La topographie d'une surface se compose généralement de plusieurs échelles, depuis l'échelle macroscopique (sa géométrie physique), jusqu'aux échelles microscopiques ou atomiques appelées rugosité. L'évolution spatiale et géométrique de la rugosité fournit une description plus complète de la surface, et une interprétation physique de certains problèmes importants tels que le frottement et les mécanismes d'usure pendant le contact mécanique entre deux surfaces. La topographie d'une surface rugueuse est de nature aléatoire, ce qui traduit par des altitudes spatialement corrélées, appelées pics et vallées. La relation entre leurs densités de probabilité et leurs propriétés géométriques sont les aspects fondamentaux qui ont été développés dans cette thèse, en utilisant la théorie des champs aléatoires et la géométrie intégrale. Un modèle aléatoire approprié pour représenter une surface rugueuse a été mis en place et étudié au moyen des paramètres les plus significatifs, dont les changements influencent la géométrie des ensembles de niveaux (excursion sets) de cette surface. Les ensembles de niveaux ont été quantifiés par des fonctionnelles connues sous le nom de fonctionnelles de Minkowski, ou d'une manière équivalente sous le nom de volumes intrinsèques. Dans un premier temps, les volumes intrinsèques des ensembles de niveaux sont calculés analytiquement sur une classe de modèles mixtes, qui sont définis par la combinaison linéaire d'un champ aléatoire Gaussien et d'un champ de t-student (t-field), et ceux d'une classe de champs aléatoires asymétriques appelés skew-t. Ces volumes sont comparés et testés sur des surfaces produites par des simulations numériques. Dans un second temps, les modèles aléatoires proposés ont été appliqués sur des surfaces réelles acquises à partir d'une cupule d'UHMWPE (provenant d’une prothèse totale de hanche) avant et après les processus d'usure. Les résultats ont montré que le champ aléatoire skew-t est un modèle mieux approprié pour décrire la rugosité de surfaces usées, contrairement aux modèles adoptés dans la littérature. Une analyse statistique, basée sur le champ aléatoire skew-t, est ensuite proposée pour détecter les niveaux des pics/vallées de la surface usée et pour décrire le comportement et la fonctionnalité de la surface usée. / Surface topography is, generally, composed of many length scales starting from its physical geometry, to its microscopic or atomic scales known by roughness. The spatial and geometrical evolution of the roughness topography of engineering surfaces avail comprehensive understanding, and interpretation of many physical and engineering problems such as friction, and wear mechanisms during the mechanical contact between adjoined surfaces. Obviously, the topography of rough surfaces is of random nature. It is composed of irregular hills/valleys being spatially correlated. The relation between their densities and their geometric properties are the fundamental topics that have been developed, in this research study, using the theory of random fields and the integral geometry.An appropriate random field model of a rough surface has been defined by the most significant parameters, whose changes influence the geometry of its excursion. The excursion sets were quantified by functions known as intrinsic volumes. These functions have many physical interpretations, in practice. It is possible by deriving their analytical formula to estimate the parameters of the random field model being applied on the surface, and for statistical analysis investigation of its excursion sets. These subjects have been essentially considered in this thesis. Firstly, the intrinsic volumes of the excursion sets of a class of mixture models defined by the linear combination of Gaussian and t random fields, then for the skew-t random fields are derived analytically. They have been compared and tested on surfaces generated by simulations. In the second stage, these random fields have been applied to real surfaces measured from the UHMWPE component, involved in application of total hip implant, before and after wear simulation process. The primary results showed that the skew-t random field is more adequate, and flexible for modelling the topographic roughness. Following these arguments, a statistical analysis approach, based on the skew-t random field, is then proposed. It aims at estimating, hierarchically, the significant levels including the real hills/valleys among the uncertain measurements. The evolution of the mean area of the hills/valleys and their levels enabled describing the functional behaviour of the UHMWPE surface over wear time, and indicating the predominant wear mechanisms.

Surface topography is, generally, composed of many length scales starting from its physical geometry, to its microscopic or atomic scales known by roughness. The spatial and geometrical evolution of the roughness topography of engineering surfaces avail comprehensive understanding, and interpretation of many physical and engineering problems such as friction, and wear mechanisms during the mechanical contact between adjoined surfaces. Obviously, the topography of rough surfaces is of random nature. It is composed of irregular hills/valleys being spatially correlated. The relation between their densities and their geometric properties are the fundamental topics that have been developed, in this research study, using the theory of random fields and the integral geometry.An appropriate random field model of a rough surface has been defined by the most significant parameters, whose changes influence the geometry of its excursion. The excursion sets were quantified by functions known as intrinsic volumes. These functions have many physical interpretations, in practice. It is possible by deriving their analytical formula to estimate the parameters of the random field model being applied on the surface, and for statistical analysis investigation of its excursion sets. These subjects have been essentially considered in this thesis. Firstly, the intrinsic volumes of the excursion sets of a class of mixture models defined by the linear combination of Gaussian and t random fields, then for the skew-t random fields are derived analytically. They have been compared and tested on surfaces generated by simulations. In the second stage, these random fields have been applied to real surfaces measured from the UHMWPE component, involved in application of total hip implant, before and after wear simulation process. The primary results showed that the skew-t random field is more adequate, and flexible for modelling the topographic roughness. Following these arguments, a statistical analysis approach, based on the skew-t random field, is then proposed. It aims at estimating, hierarchically, the significant levels including the real hills/valleys among the uncertain measurements. The evolution of the mean area of the hills/valleys and their levels enabled describing the functional behaviour of the UHMWPE surface over wear time, and indicating the predominant wear mechanisms.

The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.