Computational methods for incompressible fluid flows with reference to interface modelling by an extended finite element method. - James MacFadden (2006)
In this thesis an implicit Semi-Discrete Stabilized eXtended Finite Element formulation has been successfully developed and implemented for laminar Newtonian incompressible fluid flows. In doing so we have contributed to the research into the field of incompressible fluid flows, multiphase flow and fluid-rigid body interaction. The fluid flows are governed by the incompressible viscous Navier-Stokes equations, using a Finite Element formulation to model the fluid behaviour numerically. A Semi-Discrete time integration scheme was implemented, discretizing in space, leaving the system of ordinary differential equations to be integrated in time. Initially the classical Galerkin method is used to formulate the boundary value problem from the governing equations, however stability issues due to incompressibility and dominant advection terms force the implementation of the stabilized formulation, i.e. SUPG/PSPG. This approach gives greater flexibility in choice of velocity/pressure interpolations, such as equal order functions. The time integration schemes (Generalized alpha method and Generalized Midpoint rule) were compared and contrasted, with the Generalized alpha method demonstrating improved convergence. The highly nonlinear form of the governing equations required an implicit iterative solver and the Newton-Raphson procedure was chosen. Several tests were performed throughout the formulation of the boundary value problem to validate the implementation. The result, a robust, efficient and accurate unsteady incompressible Newtonian fluid formulation. extended FEM was introduced by adding terms to the FEM formulation in a Partition of Unity framework. With the addition of complex solution procedures X-FEM was implemented and tested for multiphase and fluid-rigid body interaction, demonstrating the attractive qualities of this method.