Search

In this chapter we sketch some rather elegant mathematical theorems that have had a significant impact on computational aspects of electrical engineering. We have used them over the years in performing eddy-current inversions and believe that they will have an expanded role to play in the future development of eddy-current NDE.

The preceding chapters of this book dealt largely with problems in which the host material and any anomalies were pure electrical conductors. The presence of magnetic permeabilities was merely an interesting side effect to the creation of the VIC-3D ® model and its solution. There are, of course, many problems in which magnetic permeability is present with electrical conductivity and must be accounted for, not only in creating the model but also in understanding the physics of the solution and its effect on the NDE process. In this chapter we consider magnetic effects in heat-exchanger tubes, focusing later on ferritic tubes, which are becoming of increasing importance in the nuclear power industry.

We will discretize (3.22) by employing Galerkin’s method, which uses the same vector functions for expansion and testing. The spatial derivatives that cause problems will be removed by the testing process. In order to test these derivatives, we introduce special vector expansion functions, called “facet elements” and “edge elements,” that comprise products of pulse and tent functions.

VIC-3D ® computes impedances from field calculations by using the reaction concept. We will follow Harrington [47, pp. 116–120], in developing this concept.

In this chapter, we describe the mathematical development of a general axisymmetric model for VIC-3D®. This model is capable of analyzing tubes with tube supports and roll-expanded transition zones. Features such as magnetite and sludge,are included, and materials may be either ferromagnetic or nonmagnetic. The model described in this chapter will include only differential (or absolute) bobbin coils.

In the last chapter, we developed a volume-integral approach to the analysis of printed-circuit probes. Such probes are finding increased use in planar and conformable arrays, because of their small size and the ability to be manufactured in various shapes. Arrays are typically composed of a set of transmitter coils and receiver coils, which allows a variety of multiplexing schemes to generate responses with given features (see [96]). For example, one can generate a two-dimensional “scan” with a suitable array, without moving it. This can be very effective when doing voxel-based inversions, as will be discussed in a later volume of this work.

Now that we have the electromagnetics down pat, let’s look ahead to see where that leads us. In this part of the book, a general inverse method process is introduced that provides a framework for incorporating the numerical forward models discussed in Part I in an algorithm that estimates an unknown set of values of the test. A diagram of a general inverse method process is presented in Fig. 11.1.

Although the development of the infinite-space and layered-space dyadic Green functions is done in the Fourier-domain (or, as it is often referred to, the spectral-domain), the further development of the integral equation, together with its discretization, is best done in the spatial domain. We will now give an example of the transformation from the Fourier-domain to the spatial domain.

By “clutter” we mean a large background signal of known origin, such as probe lift-off, or a large background signal due to systematic errors of unknown origin, such as a variation in the material properties of the sample, or variations in the subsurface structure. In any case, clutter is usually more pernicious to the reliable detection and reconstruction of a flaw than random (electronic) noise that can be often eliminated by averaging. Hence, the ability to model clutter and reject it is paramount to the successful application of inverse methods and is the subject of the first part of this chapter.

“Robust” estimators are resistant to outliers in data, contrary to the usual classical least-squares estimator such as NLSE. We will describe two robust estimators in this chapter and give an example of the application of them to pre-processing some experimental data. Both of the robust estimators are taken from [63], and we closely follow those authors in our presentation.

By a “nonstandard probe,” we mean one that is not the usual air-core circular coil with a square cross-section, whose axis is normal to a plane surface, as in the pancake coil of Appendix A.1 of Chap. 5, or with its axis coinciding with the axis of a tube, as in the bobbin coils of Chap. 9. In this chapter, we develop a theory that allows us to efficiently discretize the current density in such probes, thereby allowing us to use transfer matrices (these will be discussed in Chap. 10) and similar mathematical constructs for solving problems.

