3D direct and inverse solvers for eddy current testing of deposits in steam generator

We consider the inverse problem of estimating the shape profile of an unknown deposit from a set of eddy current impedance measurements. The measurements are acquired with an axial probe, which is modeled by a set of coils that generate a magnetic field inside the tube. For the direct problem, we validate the method that takes into account the tube support plates, highly conductive part, by a surface impedance condition. For the inverse problem, finite element and shape sensitivity analysis related to the eddy current problem are provided in order to determine the explicit formula of the gradient of a least square misfit functional. A geometrical-parametric shape inversion algorithm based on cylindrical coordinates is designed to improve the robustness and the quality of the reconstruction. Several numerical results are given in the experimental part. Numerical experiments on synthetic deposits, nearby or far away from the tube, with different shapes are considered in the axisymmetric configuration.Comment: 3

On the asymptotics of a Robin eigenvalue problem

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to $-\infty$ as the perturbation goes to zero. We prove that in this case, Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criteria to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g.~partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture's contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the $F_\sharp$-factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the well-posedness of the forward problem leads to an admissibility condition on the fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the $F_\sharp$-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM)

A generalized formulation of the Linear Sampling Method with exact characterization of targets in terms of farfield measurements

International audienceWe propose and analyze a new formulation of the Linear Sampling Method that uses an exact characterization of the targets shape in terms of the so-called farfield operator (at a fixed frequency). This characterization is based on constructing nearby solutions of the farfield equation using minimizing sequences of a least squares cost functional with an appropriate penalty term. We first provide a general framework for the theoretical foundation of the method in the case of noise-free and noisy measurements operator. We then explicit applications for the case of inhomogeneous inclusions and indicate possible straightforward generalizations. We finally validate the method through some numerical tests and compare the performances with classical LSM and the factorization methods

Asymptotic models for scattering problems from unbounded media with high conductivity

We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of $L2$ solution operators. We also discuss numerical approximation of obtained GIBC (up to order 3) and numerically test theoretical convergence rates

A robust inversion method for quantitative 3D shape reconstruction from coaxial eddy-current measurements

This work is motivated by the monitoring of conductive clogging deposits in steam generator at the level of support plates. One would like to use monoaxial coils measurements to obtain estimates on the clogging volume. We propose a 3D shape optimization technique based on simplified parametrization of the geometry adapted to the measurement nature and resolution. The direct problem is modeled by the eddy current approximation of time-harmonic Maxwell's equations in the low frequency regime. A potential formulation is adopted in order to easily handle the complex topology of the industrial problem setting. We first characterize the shape derivatives of the deposit impedance signal using an adjoint field technique. For the inversion procedure, the direct and adjoint problems have to be solved for each coil vertical position which is excessively time and memory consuming. To overcome this difficulty, we propose and discuss a steepest descent method based on a fixed and invariant triangulation. Numerical experiments are presented to illustrate the convergence and the efficiency of the method

Numerical analysis of the factorization method for EIT with a piecewise constant uncertain background

International audienceWe extend the factorization method for electrical impedance tomography to the case of background featuring uncertainty. We first describe the algorithm for the known but inhomogeneous backgrounds and indicate expected accuracy from the inversion method through some numerical tests. Then we develop three methodologies to apply the factorization method to the more difficult case of a piecewise constant but uncertain background. The first one is based on a recovery of the background through an optimization scheme and is well adapted to relatively low-dimensional random variables describing the background. The second one is based on a weighted combination of the indicator functions provided by the factorization method for different realizations of the random variables describing the uncertain background. We show through numerical experiments that this procedure is well suited to the case where many realizations of the measurement operators are available. The third strategy is a variant of the previous one when measurements for the inclusion-free background are available. In this case, a single pair of measurements is sufficient to achieve comparable accuracy to the deterministic case