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1. Theory of coating thermal noise The special role that coatings play in the thermal noise of interferometric gravitational wave detectors was pointed out by Levin. Working from the FluctuationDissipation Theorem, he showed the proximity of the coating to the laser means fluctuations from coating loss can contribute more to the noise than from loss elsewhere on the mirror. Following Levin, Gretarsson developed a formula for thermal noise in an interferometer with a Gaussian beam. This same formula was found independently by Nakagawa et ai. using a different approach. Each assumed the mirror to be a half-infinite plane and that the coating's mechanical impedance was isotropic. Gretarsson allowed for anisotropy in the loss angle of the coating.

This expository paper introduces the concept of on‐line sensor methodologies for monitoring the integrity of components in next generation power systems, and explains general benefits of the approach, while describing early conceptual developments of suitable NDE methodologies. The paper first explains the philosophy behind this approach (i.e. the design‐for‐inspectability concept). Specifically, we describe where and how decades of accumulated knowledge and experience in nuclear power system maintenance are utilized in Generation IV power system designs, as the designs are being actively developed, in order to advance their safety and economy. Second, we explain that Generation IV reactor design features call for the replacement of the current outage‐based maintenance by on‐line inspection and monitoring. Third, the model‐based approach toward design and performance optimization of on‐line sensor systems, using electromagnetic, ultrasonic, and radiation detectors, will be explained. Fourth, general types of NDE inspections that are considered amenable to on‐line health monitoring will be listed. Fifth, we will describe specific modeling developments to be used for radiography, EMAT UT, and EC detector design studies.

A method for removing the numerical instability of the conventional Green's formula in the vicinity of the boundary surface is proposed. The approach uses a pair of Green's formulae written both inside and outside the volume of interest, even when used for solving a single-phase problem. It is shown that cancellation between the pair yields an alternative, singularity-free integral representation that is amenable to numerical calculation. The derivation of the formula is given explicitly and is accompanied by several numerical test results for validation.

We report on thermal noise from the internal friction of dielectric coatings made from alternating layers of Ta2O5 and SiO2 deposited on fused silica substrates. We present calculations of the thermal noise in gravitational wave interferometers due to optical coatings, when the material properties of the coating are different from those of the substrate and the mechanical loss angle in the coating is anisotropic. The loss angle in the coatings for strains parallel to the substrate surface was determined from ringdown experiments. We measured the mechanical quality factor of three fused silica samples with coatings deposited on them. The loss angle of the coating material for strains parallel to the coated surface was found to be (4.2 +- 0.3)*10^(-4) for coatings deposited on commercially polished slides and (1.0 +- 0.3)*10^{-4} for a coating deposited on a superpolished disk. Using these numbers, we estimate the effect of coatings on thermal noise in the initial LIGO and advanced LIGO interferometers. We also find that the corresponding prediction for thermal noise in the 40 m LIGO prototype at Caltech is consistent with the noise data. These results are complemented by results for a different type of coating, presented in a companion paper., Submitted to LSC (internal) review Sept. 20, 2001. To be submitted to Phys. Lett. A

We calculate the thermal noise in half-infinite mirrors containing a layer of arbitrary thickness and depth made of excessively lossy material but with the same elastic material properties as the substrate. For the special case of a thin lossy layer on the surface of the mirror, the excess noise scales as the ratio of the coating loss to the substrate loss and as the ratio of the coating thickness to the laser beam spot size. Assuming a silica substrate with a loss function of 3x10-8 the coating loss must be less than 3x10-5 for a 6 cm spot size and a 7 micrometers thick coating to avoid increasing the spectral density of displacement noise by more than 10%. A similar number is obtained for sapphire test masses., Passed LSC (internal) review. Submitted to Phys. Rev. D. (5/2001) Replacement: Minor typo in Eq. 17 corrected

As the name implies, the pulsed eddy current (PEC) method uses a transient excitation of an eddy current probe, as opposed to the continuous-wave (CW) excitation used in conventional eddy current inspection. The PEC approach offers several advantages over the conventional method. First, because the transient field is related, by means of a Fourier transform, to the CW field at multiple frequencies, a single PEC measurement is equivalent to multiple CW measurements as a function of frequency. PEC instrumentation is, however, much simpler and less expensive, and PEC data acquisition is much faster than multiple CW measurements. The transient method also offers the possibility of time-gating, which enables the user to distinguish signals originating from deep within a specimen from signals originating nearer the surface.

