L’objectif de ce travail de thèse est de prendre en compte à la fois l’hétérogénéité, l’anisotropie et des incertitudes dans la simulation 3D de la propagation d’ondes élastiques. Pour ce faire, dans un premier temps, on modélise le champ de propriétés mécaniques, ici le champ de tenseur d’élasticité, par un modèle de champ stochastique 3D des matrices définie-positives. La construction de ce modèle de champ est essentiellement fondée sur celle de Soize [2008]. Notre modèle conserve ainsi les propriétés principales du modèle de Soize comme le paramétrage minimal contrôlant l’amplitude de la fluctuation et la taille caractéristique de la variabilités patiale, le comportement local a priori arbitrairement anisotrope (anisotropie triclinique) et les propriétés mathématiques fondamentales. De plus, un nouveau paramètre est introduit dans ce modèle pour imposer un niveau d’anisotropie moyen souhaité. Dans un deuxième temps, on effectue des adaptations du code de calcul d’éléments finis spectraux, à savoir le code parallèle SPEC3D, afin d’une part de générer les réalisations du champ stochastique du tenseur d’élasticité et d’autre part de prendre en compte l’anisotropie dans la résolution numérique du problème élastodynamique. Des études paramétriques utilisant SPEC3D sont ensuite réalisées mettant en évidence les influences de l’anisotropie et des paramètres d’hétérogénéité sur la propagation d’ondes sismiques. En particulier, elles démontrent une dépendance directe entre la longueur de corrélation du champ de propriétés et le temps caractéristique d’apparition de la diffusion. Ce régime se manifeste par l’équipartition d’énergie entre les mouvements irrotationnels et rotationnels. / The aim of this thesis is to take into account the heterogeneity, the anisotropy and the uncertainties within 3D numerical simulation of elastic waves propagation. Firstly, the elasticity tensor field is modeled by means of a stochastic tensor-valued field. Its construction is generalized from the model of Soize [2008]. Hence, our model preserves principle properties of the former : a small set of parameters controlling the whole dispersion and the characteristic size of spatial variability, a local behavior being a priori arbitrary anisotropic (triclinic anisotropy) andothers essential mathematical properties. Moreover, a new parameter is added in order to impose a desired anisotropy mean level. Secondly, we carry out adaptations of an existing spectral finite elements-based elastic waves simulation software, namely the SPEC3D parallel computing code. On the one hand a sample generator of the elasticity random field model is implemented and on the other hand anisotropic material behavior is introduced in the elastodynamic solver. Finally, numerical parametric studies are performed using SPEC3D highlighting influences of heterogeneity and anisotropy on elastic waves behavior. In particular, it is observed that the characteristic time beyond which a multiple scattering pattern can be approximated by a diffusion regime directly depends on the correlation length of elasticity tensor field model. This time is detected by an energy equipartition between rotational and irrotational movements.

Particle-surface interactions govern a myriad of interface phenomena, that span from technological applications to naturally occurring biological processes. In the present work, particle-collector DLVO interactions are computed with the grid-surface integration (GSI) technique, previously applied to the computation of particle colloidal interactions with anionic surfaces patterned with O(10 nm) cationic patches. The applicability of the GSI technique is extended to account for interactions with collectors covered with topographical and chemical nanoscale heterogeneity. Surface roughness is shown to have a significant role in the decrease of the energy barriers, in accordance with experimental deposition rates that are higher than those predicted by the DLVO theory for smooth surfaces. An energy- and force-averaging technique is presented as a reformulation of the GSI technique, to compute the mean particle interactions with random heterogeneous collectors. A statistical model based on the averaging technique is also developed, to predict the variance of the interactions and the particle adhesion thresholds. An excellent agreement is shown between the models' predictions and results obtained from GSI calculations for large number of random heterogeneous collectors. Brownian motion effects for particle-collector systems governed by nanoscale heterogeneity are analyzed by introducing stochastic Brownian displacements in particle trajectory equations. It is shown that for the systems under consideration and particle sizes usually used in experiments, it is reasonable to neglect the effects of Brownian motion entirely. Computation of appropriately defined P ́eclet numbers that quantify the relative importance of shear, colloidal and Brownian forces validate that conclusion. An algorithm for the discretization of spherical surfaces into small equal-area elements is implemented in conjunction with the GSI technique and mobility matrix calculations of particle velocities, to obtain interactions and dynamic behaviors of patchy particles in the vicinity of uniform flat collectors. The patchy particle and patchy collector systems are compared in detail, through the computation of statistical measures that include adhesion probabilities and maximum residence times per patch. The lessened tendency of the patchy particle to adhere on the uniform collector is attributed to a larger maximum residence time per patch, which precludes interactions with multiple surface nano-features at a given simulated time. Also briefly described are directions for future work, that involve the modeling of two heterogeneous surfaces, and of surfaces covered with many types of heterogeneity, such as patches, pillars and spring-like structures that resemble polymer brushes or cellular receptors.