Imaging the continuous conductivity profile within layered metal structures using inductance spectroscopy

This paper presents an inverse method for determining the conductivity distribution of a flat, layered conductor using a multifrequency electromagnetic sensor. Eddy-current sensors are used in a wide range of nondestructive testing applications. Single-frequency sensors are very common; however, the potential of an eddy-current sensor with spectroscopic techniques offers the ability to extract depth profiles and examine more fully the internal structure of the test piece. In this paper, the forward solution for a small right-cylindrical air-cored coil placed next to a layered conductor is based on the analytic solution provided by the transfer matrix approach. For an inverse solution, a modified Newton-Raphson method was used to adjust the conductivity profile to fit a set of multifrequency inductances in a least-squared sense. The approximate Jacobian matrix (sensitivity matrix) was obtained by the perturbation method. Numerical results of the forward solution are provided for cases of step, continuous conductivity profiles. Good estimates for the conductivity profile were obtained. Experimental eddy-current tests are performed by taking the difference in inductance of the coil when placed next to a reference conductor and next to a layered conductor over the range 100 kHz - 1 MHz. Inverse results based on experimental and simulated data verified this method.     

Eddy current inversion for layered conductors

By making multifrequency eddy current measurements on a layered conductor, it is possible to acquire information on the depth dependence of the conductivity. We consider an inversion problem in which coil impedance data are used to determine either the layer thicknesses or layer conductivities. The algorithm is based on a well known forward model which gives the impedance of an air cored coil above a stratified conductor from a closed form expression. In the forward calculation, estimates of the unknown material parameters are used to get tentative predictions of the measurements. Differences between these predictions and measured impedances are expressed in terms of a global error that is minimized iteratively with the aid of a descent algorithm by varying the parameters of the structure. Examples of minimization searches for layer parameters are given.This article is dedicated to Professor Bertram A. Auld on the occasion of his 70th birthday and his retirement from Stanford University.     

Image reconstruction in three-dimensional magnetostatic permeability tomography

Magnetostatic permeability tomography is an imaging technique that attempts to reconstruct the permeability distribution of an object using magnetostatic measurement data. The data for image reconstruction are external magnetic field measurements on the surface of the object due to an applied magnetostatic field. Theoretically, the normal and tangential components of the magnetic field in the surface uniquely define the internal isotropic permeability distributions. However, the inverse permeability problem is an ill-posed nonlinear problem. Regularization is needed for a stable solution. In this paper, we present a numerical method to solve the reconstruction problem in three dimensions using a regularized Gauss-Newton scheme. We have solved the forward problem using an edge finite-element method and we have employed an efficient technique to calculate the Jacobian matrix. The permeability of the object is assumed to be linear and isotropic. We present the reconstruction results for the permeability using synthetically generated data with additive noise.     

Analytical solutions in eddy current testing of layered metals withcontinuous conductivity profiles

In this paper exact analytical expressions for the impedance of a cylindrical air-core coil above a layered metal structure whose conductivity varies continuously with depth are presented. Although the model is general, attention is focused on three conductivity profiles: the linear, the quadratic and the exponential. The derived expressions for the impedance change for each profile could provide a useful tool for the solution of the inverse problem: that of determining the conductivity from variable frequency measurements of the impedance. Furthermore, the obtained final formulas contain elegant mathematical functions and show a substantially higher computational efficiency with respect to existing methods     

Reconstruction of Two-Dimensional Randomly Rough Surfaces Based on Bidirectional Reflectance Distribution Function

This article presents an inverse method for reconstructing two-dimensional randomly rough surfaces based on the available (experimental or given) data of the bidirectional reflectance distribution function (BRDF). The Maxwell’s equations of electromagnetic waves are applied to describe the light scattering process of rough surfaces by accounting for the near-field effect. Such a forward problem is numerically solved with the finite-difference time-domain algorithm. The inverse scattering problem of reconstructing the surface profile is handled by means of an optimization technique—the particle swarm optimizer algorithm. As an example, reconstruction of a Gaussian rough surface is conducted based on the experimental data of BRDFs. The retrieved results of the surface profile are compared with those measured by atomic force microscopy from the samples, which shows that the reconstruction algorithm can provide the credible prediction of surface profiles. The reconstruction approach studied in this study can make reliable predictions of the actual or required surface profiles.     

Radio-wave diagnostics of plane layered dielectrics using the inverse problem

We have developed a method for the radio-wave nondestructive testing of exfoliations in a dielectric layer. The method is based on the multifrequency sounding in 5-mm wavelength range and the solution of the inverse problem with determining the thicknesses of separate layers of a dielectric structure. We have also determined experimentally the precision and resolution of this method.     

