1 Introduction
In nuclear power plants (NPPs), due to flow accelerated corrosion (FAC) and liquid droplet impingement (LDI) of the coolant inside the pipe, there may happen local wall thinning defects on the inner surface of a pipe [1]. Concerning the material of pipes, there are mainly two types, low-carbon steel and stainless steel where the stainless steel pipes are targeted in this study. To guarantee the safety of NPPs, periodical nondestructive testing (NDT) to the pipes is mandatory. Another important issue is to check whether the wall thinning exceeds the safe tolerance range. This can not only guarantee safety but also save on unnecessary renewal of pipes. This requires a so-called quantitative NDT (QNDT) method. However, the QNDT of pipe wall thinning in NPPs has so far not been well solved.Pulsed eddy current testing (PECT) method, due to its rich frequency components and applicability of large electric current, shows outstanding features especially for the detection of defect in deep region [2]. Therefore, PECT is considered as a powerful candidate for the QNDT of wall thinning defect in NPPs.PECT is a typical transient electromagnetic field problem whose efficient forward simulation tool still has not been well solved [3]. For probe optimization and especially for the inverse analysis of PECT problems, a fast and efficient numerical solver is highly required and is crucial to the calculation of numerous forward problems [4]. The huge computational resources needed for these forward simulations are sometimes beyond the capacity of an average computer. Therefore, an efficient and high accuracy simulator for the forward signal prediction of PECT is indispensable for the further development of PECT technology. On the other hand, as introduced above, the defectLast: mik TT T 980-8577 ML AITXA 2-1-1, KF T. E-mail: takagi@ifs.tohoku.ac.jpprofile characterization of pipe wall thinning in NPPs from PECT signals has so far not been well solved, thus the inverse analysis of PECT also needs much effort. Inversion analysis scheme reconstructs the size of a defect from the measured signal, which is the reverse method to the forward analysis [5] where the development of the efficient forward simulation tool of PECT could give a good basis for the quantitative wall thinning analysis based on the inversion techniques.Based on the backgrounds described above, the objective of the present study is to discuss the feasibility of QNDT to pipe wall thinning using PECT method. The works are mainly on the following two aspects: a) to develop an efficient forward numerical tool for the PECT signal prediction; b) to develop an inversion algorithm for PECT method and apply it for the pipe wall thinning reconstruction.The contents of this paper are arranged as follows: first, an efficient forward numerical simulator for the PECT signal calculation due to volumetric defect is proposed and developed by introducing the database type fast eddy current testing (ECT) simulation scheme to the Fourier-series-based PECT signal simulation method with help of an interpolation strategy. Then, an inverse analysis scheme of PECT is proposed for the sizing of pipe wall thinning in NPPs, based on a deterministic optimization strategy and the developed efficient forward PECT signal simulator. Finally, the reconstruction examples of wall thinning from PECT signals are given to validate the feasibility of PECT for wall thinning QNDT.2 Efficient forward simulator for PECT signalsIn this work, an efficient forward numerical solver for simulation of the PECT signals was developed based on the database approach and the Fourier series method with help of interpolation strategy. At first, the Fourier series method with interpolation strategy was described for the PECT signals278prediction. Second, the fast numerical solver of database approach was upgraded in order to apply it to the ECT problem of volumetric local wall thinning defect. Finally, based on the Fourier series method and the fast simulation scheme for single frequency excitation, an efficient numerical solver was developed and validated for the simulation of PECT signals due to a local wall thinning. Through comparing the numerical results using the present fast solver and a conventional method (i.e. only using Fourier series method for PECT signal calculation), it was verified that the fast solver can predict the PECT signals accurately and over 100 times faster than the conventional one.2.1 Fourier series methodWhen applying PECT, the excitation current I(t) is usually introduced as repetitive pulses of square wave which can be considered as a summation of serial harmonic sinusoidal waves represented by Eq. (1) according to the Fourier transformation theory [6],1(1)=?Fueline-1where ir=0 means the DC (direct current) component, c, is the angular frequency of sinusoidal excitation, F, is the amplitude coefficient and j is the imaginary unit. Since the PECT problem can be considered as a low fiequency linear electro-magnetic problem, its governing equations can be written as follows after finite element method (FEM) discretization,[K]{A}+[C]{A}={M}I(1),otwhere A is the vector potential, [K], [C] and {M} are coefficient matrices of the FEM equations. Because of the linear property of Eq. (2), the response signal due to pulsed excitation in the form of Eq. (1) is also composed of sinusoidal waves of frequencies appearing in the driving current. After formulae deduction, the response magnetic flux density B can be obtained by summating the response signals of each frequency B,o as shown in Eq. (3) (the DC component is neglected as it does not induce eddy current). When each B,0 has been calculated, the field signal B(1) can be obtained by using Eq. (3) simply.