Numerical simulation of incompressible viscous flow in complex geometry with curved boundaries

In this thesis, a practical algorithm for using higher order isoparametric Finite Element Methods (FEM) to handle complex geometry with curved boundaries has been proposed, developed and implemented. Optimal convergence rates have been achieved for a Poisson problem and a Stokes problem in a simple curved domain where an analytical solution is available. In fact, the algorithm developed in this thesis can be applied to any elements of degree k (k ≥ 3).; Before this general algorithm was proposed and developed, we implemented and studied the simple (but very practical) algorithm for the cubic case proposed by Ciarlet and Raviart (1972) in detail. The empirical study of this algorithm showed that even though the “C-R” point (the interior nodal point for a cubic isoparametric element proposed by Ciarlet and Raviart.) is not the “optimal” point in the coarse mesh, it approaches the “optimal” choice in the asymptotic meshes.; For a Stokes problem in a simple curved domain, fourth-order isoparametric elements together with the “iterative penalty” method are applied. Optimal convergence rates of computational velocity and pressure fields have been obtained successfully.; To demonstrate the superiority of higher order isoparametric: FEM when handling the curved boundaries, careful comparisons of isoparametric elements and the corresponding polygonal approximations have been carried out. It is shown in this work that under the “standard” boundary conditions, which are the projection of the curved surface boundary conditions directly on to the discrete boundary elements, the higher order isoparametric elements developed in this thesis are superior to the corresponding polygonal approximations.; Finally, the algorithms developed in the Stokes problem are applied to the full Navier-Stokes Equations in a realistic domain with curved boundaries. A benchmark problem, flow past a circular cylinder in a channel, has been analyzed in detail. Based on the “converged” solution obtained with the highest spatial resolution mesh, the convergence properties of the algorithms developed in this work are examined carefully. It has been proven numerically that the higher order isoparametric elements developed in this work yield better convergence rates than the corresponding polygonal elements for both velocity and pressure solutions.