to the no uniqueness of the solution, particularly in the presence of measurement noise. Several techniques have been proposed for the issue. Model free methods [3]–[5] mainly rely on signal processing techniques to establish a relationship between specific characteristics of the signal and the size of the defect, ignoring the underlying physical process. Therefore, the profile of the irregular defect can hardly be reconstructed. Model-based methods [6]–[10] usually apply physical model as forward model to obtain MFL signal of assumed defect profile. Compared the difference between simulation MFL signal and original measurement signal, the assumed defect profile is modified iteratively in order to approach the real profile of the defect.
Manuscript received June 15, 2016; revised September 27, 2016 and November 9, 2016; accepted December 14, 2016. Date of publication December 21, 2016; date of current version March 16, 2017. Corresponding author: J. Feng (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2016.2642887
The estimated accuracy of the defect profile inversed by the model-based methods depends on two key factors. One is the stability and rapidity of the converging algorithm. The other is the similarities of magnitude and shape of MFL signal between the simulated values and the measured values by Hall-effect sensors. The signal magnitude and shape are related to many parameters, such as the coercive force of permanent magnets, the permeability of the pipe wall, sensors’ liftoff, testing velocity, and so on. Particularly, in high-speed inspection, the velocity is a more significant reason that influences the MFL signal and distorts its shape than in low-speed inspection [11]. If the testing velocity is less than a certain value vlimit and the velocity effect can be ignored, the defect profile will be reconstructed by the above-mentioned model-based methods accurately. When the testing velocity is more than the value vlimit , the MFL signal is distorted by velocity effect, and the estimated accuracy will be severely diminished. Some studies of the velocity effect on the estimated accuracy have been discussed in [12]–[16]. As the testing velocity increasing, the effect of the eddy current on the position of the defect will increase and the distortion of the MFL signal will be more serious in [12] and [13]. The faster the testing velocity is, the lower the magnetized intensity of the pipeline is in [14] and [15]. The estimated accuracy will decrease when the testing velocity increases in [16], because the fault forward model will cause some errors between the simulation values and the distorted measurement values. The qualitative analysis of the velocity effect has been considered in their studies, but how to quantitatively improve the estimated accuracy of the defect profile considering the velocity effect has not been solved. To acquire the ideal estimated result, some researchers approach signal compensate methods to remove the velocity
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effect [11], [17]–[19] and settle the fault forward model problem. Lei et al. [17] come forward a radial basis function neural network to offset the velocity effect. Mandayam et al. [11], [18] present a restoration filter, which is identified by using a learning process, and a compensation signal can be achieved by this filter. Park and Park [19] propose a nonlinear map relationship of the signal feature between the motion condition and the static condition. Although the compensation algorithms mentioned earlier are effective ways to modify the velocity-induced signals, the improvement of estimated accuracy by using these methods is limited. These compensation methods depend on a transformational relation, which is applied to convert signals considering velocity effect to equivalent signals without considering velocity effect. The relationship is closely related to the defect profile and the testing velocity. In the inversion process, the defect profile is unknown, so the exact relationship is hardly acquired. The relationship in these compensation algorithms can only be obtained by an assumed profile. Therefore, the signals converted by these relationships are deviated from the real physical signals. In the precise estimation process of the defect profile, the error conversion cannot enhance signal similarities effectively. In this paper, we come up with a novel way of dealing with the signal similarities problem. Instead of converting the original measurement values, we improve the simulation values of forward model to enhance the signal similarities. A forward model considering velocity effect compensation is established by finite-element method (FEM) to simulate the real physical process. The real physical process is mainly divided into two parts: the static process and the dynamic process. The static process has been mentioned in the first paragraph in Section I. The dynamic process is that the moving magnetic field gives rise to an eddy current field in the conductive test object. The measured magnetic field is indeed a superposed field of the static magnetic field and a secondary field generated by motion-induced eddy current. The simulation values of this new forward model will be more similar to the measured values in this superposition analysis. Compared with the signal compensation method, this model compensation method can maximally preserve the signals’ original features. Another factor that affects the estimated accuracy is the stability and rapidity of the iterative convergence algorithm. Some studies on iterative convergence algorithms are discussed in [7], [9], [10], and [20]. Amineh et al. [9] adopt a space mapping optimization algorithm to iterative update a so-called coarse model. Han et al. [10] come forward a hybrid method with cuckoo search and particle filter to generate the estimated profile of each iteration. Priewald et al. [7] apply a deterministic Gauss–Newton optimization algorithm to reconstruct the defect geometry. Chen et al. [20] update the defect profile by using a combination of gradient descent and simulated annealing in the iterative inversion procedure. These algorithms mentioned earlier can converge to global optimal solution quickly, when the MFL signals acquired without considering velocity effect. However, as the increasing of the signals’ distortion, the number of local optimum solutions
IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 4, APRIL 2017
is increased. These algorithms are easily trapped into local optimum and more time-consuming. In this paper, we propose a weighting conjugate gradient algorithm due to the particularity of the forward model. Two gradients in different directions are involved in this new algorithm, and they are multiplied by different weighting coefficients, which are depend on the signal similarities. In order to get the global optimal solution rapidly, we adopt two gradient directions to optimize. These two gradient directions can avoid the final results from dropping into local optimal solutions. The contributions of the proposed method in this paper can be summarized as follows. First, an FEM model considering velocity effect is proposed as a forward model to acquire simulation values more accurately, because it fully considered the physical structure and characteristics. This model dramatically improves the signal similarities to satisfy the requirement of inversion procedure. Second, compared with common iterative convergence algorithm, the proposed weighting conjugate gradient algorithm is easier to rapidly converge to global optimal solution. The algorithm is practical enough to estimate the defect profile satisfactorily under various testing velocities. The rest of this paper is organized as follows. The theoretical analysis and numerical implementation of magnetism modeling considering velocity effect are described in Section II. The proposed weighting conjugate gradient algorithm and detailed description of the algorithm process are given in Section III. Section IV introduces the experiment results and method comparison to demonstrate the performance of the proposed method. The conclusion is given in Section V. II. M AGNETISM M ODELING C ONSIDERING V ELOCITY E FFECT A. MFL and Velocity Effect Theoretical Analysis The MFL testing technology is mainly based on the magnetism, and theoretical analysis of the magnetism can be solved by Maxwell’s equations. The magnetostatic model can be used to calculate the magnetic field distribution in the process of MFL testing if the testing process is implemented in static or low-speed status. This is properly described by [7] 1 (1) ∇ × ∇ × A = Js μ where μ, A, and Js represent the permeability of the media, the magnetic vector potential, and the source current density, respectively. The magnetic flux density B = ∇ × A, and the permeability μ is a function of the magnetic flux density B. The nonlinear B–H characteristics B = μH can be described by B–H curve. For example, the B–H curve of the cold rolled X45 steel is shown in Fig. 1. Lorentz law can be used to analyze the velocity effect in a dynamic MFL inspection system. If relative velocity is created, the Lorentz force induces eddy currents in the specimen. The current density is expressed as follows in: Jv = σ v × B
(2)
where Jv denotes the eddy current density in the specimen, v denotes the velocity of the applied magnetic field, and σ represents the conductivity of the specimen.
LU et al.: PRECISE INVERSION FOR THE RECONSTRUCTION OF ARBITRARY DEFECT PROFILES
Fig. 1.
Nonlinear B–H -characteristics of cold rolled X45 steel.
With respect to dynamic electromagnetic systems, the governing equation deduced from Maxwell’s equations is combined with the term for eddy currents due to the movement of an applied magnetic field by considering (2). The modified equation of electromagnetic field is expressed in [18], [19] ∂A 1 ∇ × A = Js − σ + σ v × (∇ × A). (3) μ ∂t As shown in (3), the eddy currents generated by moving magnetic field are influenced both by the magnetic field and the electric field distribution of the source. Compared with the governing (1), the modified equation implies that the eddy currents influence not only the currents distribution in the conductive pipe but also the magnetic field, which results in the distortion of measurement signal [12]. In general, the analytical solution of magnetic fields cannot be exactly solved by (3). So a few numerical methods are used to calculate the distribution of magnetic fields. ∇×
B. Numerical FEM Implementation for MFL The common ways to calculate the distribution of magnetic fields are finite-difference method [11], finiteelement method (FEM) [12], [13], boundary element method (BEM) [14], and so on. Some commercial numerical simulation toolkit, such as ANSOFT [11] and ANSYS [19], are programmed to simulate electromagnetic problems with moving parts by considering the eddy currents. In order to acquire accurate signals of the forward model, in this paper, we apply ANSYS 15.0 software for simulating the magnetic field distributions under various velocity conditions. It is characterized by easy and simple application, especially for numerical simulation on frequency and time domain electromagnetic fields in complex structures. It implements FEM while allowing BEM [21] codes and has strongly coupled electromagnetic, drive circuit, and mechanical formulations. It also integrates several numerical modules for solving specific problems, such as electrostatic, magnetostatic, eddy currents, quasistatic, and transient problems [12]. 1) 2-D Modeling: In the modeling process, the 2-D model of an internal defect profile is established first. A parameter vector λ is used to represent geometric properties of the defect. λ contains m degrees of freedom (DOFs), which are chosen to be defect depth values λi along the defect [7]. The situation is shown in Fig. 2. In the profile inversion and modeling process, some parameters of DOFs are defined by practical application. The interval
Fig. 2.
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Definition of DOFs (13 DOFs).