{B(t)} =F.(vx{?no}} elint = ?F{Boo eling)n=12.2 Interpolation strategyTo simulate PECT signals based on Eq. (3), it is required to know the response signals of the necessary single frequency sinusoidal excitation, which are calculated using a sinusoidal ECT simulation code. If calculating all the signal components necessary in the summation of Eq. (3) using the full FEM code, the huge computational burden will make this method inapplicable.As the frequency response curve of single frequency ECT problem is smooth, the amplitude of the response signal of a given harmonic frequency can be calculated from the signals of the selected typical frequencies through interpolation. This ideacan shorten the simulation time of single frequency ECT problems. The total number of frequencies for signal summation and the number of selected frequencies for interpolation are very important to guarantee the precision of the Fourier series simulation method, which were carefully discussed in reference [7]2.3 Fast solver of ECT signal simulation for volumetric defectsA fast solver of ECT signal simulation due to non-volumetric defects (i.e. cracks like) has been developed by authors [8, 9]. As the wall thinning is volumetric defect of 3D geometry, the fast solver has to be upgraded in order to be applied to the ECT signal simulation of wall thinning defect. The major difference between crack and wall thinning problem is the dimension of the databases of the unflawed potentials and the way to establish the inverse matrix. The theory of the fast scheme is as follows:Through subtracting the governing equations with and without defect and conducting Galerkin FEM discretization, the following system of linear equations can be obtained, [K, K2 TA LE, OSA +AL LK21 Km 10 01 A + A ]-5where subscripts 1 and 2 denote the areas at the defect and at the other area, while superscripts f and 0 denote the potentials perturbation due to flaw existence and that of the unflawed material. [K] is the unflawed global coefficient matrix of FEM equations.From Eq. (4), we can obtain a smaller system of linear equations for solving the potential perturbation {A}},[1-1,E, ]{4{}=[H,][E,]{A}, where [H] is the inverse matrix of [K].Since the coefficient matrix [H] and the unflawed potentials {Ao are independent of the flaw geometry, they can be calculated a priori and stored as databases. In this way, calculation burden of {Af} can be greatly reduced because the number of the nodes related to defect is always much smaller than that of the whole system. In present work, the database region was extended and the 2D shifting symmetry scheme was proposed to establish databases of inverse matrix (H) and the unflawed field {A““} for treating the volumetric wall thinning defect, and the corresponding numerical code is developed.2.4 Fast solver of pulsed ECT signal simulation for volumetric defectsBased on the Fourier series method and the interpolation strategy and the fast scheme for ECT signal prediction due to volumetric defect, a fast numerical solver for simulation of PECT signals due to local wall thinning defect is developed. The basic procedure of the code is as follows: initially, the sinusoidal ECT response signals of selected frequencies from local wall thinning are calculated using the fast ECT solver described in Section 2.3. Then, the response signal from the pulsed excitation (PECT signal) can be calculated following the Fourier series method. In this way, the transient PECT signals can be obtained within a very short time.279To validate the efficient PECT simulator, an example is investigated where the inspection target of the numerical model is a plate of austenitic stainless steel 316 of length (Y direction) 120 mm, width (X direction) 120 mm, and thickness (Z direction) 10 mm. The defect is set as a cuboid OD wall thinning 12 mm in length, 33 mm in width and 4 mm in depth. The parameters of the excitation coil are: inner diameter, 5 mm; outer diameter, 10 mm; height, 5 mm and total number of turns, 296. The Lift-offs of both the excitation and pickup elements are 0.5 mm. A square wave pulse is applied to the excitation coil as the driving current whose DC component is 1.0 A, the period is 0.01s and the duty cycle is 50%. Figure 1 shows the schematic of the model. The probe is scanned along the length direction of the defect. The magnetic flux density in the vertical direction at the position of the bottom center of the excitation coil is selected as the pickup signal.Figure 2 shows the numerical comparison results of pulsed signals between the conventional method (i.e. Fourier series method combined with full FEM approach for harmonic sinusoidal ECT signal prediction) and present fast solver (i.e. Fourier series method combined with the database type fast ECT solver for harmonic sinusoidal ECT signal prediction) for the given example, when the probe is just above the center of the wall thinning defect. Figure 3 shows the numerical comparison results of peak values in pulsed signals between the conventional method