Abstract EN:

The first part of this PhD is devoted to the theoretical study and the numerical resolution of a piezoelectric problem formulated in a strongly heterogeneous periodical environment. Within a homogenization mathematical theory framework, the formalism and the numerical resolution of a periodic problem with two components, of which only one is piezoelectric, have been investigated. First, the case of a fibrous structure has been studied; then a new finite element, of piezoelectric type, has been designed and studied, allowing the resolution of auxiliary problems previously written. Finally the resolution of these problems has been programmed within a framework on two scales. These developments have been applied to the work of modelization of the human cortical bone. A second subject dealing with the modelization and the numerical simulation of transport of radon in an aquifer simultaneously in gas from and dissolved from has been studied. Then the second part of the thesis concerns the modelization of the transport of radon at the same time in gas form and dissolved form in a saturated field. When a seismic activity reaches the lower part of a surface aquifer, it generates a production of radon. Various physical phenomena occur to transport this chemical element to the surface where measurement of the concentrations proves to be relevant for the follow-up of this seismic activity. A previous work (carried out by D. Calugaru) proposed that radon was immediately dissolved and consisted in simulating the transport of this element. In the real process, radon is initially created in gas form and other gases have been associated. Therefore, the same problem is solved, introducing the presence of gas to the lower part of the aquifer.