Low-frequency scattering from perfectly conducting spheroidal bodies in a conductive medium with magnetic dipole excitation

Inductive electromagnetic means that are currently employed in the exploration of the Earth’s subsurface and embedded voluminous bodies often call for an intensive use, primary at the modeling stage and later on at the inversion stage, of analytically demanding tools of field calculation. Under the aim of modeling implementation, this contribution is concerned with some interesting aspects of the low-frequency interaction of arbitrarily orientated (i.e. three-dimensional) time-harmonic magnetic dipoles, with 3-D perfectly conducting spheroidal bodies embedded in an otherwise homogeneous conductive medium. For many practical applications involving buried obstacles such as Earth’s subsurface electromagnetic probing at low-frequency or any other physical cases (e.g. geoelectromagnetics), non-axisymmetric spheroidal geometry approximates sufficiently such kind of metallic shapes. On the other hand, our analytical approach deals with prolate spheroids, since the corresponding results for the oblate spheroidal geometry can be readily obtained through a simple transformation. The particular physical model concerns a solid impenetrable (metallic) body under a magnetic dipole excitation, where the scattering boundary value problem is attacked via rigorous low-frequency expansions for the incident, scattered and total electric and magnetic fields in terms of positive integral powers of (ik), that is (ik)ⁿ for n ? 0, where k stands for the complex wavenumber of the exterior medium. The purpose of the modeling is to contribute to a simple yet versatile tool to infer information on an unknown body from measurements of the three-component electric and magnetic fields nearby. Our goal is to obtain the most important terms of the low-frequency expansions of the electromagnetic fields, that is the static (for n = 0) and the dynamic (n = 1, 2, 3) terms. In particular, for n = 1 there are no incident fields and thus no scattered ones, while for n = 0 the Rayleigh electromagnetic expression is easily obtained in terms of infinite series. Emphasis is given on the calculation of the next two non-trivial terms (at n = 2 and at n = 3) of the aforementioned fields. Consequently, those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) Laplace equations and they are given in compact fashion, as infinite series expansions for n = 2 or finite forms for n = 3. Nevertheless, the difficulty of the Poisson’s equation that has to be solved for n = 2 is presented, whereas our analytical approach demands the use of the well-known cut-off method in order to obtain an analytical closed solution. Finally, this research adds useful reference results to the already ample library of scattering by simple shapes using analytical methods.     

Electromagnetic Modeling of a Damaged Ferromagnetic Metal Tube by a Volume Integral Equation Formulation

We propose a volume integral equation formulation for eddy-current nondestructive evaluation of ferromagnetic tubes affected by volumetric defects. We solve the system of integral equations (introduced into the model by application of Green's theorem) by the method of moments for both fictitious electric and magnetic currents within the defects, and calculate variations of impedance of the probes by the reciprocity theorem. We have made thorough comparisons of our results with finite-element-method simulations and with experimental data as well.     

Determination of conductivity profiles by time-domain reflectometry

A new numerical method for the reconstitution of the inhomogeneous media conductivity is described. The conducting medium is nonmagnetic, and its relative permittivity equals one. Its frequency-independent conductivity varies normally to its interface. The medium is normally illuminated by a transverse electromagnetic (TEM) plane wave of causal time dependence. The reconstruction process is a direct approach to the inverse problem. It is based on a space time discretization of an exact integral equation. The conductivity profile is determined step-by-step without approximations other than numerical. Some examples are given to illustrate the most interesting properties of the method; special attention is given to the simulation of experimental errors.     

Étude numérique des antennes épaisses par l’équation intégrale d’Albert et Synge

The Albert and Synge integral equation allows to determine the current on a rotationnally symmetric conducting antenna of arbitrary meridian shape in the emission mode. This equation is deduced from the use of generalized boundary conditions and therefore possesses a regular kernel. This regularity is a cause of a numerical ill-conditionning of the linear systems deduced from such equations by use of moments methods. An iterative algorithm of inversion due to Le Foll allows to overcome the difficulties linked with this illconditionning. Computations have been made on various structures (spheres, cylinders with or without sharp edges, ...) and their validity has been checked from results already available or by comparison with experimental ones. Therefore such integral approach appears particularly interesting in the case of thick structures, the longitudinal or transversal dimensions of which may attain a few wavelengths. Such structures are useful in many domains. The development of wide bandwidth electromagnetic sensors, the coupling of active or passive devices from the millimeter waves up to the infrared are two important exemples.     

Experimental investigation of a diffraction tomography technique in fluid ultrasonics

Qualitative ultrasonic imaging of cylindrical fluid targets is investigated by a diffraction tomography technique applied to experimental data. The principles of the image formulation are stated and an experimental setup is described. Experimental difficulties related to the short wavelength used and respective advantages in collecting the data, either with a mechanically scanned single transducer or with an electronically scanned array of transducers, are emphasized. Representative images of simply structured phantoms and of real biological bodies are obtained in spite of the small number of views available.     

Optimization of ultrasonic arrays design and setting using a differential evolution

Optimization of both design and setting of phased arrays could be not so easy when they are performed manually via parametric studies. An optimization method based on an Evolutionary Algorithm and numerical simulation is proposed and evaluated. The Randomized Adaptive Differential Evolution has been adapted to meet the specificities of the non-destructive testing applications. In particular, the solution of multi-objective problems is aimed at with the implementation of the concept of pareto-optimal sets of solutions. The algorithm has been implemented and connected to the ultrasonic simulation modules of the CIVA software used as forward model. The efficiency of the method is illustrated on two realistic cases of application: optimization of the position and delay laws of a flexible array inspecting a nozzle, considered as a mono-objective problem; and optimization of the design of a surrounded array and its delay laws, considered as a constrained bi-objective problem.     

Optimization techniques and inverse problems: Reconstruction of conductivity profiles in the time domain

The iterative probing of an inhomogeneous lossy dielectric slab, whose conductivity is unknown, is discussed. The probing is done in the time domain from the measurements of the field on the interface when this slab is illuminated by a known field. An exact integral formulation is used. Minimization of a cost function, characteristic of the discrepancy between the measured field and the field which would be scattered by a known slab, is specified by the optimization theory. The notion of the adjoint state of the field is introduced. The influence of some parameters of this minimization is studied. Great importance is given to the sensitivity of the probing as function of the amplitude of errors in the data. Such an iterative probing appears fast, accurate, and efficient, even in the case of large errors.     

The localized nonlinear approximation in ellipsoidal geometry: a novel approach to the low-frequency scattering problem

The localized nonlinear approximation provides a very effective method within the integral equation framework of electromagnetic scattering theory. Existing results in this direction are confined to the spherical geometry alone. In this work, we extend the known results for the sphere to the case of ellipsoidal geometry which can approximate genuine three-dimensional scattering obstacles. Reduction to prolate and oblate spheroids, where rotational symmetry is present, is also discussed.