dX = Xi − Xi−1 of each DOFs is chosen based upon the need of precision. If the defect profile needs to be calculated accurately, the DOF interval can be the same or twice as much as the MFL signal sampling interval. If the defect profile is calculated roughly, the DOF interval can be four times, eight times, or ten times as much as the MFL signal sampling interval. The boundary of DOFs depends on the MFL signals. The left boundary X1 is the origin of coordinates, which is located at the point of the left valley of MFL signal. The right boundary Xm is located at the point of the right valley. The number of DOFs m is calculated by Xm − X1 . (4) dX Next, the whole FEM model is set up in 2-D with x–y coordinates representing the cross section of the MFL detector and the pipe wall as shown in Fig. 3(a). The width of the simulation model can be decided according to the actual situation, and in this model, the width is set to 10 mm. The dimensions and properties of the detector, pipe wall, and defect are shown in Fig. 3(a) and Table I. The testing direction in simulation is from left to right, which should be defined in accordance with the real testing direction. Ignoring the velocity effect, the measured signals have nothing to do with the testing direction. If the velocity effect is considered, the MFL signals measured by two directions are completely dissimilar, because the effected point of the eddy current is different. When the testing direction is from left to right, the main effective area is focused on the right of the defect, and vice versa. Different effective areas induce different leakage magnetic fields, so the measured value of the same x-position is not the same. 2) Simulation Results and Analysis: The distributions of magnetic flux lines of the magnetic circuit as shown in Fig. 3 are completely solved by ANSYS. Around the defect, some flux lines leak out of the pipe wall and flow into the air. When the velocity is zero, the magnetic flux lines distribute symmetrically. With the increase of the velocity, the distribution domain of the flux lines is more asymmetry [15]. The asymmetry phenomenon is caused by the asymmetric distribution of the eddy current, which causes a high interference magnetic field. The intensity of eddy current in the right section of the defect is bigger than the left section, when the testing direction is from left to right. Therefore, the distribution domains of flux lines in the left and the right sections are totally different. The magnitudes of leakage magnetic field (liftoff is 2 mm) in the components of axial direction are shown in Fig. 4 under m=
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 4, APRIL 2017
Fig. 3. (a) Cross section of the simulation model (unit: mm). Magnetic circuit distributions of magnetic flux lines. (b) Testing velocity is 0 m/s. (c) Testing velocity is 0.5 m/s. (d) Testing velocity is 1 m/s. (e) Testing velocity is 2 m/s. (f) Testing velocity is 5 m/s. TABLE I P ROPERTIES OF THE S IMULATION M ODEL
Fig. 4. Simulated signals of leakage magnetic field in the components of axial direction under different velocity conditions (one time per millimeter).
different velocity conditions. With the increase of the velocity, the distortion of the MFL signal is more serious. If the MFL curve appears two peaks, the distortion can be summarized as follows: 1) the signal reference value, which means the magnetic field intensity in the non-defect region, is decreased; 2) the value of the left peak decreases a little; 3) the values of the right peak and the right valley decrease obviously; and 4) the value of the left valley is nearly invariable. As can be seen in Figs. 3 and 4, for the above-mentioned physical model, the minimum velocity value vlimit is 0.5 m/s. If the testing velocity is faster than vlimit , the measured signals will be distorted by velocity effect. In addition, this modeling approach can also simulate external defects condition.
Fig. 5. Measured signals and simulated signals in different velocities (one time per millimeter in simulation and one time per two millimeters in experiment). (a) Signal i is the simulated signal in the velocity of 0.5 m/s; signal ii is the measured signal in the velocity of 0.5 m/s; signal iii is the simulated signal in the velocity of 2 m/s; signal iv is the measured signal in the velocity of 2 m/s. (b) Signal v is the simulated signal in the velocity of 1 m/s; signal vi is the measured signal in the velocity of 1 m/s; signal vii is the simulated signal in the velocity of 5 m/s; signal viii is the measured signal in the velocity of 5 m/s.
The simulation signals of external the defect will be obtained by changing the defect location. The values of the simulated results and the measured results are described as curves in Fig. 5. Under different velocity conditions, the simulated results are all similar to the measured results. The signal standard deviation (SSD) in (5) between simulated signals and measured signals is used to appraise the accuracy of simulation results p 1 [Bx(i ) − Bx(i )]2 (5) SSD = p i=1
LU et al.: PRECISE INVERSION FOR THE RECONSTRUCTION OF ARBITRARY DEFECT PROFILES
Fig. 6.
SSD results under different velocity conditions.
where Bx and Bx represent simulation result of the forward model and the measurement result, respectively. The closer the SSD is to 0, the higher accuracy of the model is. If the SSD is less than a value MSSD , it can be considered as that the simulation results are ideal. The value MSSD is related to the repeatability of test system. In many times of tests for a certain defect, the measured signals in each test are different, even it seems that the tests are conducted under the same conditions. Compared with each measured signals, the maximum of SSD could be considered as MSSD , after wiping off the abnormal SSDs by using Pauta criterion (the values greater than the triple of standard deviation are wiped off, and Pauta criterion is also called 3S technology). There is usually a difference between MSSD of two test systems. In our test system, MSSD is 5 Gs. Some SSD results, which are compared the simulated values with the measured values of various defect sizes under different velocity conditions, are shown in Fig. 6. It shows that although the SSD results decline with the velocity increasing, they are all less then 5 Gs. Therefore, it is entirely feasible that the 2-D FEM model considering velocity effect is used as the forward model, and the simulated MFL values Bx can be used to reconstruct defect profile in the following defect inversion process.
Fig. 7.
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Inversion algorithm flowchart.
the drawbacks of using this gradient descent for minimization is that the algorithm may converge to a local minimum. An improved gradient descent method called weighting conjugate gradient algorithm has been proposed in this paper. By using this new algorithm, the inversion procedure could escape from local optima and, therefore, find the global optimal solution. A. Conjugate Gradient Algorithm for Defect Inversion 1) Conjugate Gradient Algorithm Procedure: The estimated defect profile is expressed as λ = (λ1 , λ2 , . . . , λm ) with m DOFs. The MFL values Bx(λ) in axial direction are obtained by FEM simulation from the forward model, where there is a defect with the profile λ. By contrast, the real defect profile is expressed as λ. The MFL values Bx(λ) = Bx(λ, 1), Bx(λ, 2), . . . , Bx(λ, p) (a total of p sampling points) are measured by the hall sensor. In the ideal case without signal noise and model error, the following assumption (6) is established: if λ = λ, then Bx(λ) = Bx(λ).
