Electromagnetic NonDestructive Evaluation II,
Proceedings of the 3rd International Workshop on E'NDE,
Reggio Calabria, Italy, September 1997.
ISBN: 90 5199 375 7  published by
IOS Press
TABLE OF CONTENTS 
In order to evaluate crack profiles from testing signals. it is necessary to solve an inverse problem of physical phenomenon under the test, and it is difficult to obtain a unique solution. Studies of crack reconstruction are being performed based on crack modeling, using the Green function[3,4]. The gradient of errors between experiment and theory is used to find the optimal shape until the errors are minimized. Later, for the application of this approach to arbitrary geometries, finite elements play the part for an integral formulation in halfspace[5]. The algorithm of the optimization method used has an advantage that it is straightforward, while its convergence is considered slow. Instead of the above iterative procedure, a method for finding crack shape is proposed with the trust region method[6] for parameter estimation, together with the nodal finite elementboundary element hybrid method[7]. However this requires the use of a large amount of computational memory and time.
In this study, a method for the estimating algorithm of the crack depths is developed by using the reduced magnetic vector potential method with edge based finite elements and the trust region method. The reduced magnetic vector potential method allow us to avoid the recomputation to form an ECT model with the moving coil. Again edge based finite elements do not require the use of a large amount of the computational memory. This paper describes simulated ECT with respect to the steam generator tubes in nuclear plants. Four different crack profiles are characterized by the developed estimator from laboratory data, and the validity is demonstrated by comparing between estimated and true profiles.
A. Eddy current signal prediction
For simplification of the modeling task for the moving coil, the reduced A method[2] (referred to as the A_{r} method) is used. According to this method, the whole problem domain to be dealt with is divided into two parts as illustrated in Fig. 1. W_{t} includes conductive or ferromagnetic material with the standard magnetic potential A. W_{r} is the remainder of the modeled domain with the reduced magnetic vector potential Ar which differs from A when eddy currents or magnetization exists in the material in W_{t}. The governing equations are then
The use of edge based finite elements makes it possible that f is set to be zero. The boundary conditions on magnetic flux density and field at the interface between two regions require that the tangential component of the magnetic vector potential be continuous if the material in t has finite conductivity. As a result, an inconsistency arises in the boundary Gtr between the regions W_{t} and W_{r}. For conjunction of these two regions, the continuity of the flux density and field is represented by
where, n is a unit normal vector on G_{tr} and A_{s} and H_{s} are the magnetic vector potential and magnetic field due to a current source located in W_{r} or outside the whole domain. Equations (1) and (2) are coupled by these conditions. A_{s} and H_{s} can be computed by the Biot  Sarvart's law independent of the finite elements.
When a sinusoidal current of frequency w is passed through the coil yielding a current density of j_{s}, the magnetic flux penetrating the coil is given by
(5)
where, the variables are now complex amplitudes. The coil impedance is obtained from
Z = V / I =J w F / I (6)
This impedance does not include wire resistance and capacitance.
B. Parameter Estimate
The cracks evaluated here are modeled as nonconductivity regions in conductors which have uniform finite thickness. The lengths of the cracks and the planes in which they lie are considered known since these can be easily determined from signal map. In order to represent the crack profiles, it is straightforward to use spline functions as shown in Fig. 2[7].
(7)
where, q_{i} and depict and a bilinear spline function and its coefficient; the resolution of crack profiles depends on how high order is chosen. When these parameters characterizing the crack depth profiles are defined in Fig. 2, coefficient matrix of an algebraic equation by the FEM implicitly has the information on the parameters as follows,
[A(q)], {x^{i}} = {b^{i}}[A], (8)
where, [A(q)], {b^{i}} and {x^{i}} depict the coefficient matrix, a righthand vector and a solution vector, respectively. These superscripts express a measuring position of coil impedance as the sensor coil moves above a test specimen. Using solutions of (7), the impedance associated with the detecting signal is given as
Y^{i} = [C]{x^{i}} , (9)
where Y^{i} is the predicted impedance value and [C] is a matrix to obtain Y^{i} from {x^{i}} . The crack signals are treated as the impedance only due to disturbance of the eddy current by the presence of the crack, the signal is obtained from
DY^{i} = Y^{i}_{crack}  Y^{i}_{no} , (10)
where Y^{i}_{crack} is the impedance which includes the crack information Y^{i}_{crack} and also is a function of parameters. On the other hand, Y^{i}_{no} can be obtained by solving the equations (8) and (9) of a crackfree model as well as . In order to predict the depth parameters matching realistic measured data, the least square function J defined below[7] is formulated as
(11)
The optimal profiles are obtained by minimizing the above equation. The trust region method[6] is used because this is a nonlinear equation.
