Inverse Eddy Current Problem Research Articles

An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

Mathematical and Numerical Study of a Three-Dimensional Inverse Eddy Current Problem

We study an inverse problem associated with an eddy current model. We first address the ill-posedness of the inverse problem by proving the compactness of the forward map with respect to the conductivity and the nonuniqueness of the recovery process. Then by virtue of nonradiating source conceptions, we establish a regularity result for the tangential trace of the true solution on the boundary, which is necessary to justify our subsequent mathematical formulation. After that, we formulate the inverse problem as a constrained optimization problem with an appropriate regularization and prove the existence and stability of the regularized minimizers. To facilitate the numerical solution of the nonlinear nonconvex constrained optimization, we introduce a feasible Lagrangian and its discrete variant. Then the gradient of the objective functional is derived using the adjoint technique. By means of the gradient, a nonlinear conjugate gradient method is formulated for solving the optimization system, and a Sobolev gradient is incorporated to accelerate the iterative process. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.

Thickness measurement of metal pipe using swept-frequency eddy current testing

Swept-frequency eddy current measurement of pipe thickness is studied in this paper. First, suitable frequency range of swept-frequency eddy current testing is determined by comparing sensitivities of relative reactance change with respect to pipe thickness and other parameters at different frequencies. Based on analytical solutions to pipe eddy current field, Levenberg–Marquardt algorithm and variable transformation, a method for solving inverse eddy current problem is developed. Finally, several inversion calculations are carried out and the results are close to the truth values. The low errors reveal that the method presented in this paper is appropriate.

Inverse Problems Using Neural Networks for Cracks Characterization in Materials

Inverse eddy current problem using Artificial Neural Networks (ANN) approach for the localization and the classification shape of defects is considered. The task of reconstructing the cracks and damage in the plate profile of an inspected specimen in order to estimate its material properties can be described in this research work. This is accomplished by inverting eddy current probe impedance measurements that are recorded as a function of probe position, excitation frequency or both. In eddy current nondestructive evaluation, this is widely recognized as a complex theoretical problem whose solution is likely to have a significant impact on the characterization of cracks in materials .

Evaluating the conductivity distribution in isotropic polycrystalline graphite using spectroscopic eddy current technique for monitoring weight loss in advanced gas cooled reactors

Recently studies have shown correlation between the conductivity of graphite and its weight loss (density). This paper presents an eddy current technique for evaluating the conductivity profiles within 100mm thick layered graphite sections. Flat graphite plates with three different grades of density for which the corresponding conductivities were measured, have been used to construct these sections. An initial test was performed to examine the capability of the eddy current technique to penetrate to a 100mm depth in graphite. Then spectroscopic inductance measurements were acquired from different test profiles and compared with finite element method (FEM) simulated data. Conductivity profiles were reconstructed from the experimental data via the solution of an inverse eddy current problem. For an inverse solution, direct Tikhonov regularization and iterative Gauss Newton reconstruction schemes have been employed. Results show that the inverse methods are able to produce reasonable estimates of the conductivity profiles.

Two approaches to coil impedance calculation of eddy current sensor

Eddy current (EC) sensor detects measurand by the change of coil impedance, which is introduced by coupling the magnetic energy between coil and measurement target. Therefore, it is of great significance for both the forward and inverse eddy current problems to study on the approaches to coil impedance calculation. In the current paper, two approaches are presented to calculate coil impedance of EC sensor. First, the theoretical calculation model of coil impedance is simplified as an air-cored cylindrical coil, over an infinite measurement target. And the integral expression of coil impedance is derived according to the sensor operation principle and electromagnetic field theory. Then, the integral expression is expanded as series form in cylindrical coordinate system in order to compute the coil impedance easily, and computed using MathematicaTM. Next, based on the finite element modeling theory, two different finite element models of eddy current measurement system are built. The infinite boundary in mode1 is truncated, while in model 2, it is simulated completely by an infinite element. Furthermore, the appropriate location where the air domain is truncated at is determined. Finally, the coil impedance and the magnetic distribution of eddy current testing system are obtained by finite element method, respectively. By comparison, the results from the theoretical calculation approach and finite element method are in very good agreement, and the accuracy of two approaches is validated with each other.

Absolute Conductivity Reconstruction in Magnetic Induction Tomography Using a Nonlinear Method

Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A - V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added noise.

Single-step 3-d image reconstruction in magnetic induction tomography: theoretical limits of spatial resolution and contrast to noise ratio.

Magnetic induction tomography (MIT) is a low-resolution imaging modality for reconstructing the changes of the complex conductivity in an object. MIT is based on determining the perturbation of an alternating magnetic field, which is coupled from several excitation coils to the object. The conductivity distribution is reconstructed from the corresponding voltage changes induced in several receiver coils. Potential medical applications comprise the continuous, non-invasive monitoring of tissue alterations which are reflected in the change of the conductivity, e.g. edema, ventilation disorders, wound healing and ischemic processes. MIT requires the solution of an ill-posed inverse eddy current problem. A linearized version of this problem was solved for 16 excitation coils and 32 receiver coils with a model of two spherical perturbations within a cylindrical phantom. The method was tested with simulated measurement data. Images were reconstructed with a regularized single-step Gauss–Newton approach. Theoretical limits for spatial resolution and contrast/noise ratio were calculated and compared with the empirical results from a Monte-Carlo study. The conductivity perturbations inside a homogeneous cylinder were localized for a SNR between 44 and 64 dB. The results prove the feasibility of difference imaging with MIT and give some quantitative data on the limitations of the method.