The mean square error ε(λ) of the magnetic field distribution between estimated results and measured results is expressed as p
III. MFL I NVERSION A LGORITHM C ONSIDERING V ELOCITY E FFECT The flowchart of the iterative inversion algorithm for MFL signal as shown in Fig. 7 is implemented using the FEM model as the forward model to reconstruct the defect profile. The algorithm starts with an initial estimate of the defect profile and computes the MFL signal for this profile. And then, the signal is compared with the measured signal. The basic principle of this algorithm is that, if the predicted signal is similar to the measured signal, then the corresponding defect profile is close to the desired defect profile. If the signals are not similar, then the defect profile is updated iteratively to minimize the error. Sometimes, it may be similar that two MFL signals are measured by an external defect and an internal defect with different profiles. In consideration of this issue, a recognition system should be used to identify the inner or outside defect before the inversion process. The key element in the proposed approach is the method how the defect profile is updated. A common method is using conjugate gradient algorithm to minimize the error. One of
(6)
ε(λ) = i=1 [Bx(λ, i ) − Bx(λ, i )]2 .
(7)
The objective of the inversion process is to find an estimated profile λn (the nth inversion profile) to near the real profile, and minimize the object function ε(λn ). In the conjugate gradient algorithm, at each iteration, the input defect profile is changed according to the profile iterative function, as shown in [20] λn = λn−1 − an ∇ε(λn−1 )
(8)
where an represents the nth parameter of the iteration stepsize, and ∇ε(λn−1 ) represents the gradient of the error ε(λn−1 ). 2) Application for Defect Profile Inversion: In order to verify the performance of conjugate gradient algorithm in the MFL testing, we simulate an irregular defect and the defect profile is shown in Fig. 8. Through two simulation tests in the velocity of 0.5 and 5 m/s, the MFL values in axial direction are simulated and described by curves in Fig. 9(a). The number of this defect’s DOFs is 4, and the interval of each DOFs is 10 mm. For the sake of convenience in researching the inversion problems, λ1 and λ4 are supposed
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Fig. 8. Real defect profile, initialized defect profile, and estimative defect profile by conjugate gradient algorithm under different velocities.
Fig. 9. (a) Simulation signals of defect profile. Signal i and signal ii are the simulation MFL values of real defect profile under 0.5 and 5 m/s conditions, respectively. Signal iii and signal iv are the simulation MFL values of estimated defect profile by conjugate gradient algorithm under 0.5 and 5 m/s conditions, respectively. (b) Two curves of iteration process of conjugate gradient algorithm under 0.5 and 5 m/s conditions.
to be 0 during the whole inversion process. Meanwhile, λ2 and λ3 are the targets, which will be updated in each iterative process. We reconstruct the defect profile by using the FEM model considering velocity effect as the forward model and the conjugate gradient algorithm as iterative convergence algorithm. The estimated profiles after inversion are shown in Fig. 8, and the MFL values simulated under different velocity conditions are described by curves in Fig. 9(a). As can be seen from Figs. 8 and 9(a), the estimated profile of the 0.5 m/s test is inversed accurately, but the estimated profile of the 5 m/s test has some errors with the original profile. The mainly reason of the errors produced is that the convergence result is trapped into local optimal solution. With the increasing of the testing velocity, the influences of eddy current on MFL values increase, and the distortion degree of MFL signal waveform increases as well, and also the object function ε of iterative convergence algorithm is likely to exist one or more local minimum. We simulate all MFL values based on the FEM model with the range of λ2 and λ3 from 0 to 10 mm, and calculate all the object function ε (MSE) for researching the optimization problem. As can be seen in Fig. 10(a) and 11(a), when the testing velocity is 0.5 m/s, there is only one optimum in the distribution of convergence domain. So the result of defect profile can be converged to the optimum point by the algorithm directly, and the detailed convergence curves are shown in Figs. 9(b) and 11(a). On the contrary, there is not only one optimum in the distribution of convergence domain under the velocity of 5 m/s condition. There exist one global optimum and two
IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 4, APRIL 2017
Fig. 10. 3-D view of the relationship between MSE and DOFs. (a) v = 0.5 m/s. (b) v = 5 m/s.
Fig. 11. Convergence curves of conjugate gradient algorithm. (a) v = 0.5 m/s. (b) v = 5 m/s.
local optima, as shown in Fig. 10(b) and 11(b). So the result is easily converged to the local optimum point and has error with the real result. The detailed convergence curves are shown in Figs. 9(b) and 11(b). According to a series of simulation, we find that the number of local minima is increased with the number of DOFs and the testing velocity. It results in the optimal solutions of convergent algorithm dropping into local optimum easily. So an improved method should be applied to overcome this shortcoming. B. Weighting Conjugate Gradient Algorithm for Defect Profile Considering Velocity Effect 1) Calculation of Equivalent Signal: The distortion of the measured signal is regarded as a kind of noise, which can be called eddy current noise (ECN) when the testing velocity is faster than v limit . The magnitude and frequency of ECN are closely related to the defect profile and the testing velocity. The ECN can be eliminated by the Fourier transform of MFL signals for a certain defect. The details are as follows. (v) Step 1: Acquire MFL signals Bx (λ, i ) for a certain defect, whose profile is λ, and the testing velocity is v (m/s). Step 2: Establish the model according to the defect profile λ, and simulate MFL signals Bx(0) (λ, i ) and Bx(v) (λ, i ) separately under the velocity of 0 and v (m/s) conditions. (v) Step 3: Calculate the signals Bx (λ, ω), Bx(0) (λ, ω), and (v) Bx (λ, ω) in frequency domain through the Fourier trans(v) form of the signals Bx (λ, i ), Bx(0) (λ, i ), and Bx(v) (λ, i ), respectively.
