Non-destructive techniques are used widely in the metal industry in order to control the quality of materials. Eddy current testing is one of the most extensively used non-destructive techniques for inspecting electrically conductive materials at very high speeds that does not require any contact between the test piece and the sensor. This paper includes an overview of the fundamentals and main variables of eddy current testing. It also describes the state-of-the-art sensors and modern techniques such as multi-frequency and pulsed systems. Recent advances in complex models towards solving crack-sensor interaction, developments in instrumentation due to advances in electronic devices, and the evolution of data processing suggest that eddy current testing systems will be increasingly used in the future.

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Non-Destructive Techniques Based on Eddy Current Testing

https://physicstoday.scitation.org/doi/pdf/10.1063/1.3047693

Introduction to the Theory of Coherence and Polarization of Light

Ground penetrating radar (GPR) is a tool for indirectly looking at underground objects (such as graves), gravel and sand layers, and other underground structures. The information or data received by GPR is like an x-ray or map of the underground. In fact, GPR uses electromagnetic (EM) waves, as xray machines do, but GPR uses radio waves, which have a longer wavelength (see Figure A1). The wavelength, or the length of one wave, is the fundamental difference between the forms of electromagnetic energy. For example, the wavelength of x-rays range from about 10 billionths of a meter to about 10 trillionths of a meter, whereas radio waves can be a few meters long.

Ground Penetrating Radar

This review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks. These inverse problems are considered mainly for three-dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e., fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.

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Inverse problems in elasticity

We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using different popular models, such as for example Maxwell's equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion approximation. These models are also representative of a broader class of inverse problems. Starting out from the original binary approach of Santosa for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, such as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in two dimensions and in three dimensions for demonstrating the performance of level set techniques in realistic applications.

https://iopscience.iop.org/article/10.1088/0266-5611/22/4/R01/pdf

Level set methods for inverse scattering

A family of new integral equations (NIE) is proposed in this paper, which are transformed from the original Lippmann–Schwinger integral equation. It can be shown that the NIE can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast. Equipping the previously proposed twofold subspace-based optimization method with such NIE, the new inversion method is able to solve inverse scattering problems with strong scatterers, like with high contrast and/or large dimensions (in terms of wavelength) ones. Furthermore, such a family of NIE could provide a convenient tool to appraise reconstructed results. Several representative numerical tests are carried out, using both synthetic and experimental data, to verify the efficacy of the new inversion method.

A New Integral Equation Method to Solve Highly Nonlinear Inverse Scattering Problems

A new technique is proposed to reconstruct faulty wiring networks from the time-domain reflectometry (TDR) response. The developed method is also for characterization of defects in the branches of the network. The direct problem (propagation along the cables) is modeled by RLCG circuit parameters computed by finite element method (FEM) and the finite-difference time domain (FDTD) method. Genetic algorithms (GAs) are used to solve the inverse problem. The proposed method allows to accurately locate wire faults. Some examples are presented to validate and illustrate the ability of this reconstruction method.

/pdf/detection-of-defects-in-wiring-networks-using-time-domain-1xm09fjwwu.pdf

Detection of Defects in Wiring Networks Using Time Domain Reflectometry

The nonlinearized reconstruction of the cross-sectional contour of a homogeneous, possibly multiply connected obstacle buried in a half-space from time-harmonic wave field data collected above this half-space in both transverse magnetic (TM) and transverse electric (TE) polarization cases is investigated. The reconstruction is performed via controlled evolution of a level set that was pioneered by Litman et al (Litman A, Lesselier D and Santosa F 1998 Inverse Problems 14 685-706) but at this time restricted to free space for TM data collected all around the sought obstacle. The main novelty of the investigation lies in the following points: from the rigorous contrast-source domain integral formulation (TM) and integral-differential formulation (TE) of the direct scattering problems in the buried obstacle configuration, and from appropriately cast adjoint scattering problems, we demonstrate, by processing min-max formulations of an objective functional J made of the data error to be minimized, that its derivatives with respect to the evolution time t are given in closed form. They are contour integrals involving the normal component of the velocity field of evolution times the product of direct and adjoint fields (TM), or of the scalar product of gradients of such fields (TE) at t. This approach only calls for the analysis of the well posed direct and adjoint scattering problems formulated from the TM and TE Green systems of the unperturbed layered environment and, unusually, it avoids the differentiation of state fields. Other contributions of the paper come from exhibiting and analysing via comprehensive numerical experimentation how and under which conditions evolutions of level sets involving velocities opposite to shape gradients perform in demanding configurations including two disjoint obstacles, constitutive materials strongly less or more refractive than the embedding material, aspect-limited and noisy monochromatic data, in the severe TE case as well as in the more menial TM case. A comparison with a binary-specialized modified-gradient solution method is also led for several, more and more lossy embedding half-spaces. Rules of thumb for effectiveness of the inversions and pending theoretical and computational questions are outlined in conclusion.

https://iopscience.iop.org/article/10.1088/0266-5611/17/4/335/pdf

Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation

In the framework of inverse scattering techniques for microwave imaging, this paper proposes an approach based on the integration between a multiscaling procedure and the level-set-based optimization in order to properly deal with the shape reconstruction of multiple and disconnected homogeneous scatterers. The effectiveness and robustness of the proposed approach is assessed against both synthetic and experimental data. A selected set of results concerned with complex shapes is presented and discussed.

/pdf/multiple-shape-reconstruction-by-means-of-multiregion-level-3urlmvy391.pdf

Multiple-Shape Reconstruction by Means of Multiregion Level Sets

In the framework of nondestructive eddy-current testing, an inversion method is proposed. It relies on a metamodel-based optimization method: a fast optimization by means of a particle swarm algorithm is performed wherein the exact forward computations are replaced by an appropriate metamodel. The latter is obtained by radial basis function interpolation over a database previously generated by sequential design. Both synthetically generated and laboratory-controlled experimental data are used to illustrate its performances. Comparisons to inversions by support vector machines, now common for defect characterization, are carried out. The proposed method appears to be an useful tool for decision analysis, highlighting the likeliest results.

Marc Lambert

Papers

A New Integral Equation Method to Solve Highly Nonlinear Inverse Scattering Problems

Detection of Defects in Wiring Networks Using Time Domain Reflectometry

Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation

Multiple-Shape Reconstruction by Means of Multiregion Level Sets

Adaptive Metamodels for Crack Characterization in Eddy-Current Testing