and the fast solver, when the probe is scanning along the length direction shown in Fig. 1. In addition, in the Fourier series method, the first 450 harmonic components are used for summation and 30 selected harmonic components are used for interpolation. The relative error (difference) of the PECT signal in Fig. 3 between the conventional method and the fast solver is about 2.1%.Though all of the above results show very good agreement, the conventional method takes about 9 hours to obtain the results in Fig. 3, while the fast solver needs only about 4 minutes, which is more than 100 times faster. The computation is carried out on a personal computer, consisting of an Intel (R) Core (TM) i7 CPU 960 @ 3.2 GHz, Memory 6 GB, OS Fedora 12 and Intel Fortran Compiler 10.1.Concerning establishing the databases, in this case it takes about 43 hours to build the databases for the unflawed fields and the inverse matrix. Although this time is rather long, it is only necessary to be established once for inverse analysis of the PECT signals of a given inspection target. Once the database is well established, each forward analysis only takes several minutes compared with the almost intolerable several hours for the conventional method. Thus the proposed fast solver for signal prediction of PECT gives a good basis for the inversion problem in PECT technology.Measuring pointExciting coilScanning direction30mm Starting pointDepth Origin Wall thinning?.Figure 1. The schematic model for defect detectionPulse signal (Fast solver) --- Pulse signal (Conventional)Pulse signal, Bz (G)Peak value0.0050 0.0052 0.0054 0.0056Time (s) Figure 2. Pulsed signal when probe is above the center of the wall thinning, for comparison of fast solver and conventional method simulation of wall thinning in the considered example under pulsed excitationPeak value (Fast solver) Peak value (Conventional)Peak value in pulse signal, Bz (G)-30 -20 -10 0 10 20 30Scanning position, y direction (mm) Figure 3. Peak value of pulsed signal when probe is scanning along the length direction, for comparison of fast solver and conventional method simulation of wall thinning in the considered example under pulsed excitation3 Inversion algorithm of PECT for sizing of wall thinningIn this work, based on the above developed efficient forward PECT signal simulator, a deterministic optimization strategy based reconstruction scheme with the help of Conjugate Gradient (CG) method is proposed and developed to deal with the sizing of local wall thinning in pipes of NPPs.3.1 Model of wall thinning for reconstructionAs shown in Fig. 4, local wall thinning is modelled as a group of planar slit defects (rows) of given width but of differing length (for example, in the w-th row, the length of wall thinning equals bu-aw) and depth (in the w-th row, the maximum depth of wall thinning to the total thickness of specimen is dw (%)). These are selected as the defect shape parameters to be reconstructed. As these parameters have to be simultaneously reconstructed, two-dimensional signals (probe scans along the length and width direction in Fig. 4) scanned over the wall thinning on the far side are used for the reconstruction. In cases where the defect length and depth are close to zero in some rows after reconstruction, then these rows will be treated as an unflawed region. In this way, the width information of the wall thinning can also be properly reconstructed [10].280Z: ThicknessSuspect regionY: LengthX: Width? Wall thinningSuspect regional d1(%Row 1d21%)Row 2Width directiondw%)bwRow wLength directionFigure 4. Reconstruction strategy schematic for 3D profile of wall thinning in suspect region3.2 Principle of inversion algorithmAn inverse analysis method, based on the deterministic optimization algorithm, is used to reconstruct the profile of the 3D wall thinning. This means that the sizing process is converted to an optimization problem of minimizing the objective function,66c““) = ??(P.(co) ? Polis) E? ()““, m1=1 =1I=m=1where k is the iteration step, I and m represent the position of 2D scanning point, c* the shape parameter vector of wall thinning after k-th iteration and E(c““) the objective function (residual error). P.?(c) is the feature parameter extracted from the PECT signal B(t) at (I, m) scanning point for defect shape c* and Pro is the corresponding feature parameter extracted from the measured signal. In this study, peak value is employed for the feature parameter of the PECT signal, as shown in Eq. (7), where to represents peak time.P(r)= B(1,1)I=10The conjugate gradient (CG) based reconstruction scheme is used to predict the length and depth of the wall thinning in each row. In this study, Eq. (8) is adopted for the direct calculation of the gradient vector from the calculated electric fields for the transmitter-receiver type PECT probe [11].OB(r,t)_OCwiO?E:(055-8' + E!) Osa(0,1““) des bonnesaw)In Eq. (8), or is the conductivity of the specimen in unflawed region, E, is the unflawed electric field generated by unit current in the pickup coil, E““ +E is the flawed electric field generated by the excitation coil (including unflawed and perturbed field), a is a coefficient to correspond the output of pickup coil to the PECT signal, Cywi is the i-th wall thinning shape parameter at the w-th planar row in width direction and s c,r) = 0 is the equation of the defect boundary surface in w-th row. As for the PECT signal, the feature parameter ?P(r) / Ocvi in the signal of the gradient vector is extracted from Eq. (9), where to (peak time obtained in Eq. (7)), as shown in Eq. (9).?P(r) _OB(r,1)OCw, OC, L With the CG based optimization method, the shape parameter vector of wall thinning c* can be obtained through the following iteration procedure, ck = ck-1 +ak x(Sc)““,-101towhere (c)* is the updating direction in the k-th iteration, which is chosen as the direction of the conjugate gradient vector in case of the CG algorithm. This can be obtained using Eq. (11), according to the definition of the CG algorithm.