Méthodes de domaine fictif pour des problèmes elliptiques avec conditions aux limites générales en vue de la simulation numérique d'écoulements diphasiques.

摘要

This work is dedicated to the introduction of two original fictitious domain methods for the resolution of elliptic problems (mainly convection-diffusion problems) with general and eventually mixed boundary conditions: Dirichlet, Robin or Neumann. The originality lies in the approximation of the immersed boundary by an approximate interface derived from the fictitious domain Cartesian mesh, which is generally not boundary-fitted to the physical domain. The same generic numerical scheme is used to impose the embedded boundary conditions. Hence, these methods require neither a surface mesh of the immersed boundary nor the local modification of the numerical scheme. We study two modellings of the immersed boundary. In the first one, called spread interface, the approximate immersed boundary is the union of the cells crossed by the physical immersed boundary. In the second one, called thin interface, the approximate immersed boundary lies on sides of mesh cells. Additional algebraic transmission conditions linking both flux and solution jumps through the thin approximate interface are introduced. The fictitious problem to solve as well as the treatment of the embedded boundary conditions are detailed for the two methods. A Q1 finite element scheme is implemented for the numerical validation of the spread interface approach while a new cell-centered finite volume scheme is derived for the thin interface approach with immersed jumps. Each method is then combined to multilevel local mesh refinement algorithms (with solution or flux residual) to increase the precision of the solution in the vicinity of the immersed interface. A convergence analysis of a Q1 finite element method with non-boundary fitted meshes is also presented. This study proves the convergence rates of the present methods.Among the various industrial applications, the simulation on a model of heat exchanger in french nuclear power plants enables us to appreciate the performances of the fictitious domain methods introduced here.