LU et al.: PRECISE INVERSION FOR THE RECONSTRUCTION OF ARBITRARY DEFECT PROFILES
Fig. 12. Real defect profile, initialized defect profile, and estimative defect profile by weighting conjugate gradient algorithm under different velocities.
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Fig. 16.
Definition of surface errors PSD and PDE.
Fig. 17.
Defect with complex profile.
Fig. 13. (a) Simulation signals of the defect with different profiles. Signal i and signal ii are the simulation MFL values of real defect profile under 0.5 and 5 m/s conditions, respectively. Signal iii and signal iv are the simulation MFL values of estimated defect profile by weighting conjugate gradient algorithm under 0.5 and 5 m/s conditions, respectively. (b) Two curves of iteration process of weighting conjugate gradient algorithm under 0.5 and 5 m/s conditions.
Fig. 14. Convergence curves of weighting conjugate gradient algorithm. (a) v = 0.5 m/s. (b) v = 5 m/s.
Fig. 18. Original measured signals and final inversed signals in different velocities. (a) 0.5 m/s. (b) 1 m/s. (c) 2 m/s. (d) 5 m/s.
Step 5: Calculate equivalent signals Bx ECN by using (9) Bx Fig. 15. Actual platform photograph. (a) Whole graph of the experiment platform. (b) Detail of the detector.
Step 4: Set up the transfer function (9) in frequency domain H(v) (λ, ω) =
Bx(0) (λ, ω) Bx(v) (λ, ω)
.
(9)
(0)
(0)
(λ, ω) without
(v)
(λ, ω) = H(v) (λ, ω) × Bx (λ, ω).
(10) (0)
Step 6: Reconstruct equivalent MFL signals Bx (λ, i ) through the inverse Fourier transform of (10). The above-mentioned transform algorithm depends on the given defect profile. In the inversion process, however, the defect profile is unknown and needs to be solved. The equiv(0) alent signals Bx (λ, i ) are calculated by establishing the model with an estimated defect profile λ. Although they have
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Fig. 19.
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Real profile and estimative profile (13 DOFs) in different velocities. (a) 0.5 m/s. (b) 1 m/s. (c) 2 m/s. (d) 5 m/s. (0)
some errors compared with Bx (λ, i ), they can still reflect the features of the denoising signals. When the estimated defect profile λ is closed to the real profile λ, the features they reflected are more accurate. Therefore, they can be considered as another reference signals for the iterative convergence algorithm. 2) Weighting Conjugate Gradient Algorithm Procedure: The weighting conjugate gradient algorithm proposed in this paper can increase the convergence speed and escape the local optimum. The major improvement of the iterative convergence algorithm is that the original iterative equation (8) is replaced by a new equation (11) to update the defect profile in each iteration λn+1 = λn − an(v) ∇ε(v) (λn ) · (1 − wn ) − an(0)∇ε(0) (λn ) · wn (11) where wn ∈ [0, 1] represents the weight coefficient, which is related to the step-size parameter an , as shown in
⎧ (v) π∇a ⎪ n (v) (v) ⎪cos2 ⎪ , an−1 ≥ an , n > 1 ⎪ (v) ⎪ ⎪ ⎨ 2an−1
(v) wn = (12) π∇an (v) 2 ⎪ , an−1 cos < an(v) , n > 1 ⎪ (v) ⎪ ⎪ 2an ⎪ ⎪ ⎩ 0, n=1 (0) where a(v) n and an represent the nth parameter of the iteration step-size under the velocity of v and 0 m/s conditions, respectively. They can be calculated by p (v) ∂Bx(v) (λn ,i) (v) (v) i=1 [Bx (λn , i ) − Bx (λ, i )] · ∂an a(v) . (13) 2 n = p ∂Bx(v) (λn ,i) i=1 (v) ∂an
The updating function of the defect profile mainly consists of three parts. Compared with (8), the second term, which is the value of gradient variation in the velocity of v, is multiplied by a weighing coefficient. What’s more, another gradient variation value in the velocity of 0 is introduced into the modified function with its weighing coefficient. At each iteration process, the two terms always produce two optimal directions, and they may be the same or not the same. When the solution along an optimal direction nearly traps into the local optimum, the weighing coefficient of this optimal direction will decrease, and the other weighing coefficient will increase. The whole optimal direction will deviate from the direction with a bigger weighing coefficient, so that it can avoid the local optimum solution being found. Because of the particularity of the equivalent signal (0) Bx (λ), when the estimated profile λ is similar to the real profile λ, these two optimal directions can only be the same. Therefore, there always exists an optimal direction to ensure that the value of the object function falls below the previously set threshold and the final result is converged to the global optimum during the optimization process. 3) Application for Defect Profile Inversion: In order to verify the performance of weighting conjugate gradient algorithm in the MFL testing, we reevaluate defect profiles of simulated MFL signals in Section III-A2. The new profiles after inversion by weighting conjugate gradient algorithm are shown in Fig. 12, and the MFL signals simulated by new profiles are described in Fig. 13(a). As can be seen in Figs. 12 and 13(a), the profiles are both inversed accurately in the velocity of 0.5 and 5 m/s. The two convergence curves in Figs. 13(b) and 14 show that the final results are all converged to the globally optimal minimum.