Characterizing surfaces and coatings has long been a staple of eddy-current technology, but with the advent of inverse methods that technology has become even more powerful. In this chapter we describe a case in which model-based inversion, coupled with computational electromagnetics, can be effectively used to solve complex coating problems.

Given the material properties of typical aircraft structures and environmental conditions experienced over a service life, corrosion in aircraft structures is a major issue. For the United States Air Force alone, the total annual cost for corrosion management is on the order of a billion dollars. Avoidance of even a small portion of this annual expenditure would result in significant savings. To improve the management of corrosion, limits of existing inspection tools must be addressed.

To validate the performance of inspection techniques, probability of detection (POD) studies are performed. The preparation of POD samples with real fatigue cracks and the process to acquire a statistically significant number of measurements is often very time consuming and expensive, providing a significant burden for the validation of new inspection techniques.

Nondestructive evaluation (NDE) is to materials and structures what CAT scanning is to the human body—an attempt to look inside without opening up the body. As in CAT scanning, modern NDE requires sophisticated mathematical software to perform its function. This is especially true with regard to quantitative NDE, wherein we attempt to quantify defects, that is, determine their size, location, even shape, rather than just to detect their presence. Low-frequency electromagnetic methods using eddy-currents are a traditional mode of doing NDE (approximately 35% of NDE uses eddy-currents, depending upon the specific application), but the technology still suffers from a lack of algorithms and software to allow its full potential to be realized. The electromagnetics code, VIC-3D ®, was developed to alleviate that problem, and in this book we apply VIC-3D ®to solve forward and inverse problems in eddy-current NDE.

A problem of particular interest in the field of nondestructive evaluation is the inspection of fastener sites in aircraft structures for fatigue cracks. An important class of structures comprises plane-parallel layered media representing joints in both fuselage and wing locations. The layer stackup typically consists of two to four panels and often includes thin layers of a nonconducting sealant or adhesive between the panels.

The Green’s dyad, which is the electric-field response to a delta-function vector current source, plays a principal role in volume-integral equations, as we shall see later. In this chapter we develop the theory of the Green’s dyad for plane-parallel layered media. In Chap. 9 we extend the development to multilayered media with cylindrical geometry.

Chapter 11 introduced us to the notion of an inverse problem and gave us some examples of the value of this idea to the solution of realistic industrial problems. The basic inversion algorithm described in Chap. 11 was based upon the Gauss–Newton theory of nonlinear least-squares estimation and is called NLSE in this book. In this chapter we will develop the mathematical background of this theory more fully, because this algorithm will be the foundation of inverse methods and their applications during the remainder of this book. We hope, thereby, to introduce the reader to the application of sophisticated mathematical concepts to engineering practice without introducing excessive mathematical sophistication.

Eddy-current measurements of conductivity spectra are needed to gauge the near-surface residual stress in surface-treated turbine engine components for life extension, where conductivity and stress are related via the piezoresistive effect. Broadband (0.1–100 MHz) planar spiral-coil eddy-current probes are currently being developed using photolithography for the purpose of efficiently measuring the conductivity spectra for subsequent inversion to residual stress versus depth profiles.

The nuclear-power industry faces the serious challenge of convincing a skeptical public and regulatory agencies that it can operate safely and efficiently. Nondestructive evaluation (NDE) plays a significant role in this task, and computer modeling is playing a significant role in NDE. The industry now realizes the value of using such modeling to replace expensive experimental tests, as well as to design equipment, and interpret results. Eddy-currents have a traditional place in the inspection of steam generator tubing, and the industry seeks improved tools for such inspections.

“Spatial decomposition” is a generic term for the first step in transforming a functional equation into its discrete version prior to numerical computation. Umashankar [51] introduced the term “spatial decomposition technique” for a method of solving two-dimensional scattering problems from electrically large bodies using boundary-integral equations. As far as we know, we are the first to apply the “spatial decomposition algorithm” to the solution of three-dimensional scattering problems when the scatterer (the “flaw”) occupies several regions with different electrical properties using volume-integral equations.