The efforts of past several years have resulted in development of an eddy current model [1–8], using the boundary element method (BEM). As of last year, the BEM algorithm based on the Hertz potential approach [1–3] was shown to be effective in dealing with complex part and probe geometry [4–6], and particularly in modeling crack signals [7–9]. Previously, the modeling capabilities were demonstrated mostly with absolute probes. This year, the focus has been shifted toward on crack signals of differential and reflection probes.

Eddy current techniques have been widely used in the NDE inspection of aircraft engine components. Depending on the flaw characteristics and specimen composition, various EC probe designs have been employed to achieve the maximum probability of detection (POD). Traditionally, the effectiveness of a probe design for a given inspection is determined experimentally. In particular, parameters such as probe types, operating frequency, scan spacing, etc. are evaluated experimentally in terms of POD. It is obvious that this is a costly way of defining inspection parameters. A more cost-effective alternative is to evaluate the test parameters through the use of numerical simulation. This can be done by casting the entire EC inspection process in terms of a numerical model governed by a set of integral equations. By computing the solutions to the integral equations, outputs in the form of impedance changes due to flaws can be used to generate the POD. Previously, we have introduced a modified version of the Hertzian magnetic potential approach for eddy current probe design [1]–[3]. In those papers, it was shown that the formulation can be used to solve problems with arbitrary geometries including geometrical singularities such as edges and corners. In the present paper, we have modified the boundary integral equations (BIEs) formulation for computing the impedance change in the presence of ideal tight cracks. Some unique features of this model include the allowance for arbitrarily shaped air core probes and test components that include singular geometries.

For the past several years, we have been developing an eddy current model, using the boundary element method (BEM). Last year, in particular, a BEM algorithm based on the Hertz potential approach was found and shown to be effective in dealing with complex part geometry, while keeping the computational resource requirement to a minimum [1–3]. This paper concerns a further extension of the model to include cracks.

This paper reports on development of simulation software for eddy current (EC) inspections.

The inspection of aircraft engine components using eddy current (EC) techniques has played a vital role in the nondestructive evaluation industry. The objective of the inspection is to determine the structural integrity of the components in a noninvasive manner. The effectiveness of an EC probe design is often evaluated in terms of the probability of detection (POD) of flaws. Many factors need to be considered when estimating the POD. These factors include scan and index spacings, operating frequencies, and flaw morphologies. Traditionally, the EC probe design cycle is iteratively performed experimentally until one either meets or exceeds the minimum required POD. Undoubtedly, this is a time consuming and expensive process since a new probe has to be constructed and tested every time the design is changed. A more sensible approach is to numerically simulate the functionality of probes so design improvements can be done iteratively using a computer under a CAD environment. The numerical probe design model is developed using the boundary integral equation (BIE) approach. By solving the BIEs numerically using the boundary element method (BEM), electromagnetic fields produced by the EC probes can be easily obtained through simple numerical integration.

This paper presents a comparative study of two different boundary integral equation (BIE) formulations applicable to eddy current (EC) problems.

In-service eddy current (EC) inspection of aircraft engine components is one of the most demanding NDE applications. Any defect in such high-performance components will have serious consequences, and hence reliable periodic inspections are required. Yet, it is known that significant improvements to the current implementation are desirable in various aspects. One of our recent modeling activities is directed toward developing a software simulator that is able to predict the performance of inspections and, hence, to examine any improvement ideas such as new probe designs on computers.

