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Electromagnetic fields due to excitation coils in boreholes containing cracks have been calculated using a boundary element method formulated in terms of a scalar decomposition of the field. By using a Green's kernel that ensures continuity of the tangential electric and magnetic field at the cylindrical surface, it is only necessary to render the flaw region in a discrete form. The effect of an ideal crack, one with zero opening but acting as a perfect barrier to eddy currents, is here represented by a current dipole layer orientated normal to the crack surface. An integral equation determines the source density on the crack and a numerical solution is found using boundary elements. The coil impedance change due to the crack is then computed using an expression based on a reciprocity principle. Index Terms--Boundary elements, cracks, eddy current, integral equation, nondestructive evaluation.

We have determined the quasi-static time-domain dyadic Green's function for an electric source in a conductive plate for use in electric field integral equations. Starting with a frequency-domain representation, we constructed the dyadic kernel from electric and magnetic scalar potentials defined with respect to a preferred direction normal to plate. The final time-domain expression has three parts: a free-space term, multiple image terms, and partial reflection terms. The free-space fundamental solution is expressed in terms of a three-dimensional Gaussian bell curve satisfying the diffusion equation. Similarly, the image terms are expressed in the same form with coordinates shifted so that each Gaussian curve centers on an image point. In order to carry out the inverse Laplace transform from the frequency domain to the time domain, we expand the partial reflection functions as asymptotic series before transforming them analytically. The resulting expression for dyadic Green's kernel can be evaluated efficiently and with well-controlled accuracy. Index Terms--Conductive plate, dyadic Green's function, transient eddy current.

We describe the calculation of eddy currents in a two-layer conducting rod of finite length excited by a coaxial circular coil carrying an alternating current. The calculation uses the truncated region eigenfunction expansion (TREE) method. By truncating the solution region to a finite length in the axial direction, we can express the magnetic vector potential as a series expansion of orthogonal eigenfunctions instead of as a Fourier integral. The restricted domain can be arbitrarily large to yield results that are as close to the infinite domain results as desired. Integral form solutions for an infinite rod are well known and relatively simple. For a finite length cylindrical conductor, however, additional boundary conditions must be satisfied at the ends. We do this by comparing series expansions term by term to match the solutions across the end of the cylinder. We derive closed-form expressions for the electromagnetic field in the presence of a finite two-layer rod. A special case of the solution is that for a conductive tube. We illustrate the end effect by calculating the coil impedance variation with respect to the axial location of the coil. The results are in very good agreement with those obtained by using a two-dimensional finite-element code. Index Terms--Coil impedance, eddy current, eigenfunction expansion, finite rod.

An inversion process uses eddy-current probe computations to determine the size and shape of ideal cracks in eddy-current interactions. An optimization scheme based on a descent algorithm minimizes the global error function using the flow structure-based error gradient. An integral formulation of an ideal crack problem derives an equation for the error gradient. Numerical computations and measured slot shapes of probe impedances and error gradient are consistent.

A new iterative least-squares algorithm explaining the mechanics of eddy current inversion was presented. The algorithm minimized the mean-square difference between the predicted and actual impedance values by using the conductivity distribution. The gradient of the impedance showed how the impedance gradient can be updated such that a reduction in the mean-square difference is ensured.

An analytical model of an axisymmetric eddy current ferrite-cored probe above a multi-layered conducting half-space has been developed using a procedure in which the domain of the problem is truncated radially. This means that solutions can be expressed in the form of generalized Fourier-Bessel series. The expansion coefficients are found by matching the field across the interfaces between the subregions of the problem. Initially, the magnetic vector potential of a simple circular current filament is expanded in a series form. The solution is then modified to accommodate an infinitely long coaxial ferrite core, and the principle of superposition is invoked to derive a coil field from the filament field in the presence of the core. Next, we consider a semi-infinite core and then one of finite length. Finally, the effects of a multi-layered conductor are included. Numerical predictions of probe impedance have been compared with experimental data showing excellent agreement. [ABSTRACT FROM AUTHOR]

The electromagnetic field of a cylindrical eddy current probe coil near the open end of a borehole in a conductor has been calculated analytically accounting for edge effects. Calculations of the coil impedance as a function of position and excitation frequency have been made allowing theoretical results to be compared with experimental measurements. Comparisons have been carried out for special cases in which a cylindrical coil has its axis either perpendicular or parallel to the axis of the hole. In the approach used, the field is expressed in terms of transverse electric and transverse magnetic potentials defined with respect to the axis of the hole. The domain of the problem is truncated in the axial direction in order to express the solution in the form of eigenfunction expansions. The truncation modifies the original unbounded domain problem, but the additional boundaries can be made as remote from the coil as desired so that they have a negligible effect on numerical estimates of the coil field. The truncated region approach has proved to be accurate and computationally efficient but more significantly, it allows new solutions to be found for problems that are otherwise analytically intractable. The results model eddy current inspections of boreholes including edge effects at the opening of the hole. [ABSTRACT FROM AUTHOR]

Presents a study that described an inversion scheme of eddy-current interaction in an ideal crack metal. Analysis of the impedance variation; Details of the local edge solution; Evaluation of the impedance gradient.

A boundary integral vector potential formulation has been developed to evaluate eddy-current interactions with three-dimensional finite cracks in conductors. The approach is compared with an electric field integral equation method also used for solving crack problems in eddy-current nondestructive evaluation. An important advantage of the vector potential integral formulation is that the kernel has a weak singularity, but a drawback is that two unknown functions must be found on the crack surface. One of these functions, the current dipole density, represents the effect of the crack in terms of an induced source, and the other function is a solution of the two-dimensional Laplace equation, By contrast, the source density alone is needed for a complete solution of the electric field integral equation. In order to determine the surface Laplacian for finite cracks of arbitrary shape, a general numerical solution utilizing the boundary element technique is introduced. Numerical predictions of the eddy-current probe response to a crack give good agreement with experimental measurements, supporting the validity of the formulation. Index Terms--Eddy-current testing, forward problem, integral formulation, probe response, vector potential.