Isogeometrisk analyse av Boussinesq ligningene

Abstract

In this master thesis, I have solved the Boussinesq equations with Isogeometric Analysis to study heat transfer coupled with air circulation. The Boussinesq equations consist of the Navier-Stokes equations and the convection-diffusion equation, and they are coupled together through the Boussinesq-Oberbeck approximation. The equation system is solved on a closed domain with mixed boundary conditions. The Navier-Stokes equations model the air motion, and the convection-diffusion equation models the temperature distribution.

In order to discretize this system of partial differential equations and to obtain sufficient stability over long time, we combine isogeometric analysis with the mixed and multiscale finite element methods. The global computational complexity and quasilinear structure of the model are managed by using different A-stable time-integrators separately on the sub-equations. Thus, the running time is reduced, and we avoid restrictions on the time steps. We illustrate the advantages which isogeometric analysis has compared with the classical finite element method, and show that it can be used together with already exist- ing algorithms for computational fluid dynamics. An essential description of isogeometric analysis and mesh generation is presented in the beginning.

We perform a systematic analysis of the suitable numerical methods for the Boussinesq equations, with emphasis on h- and p-refinement with a simple tensor mesh on a square domain. We also discuss which time-integrator is best. To verify whether the simulations are correct, we have constructed manufactured reference solutions. Since the domain is simple, we use B-splines as basis functions. We have written complete object-oriented MATLAB codes for the simulations, and GLview Inova is used for visualizing dynamic simulations.