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The mathematical theory of functional integration was developed in the fifties to be applied to problems in probability and quantum field theory. We’re not interested in the formal theory, but in its more practical form which can actually be used to generate numbers. In particular, it leads to the probabilistic collocation method (PCM) and other techniques for high-dimensional model representation (HDMR), which we intend to apply to eddy-current nondestructive evaluation (NDE), using the volume-integral code, VIC-3D®, as our vehicle.

Our goal in this chapter is to develop electromagnetic models for the inspection of carbon-nanotube reinforced polymers (CNRPs), along the lines of our work with CFRPs. Electromagnetic models for carbon nanotube structures are of considerable current interest.

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. In this chapter we sketch the theory of the Green’s dyad for plane-parallel layered media as it is applied to titanium and titanium-like alloys that posses 6 mm symmetry.

Titanium is noisy. Contrary to the situation discussed in Chap. 7 in which the surface of an isotropic body, copper, is made artificially noisy through shot-peening, titanium is intrinsically noisy because of its microstructure consisting of a random distribution of anisotropic crystallites.

Standard methods for minimizing a real-valued function of several variables can be divided into two general classes: those that require second derivative information, usually referred to as Newton-type methods, and those requiring only first derivative information, referred to as gradient methods. There are several excellent texts which, in addition to discussing many of these methods in detail, also give suggestions on when to use certain techniques.

Figure 10.1 shows a planar model of a type Vb lesion that could be situated in the coronary arteries. This lesion is the largest and most complex stable lesion that can form, and is characterized by the formation of calcium of the outer cap, which gives rise to the expression ‘hardening of the arteries.’ Because the vulnerability of plaque is not closely correlated to plaque size, but rather to the overall composition, we are interested in determining the amount of fibrous tissue, calcium, lipid core, and smooth muscle that exists in the lesion, and to do this we return to Fig. 10.1.

Composite materials in the form of fiber-reinforced matrix materials as, for example, graphite-epoxy, are being increasingly used in critical structures and structural components because of their high strength-to-weight ratio. In order to assess the integrity of these structures, it is necessary to employ suitable methods for quantitative NDE. One method uses eddy-currents; composite materials, however, are inherently anisotropic, which means that many of the classical eddy-current technology and design procedures are not applicable. In addition, composite materials vary widely in their permittivities and conductivities, which means that new analyses must be carried out to develop effective strategies for using eddy-currents in quantitative NDE.

A major milestone that we have demonstrated is that the set-theoretic algorithm works well with anisotropic media, such as CFRPs. Such problems require at least two variables to be reconstructed at each voxel, such as the longitudinal and transverse conductivities. The details are described in Sects. 4.3.1–4.3.2 in the context of measuring FAWT (Fiber Areal-Weight), which is an important parameter during the manufacture of CFRP prepregs. An interesting corollary is shown in Sect. 4.3.2, in which a single transmitting coil is used with the same receiver array as before, and very good results were achieved. This suggests that less extreme T/R-arrays can be used in the algorithm, with the possibility of using sparse-grid interpolation to fill in gaps in the measured data.

Recent work has demonstrated the capability of applying inverse methods to automated eddy current (EC) data of surface breaking cracks and notches of various sizes, orientations and aspect ratios. However, not all eddy current indications in turbine engine component inspections originate from cracks, which can result in the unnecessary removal of engine components from service. For powder metallurgy nickel-based superalloys, non-metallic inclusions (NMIs) and non-metallic particles are frequently present. If an EC inspection can reliably classify NMI indications from crack indications, there would be great payoff for the USAF. In this work, simulated results are presented to highlight differences in eddy current signals from cracks and NMIs. Progress is presented on the development of a new model-based inversion scheme highlighting enhancements to the numerical model VIC-3D®, improved indication registration in noisy scans, and the fitting and evaluation of multiple surrogate model classes. Lastly, inversion results demonstrate the ability to distinguish cracks and NMIs, and the potential to characterize the approximate dimensions and depth of NMIs.

A model-based calibration process is introduced that estimates the state of the eddy current probe. First, a carefully designed surrogate model was built using VIC-3D® simulations covering the critical range of probe rotation angles, tilt in two directions, and probe offset (liftoff) for both transverse and longitudinal flaw orientations. Some approximations and numerical compromises in the model were made to represent tilt in two directions and reduce simulation time; however, this surrogate model was found to represent the key trends in the eddy current response for each of the four probe properties in experimental verification studies well. Next, this model was incorporated into an iterative inversion scheme during the calibration process, to estimate the probe state while also addressing the amplitude/phase fit and centering the calibration notch indication. Results are presented showing several examples of the blind estimation of tilt and rotation angle for known experimental cases with reasonable ag...

