Note publique d'information : La première partie de la thèse est relative à l’étude théorique et la résolution numérique
d’un problème piézoélectrique posé dans un milieu fortement hétérogène périodique.
Un problème périodique à deux composants dont seul l'un d'eux est piézoélectrique
a été étudié, dans le cadre de la théorie mathématique de l’homogénéisation, d'abord
dans son formalisme puis sous l'aspect résolution numérique.
Note publique d'information : 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.
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