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

A major problem in the use of ultrasound or microwaves for purposes of nondestructive testing or medical imaging is the computational complexity of solving the inverse scattering problem that arises in such applications. This is due to the fact that in order to achieve satisfactory resolution and sufficient penetration of the wave into the material it is often necessary to use frequencies in the resonance region. In this case the inverse scattering problem is not only improperly posed but also nonlinear and even in the case of two dimensions the time needed to solve such problems can be prohibitive. To date the time consuming nature of the problem has mainly been dealt with by the introduction of various innovative schemes that avoid the use of volume integral equations and instead rely on finite difference or finite element methods (cf. [5], [8]). However, for large scale problems (for example those involving imaging of the human body) the problem of computational complexity remains a serious problem for any practitioner. In this paper we would like to propose a different approach to this problem that, although still in its infancy, has the promise of providing rapid solutions to a number of inverse scattering problems of practical significance.

qualitative-methods-in-inverse-scattering-theory

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in and . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

https://iopscience.iop.org/article/10.1088/0266-5611/19/6/057/pdf

The linear sampling method in inverse electromagnetic scattering theory

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.

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Handbook of mathematical methods in imaging

This book for the first time unifies the theories of acoustic, electromagnetic, and elastic waves and discusses the many physical and geometric aspects of their interactions with obstacles It offers a complete introduction to scattering theory, and goes on to discuss low frequency scattering in particular A significant number of results are previously unpublished, and the book fills many gaps in the existing literature Included is an extended bibliography covering the whole existing literature on low frequency scattering, making this an invaluable reference for researchers

Low Frequency Scattering

The present work is concerned with the extension of the factorization method to the inverse elastic scattering problem by penetrable isotropic bodies embedded in an isotropic host environment for time-harmonic plane wave incidence. Although the former method has been successfully employed for the shape reconstruction problem in the field of elastodynamic scattering by rigid bodies or cavities, no corresponding results have been recorded, so far, for the very interesting (both from a theoretical and a practical point of view) case of isotropic elastic inclusions. This paper aims at closing this gap by developing the theoretical framework which is necessary for the application of the factorization method to the inverse transmission problem in elastodynamics. As in the previous works referring to the particular reconstruction method, the main outcome is the construction of a binary criterion which determines whether a given point is inside or outside the scattering obstacle by using only the spectral data of the far-field operator.

http://iopscience.iop.org/article/10.1088/0266-5611/23/1/002/pdf

The factorization method in inverse elastic scattering from penetrable bodies

In this paper the sampling method for the shape reconstruction of a penetrable scatterer in three-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. We establish the interior transmission problem in the weak sense and consider the case where the nonhomogeneous boundary data are generated by a dyadic source point located in the interior domain. Assuming that the inclusion has absorbing behaviour, we prove the existence and uniqueness of the weak solution of the interior transmission problem. In this framework the main theorem for the shape reconstruction for the transmission case is established. As for the cases of the rigid body and the cavity an approximate far-field equation is derived with the known dyadic Green function term with the source point an interior point of the inclusion. The inversion scheme which is proposed is based on the unboundedness for the solution of an equation of the first kind. More precisely, the support of the body can be found by noting that the solution of the integral equation is not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interior points.

The linear sampling method for the transmission problem in three-dimensional linear elasticity

Une onde plane longitudinale ou transversale est diffusee par une region limitee immergee dans un milieu elastique isotrope et homogene infini. La region peut etre un diffuseur rigide ou une cavite. Representation integrale du champ de deplacement; representation de la section efficace de diffusion que l'onde incidente soit longitudinale ou transversale. A l'aide des potentiels de Papkovich et des techniques de basse frequence les problemes de diffusion sont reduits a une sequence iterative de problemes de potentiel qui sont resolus par developpements en fonctions harmoniques appropriees

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The low-frequency theory of elastic wave scattering

In this paper the linear sampling method for the shape reconstruction of a penetrable non-dissipative scatterer in two-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. Assuming that the inclusion has non-absorbing behaviour, we consider this problem as a degenerate case of a non-dissipative anisotropic inclusion. Results for the existence and uniqueness of the weak solution of the interior transmission problem are obtained. In this framework the main theorem for the shape reconstruction for the transmission case is established. As in the previous works referring to the linear sampling method in acoustics and linear elasticity, the inversion scheme which is proposed is based on the unboundedness of the solution of an equation of the first kind having as the known term the far-field of the free-space Green dyadic, generated by a source inside the inclusion approaching the boundary. Numerical results are presented for different inclusion geometries, assuring the simple and efficient implementation of the algorithm, using synthetic data derived from the boundary element method.

http://iopscience.iop.org/article/10.1088/0266-5611/19/3/305/pdf

The linear sampling method for non-absorbing penetrable elastic bodies

Reciprocity and scattering theorems for the normalized spherical scattering amplitude for elastic waves are obtained for the case of a rigid scatterer, a cavity and a penetrable scattering region. Depending on the polarization of the two incident waves reciprocity relations of the radial-radial, radial-angular, and angular-angular type are established. Radial and angular scattering theorems, expressing the corresponding scattering amplitudes via integrals of the amplitudes over all directions of observation, as well as their special forms for scatterers with inversion symmetry are also provided. As a consequence of the stated scattering theorems the scattering cross-section for either a longitudinal, or a transverse incident wave is expressed through the forward value of the radial, or the angular amplitude, correspondingly. All the known relative theorems for acoustic scattering are trivially recovered from their elastic counterparts.

Kiriakie Kiriaki

Papers

The factorization method in inverse elastic scattering from penetrable bodies

The linear sampling method for the transmission problem in three-dimensional linear elasticity

The low-frequency theory of elastic wave scattering

The linear sampling method for non-absorbing penetrable elastic bodies

On the scattering amplitudes for elastic waves