The objective of this work is to expand eddy current inversion methods to enable estimation of flaw dimensions and orientations for data collected using a variety of scan resolutions and filter configurations. Prior work [1, 2] has demonstrated the capability of applying inverse methods to unfiltered, high-fidelity, automated eddy current data. This has shown advantages over a simple amplitude-based analysis of the data. However, to apply this approach to data collected using standard depot inspection settings, additional algorithms must be developed. In addition to including a high-pass filter, standard depot surface inspections often involve a part rotating at a constant angular velocity while a probe moves along a linear path. To expand the current inversion capabilities, two options were investigated. The first option involves re-sampling and filtering the forward model prior to inversion. The second option focuses on post-processing the flaw signal to a) effectively remove or minimize the effect of the filter and b) re-sample the data to have a pre-determined, uniform sample-point spacing. To validate the developed algorithms, data were collected by rotating a part while sampling along a linear path, and a 20 Hz high-pass filter was applied. The complete set of inversion algorithms was then used, and the resulting estimates of flaw length, depth, width, and orientation were in good agreement with the corresponding estimates determined using the original, high-fidelity, unfiltered data.

In model-based inverse problems, the unknown parameters (such as length, width, depth) need to be estimated. When the unknown parameters are few, the conventional mathematical methods are suitable. But the increasing number of unknown parameters will make the computation become heavy. To reduce the burden of computation, the sparse grid algorithm was used in our work. As a result, we obtain a powerful interpolation method that requires significantly fewer support nodes than conventional interpolation on a full grid.

A comprehensive approach to NDE and SHM characterization error (CE) evaluation is presented that follows the framework of the ‘ahat-versus-a’ regression analysis for POD assessment. Characterization capability evaluation is typically more complex with respect to current POD evaluations and thus requires engineering and statistical expertise in the model-building process to ensure all key effects and interactions are addressed. Justifying the statistical model choice with underlying assumptions is key. Several sizing case studies are presented with detailed evaluations of the most appropriate statistical model for each data set. The use of a model-assisted approach is introduced to help assess the reliability of NDE and SHM characterization capability under a wide range of part, environmental and damage conditions. Best practices of using models are presented for both an eddy current NDE sizing and vibration-based SHM case studies. The results of these studies highlight the general protocol feasibility, em...

A comprehensive approach is presented to perform model-based inversion of crack characteristics using bolt hole eddy current (BHEC) techniques. Data was acquired for a wide range of crack sizes and shapes, including mid-bore, corner and through-thickness crack types, and from both standard eddy current hardware and a prototype BHEC system with z-axis position encoding. Signal processing algorithms were developed to process and extract features from the 2D data sets, and inversion algorithms using VIC-3D generated surrogate models were used for inverting crack size. New model results are presented, which now address the effect of having a corner crack at an edge and a through crack adjacent to two edges. A two-step inversion process was implemented that first evaluates the material layer thickness, crack type and location, in order to select the most appropriate VIC-3D surrogate model for subsequent crack sizing inversion step. Inversion results for select mid-bore, through and corner crack specimens are presented where sizing performance was found to be satisfactory in general, but also depend on the size and location of the flaw.

The use of coupled integral equations and anomalous currents allows us to efficiently remove ‘background effects’ in either forward or inverse modeling. This is especially true when computing the change in impedance due to a small flaw in the presence of a larger background anomaly. It is more accurate than simply computing the response with and without the flaw and then subtracting the two nearly equal values to obtain the small difference due to the flaw. The problem that we address in this paper involves a ’SplitD’ probe that includes complex, noncircular coils, as well as ferrite cores, inserted within a bolt hole, and exciting both the bolt hole and an adjacent flaw. This introduces three coupled anomalies, each with its on ’scale.’ The largest, of course, is the bolt hole, followed (generally) by the probe, and then the flaw. The overall system is represented mathematically by three coupled volume-integral equations. We describe the development of the model and its code, which is a part of the gener...

