# Identification of material damage in two-dimensional domains using the SQUID-based nondestructive evaluation system

TL;DR: In this article, the identification of two-dimensional spatial domains arising in the detection and characterization of material damage is considered, and approximation schemes are developed applying a finite element Galerkin approach.

Abstract: Problems on the identification of two-dimensional spatial domains arising in the detection and characterization of material damage are considered. For electromagnetic nondestructive evaluation systems, observations of the magnetic flux from the front surface are used in a output least-squares approach. Parameter estimation techniques based on the method of mappings are discussed and approximation schemes are developed applying a finite-element Galerkin approach. Theoretical convergence results for computational techniques are given and results are applied to numerical experiments to demonstrate the efficacy of the proposed schemes.

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TL;DR: In this article, the forward and inverse simulation of electromagnetic nondestructive evaluation (ENDE) signals are introduced, where a database type strategy using precalculated unflawed potential field data is introduced.

Abstract: In this paper, some progresses in numerical techniques mainly made in our research group for the forward and inverse simulation of electromagnetic nondestructive evaluation (ENDE) signals are introduced. For the first part, efficient forward analysis schemes for the simulation of eddy current testing (ECT), remote field ECT (RFECT) and magnetic flux leakage testing (MFLT) signals are described respectively, in addition to some numerical examples. Fast and accurate ECT signal simulation is realised by introducing a database type strategy using precalculated unflawed potential field data. To meet the high accuracy requirement of the simulation of RFECT signals, a hybrid scheme using 2D and 3D geometry and a new formula for pickup signal are proposed. To improve the efficiency of MFLT signal simulation, a fast scheme is developed based on a FEM–BEM hybrid code of polarisation method. In addition, a phenomenological method is also described in the first part, which is developed for the qualitative estimation ...

53 citations

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TL;DR: This exposition considers an inverse Dirichlet problem for harmonic functions that arises in the mathematical modelling of electrostatic imaging methods and outlines a recently developed method that is based on conformal mapping techniques.

46 citations

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TL;DR: In this paper, the authors present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce magnetic fields either above or below the sample, or that are magnetically silent (i.e. no magnetic field anywhere outside the sample).

Abstract: Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique inverse problem. This problem leads to an analysis of the kernel of the related magnetization operators, which also has relevance to the 'equivalent source problem' in the case of measurements taken from just one side of the magnetization. We characterize the kernel of the operator relating planar magnetization distributions to planar magnetic field maps in various function and distribution spaces (e.g., sums of derivatives of Lp (Lebesgue spaces) or bounded mean oscillation (BMO) functions). For this purpose, we present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce a magnetic field either above or below the sample, or that are magnetically silent (i.e. no magnetic field anywhere outside the sample). For example, we show that a thin-plate magnetization is silent (i.e. in the kernel) when its normal component is zero and its tangential component is divergence free. In addition, we show that compactly supported magnetizations (i.e. magnetizations that are zero outside of a bounded set in the source plane) that do not produce magnetic fields either above or below the sample are necessarily silent. In particular, neither a nontrivial planar magnetization with fixed direction (unidimensional) compact support nor a bidimensional planar magnetization (i.e. a sum of two unidimensional magnetizations) that is nontangential can be silent. We prove that any planar magnetization distribution is equivalent to a unidimensional one. We also discuss the advantages of mapping the field on both sides of a magnetization, whenever experimentally feasible. Examples of source recovery are given along with a brief discussion of the Fourier-based inversion techniques that are utilized.

40 citations

01 Jan 2013

TL;DR: In this paper, the authors present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce magnetic fields either above or below the sample, or that are magnetically silent (i.e. no magnetic field anywhere outside the sample).