Multifrequency near-field acoustic tomography and holography of 3D subbottom inhomogeneities

A method of coherent multifrequency acoustic tomography and holography of spatially localized subbottom inhomogeneities in shallow seas is proposed. This method is based on solving of the near-field inverse scattering problem that makes it possible to realize a subwavelength resolution. It involves the analysis of measurement data obtained by the 2D transversal scanning with the source-receiver system along the sea bottom, over the area of sounded inhomogeneities. The solution begins with the Born approximation, where the original 3D integral equation for the scattered field is reduced to the 1D Fredholm equation of the first kind relative to the depth profile of the lateral spectrum of inhomogeneities. When solving this integral equation for each pair of spectral components, the generalized discrepancy method is in use. Then, corrections to the Born approximation can be obtained in the proposed iterative procedure. For distributed inhomogeneities, the inverse Fourier transform of the retrieved spectrum gives their 3D distribution that can be visualized as tomography images. For solid targets, this spectrum is used to obtain their shape (i.e. to solve the problem of computer holography). Corresponding results of the numerical simulation are presented.     

层状地基一维固结电阻网法解答

考虑层状地基一维固结问题的复杂性,现有解答较为繁杂同时不便应用。该文基于Liebmann 电阻网原理,推导了初始孔压沿深度任意分布、荷载随时间任意变化作用下的孔压、平均固结度的解答。该解对于各种复杂条件均具有统一的矩阵表达式,便于编程计算和工程应用。通过与精确解的比较论证了该文方法的有效性。最后,对三层地基一维固结问题进行了分析和讨论。     

How to characterize soft magnetic materials by measuring magneticflux density in a rotating field apparatus

This paper deals with the characterization of the magnetic permeability of soft magnetic materials under a rotating magnetic field. The paper reviews the principle of the rotating-flux-density device used for measurement of flux density, then describes the mathematical method used to calculate permeability from the measurements. The method combines direct and inverse solutions and is based on a functional minimization sequence. An example demonstrates the validity of the method. Finally, the paper discusses the uniqueness of the solution and its sensitivity     

Basic equations for incompressible,non-Newtonian fluids in curvilinear,non-orthogonal and accelerated coordinate systems

Summary The analytical solution for the average displacements and stresses, in the case of in-plane harmonic motion for a randomly inhomogeneous layered half-space, is derived. The method used is based on the fundamental matrix and neglect of third and higher order correlations. The resulting system of integrodifferential equations is solved by Laplace transform. The results are used to calculate the dynamic-stiffness matrix. The closed form solution for the case of exponential correlation function is obtained.     

Thermal Transport Properties of Layered Materials: Identification by a New Numerical Algorithm for Transient Measurements

Transient methods are widely used to determine thermal transport properties. In some situations they can be used for homogeneous media to measure several properties either simultaneously or separately. In this context an analytic model is available and a well-posed inverse problem of parameter identification has to be solved. The examination of composite media is more complicated. The algorithm proposed here allows simultaneous determination of the thermal conductivity and thermal diffusivity of layered dielectrics by transient measurements. It is based on a plane source that acts both as a resistive heater and temperature sensor. For the technique to be successful two essential aspects have to be considered: firstly, the mathematical modeling of the measured data (the forward problem) and secondly, the problem of ill-posedness of the inverse problem. For the proposed measurement configuration, a new fast data analysis algorithm based on an analytic solution for the forward problem is presented. In principle, a numerical solution such as an FEM solution of the heat conduction equation can be used instead of the analytical one, but the computational effort is much greater. The inverse problem is formulated as an output-least-squares problem, which leads to a transcendent algebraic system of equations. The method was successfully tested for different situations.Paper presented at the Fifteenth Symposium on Thermophysical Properties, June 22--27, 2003, Boulder, Colorado, U.S.A.     

A Gauss–Newton full-waveform inversion for material profile reconstruction in viscoelastic semi-infinite solid media

An inversion framework employing a Gauss–Newton method is developed to reconstruct material profiles in heterogeneous, viscoelastic, semi-infinite domains. In particular, a full-waveform inversion approach is investigated to image the elastic and attenuating parameters of a layered media. To account for the viscoelasticity of the medium, a Generalized Maxwell Body with one spring and two Maxwell elements in parallel (GMB2) is adopted in the forward and inverse wave propagation problems. Perfectly-matched-layers were introduced as wave absorbing buffers to simulate the semi-infinite extent of the domain. Using transient wave equations endowed with the GMB2 constitutive relation and the PML, a partial-differential-equations-constrained optimization scheme was implemented that lead to classic KKT (Karush–Kuhn–Tucker) conditions including time-dependent state, adjoint, and time-invariant control problems. An optimal solution of the viscoelastic parameters was obtained using a reduced-space approach based on a line search algorithm where the search direction was computed by the Gauss–Newton method. Considerable improvements on the accuracy and convergence rate of solutions were made by the developed Gauss–Newton inversion procedure compared to previous research using the Fletcher–Reeves method.     