(8cw;)* =(8cm;)*+1 (11)+(elcwi)*a' is a step size parameter selected as the value which can reduce e(ck) most efficiently. After formulation it can be obtained as shown in Eqs. (12) and (13).2 k-1 _ pobs xP, k-11,2““1,m^““ - pobs1,7m)XI=m=1,(12)k21=l m=1-131 LOP,*ocmir OP,*-( de ) in domo dar - Mit demi loomi)Based on the above algorithm, the flowchart of the wall thinning reconstruction procedure is shown in Fig. 5. During2 (0:1)““}er.. an,(12)281iteration when the residual error is small enough, the computational feature extracted from PECT signals will be very close to the objective feature extracted from the measured PECT signals, which means that the iterative wall thinning shape parameters are near to the real ones. Therefore, the stop conditions are that the residual error is smaller than a certain value E, or that the number of iterations is larger than a set value N. In the flowchart, the forward simulation part needs to be repeated many times to calculate the PECT signals under different wall thinning parameters. Therefore, the efficient forward simulator is employed here and can significantly reduce the computational burden. According to the flowchart, an inverse analysis code has been developed for the reconstruction of wall thinning from the measured PECT signals.StartInitial shape of wall thinningForward calculation of PECT signal using databasesCalculate residual error ?()=Calculate residual error ele“) = E?(.(eo) ? PIN) E?(R)I=1IYesResidual error < E or Number of iteration>NStop & OutputI No Calculate (8c)”, ak by CG algorithmUpdate defect shapeFigure 5. Program flow chart of inverse problem3.3 Reconstructed examplesThe inversion algorithm preceding and the corresponding developed inverse analysis code are validated by 3D wall thinning reconstruction from the simulated PECT signals. A block of austenitic stainless steel 316, with conductivity of 1.35 X 10S/m, is employed to simulate the big diameter pipe in NPPs as the host conductor, with length 120 mm, width 120 mm, and thickness 10 mm. Wall thinning is located in the bottom side of the specimen. To establish the database for the fast forward solver, the selected possible wall thinning region (search region) is taken as 30 mm in length, 33 mm in width and 10 mm in depth, and subdivided into 550 (10 X 11 X 5) wall thinning cells.A pancake excitation coil (inner diameter, 5 mm; outer diameter, 10 mm; height, 5 mm; and total number of turns, 296) is applied as the inspection probe. The magnetic flux density in the vertical direction at the position of the bottom center of the excitation coil is selected as the pickup PECT signal. The peak value is extracted from the transient PECT signals as the feature parameter for the wall thinning reconstruction. This is because peak value is more stable in the experimental case (that is, with noisy signals) than the other typical features in PECT signals (peak time, zero crossing time, rising point etc.) [2]. Lift-off is 0.5mm. A square wave pulse is applied to the excitation coil as the driving current whose DC component is 1.0 A, the period is 0.01s and the duty cycle is 50%. The probe is scanned over the possible wall thinning region part along the length and width direction. Figure 6 shows the schematic of the model.The computation is carried out on a personal computer, consisting of an Intel (R) Core(TM) i7 CPU 960 @ 3.2 GHz, Memory 6 GB, OS Fedora 12 and Intel Fortran Compiler 10.1.Excitation coilMeasuring point60 MHZ30 mmScanningiZ: thicknessOriginall thinninY: length2D scarming region for probeX; widthFigure 6. The schematic model for wall thinning detectionAccording to the wall thinning model introduced in Section 3.1, there are 3 parameters in every row which are: (i) coordinate of left edge of wall thinning in length direction aw, (ii) coordinate of right edge of wall thinning in length direction bw and (iii) depth dw. In our model, the search region is 33 mm in width, subdivided into 11 cells, so the range of w is from 1 to 11. Therefore, originally, there are 33 independent parameters to indicate the shape of 3D wall thinning. However, as general knowledge and our experience with the CG method grows, more parameters will lead to more local mininia and the searching difficulty will increase dramatically. However, some artificial restrictions can be applied to the 33 parameters to decrease the degrees of freedom and express a certain 3D wall thinning shape.In this section, a half ellipse-column wall thinning model is constructed and the reconstruction effect is investigated. Three parameters: aw, bw and dw are used to represent a half ellipse shape in the w-th row, where the half long axis of the ellipse is (bw-a)/2 and the half minor axis of the ellipse is dw. Two more parameters of wl and w2 (1