LU et al.: PRECISE INVERSION FOR THE RECONSTRUCTION OF ARBITRARY DEFECT PROFILES
Fig. 20.
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Signal error, PSD, and PDE in the process of each iteration. (a) SSD. (b) PSD. (c) PDE.
The convergence curve of 5 m/s testing in Fig. 14(b) clearly represents the situation of each iterative process. The optimal solution of the fifth iteration is nearly going into one of the local optima, but it jumps out of the local optimum and goes closely to the global optimum after the sixth iteration. This jump action is benefited from the novel algorithm, which involves two gradient directions to optimize. IV. E XPERIMENT R ESULTS For verifying the algorithm performance in practical application, a series of tests on semi-pipe using MFL technique are carried out in different testing velocities. A specially designed test platform and experimental models are employed to produce the real MFL signals. Through processing the signals, the defect profiles can be reconstructed by using the method proposed in this paper. The accuracy of estimated defect profiles reflects the capabilities of the MFL inversion algorithm considering velocity effect. A. Experimental Setup The samples are some semi-pipe specimens, which are made by cutting open the 8 in oil pipelines in half. The material of test specimens is cool rolled X45 steel and its B–H curve is shown in Fig. 1. The thickness of pipes is 9.5 mm, and each semi-pipe is 3 m long with ten artificial surface defects of different types. These defects are at 270 mm intervals on the bottom of the pipes. The structure and material property of the testing detector are shown in Fig. 3(a) and Table I. An array of Hall sensors, eddy current sensors, and signal processing unit is installed between the two poles of the magnetizer. The eddy current sensor is used to identify the inner or outside defect. (If the eddy current sensors and Hall sensors both find the flaws, the defects are identified as inside defects. If the Hall sensors find the flaws only, the defects are identified as outside defects.) The liftoff of the sensors is 2 mm. The signal processing unit is mainly composed of RC filter, voltage amplifying circuit, and A/D converter. The controlling unit is digital signal processor (DSP). The velocity of the detector is within the range of 0.1–5 m/s by the drag of the motor and the conveyor belt in the testing platform. The sensing elements
Fig. 21. Some defects with different complex profiles (length × width × depth). (a) 10 mm × 20 mm × 3 mm. (b) 60 mm × 20 mm × 5 mm. (c) 40 mm × 20 mm × 5 mm. (d) 20 mm × 10 mm × 7 mm. (e) 10 mm × 10 mm × 1 mm. (f) 20 mm × 40 mm × 1 mm.
sampling interval is 2 mm. The actual platform photograph is shown in Fig. 15. B. Error Definitions Due to the nature of the MFL problem, it is useful to define two error measures for profile reconstruction, as shown in Fig. 16, which have different practical importance. In a practical situation, for example, it is important to predict the maximum depth of any defect with high accuracy. It is also highly desirable to have only one small underestimation of the depth profile, because such errors can lead to wrong management decisions regarding maintenance scheduling. Therefore, profile standard deviation (PSD) and peak depth error (PDE) are adopted to describe the depth error types, and they are defined as m 1 (λi − λi )2 (14) PSD(λ) = m i=1
PDE = |λmax − λmax |
(15)
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Fig. 22.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 4, APRIL 2017
Signals error, PSD, and PDE in the process of each iteration. (a) SSD. (b) PSD. (c) PDE.
where λmax and λmax are the maximum values of the reconstructed and referenced profile, respectively.
accurately from the distortion signals by using the method proposed in this paper.
C. Inversion Results and Discussion
D. Comparison and Discussion
To examine the accuracy of the proposed method, we test a defect with complex profile on the semi-pipe along its centerline, as shown in Fig. 17. The thickness of the pipe is 9.5 mm. The maximum and minimum width of the defect are 8.9 and 6.6 mm along the direction perpendicular to the testing path. The influence on the MFL signals caused by the variation of width can be ignored, because the width variation is not obvious. The experiments are carried out by four times in different velocities (0.5, 1, 2, and 5 m/s). According to the characteristics of measured MFL signals and the requirement of inversive accuracy, the number and the interval of DOFs are, respectively, set to 13 and 2 mm. The initial defect profile is chosen as the profile of a flat (λ = 0), and then, the estimative defect profile is reconstructed by the proposed inversion algorithm under four kinds of velocity conditions. The original measured signals and the final inversed signals are described in Fig. 18, and the real, initial, and final defect profiles are shown in Fig. 19. The profile curve is the profile of the defect along the testing path, as shown by dotted line in Fig. 17. If the testing path changes, the measured signal and the inversion profile will change as well. Although measured signals of different testing paths are all dissimilar, the defect profiles can still be estimated accurately and these profiles curves are the profiles of the defect along the corresponding testing paths. As the results in Fig. 19 clearly illustrate, the defect profiles are acquired accurately by inversing the MFL signals in different velocities. Through summarizing the SSD, PSD, and PDE of each iterative process, the results of different velocities are described by curves in Fig. 20. It shows that the iterations are increased and the error decrements are decreased with the testing velocity increasing. Furthermore, the SSD, the PSD, and PDE of the estimated profile fall below 5 Gs, 0.3 mm, and 0.5 mm, respectively, after multiple iterations (maximum of 12). So the estimated profile of the last update is similar to the profile of the real defect. Through the above-mentioned experiments, it proves that the defect profile can be estimated