Detection of cracks close to an edge by conventional eddy current techniques is difficult, owing to the significant background signal from the edge. It is necessary to develop methods for minimizing the effect of the background signal, thereby increasing the probability of detection. The edge signal is known to be influenced by a number of factors and it is essential to characterize these in order to minimize its influence. This study aims at developing a good understanding of these factors so as to facilitate the development of such techniques. A boundary element method (BEM) approach was used to model the signal due to the edge and to compare with experimental measurements. Experiments were conducted on electro-discharge machined (EDM) slots in the vicinity of an edge using both absolute and differential probes. The influence of orientation of a differential probe on the signal from the crack and the edge was also studied. We report on the development of improved methods to reduce the influence of signal due to the edge by appropriate use of differential probes and with the aid of signal processing. An inexpensive physical technique which results in a improved detectability was also developed.

For many years, the inspection of components using eddy current (EC) techniques has been playing an important role in the nondestructive evaluation (NDE) industry. There are many factors which affect the probability of detection (POD) of flaws. One important consideration is how to select suitable EC probes to inspect a given test component with certain class of flaws. Commonly, the performance of an EC probe design for a scan is evaluated experimentally by physically constructing prototype probes and perform test scans. This approach is both time-consuming and expensive. In particular, the EC probe has to be reconstructed each time the probe design is changed and the design cycle typically requires a number of iterations before a satisfactory performance is achieved.

Eddy current inspection is highly sensitive to the position of the probe relative to the component being scanned. Ensuring that the probe maintains a constant lift-off and remains perpendicular to the surface being scanned is very import to the scanning process, as the signal strength is best under the these constraints. Variations of tilt or lift-off during a scan can introduce large, spurious signals that may mask flaw indications. Earlier work on scan automation used the probe signal to control the lift-off variations so that a constant lift-off could be maintained over a non-planar surface[1]. The present study focuses on achieving probe perpendicularity and constant lift-off all through a given scan, to attain accurate and rep eatable eddy current measurements.

A modified boundary integral equation/boundary element method (BIE/BEM) is being developed for eddy current problems in three dimensions. Maxwell’s equations governing the eddy current problems are formulated in two sets of BIE’s, one for the electric field and the other for the magnetic field. These BIE’s involve both the field and the normal derivative of the field, for both exterior (air) and interior (metal) regions. In addition to the usual set of interface conditions involving only the field, a set of interface conditions involving the normal derivatives of the field is derived by applying Maxwell’s equations near the interface. The present approach represents a departure from the existing BIE formulation for eddy current problems (see, e.g. [1–3]) in which normal derivatives of the field do not explicitly appear.

The Westinghouse Science and Technology Center has successfully demonstrated that its patented mercury modeling technique provides a unique experimental bridge between tubing eddy current inspection conditions and analytical modeling efforts [1,2,3]. This approach is particularly useful for developing field inspection applications in nuclear steam generator tubing where the characterization of primary water stress corrosion cracking (PWSCC) has presented a challenge for eddy current inspection technology. Significant progress has been reported recently toward developing analytical models of the eddy current signal structure [4,5] and toward new techniques for inverting crack size from the eddy current signal structure. These models are presently limited in ability to represent real-life complexities, and they also must be experimentally validated. The mercury modeling approach permits the controlled study of an unlimited range of discontinuity morphologies and sizes, as well as steam generator structural and geometric factors. The mercury model is used in this study to investigate the effects of crack morphology on newly proposed eddy current length sizing techniques compared with current industry practices. The primary emphasis is to model cracks that more closely represent the morphology of PWSCC compared with ideal notch-like shapes. Effects of crack shape, crack face connectivity, and the presence of multiple initiation-sites are demonstrated.

In this paper we report on the development of an eddy-current measurement model, which is a generalization of the one reported earlier [1-2]. Our objective is to establish a computer model that is capable of simulating eddy current NDE in generic inspection geometries. To achieve this goal, we started with an earlier version of the model applicable to a flat-plate geometry and a tightly closed crack [3]. The model has been generalized so that the current version can handle more general specimen geometries, including curved surfaces and corners. Other features of the original model were kept unchanged; for instance, it uses the boundary integral equation method, can handle tight cracks, and works for an air-core coil driven at arbitrary frequencies.

