<p>The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.</p>

The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.

Cette thèse concerne l’étude de la propagation d’ondes acoustiques dans des structures hétérogènes. Le but essentiel de ces travaux est de confronter des résultats d’expériences numériques effectuées dans le domaine physique (espace, temps) à des prédictions analytiques pour la propagation des ondes de surface SH le long d’un demi-espace stratifié périodique produisant des spectres discontinus de dispersion pour les ondes, ainsi que pour la diffusion multiple dans des milieux aléatoires inclusionnaires (fissures, cavités). Le code numérique FDTD développé lors de cette étude a permis, en autres choses, de corroborer quantitativement les fenêtres spectrales théoriques d’existence des ondes de surface dans les demi-espaces périodiques,ainsi que de montrer des zones de validité fréquentielles des approches analytiques de diffusion multiple concernant les propriétés effectives de milieux aléatoires. / The study is concerned with acoustic waves in elastic media with a different nature of in homogeneity consisting in either periodically continuous or piece wise variation of material properties, or in random sets of defects embedded into a homogeneous matrix, with a given statistical distribution. The scope of problems is topical in non-destructive testing and other applications of ultrasound.Theoretical methods describing involved acoustic phenomena (complex dispersion features, coherent wave in random media, ensemble average techniques) often rely on certain a priori assumptions which render numerical verification especially important.The thesis presents results of analytical modelling of the propagation of surface acoustic waves along periodic half-space, for which the dispersion spectrum is rather complex (discontinuous spectrum of propagation for the surface waves). A 2nd order FDTD numerical code has been developed in order to perform numerical experiments in the space and time domains, and to corroborate the analytical predictions in the frequency domain. A good agreement of simulated results with analytical modelling demonstrates applicability and consistency of the numerical tool. Finally, the code has been used for extracting numerically the coherent wave regime (mean wave over ensemble averaging of the positions of scatterers) for the acoustic propagation in different types of populations of randomly distributed scatterers. The results indicate ranges of validity of some multiple scattering analytical techniques.