Isogeometric Methods for Computational Fluid Dynamics: Divergence-conforming Discretizations for the 2D Stokes Equations

Abstract

In this thesis we look at how boundary value problems for partial differential equations can be solved numerically using B-splines, or more generally NURBS, both to express the geometry of the problem exactly and as a basis for a finite element approximation. This is called isogeometric analysis, and we consider the theory behind the method as well as aspects regarding implementation. We take a close look at the construction of B-spline basis functions and geometries, and how the basis can be refined, leading up to the construction of NURBS basis functions and geometries. The ubiquitous Poisson problem is considered as a model problem, and a numerical solver for this problem is implemented in MATLAB using Galerkin s finite element method. We finally consider a method for numerically solving the Stokes problem for incompressible fluid flow, using divergence-conforming B-splines in an isogeometric setting. This method gives a discrete velocity which is pointwise divergence-free, making the numerical solution satisfy mass conservation in an exact sense. Numerical tests are performed, showing that isogeometric analysis makes it possible to use exact geometry throughout the analysis and provides great flexibility regarding refinement. The convergence properties of the method for the Stokes problem are investigated numerically, with very good results for the numerical velocity solution, but with a reduced convergence rate for the pressure solution that is accounted for. The method is also tested on benchmark problems, the results confirming the stability of the method.