4 research outputs found
Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework
This study deals with elastic-wave identification of discrete heterogeneities (inclusions) in an otherwise homogeneous ``reference'' solid from limited-aperture waveform measurements taken on its surface. On adopting the boundary integral equation (BIE) framework for elastodynamic scattering, the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary, elastic moduli, and mass density. For an optimal performance of the gradient-based search methods suited to solve the problem, explicit expressions for the shape (i.e. boundary) and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing BIEs. Making use of the message-passing interface, the proposed sensitivity formulas are implemented in a data-parallel code and integrated into a nonlinear optimization framework based on the direct BIE method and an augmented Lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible (or overly distorted) trial inclusion configurations. Numerical results for the reconstruction of an ellipsoidal defect in a semi-infinite solid show the effectiveness of the proposed shape-material sensitivity formulation, which constitutes an essential computational component of the defect identification algorithm
DĆ©veloppement et utilisation de mĆ©thodes asymptotiques d'ordre Ć©levĆ© pour la rĆ©solution de problĆØmes de diffraction inverse
The purpose of this work was to develop new methods to address inverse problems in elasticity, taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptotic expansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity B in a 3D elastic domain. In this goal, we focus on the study of functionals J(Ba) quantifying the misfit between B and a trial homogeneity Ba . Such functionals are to be minimized w.r.t. some or allthe characteristics of the trial inclusion Ba (location, size, mechanical properties ...) to find the best agreement with B. To this end, we produce an expansion of J with respect to the size of Ba , providing a polynomial approximation easier to minimize. This expansion, established up to the sixth order in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space.The main purpose of this second part is to characterize a microstructured two-phases layered 1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigenvalues of the so-called interior transmission problem (ITP). To providea convenient invertible model, while accounting for the microstructure effects, we rely on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method to recover the macroscopic parameters (length and material contrast) of such inclusion. To access the key features of the microstructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.L'objectif de ce travail est le dĆ©veloppement de nouvelles mĆ©thodes pour aborder certains problĆØmes inverses en Ć©lasticitĆ©, en tirant parti de la prĆ©sence d'un petit paramĆØtre dans ces problĆØmes pour construire des approximations asymptotiques d'ordre Ć©levĆ©.La premieĢre partie est consacreĢe aĢ lāidentification de la taille et la position dāune inhomogeĢneĢiteĢ B enfouie dans un domaine eĢlastique tridimensionnel. Nous nous concentrons sur lāeĢtude de fonctions-couĢts J(Ba) quantifiant lāeĢcart entre B et une heĢteĢrogeĢneĢiteĢ ātestā Ba . Une telle fonction-couĢt peut en effet eĢtre minimiseĢe par rapport aĢ tout ou partie des caracteĢristiques de lāinclusion ātestā Ba (position, taille,proprieĢteĢs meĢcaniques ...) pour eĢtablir la meilleure correspondance possible entre Ba et B. A cet effet, nous produisons un deĢveloppement asymptotique de J en la taille de Ba , qui en constitue une approximation polynomiale plus aiseĢe aĢ minimiser. Ce deĢveloppement, eĢtabli jusquāaĢ lāordre six, est justifieĢ par une estimation du reĢsidu. Une meĢthode dāidentification adapteĢe est ensuite preĢsenteĢe et illustreĢe par des exemples numeĢriques portant sur des obstacles de formes simples dans lāespace libre.Lāobjet de la seconde partie est de caracteĢriser une inclusion microstructureĢe de longueur L, modeĢliseĢe en une dimension, composeĢe de couches de deux mateĢriaux alterneĢs peĢriodiquement, en supposant que les plus basses de ses freĢquences propres de transmission (TEs) sont connues. Ces freĢquences sont les valeurs propres dāun probleĢme dit de transmission inteĢrieur (ITP). Afin de disposer dāun modeĢle propice aĢ lāinversion, tout en prenant en compte les effets de la microstructure, nous nous reposons sur des approximations de lāITP exact obtenues par homogeĢneĢisation. A partir du modeĢle homogeĢneĢiseĢ dāordre 0, nous eĢtablissons tout dāabord une meĢthode simple pour deĢterminer les parameĢtres macroscopiques(longueur et contrastes mateĢriaux) dāune telle inclusion. Pour avoir acceĢs aĢ la peĢriode de la microstructure, nous nous inteĢressons ensuite aĢ des modeĢles homogeĢneĢiseĢs dāordre eĢleveĢ, pour lesquels nous soulignons le besoin de conditions aux limites adapteĢes