In order to demonstrate the algorithm’s performance for various defects, 50 artificial defects with different complex profiles in five semi-pipes are detected in four kinds of velocity (0.5, 1, 2, and 5 m/s). The pictures of some defects are shown in Fig. 21. We reconstruct the profile from these 200 groups of MFL signals by using four methods as followed. Method A: the forward model is the 2-D finite element model without considering velocity effect by using the ANSYS software, and the profile iteration method is conjugate gradient algorithm. Method B: first, pre-process the original measured signals by using the invariance transformation method in [18], and then use Method A to reconstruct the profile. Method C: the forward model is the 2-D finite-element model considering velocity effect, and the profile iteration method is the same as Method A. Method D: reconstruct the profile by using the algorithm proposed in this paper. The averages of the inversion results, including the iterations, PSD, and PDE of the 50 defects, are plotted as histograms in Fig. 22. Some conclusions are summarized as follows: 1) when the testing velocity is 0.5 m/s and the velocity effect can be ignored, the iterations and the inversion accuracy of the four methods are the same; 2) with the testing velocity increases, the inversion accuracy of Method A is decreased, because the velocity effect is not considered in the forward model; 3) the inversion accuracy by using the algorithm proposed in this paper (Method D) is better than it by using Method B, because the invariance transformation method applied in Method B destroys the original signals partly and has error compensation; and 4) compared Method D with Method C, the number of iterations is less and the inversion accuracy is higher. Because the weighting conjugate gradient algorithm used in Method D exhibits better performance of higher global convergence and speed of optimization. V. C ONCLUSION This paper presents a novel inversion approach for estimating the profile of an arbitrary defect from MFL signals
LU et al.: PRECISE INVERSION FOR THE RECONSTRUCTION OF ARBITRARY DEFECT PROFILES
under various velocity conditions. In this method, first, an FEM based on ANSYS considering velocity effect is applied as the forward model to simulate comparison MFL signals. This reduces the errors between the measured MFL values by Hall sensors and the FEM simulated values, and improves the signal similarities. The weighting conjugate gradient algorithm is then used to estimate the defect profile iteratively. The proposed method is first tested and verified for a defect (four DOFs) with simulated MFL signals under two different velocity conditions. The results demonstrate that this inversion technique can estimate the defect profile more accurately than traditional conjugate gradient algorithm not only in low velocity but also in high velocity. To examine the performance of the proposed method in practical examples, we have applied it to reconstruction of an artificial complex-shaped defect (13 DOFs) whose MFL signals are measured under four different velocity conditions. The results demonstrate that it can reconstruct the defect profile accurately in the velocity from 0.5 to 5 m/s, and the precision of defect profiles reaches to 0.18 and 0.28 mm for PSD and PDE at most and 0.30 and 0.52 mm at least. Furthermore, we test 50 artificial defects with different complex profiles in four kinds of velocity, and reconstruct these defects by using four different methods. The results demonstrate that the FEM considering velocity effect applying as the forward model can improve 47.2% and 71.4% for PSD and PDE, and the weighting conjugate gradient algorithm used to estimate the defect profile can improve 6.5% and 29.7% for PSD and PDE in the velocity of 5 m/s. From the above-mentioned simulations and experiments, if the variation of the defect width is less than 5 mm [e.g., the defects shown in Figs. 17 and 21(a)–(f)], the defect profiles can still be estimated accurately. If the variation of the defect width is more than 5 mm [e.g., the defect shown in Fig. 21(d)], the variation of MFL signals will be from 5 to 30 Gs, and the estimated accuracy will decline nonlinearity with the variation of the defect width increasing. This problem can be settled by establishing a 3-D FEM model, and it is the direction of our future research. Besides, the usage of the FEM model through the implementation of commercial software ANSYS is time-consuming. A novel method needs to be developed to reduce the simulation time and ensures the simulating precision. Efforts in that direction are currently underway. ACKNOWLEDGMENT This work was supported in part by the National Natural Science Foundation of China under Grant 61673093, Grant 61273164, Grant 61627809, Grant 61621004, Grant 61433004, and Grant 61473069, and in part by the Fundamental Research Funds for the Central Universities of China under Grant N130104001. R EFERENCES [1] S. M. Dutta, F. H. Ghorbel, and R. K. Stanley, “Dipole modeling of magnetic flux leakage,” IEEE Trans. Magn., vol. 45, no. 4, pp. 1959–1965, Apr. 2009. [2] M. Afzal and S. Udpa, “Advanced signal processing of magnetic flux leakage data obtained from seamless gas pipeline,” NDT E Int., vol. 35, no. 7, pp. 449–457, 2002.
