Διάδοση κυμάτων σε ανομοιογενή μέσα

Abstract

The scope of this doctoral thesis is, first, to develop an analytical, in principle, method for the solution of the two-dimensional scattering problem of time-harmonic elastic plane waves by a homogeneous orthotropic scatterer, second, to establish the complete theoretical framework, which is necessary for the application of the Linear Sampling Method (LSM) to the problem of reconstructing the support of twodimensional elastic anisotropic inclusions embedded in isotropic media and, third, to derive an extension of the Factorization Method (FM) to the inverse elastic scattering problem by penetrable isotropic bodies for time-harmonic plane wave incidence. Aconcise description of the contents of the thesis is outlined below. Chapter one contains a detailed bibliographical search, which is related to the analytical and numerical methods (with emphasis on the former) usually employed for the solution of the direct scattering problem by anisotropic elastic bodies as well as to those inverse scattering techniques, which are usually referred to as sampling and probe methods and, in particular, the LSM and the FM. Chapter two commences with a brief discussion of some fundamental results from the linearized theory of dynamic elasticity. The problem of a rigorous analysis of the elasticity equation governing the elastic behaviour of an orthotropic material in two dimensions is then addressed. This analysis, which is based on a suitable diagonalization applied to the underlying differential system and a plane wave expansion of the sought field, results in a Fourier series expansion for the displacement field describing the elastic deformations of the orthotropic medium and is complemented by the results of appendix A. A mathematical model for the solution of the associated transmission scattering problem, taking advantage of the aforementioned expansion, is also settled and analyzed. The details of its numerical treatment can be found in appendix B. Finally, numerical results for several inclusion geometries and a system thereof with material properties characterized by the cubic symmetryclass -a special case of the orthotropic class of symmetry- are presented. In chapter three, the LSM is extended to the case of a two-dimensional homogeneous anisotropic inclusion embedded in an isotropic background medium. The concepts of the elastic Herglotz function, the elastic far-field operator and the corresponding far-field equation, on which the formulation of the LSM heavily relies, are first introduced. Then, the proposed inverse scattering scheme is introduced and discussed in detail. By means of an appropriate operator decomposition of the far-field operator,the main theorem of the method, concerning the characterization of the behaviour of an approximate solution to the far-field equation at the boundary of the scatterer, is proved. In the end of the third chapter, the performance of the LSM is examined by applying it to a set of different geometric configurations of the elastic inclusion, filled with a cubic anisotropic material. An investigation of the effect of the various parameters entering the problem, such as the scatterer’s degree of anisotropy, the polarization of the elastic point source located at the sampling point and the noise level in the synthetic far-field data, on the reconstructed geometric profiles’ quality,is carried out. In the fourth chapter, the FM is elaborated for the shape reconstruction of a penetrable isotropic elastic body from the knowledge of the far-field pattern of the scattered fields for plane incident waves. The theoretical analysis is conducted in three dimensions and focuses on deriving a factorization of the far-field operator, which is the cornerstone for the applicability of the particular inversion scheme, and investigating thorougly the properties of the involved operators. This investigation gave birth to a number of interesting by-products and one of them, namely, a regularity estimate for the solution of a particular form of the corresponding interior transmission problem, is the subject matter of appendix C. By means of the proposed factorization, a series of theorems, which finally lead to an explicit characterization of the scattering obstacle, is then proved. In the end of the chapter, the performance of the investigated inverse scattering technique is demonstrated by applying it to specific two-dimensional elastic scatterer reconstruction problems involving different scatterer configurations and various choices for their constitutive parameters. The effect of using different levels of additive random noise in the forward synthetic data and combining results obtained for different polarizations of the elastic point source located at the sampling point, on the quality of the reconstructed profiles, is also examined. Finally, chapter five draws the conclusions that flow from the foregoing chapters and discusses the contribution of this doctoral thesis. A brief discussion about possible future studies is also included.