Abstract: Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique inverse problem. This problem leads to an analysis of the kernel of the related magnetization operators, which also has relevance to the 'equivalent source problem' in the case of measurements taken from just one side of the magnetization. We characterize the kernel of the operator relating planar magnetization distributions to planar magnetic field maps in various function and distribution spaces (e.g., sums of derivatives of Lp (Lebesgue spaces) or bounded mean oscillation (BMO) functions). For this purpose, we present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce a magnetic field either above or below the sample, or that are magnetically silent (i.e. no magnetic field anywhere outside the sample). For example, we show that a thin-plate magnetization is silent (i.e. in the kernel) when its normal component is zero and its tangential component is divergence free. In addition, we show that compactly supported magnetizations (i.e. magnetizations that are zero outside of a bounded set in the source plane) that do not produce magnetic fields either above or below the sample are necessarily silent. In particular, neither a nontrivial planar magnetization with fixed direction (unidimensional) compact support nor a bidimensional planar magnetization (i.e. a sum of two unidimensional magnetizations) that is nontangential can be silent. We prove that any planar magnetization distribution is equivalent to a unidimensional one. We also discuss the advantages of mapping the field on both sides of a magnetization, whenever experimentally feasible. Examples of source recovery are given along with a brief discussion of the Fourier-based inversion techniques that are utilized.

37 citations

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Kobe University

^{1}TL;DR: In this paper, a stochastic inverse methodology arising in electromagnetic imaging is proposed for the identification of electromagnetic material parameters and emphasis is on one dimensional scattering of a dielectric slab, which can be solved numerically using the finite-difference time-domain method (FDTD).

Abstract: : This paper is concerned with a stochastic inverse methodology arising in electromagnetic imaging. Nondestructive testing using guided microwaves covers a wide range of industrial applications including early detection of anomalies in conducting materials. The focus of this paper is the identification of electromagnetic material parameters and emphasis is on one dimensional scattering of a dielectric slab. The direct problem can be solved numerically using the finite-difference time-domain method (FDTD). The Markov Chain Monte Carlo method (MCMC) is applied to the inversion problem. Some successful results of computational experiments are demonstrated in order to show the feasibility and applicability of the proposed method.

15 citations

### Cites background from "Identification of material damage i..."

...Although prior work exists using nonlinear least square methods [2, 3, 4, 5, 6], it is well known that the problem mentioned above has many solutions due to the fact that it is ill-posed....

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##### References

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01 Jan 1978

TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.

Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.

10,258 citations

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TL;DR: The main purpose of this paper is to advocate the use of the graph associated with Tikhonov regularization in the numerical treatment of discrete ill-posed problems, and to demonstrate several important relations between regularized solutions and the graph.

Abstract: When discrete ill-posed problems are analyzed and solved by various numerical regularization techniques, a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector. In particular, the graph associated with Tikhonov regularization plays a central role. The main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill-posed problems. The graph is characterized quantitatively, and several important relations between regularized solutions and the graph are derived. It is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic L-shaped “corner” of the graph.

3,585 citations

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TL;DR: The gradient projection method was originally presented to the American Mathematical Society for solving linear programming problems by Dantzig et al. as discussed by the authors, and has been applied to nonlinear programming problems as well.

Abstract: more constraints or equations, with either a linear or nonlinear objective function. This distinction is made primarily on the basis of the difficulty of solving these two types of nonlinear problems. The first type is the less difficult of the two, and in this, Part I of the paper, it is shown how it is solved by the gradient projection method. It should be noted that since a linear objective function is a special case of a nonlinear objective function, the gradient projection method will also solve a linear programming problem. In Part II of the paper [16], the extension of the gradient projection method to the more difficult problem of nonlinear constraints and equations will be described. The basic paper on linear programming is the paper by Dantzig [5] in which the simplex method for solving the linear programming problem is presented. The nonlinear programming problem is formulated and a necessary and sufficient condition for a constrained maximum is given in terms of an equivalent saddle value problem in the paper by Kuhn and Tucker [10]. Further developments motivated by this paper, including a computational procedure, have been published recently [1]. The gradient projection method was originally presented to the American Mathematical Society

1,142 citations

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01 Jan 1996

TL;DR: In this article, the authors present a survey of the well-posedness of Abstract Structural Models, including Shells, Plates and Beams, and their application in smart materials technology and control applications.

Abstract: Smart Materials Technology and Control Applications. Modeling Aspects of Shells, Plates and Beams. Patch Contributions to Structural Equations. Well-Posedness of Abstract Structural Models. Estimation of Parameters and Inverse Problems. Damage Detection in Smart Material Structures. Infinite Dimensional Control and Galerkin Approximation. Implementation of Finite-Dimensional Compensators. Modeling and Control in Coupled Systems. Bibliography. Notation. Index.

391 citations