反自反矩阵的二次特征值反问题及其最佳逼近

二次特征值反问题是二次特征值问题的一个逆过程,在结构动力模型修正领域中应用非常广泛．本文由给定的部分特征值和特征向量,利用矩阵分块法、奇异值分解和Moore-Penrose广义逆,分析了二次特征值反问题反自反解的存在性,得出了解的一般表达式．然后讨论了任意给定矩阵在解集中最佳逼近解的存在性和唯一性．最后给出解的表达式和数值算法,由算例验证了结果的正确性．     

The boundary-element method for the determination of a heat source dependent on one variable

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions of the direct problem and a supplementary condition, called an overdetermination. In this problem, if the heat source is taken to be space-dependent only, then the overdetermination is the temperature measurement at a given single instant, whilst if the heat source is time-dependent only, then the overdetermination is the transient temperature measurement recorded by a single thermocouple installed in the interior of the heat conductor. These measurements ensure that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method with the discrepancy principle or the L-curve criterion for the choice of the regularization parameter. The boundary-element method (BEM) is developed for solving numerically the inverse problem and numerical results for some benchmark test examples are obtained and discussed     

Computation of electromagnetic wave scattering from an arbitrarythree-dimensional inhomogeneous dielectric object

A scattering matrix formulation for scattering of electromagnetic fields by an arbitrary three-dimensional dielectric objects is presented. It is based on an extended moment method that digitizes the scattering equation by dividing the scatterer as well as the measurement region into small cells. The Green's function within each cell is derived analytically by integrating over the cell. The accuracy of the method is demonstrated by computing vector scattering fields from a uniform dielectric sphere and comparing the results with those calculated using the Mie scattering formula. The formulation also provides an inverse solution, namely, determination of the dielectric profile of the scatterer from the scattering field measured in a finite region outside the scatterer     

Multifrequency analysis for the Helmholtz equation

In this paper a new numerical method for the multifrequency analysis of the three-dimensional Helmholtz equation is introduced. The collocation boundary element method (BEM) is used for the discretisation of the problem. The identity of the Fourier transform with respect to the wave number is applied to the matrix of the resulting linear system. The analytical form and some important properties are derived. Some numerical examples for the solution are presented.     

Electrical resistance tomography for locating inclusions using analytical boundary element integrals and their partial derivatives

Electrical resistance tomography is a novel non-intrusive technique to image the electrical conductivity distribution. The technique is characterized as a nonlinear, ill-posed inverse problem with many unknowns. A numerical method based on boundary element method is presented in this paper, aiming to obtain the boundary of inclusions with a constant conductivity. The forward problem employs the complete electrode model and the size of system matrix is reduced. The Jacobian matrix is established through the partial derivatives of element integrals and linearization of the system. The analytical integrations for quadratic elements are presented, where higher-order integrals are expressed by lower-order ones for rapid calculation. The iterative Levenberg–Marquardt method is adopted to solve the inverse problem.     

Performance-based assessment of durability and prediction of RC structure service life: transport properties as input data for physical models

This paper deals with an approach based on durability indicators (DIs), intended for the assessment of the durability and the prediction of the service life (SL) of reinforced concrete structures with respect to reinforcement corrosion. The methods of assessment of the DIs, more specifically of the transport properties effective chloride diffusion coefficient and “intrinsic” liquid water permeability, are first investigated. Direct experimental methods, indirect methods based on simple analytical formulas, as well as numerical inverse analysis which involves the same physical models as used for prediction, are compared. Very good agreement is pointed out between the results obtained by the various methods for a broad range of materials. The effect of SCM, as well as of accelerated carbonation in the case of permeability, is also investigated. 1-D numerical models of moisture or ionic isothermal transport are then introduced, where the selected DIs are the main input data. Simulations carried out with the moisture transport model are compared to moisture profiles measured by gamma-ray attenuation, in order to validate the model and the input data (permeability). With regard to chloride ingress, examples of application of the multispecies transport model (saturated conditions) are provided for concentration profile prediction, in lab and in field conditions, as well as SL prediction.     

Inverse scattering of buried inhomogeneous biaxial dielectric cylinders coated on a conductor

The inverse scattering of buried inhomogeneous biaxial dielectric cylinders coated on a conductor with known cross section is investigated. Dielectric cylinders with known cross section coated on a conductor of unknown permittivities are buried in one half space and scatter a group of unrelated waves incident from another half space where the scattered field is recorded. By proper arrangement of the various unrelated incident fields, the difficulties of ill‐posedness and nonlinearity are circumvented, and the permittivity distribution can be reconstructed through simple matrix operations. The algorithm is based on the moment method and the unrelated illumination method. Numerical results are given to demonstrate the capability of the inverse algorithm. Good reconstruction is obtained both with and without Gaussian noise in measured data. In addition, the effect of noise on the reconstruction result is also investigated. © 2008 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 18, 228–236, 2008; Published online in Wiley InterScience (www.interscience.wiley.com).     

Synthesis of symmetric and asymmetric planar optical waveguides

A novel and accurate refractive index profile synthesis method for planar optical waveguides is presented and demonstrated using the transmitted near-electric-field-data. This method is based on the inverse transmission-line (TL) technique. From Maxwell's equations, a TL equivalent circuit (electric T-circuit) for the refractive index profile of a planar optical waveguide is derived. The authors demonstrate how to use this model to carry out the inverse problem and synthesise the exact refractive index profile numerically from near-field-data. The TL method can reconstruct arbitrary refractive index profiles for planar optical waveguides that support singlemode or multi- modes. The cases of both symmetric and asymmetric arbitrary refractive index profile planar waveguides are discussed. The accuracy of the reconstructed waveguides is examined numerically.