In past work we have developed a rigorous electromagnetic model and an inversion algorithm for the three-dimensional NDE of advanced composite materials. This approach extends Victor Technologies’ work in eddy-current NDE of conventional metals, and allows one to determine in localized regions the fiber-resin ratio in graphite-epoxy, and to determine those anomalies, e.g., delaminations, broken fibers, moisture content, etc., that can be reconstructed by our inversion method. In developing the model, we applied rigorous electromagnetic theory to determine a Green’s function for a slab of anisotropic composite material, and then determine the integral relations for the forward and inverse problems using the Green’s function. In addition, we have given examples of the solution of forward and inverse problems using these algorithms.

The objective of this work is to demonstrate and validate model-based inversion techniques to characterize length, depth, width and orientation of surface-breaking cracks using eddy current NDE under varying probe conditions. A series of parametric studies of probe characteristics are presented for a fixed set of well-characterized flaws with varying length, depth, opening width and orientation angle. Results show inversion performance differences between probes with the same design specifications. Inversion results were also evaluated for a probe that was selectively controlled for varying probe liftoff, varying tilt in two directions, and orientation. Certain levels of probe tilt and liftoff were found to degrade the performance of the inversion technique. By using a model calibration process that incorporates the matching probe calibration data, better inversion results can be achieved, to a limited degree. There is a need to more appropriately adapt the model through the calibration fit to compensate ...

The objective of this work is to demonstrate model-based inversion techniques to characterize the length, depth, width, and orientation of surface-breaking cracks using eddy current (EC) NDE. The paper presents experimental testing to acquire high fidelity automated eddy current data, enhancements made in VIC-3D® to improve both speed and accuracy, benchmark studies demonstrating model accuracy, improved data registration and reduction methods, and surrogate models and model calibration schemes to ensure the fastest and highest quality models are used for inversion. Initial inversion results indicate the potential to accurately size cracks and EDM notches over a wide range of flaw characteristics and probe orientations. Insight into EC variability for POD crack sets is presented using inversion results for crack length and depth.

We have developed a rigorous electromagnetic model and an inversion algorithm for the three-dimensional NDE of advanced composite materials. This approach extends Victor Technologies' work in eddy-current NDE of conventional metals, and allows one to determine in localized regions the fiber-resin ratio in graphite-epoxy, and to determine those anomalies, e.g., delaminations, broken fibers, moisture content, etc., that can be reconstructed by our inversion method. In developing the model, we apply rigorous electromagnetic theory to determine a Green's function for a slab of anisotropic composite material, and then determine the integral relations for the forward and inverse problems using the Green's function. We will give examples of the solution of forward problems using this model.

This paper introduces different data analysis strategies and characterization metrics for (1) designing and optimizing NDE characterization techniques, (2) validating NDE characterization performance and (3) ensuring in-service quality control. Rather than discard data that lack perfect knowledge of the flaw state, it is proposed to use it appropriately in order to optimize the inversion process. For example, such cracks of uncertain depth can be repeatedly scanned at different orientations and with different probe, and the inversion process can be designed to minimize the variance in the depth estimates under the varying conditions. To address validation of NDE characterization capability, the use of a finite number of real cracks that are destructively characterized in conjunction with a broader notch specimen set and simulated data is proposed. Lastly, inversion metrics are reviewed when no prior knowledge of the crack state is available.

When performing inverse-methods for the characterization of cracks, there is a need to assess the quality of the result without prior knowledge of the flaw state. This paper investigates the use of two estimation metrics, confidence intervals on parameter estimates and the frequency that the global solution was achieved, for assessing inversion performance. A case study on eddy current inversion of surface-breaking cracks was used to evaluate the viability of these metrics. While these metrics were found to have benefit, the parameter confidence intervals were found to not be conservative. As well, when there are problems with convergence to the global minima, residual error plots and mean square error maps were found to provide more information on model quality and the true bounds on the parameter estimates.

Forward uncertainty propagation has been a topic of interest to NDE researchers for several years. To this point, the purpose has been to gain an understanding of the uncertainties that can be seen in signals from NDE sensors given uncertainties in the geometric and material parameters of the problem. However, a complex analysis of an inspection scenario with high variability has not been performed. Furthermore, these methods have not seen direct practical application in the areas of model assisted probability of detection or inverse problems. In this paper, uncertainty due to spatial heterogeneity in material systems that undergo NDE inspection will be discussed. Propagation of this uncertainty through forward models of inspection scenarios will be outlined and the mechanisms for representing the spatial heterogeneity will be explained in detail. Examples will be provided that illustrate the effect of high variability in uncertainty propagation in the context of forward modeling.