Solution of the inverse problem of magnetic induction tomography (MIT) with multiple objects: analysis of detectability and statistical properties with respect to the reconstructed conducting region

Magnetic induction tomography (MIT) is a technique to image the passive electrical properties (i.e. conductivity, permittivity, permeability) of biological tissues. The inverse eddy current problem is nonlinear and ill-posed, thus a Gauss–Newton one-step method in combination with four different regularization schemes is used to obtain stable solutions. Simulations with 16 excitation coils, 32 receiving coils and different spherical perturbations inside a homogeneous cylinder were computed. In order to compare the statistical properties of the reconstructed results a Monte Carlo study with a SNR of 40 dB and 20 dB was carried out. Simulated conductivity perturbations inside a homogeneous cylinder can be localized and resolved and the results prove the feasibility of difference imaging with MIT.

Solution of the inverse problem of magnetic induction tomography (MIT)

Magnetic induction tomography (MIT) of biological tissue is used to reconstruct the changes in the complex conductivity distribution inside an object under investigation. The measurement principle is based on determining the perturbation ΔB of a primary alternating magnetic field B0, which is coupled from an array of excitation coils to the object under investigation. The corresponding voltages ΔV and V0 induced in a receiver coil carry the information about the passive electrical properties (i.e. conductivity, permittivity and permeability). The reconstruction of the conductivity distribution requires the solution of a 3D inverse eddy current problem. As in EIT the inverse problem is ill-posed and on this account some regularization scheme has to be applied. We developed an inverse solver based on the Gauss–Newton-one-step method for differential imaging, and we implemented and tested four different regularization schemes: the first and second approaches employ a classical smoothness criterion using the unit matrix and a differential matrix of first order as the regularization matrix. The third method is based on variance uniformization, and the fourth method is based on the truncated singular value decomposition. Reconstructions were carried out with synthetic measurement data generated with a spherical perturbation at different locations within a conducting cylinder. Data were generated on a different mesh and 1% random noise was added. The model contained 16 excitation coils and 32 receiver coils which could be combined pairwise to give 16 planar gradiometers. With 32 receiver coils all regularization methods yield fairly good 3D-images of the modelled changes of the conductivity distribution, and prove the feasibility of difference imaging with MIT. The reconstructed perturbations appear at the right location, and their size is in the expected range. With 16 planar gradiometers an additional spurious feature appears mirrored with respect to the median plane with negative sign. This demonstrates that a symmetrical arrangement with one ring of planar gradiometers cannot distinguish between a positive conductivity change at the true location and a negative conductivity change at the mirrored location.

Simple analytical approximations for eddy current profiling of the near-surface residual stress in shot-peened metals

Because of their frequency-dependent penetration depth, eddy current measurements are capable of mapping the near-surface depth profile of the electrical conductivity. This technique can be used to nondestructively characterize the subsurface residual stress distribution in certain types of shot-peened metals, e.g., in nickel-base superalloys. For quantitative evaluation of the experimental results, analytical and computational techniques are needed to solve the direct and inverse problems, i.e., to predict the frequency-dependent apparent eddy current conductivity from the depth profile of the frequency-independent intrinsic electrical conductivity of the specimen and vice versa. Simple analytical approximations are presented for both the direct and inverse eddy current problems by exploiting two specific features of the subsurface electrical conductivity variation caused by near-surface residual stresses in shot-peened metals. First, compressive residual stresses are limited to a shallow surface region of depth, much less than typical probe coil diameters. Second, the change in electrical conductivity due to residual stresses is always very small, typically less than 1%. The proposed approximations are verified by numerical comparison to much more complicated numerical solutions.

Application of surface impedance concept to inverse problems of reconstructing transient currents

The inverse eddy-current problem of fast transients is solved by a new boundary element formulation employing time domain surface impedance boundary conditions. The integral equation is transformed to invariant form and is solved only once for a given geometry of the problem. Numerical results are in good agreement with measured data.

Theory of eddy current inversion

The inverse eddy current problem can be described as the task of reconstructing an unknown distribution of electrical conductivity from eddy-current probe impedance measurements recorded as a function of probe position, excitation frequency, or both. In eddy current nondestructive evaluation, this is widely recognized as a central theoretical problem whose solution is likely to have a significant impact on the characterization of flaws in conducting materials. Because the inverse problem is nonlinear, we propose using an iterative least-squares algorithm for recovering the conductivity. In this algorithm, the conductivity distribution sought minimizes the mean-square difference between the predicted and measured impedance values. The gradient of the impedance plays a fundamental role since it tells us how to update the conductivity in such a way as to guarantee a reduction in the mean-square difference. The impedance gradient is obtained in analytic form using function-space methods. The resulting expression is independent of the type of discretization ultimately chosen to approximate the flaw, and thus has greater generality than an approach in which discretization is performed first. The gradient is derived from the solution to two forward problems: an ordinary and an ‘‘adjoint’’ problem. In contrast, a finite difference computation of the gradient requires the solution of multiple forward problems, one for each unknown parameter used in modeling the flaw. Two general types of inverse problems are considered: the reconstruction of a conductivity distribution, and the reconstruction of the shape of an inclusion or crack whose conductivity is known or assumed to be zero. A layered conductor with unknown layer conductivities is treated as an example of the first type of inversion problem. An ellipsoidal crack is presented as an example of the second type of inversion problem.