78 research outputs found
Inverse problems in elasticity
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
Recovering source location, polarization, and shape of obstacle from elastic scattering data
We consider an inverse elastic scattering problem of simultaneously
reconstructing a rigid obstacle and the excitation sources using near-field
measurements. A two-phase numerical method is proposed to achieve the
co-inversion of multiple targets. In the first phase, we develop several
indicator functionals to determine the source locations and the polarizations
from the total field data, and then we manage to obtain the approximate
scattered field. In this phase, only the inner products of the total field with
the fundamental solutions are involved in the computation, and thus it is
direct and computationally efficient. In the second phase, we propose an
iteration method of Newton's type to reconstruct the shape of the obstacle from
the approximate scattered field. Using the layer potential representations on
an auxiliary curve inside the obstacle, the scattered field together with its
derivative on each iteration surface can be easily derived. Theoretically, we
establish the uniqueness of the co-inversion problem and analyze the indicating
behavior of the sampling-type scheme. An explicit derivative is provided for
the Newton-type method. Numerical results are presented to corroborate the
effectiveness and efficiency of the proposed method.Comment: 29 pages, 11 figure
Approximation by multipoles of the multiple acoustic scattering by small obstacles and application to the Foldy theory of isotropic scattering.
50 (avec 1,5 interligne)International audienceThe asymptotic analysis, carried out in this paper, for the problem of a multiple scattering of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy's method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and taking into account of self-interactions
The Ellipsoidal Harmonics in Solving Inverse Scattering Problems
Η παρούσα διπλωματική εργασία έχει ως κεντρικό θέμα την επίλυση αντστρόφων προβλημάτων σκέδασης ακουστικών και ηλεκτρομαγνητικών κυμάτων για ελλειψοειδή με χρήση των ελλειψοειδών αρμονικών συναρτήσεων. Περγράφονται τα προβλήματα σκέδασης επίπεδων ακουστικών και ηλεκτρομαγνητικών κυμάτων με αρμονική χρονική εξάρτηση για ελλειψοειδές σκεδαστή με διάφορες συνοριακές συνθήκες. Η μελέτη του ελλειψοειδούς συστήματος συντεταγμένων, οδηγεί στον ορισμό των ελλειψοειδών αρμονικών συναρτήσεων, οι οποίες υπεισέρχονται στα προβήματα σκέδασης μέσω της θεωρίας χαμηλών συχνοτήτων. Παρουσιάζεται η διαδικασία που ακολουθείται για τον υπολογισμό των προσεγγίσεων χαμηλών συχνοτήτων για ελλειψοειδή. Περιγράφονται τα αντίστροφα προβλήματα σκέδασης για ελλειψοειδείς σκεδαστές. Πεπερασμένος αριθμός μετρήσεων δεδομένων μακρινού πεδίου ή κοντινού πεδίου, οδηγούν στον προσδιορισμό του μεγέθους και του προσανατολισμού ενός αγνώστου ελλειψοειδούς σκεδαστή. Στην περίπτωση διαπερατού σκεδαστή, προσδιορίζονται επιπλέον φυσικές παράμετροι του εσωτερικού του. Αντίστοιχα αποτελέσματα για την περίπτωση της σφαίρας και του σφαιροειδούς υπολογίζονται θεωρώντας τα σχήματα αυτά ως γεωμετρικούς εκφυλισμούς του ελλειψοειδούς για κατάλληλες τιμές των γεωμετρικών παραμέτρων του.The main subject of this study is the solution of inverse acoustic and electromagnetic scattering problems for ellipsoids using the ellipsoidal harmonics. The scattering problems of time-harmonic acoustic and electromagnetic plane waves by an ellipsoidal scatterer for various boundary conditions imposed on its surface are considered. The study of the ellipsoidal coordinate system, leads to the definition of the ellipsoidal harmonics, which enter in the scattering problems via the low-frequency theory. The methodology which leads to the derivation of low-frequency approximations for ellipsoids is presented. Inverse scattering problems for acoustic and electromagnetic waves for an ellipsoidal scatterer are described. A finite number of measurements of far-field data or near-field data leads to the specification of the size and the orientation of an unknown ellipsoidal scatterer. For the case of penetrable scatterer, physical parameters of its interior are also obtained. Corresponding results for the cases of the sphere and the spheroid are derived, considering them as geometrically degenerate cases of the ellipsoid for appropriate values of its geometrical parameters
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Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University
This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield
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