In past work, we introduced model-based inverse methods, and applied them to problems in which the anomaly could be reasonably modeled by simple canonical shapes, such as rectangular solids. In these cases the parameters to be inverted would be length, width and height, as well as the occasional probe lift-off or rotation. We are now developing a formulation that allows more flexibility in modeling complex flaws. The idea consists of expanding the flaw in a sequence of basis functions, and then solving for the expansion coefficients of this sequence, which are modeled as independent random variables, uniformly distributed over their range of values. There are a number of applications of such modeling: 1. Connected cracks and multiple half-moons, which we have noted in a POD set. Ideally we would like to distinguish connected cracks from one long shallow crack. 2. Cracks of irregular profile and shape which have appeared in cold work holes during bolt-hole eddy-current inspection. One side of such cracks i...

While parametric uncertainty quantification for NDE models has been addressed in recent years, the problem of stochastic field parameters such as spatially distributed electrical conductivity has only been investigated minimally in the last year. In that work, the authors treated the field as a one-dimensional random process and Karhunen-Loeve methods were used to discretize this process to make it amenable to UQ methods such as ANOVA expansions. In the present work, we will treat the field as a two dimensional random process, and the eigenvalues and eigenfunctions of the integral operator will be determined via Galerkin methods. The Karhunen-Loeve methods is extended to two dimensions and implemented to represent this process. Several different choices for basis functions will be discussed, as well as convergence criteria for each. The methods are applied to correlation functions collected over electron backscatter data from highly micro textured Ti-7Al.

We extend our previous work, in which we applied high-dimensional model representation (HDMR) and analysis of variance (ANOVA) concepts to the characterization of a metallic surface that has undergone a shot-peening treatment to reduce residual stresses, and has, therefore, become a random conductivity field. That example was treated as a onedimensional problem, because those were the only data available. In this study, we develop a more rigorous two-dimensional model for characterizing random, anisotropic grain noise in titanium alloys. Such a model is necessary if we are to accurately capture the 'clumping' of crystallites into long chains that appear during the processing of the metal into a finished product. The mathematical model starts with an application of the Karhunen-Loeve (K-L) expansion for the random Euler angles, θ and φ, that characterize the orientation of each crystallite in the sample. The random orientation of each crystallite then defines the stochastic nature of the electrical conductivity tensor of the metal. We study two possible covariances, Gaussian and double-exponential, which are the kernel of the K-L integral equation, and find that the double-exponential appears to satisfy measurements more closely of the two. Results based on data from a Ti-7Al sample will be given, and further applications of HDMR and ANOVA will be discussed.

This work presents a comprehensive approach for model-based inversion of crack characteristics using eddy current nondestructive evaluation (NDE) and includes a demonstration using a bolt-hole eddy-current (BHEC) technique. Data were acquired using standard eddy current hardware for crack and notch samples of varying size and profile. The inversion results were found to be accurate within 10% for sizing the depth of through cracks and able to accurately bin mid-bore cracks by size.

A comprehensive approach to NDE characterization error evaluation is presented that follows the framework of the ‘ahat-versus-a’ model evaluation process for probability of detection (POD) assessment. Before characterization error model building is performed, an intermediate step must evaluate the presence and frequency of several possible classes of poor characterization results. A case study is introduced based on the estimation the length, depth and width of surface breaking cracks using bolt hole eddy current (BHEC) NDE. This study highlights the importance of engineering and statistical expertise in the model-building process to ensure all key effects and possible interactions are addressed.

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.

Generalized polynomical chaos (gPC), the probabilistic collocation method (PCM), and analysis of variance (ANOVA) are finding considerable application to problems of interest to engineers in which random parameters are an essential feature of the mathematical model. So far the applications have been mainly to stochastic partial differential equations, but we extend the method to volume-integral equations, which have met great success in electromagnetic nondestructive evaluation (NDE), especially with eddy-currents. The problems of main interest to the NDE community in this connection are concerned with the issue of ‘propagation of uncertainty’ when the relevant parameters are not well characterized, or are known simply as random variables. We demonstrate the ideas by considering a metallic surface that has undergone a shot-peening treatment to reduce residual stresses, and has, therefore, become a random conductivity field. In particular, we show how ANOVA becomes a very promising method for high-dimensional model representation (HDMR) when there are a large number of random variables present in the problem.

“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.