Fig. 3 A coil and a test specimen for simulated ECT 
Fig.4 Geometry and dimensions of cracks 
Table I Configuration of the coil and the test piece  
Coil  Height: 0.8 mm, Width: 1.0 mm Inner diameter: 1.2 mm Outer diameter: 3.2 mm Prescribed current: 1/140 A Frequency: 150 kHz, 300 kHz 
Test piece  Size: 140x140x1.25mmt Conductivity: 1.0x10^{6} S/m Relative permeability: 1 
EDM crack  Length: 10 mm, Opening: 0.22 mm (elliptical) Depth: 60% (maximum) 
Let us define the origin be the center of each crack and the coil path is along the x axis above the crack; the crack edges are 5 mm and 5 mm. By symmetry, a halfmodel of the problem was constructed using the finite elements shown in Fig. 5. Table II summarizes the discrete data and the information about the ICCG method. The sampling points of the impedance signals as the coil moves parallel to the cracks are set to 9 from y= 8 mm to 8 mm, and six depth parameters are located from 5 mm to 5 mm at regular interval, respectively. The admissible parameters are taken as 0.1 mm 1.15 mm. For the implementation of the trust region method, a FORTRAN package " OPT2"[9] is used.
Table II Discrete data and computational costs  
Elements  3,380 
Nodes  4158 
Edges  11,645 
Unknowns  8,733 
Memory (MB)  19 
Calculating precession: double Sampling points in Gauss  Legendre quadrature in an element: 27(=3^{3}) points Convergence criterion in CG method : 10^{5} Acceleration factor of ICCG method: 1.02 
Fig.5 A finite element model by using first order hexahedral edge elements 
Predictions of impedance signal due to the crack are compared with experimental measurements[10] in Fig. 6. Excellent agreement demonstrates the effectiveness of numerical modeling for the forward problem. Both results are high accurate because these values are less than 2 % of the impedance in air.
Giving the length, opening, location, and breaking side of each crack is mere assumption for simplification of the modeling. However the location is presumed from the distribution on the impedance map in Fig. 7, and the side is predicted in the phases of the signals.


Fig.6 Comparison between experimental measurement and numerical simulation (inner elliptical crack, 300kHz)  Fig.7 Experimental measurement of eddy current signal (inner elliptical crack, 300kHz) 
Output least square errors are shown in Fig. 8 until convergent parameters are obtained from initial conditions. It can be seen that they decrease at every iteration in each case. The finally estimated crack profiles with the true profiles are shown in Fig. 9 when those of initial guesses are taken as rectangular cracks 0.65mm (50%) deep. The estimated results are in good agreement with the true and the estimated profiles in maximum depth, irrespective of shape symmetry of the cracks.
Fig.9 Results of estimated crack profiles 
(a) Inner cracks 
(b) Outer cracks 
Fig.8 Output least square errors (300kHz) 
At the shallow parts of the cracks, it is found that there are differences between the true and estimated profile. One of issues arises from low sensitivity at the scanning path above the shallow parts, in comparison to the area where maximum signal strength can be observed around there. The other is that there is no experimentally measured value to be referred closest to the parts on the path. However it dose not result in improvement although more measuring points are used to cover the parts. This can be considered it is a matter of sensitivity at the edges. The detectablity of ECT depends on test frequencies by nature of the skin effect. In the cases of sloping and stepwise cracks from the viewpoint of the sensitivity, this also results in unexpected profiles at high frequency at the corner of shallow depth.
The large amount of computation to predict eddy current signals is required to reach the final results; equation (8) is solved 940 times in the longest case. It definitely makes it difficult to apply this approach to realistic diagnosis. Further work is devoted to the development of a fast inversion technique in this method.
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