Multimodels for incompressible flows

The Navier-Stokes equations for incompressible fluid are coupled to models of reduced complexity, such as Oseen and Stokes equations, and the transmission conditions are investigated. We carry out the mathematical analysis of the corresponding problem and present numerical results obtained by finite elements and spectral elements for several flows of physical interest.

Iterative solutions of multimodels for incompressible flows

Alfio Quarteroni

Summary: For the simulation of two-dimensional incompressible flows, the Navier-Stokes equations can be coupled with models of reduced complexity, such as Oseen and Stokes models. This approach is briefly reviewed here. Then we propose a numerical solution of the coupled problem by subdomain splitting. We give numerical results obtained by finite element and spectral element approximations on a couple of test problems of physical interest.

World Scientific2000

Analysis and numerical simulation of free surface flows

Alexandre Caboussat

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

EPFL2004

Finite element methods with patches and applications

Joël Wagner

Theoretical and numerical aspects of multi-scale problems are investigated. On one hand, mathematical analysis is done on a new method for numerically solving problems with multi-scale behavior using multiple levels of not necessarily nested grids. A particularly flexible multiplicative Schwarz method is presented, requiring no conformity between the meshes at the different scales. The relaxed iterative method consists in calculating successive corrections to the solution in regions where the variations of a problem are too strong to be captured by a coarse initial mesh. In these sub-domains patches of finite elements are applied. A priori and a posteriori error estimates are given and an exact spectral analysis of the iteration operator describing the algorithm is presented. Computational issues are addressed and numerical methods to obtain optimal convergence are given. Crucial implementation matters are discussed with special regard to usage of memory and CPU-time. On the other hand, the efficiency of the introduced correction method is demonstrated on Laplace model problems, either with changing Dirichlet-Neumann boundary conditions or in a polygonal domain with entrant corner. The regularity of the solutions is studied as well as the improvement of the convergence order in the mesh size using various sizes of patches. The correction algorithm is also applied to improve the accuracy in the simulation of the stress field in glacier modeling. A simple model to obtain the effective stress field in the ice mass of a glacier is presented and concluding results are obtained using patches in the regions where changes in the basal boundary conditions are involved.

EPFL2006

Analysis and numerical simulation of free surface flows

Alexandre Caboussat

EPFL2004