Model based POD estimates constitute a powerful tool for addressing a wide variety of issues relating to NDE reliability. These estimates can, for example not only provide insight into factors affecting detectability but also assist in determining optimum test parameters. Model based POD estimates can also play a crucial role as a vehicle for interpolating and extrapolating results obtained from experimental POD models. Such use can lead to significant cost benefits particularly in situations involving defects that are difficult and expensive to replicate in a laboratory in large numbers.

The purpose of this paper is to report on the development of an eddy current (EC) measurement model applicable to corner crack inspections. Naturally, corner cracks are more difficult to detect than those on flat surfaces, because the specimen edge itself gives a large response to the EC probe. The flaw signal, if any, tends to be obscured by the large edge signal. Thus, probe impedance should be determined more accurately than usual in order to extract flaw signals out of the background. Experimentally, this requires high-accuracy impedance measurements with rigid control over probe motion. In modeling point of view, this means that predictions should be made from an exact model, or at least from a model which can achieve the required level of accuracy [1–3].

We have developed an automated eddy current measurement system in our laboratory for quantitative nondestructive evaluation applications. The heart of the measurement system is a precision impedance analyzer capable of measuring impedance or any impedance related quantity over a wide range in frequency (102–108 Hz). Data acquisition, processing, analysis, and display is accomplished with a personal computer. Computer-controlled x-y positioning stages permit measurements to be obtained for either one- or two-dimensional scans of the specimen. In this article we describe the measurement system and give examples of its use to measure flaw signals with a uniform-field eddy current probe [1].

Among current NDE investigations, one of the important topics is the development of so-called probability-of-detection (POD) models. This activity is important because, given a POD model, one can examine inspection systems quantitatively in terms of flaw detectability. In the area of the eddy-current (EC) testing, there are a few applicable POD models in the literature [1,2]. In this paper, we will report on a generalization of the model constructed by one of the present authors [2]. After the generalization is done, the model becomes applicable to a wider variety of flaws than before.

Photoinductive imaging is a unique dual-mode NDE technique that combines eddy current and thermal wave methods. The photoinductive effect, upon which this method is based, is the thermally induced change in the impedance of an eddy current probe in proximity to a conducting surface that is illuminated with a modulated light source. The change in probe impedance is caused by the temperature-induced change in the conductivity of the specimen. Typical changes in probe impedance are small, on the order of a few ppm, but because they are synchronous with the light-beam modulation, lock-in techniques can be used to detect the signals, which can then be used to image surface or near-surface defects, voids, inclusions, or other thermal or structural inhomogeneities.

We determine the eigenstates of the bound-state Hamiltonian proposed in the preceding article using the covariant Bethe-Salpeter (BS) formalism. The interaction Hamiltonian is the invariant direct four-fermion interaction, and we use the invariant cutoff function. We evaluate the two-body irreducible kernel, considering a very wide class of Feynman diagrams including the ones with the infinite order, which we approximate and express in terms of parameters. This kernel is expected to be adequate for the $S$-wave bound states. We then solve the BS equation for the bound state exactly. We derive in this way a set of eigenvalue equations, which is formally identical to the set obtained in the preceding article. There appears no difficulty in explaining all the observed pseudoscalar and vector mesons by adjusting the parameters in the eigenvalue equations. These mesons include $\ensuremath{\pi}$, $\ensuremath{\rho}$, and all the way up to ${\ensuremath{\eta}}_{C}$ and $\ensuremath{\psi}(3100)$. In particular we can achieve this fitting independently of the cutoff parameters. Assuming that $\ensuremath{\Upsilon}$ is also a two-body bound state of $B\overline{B}$, where $B$ is the underlying bottom fermion, we determine not only the physical mass of $B$ but also the masses of all the expected pseudoscalar and vector bottom mesons. The confirmation of these masses is to be considered as a strong support to this approach.