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[3] M. R. Kandroodi, F. Shirani, B. N. Araabi, M. N. Ahmadabadi, and M. M. Bassiri, “Defect detection and width estimation in natural gas pipelines using MFL signals,” in Proc. 9th Asian Control Conf. (ASCC), Jun. 2013, pp. 1–6. [4] H. M. Kim and G. S. Park, “A study on the estimation of the shapes of axially oriented cracks in CMFL type NDT system,” IEEE Trans. Magn., vol. 50, no. 2, pp. 109–112, Feb. 2014. [5] D. Mukherjee, S. Saha, and S. Mukhopadhyay, “Inverse mapping of magnetic flux leakage signal for defect characterization,” NDT E Int., vol. 54, pp. 198–208, Mar. 2013. [6] M. Ravan, R. K. Amineh, S. Koziel, N. K. Nikolova, and J. P. Reilly, “Sizing of 3-D arbitrary defects using magnetic flux leakage measurements,” IEEE Trans. Magn., vol. 46, no. 4, pp. 1024–1033, Apr. 2010. [7] R. H. Priewald, C. Magele, P. D. Ledger, N. R. Pearson, and J. S. D. Mason, “Fast magnetic flux leakage signal inversion for the reconstruction of arbitrary defect profiles in steel using finite elements,” IEEE Trans. Magn., vol. 49, no. 1, pp. 506–516, Jan. 2013. [8] K. C. Hari, M. Nabi, and S. V. Kulkarni, “Improved FEM model for defect-shape construction from MFL signal by using genetic algorithm,” IET Sci., Meas., Technol., vol. 1, no. 4, pp. 196–200, Jul. 2007. [9] R. K. Amineh, S. Koziel, N. K. Nikolova, J. W. Bandler, and J. P. Reilly, “A space mapping methodology for defect characterization from magnetic flux leakage measurements,” IEEE Trans. Magn., vol. 44, no. 8, pp. 2058–2065, Aug. 2008. [10] W. Han et al., “Cuckoo search and particle filter-based inversing approach to estimating defects via magnetic flux leakage signals,” IEEE Trans. Magn., vol. 52, no. 4, pp. 1–11, Apr. 2016. [11] S. Mandayam, L. Udpa, S. S. Udpa, and W. Lord, “Signal processing for in-line inspection of gas transmission pipelines,” Res. Nondestruct. Eval., vol. 8, no. 4, pp. 233–247, 1996. [12] Y. Li, G. Y. Tian, and S. Ward, “Numerical simulation on magnetic flux leakage evaluation at high speed,” NDT E Int., vol. 39, no. 5, pp. 367–373, 2006. [13] P. Wang, Y. Gao, G. Tian, and H. Wang, “Velocity effect analysis of dynamic magnetization in high speed magnetic flux leakage inspection,” NDT E Int., vol. 64, pp. 7–12, Jan. 2014. [14] Z. Chen, J. Xuan, P. Wang, H. Wang, and G. Tian, “Simulation on high speed rail magnetic flux leakage inspection,” in Proc. IEEE Instrum. Meas. Technol. Conf. (I2MTC), May 2011, pp. 1–5. [15] Z. Gan and X. Chai, “Numerical simulation on magnetic flux leakage testing of the steel cable at different speed title,” in Proc. Int. Conf. Electron. Optoelectron. (ICEOE), vol. 3. Jul. 2011, pp. V3-316–V3-319. [16] L. Zhang, F. Belblidia, I. Cameron, J. Sienz, M. Boat, and N. Pearson, “Influence of specimen velocity on the leakage signal in magnetic flux leakage type nondestructive testing,” J. Nondestruct. Eval., vol. 34, no. 2, pp. 1–8, 2015. [17] L. Lei, C. Wang, F. Ji, and Q. Wang, “RBF-based compensation of velocity effects on MFL signals,” Insight, vol. 51, pp. 508–511, Sep. 2009. [18] S. Mandayam, L. Udpa, S. S. Udpa, and W. Lord, “Invariance transformations for magnetic flux leakage signals,” IEEE Trans. Magn., vol. 32, no. 3, pp. 1577–1580, May 1996. [19] G. S. Park and S. H. Park, “Analysis of the velocity-induced eddy current in MFL type NDT,” IEEE Trans. Magn., vol. 40, no. 2, pp. 663–666, Mar. 2004. [20] J. Chen, S. Huang, and W. Zhao, “Three-dimensional defect inversion from magnetic flux leakage signals using iterative neural network,” IET Sci. Meas. Technol., vol. 9, pp. 418–426, Jul. 2015. [21] R. V. Sabariego, J. Gyselinck, C. Geuzaine, P. Dular, and W. Legros, “Application of the fast multipole method to hybrid finite element–boundary element models,” J. Comput. Appl. Math., vol. 168, nos. 1–2, pp. 403–412, 2004.
Senxiang Lu was born in Shenyang, China, in 1988. He received the B.S. degree in measurement and control technology and instrument and the M.S. degree in electrical engineering from Northeastern University, Shenyang, China, in 2011 and 2013, respectively. He is currently pursuing the Ph.D. degree with the School of Information Science and Engineering, Northeastern University. His current research interests include nondestructive testing and its application.
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Jian Feng received the B.S., M.S., and Ph.D. degrees in control theory and control engineering from the Northeastern University of China, Shenyang, China, in 1993, 1996, and 2005, respectively. He is currently a Professor, a Doctoral Supervisor, and the Vice Head of the Electric Automation Institute, Northeastern University. His current research interests include fault diagnosis, signal processing, neural networks, and their industrial application. Dr. Feng was awarded the “New Century Excellent Talents in University,” nominated by Ministry of Education China.
Fangming Li received the B.S. degree in automation control from Northeastern University, Shenyang, China, in 2012, where he is currently pursuing the Ph.D. degree. His current research interests include nondestructive testing, magnetic flux leakage inspection, signal processing, and convolutional neural network.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 4, APRIL 2017
Jinhai Liu received the B.S. degree in automation from the Harbin Institute of Technology, Harbin, China, in 2002, and the M.S. degree in power electronics and power transmission and the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2005 and 2009, respectively. He is currently an Associate Professor and a Doctoral Supervisor with Northeastern University. His current research interests include data driven, fault diagnosis, neural networks, and safety technology of long pipelines.