The response of capacitive array sensors in the presence of flawed solid materials is simulated using finite elements and infinite elements with exponential decay. Conventional finite elements are used to model the critical regions near the probe and the surface of the solid. Infinite elements are used to represent the farfield conditions of the space surrounding the probe and the solid. The method is first applied to problems with analytic solutions to determine the accuracy of the results obtained using the infinite elements. The response of a capacitive array sensor is then simulated using a line integral which measures the relative change in admittance between flawed and unflawed solids. Examples of capacitive probe responses are given for several parametric variations of the flaw size and dielectric constant of the solid.

The non·abelian gauge field theory is formulated under a gauge fixing condition of non· differential form. The condition is characterized by two properties: (i) Time· like components of gauge fields are completely determined by the other components and source fields, as in the radiation gauge. (ii) Gauge functions are timely constant arbitrary functions, and then independent of field operators. The theory is proved to be "form-invariant" under local gauge transformations. Our formulation can be regarded as a legitimate formulation of an expedience in the radiation gauge formulation, where the field operator dependence of gauge functions is completely disregarded. Property (ii) makes our formulation deviate from the radiation gauge formulation to a large extent.

Capacitive array sensors are one among many electromagnetic techniques that can be used to detect flaws or other irregularities at or near the surface of materials. Capacitive sensors have an advantage over inductive sensors in that insulating materials may be interrogated as well as conducting materials. These sensors have seen some application in nondestructive evaluation, including flaw detection, the monitoring of porosity and thickness of thermal barrier coatings, dielectric cure monitoring, and robotic proximity sensing [1, 2]. Only surface features can be examined on metallic plates because the accumulation of surface charges blind the capacitive probe to interior features. In dielectric materials, both surface and subsurface features can be examined.

The eddy-current (EC) NDE method has been in use for quite some time, and efforts have been made to make it a fully quantitative method. To evaluate impedance signals for a given EC inspection system, one has to characterize the system as a whole, including both probes and specimens. In particular, until probes are characterized, the electromagnetic fields cannot be calculated. Naturally, the amount of numerical computation becomes a serious issue during the course of development. It is necessary to choose probes carefully so as to maximize the flaw-characterization capability, while keeping numerical tasks within a reasonable size. Probes that are suitable for quantitative assessment are presumably different in nature from those with maximum detection capability. Among all kinds of existing probes, the simplest characterizable probe is the uniform-field-eddy-current (UFEC) probe. In fact, a series of studies, both theoretical and experimental, were devoted to demonstrating potential capabilities of UFEC probes [1–9]. The present theoretical work is another entry in this series. The numerical procedure developed in this work is limited to the case where cracks are tightly closed. The procedure is nevertheless capable, in principle, of dealing with an arbitrary range of frequencies.

Two approaches to the problem of the AC magnetic field distribution inside conducting media are currently used. One consists of solving the quasi-stationary Maxwell differential equations subject to certain boundary conditions. Another one is based upon certain boundary integral equations that involve only the field components at the boundary surface [1,2]. Aside from some numerical advantage in treating unknowns of a lower dimension, the second method fits better the common formulation of the eddy current NDE, which aims at determination of the total impedance of a metal part (or its change due to a defect). The impedance is related to the integrated energy flux through the metal surface and as such it can be expressed in terms of the field components at the surface exclusively.

We describe a new NDE imaging technique based on eddy current detection of thermal waves. NDE applications of thermal-wave imaging techniques using detection by photoacoustic, piezoelectric, radiometric, and optical beam deflection methods are well known [1,2]. Saniie and Luukkala [3] previously reported generating thermal waves in metals with eddy currents, but we believe this is the first instance of detecting thermal waves with an eddy current probe. Since the method depends on detecting photothermally induced changes in the inductance of an eddy current coil, we have coined the term “photoinductive imaging” to describe this new technique.