Numerical Solution og Coupled CFD Problems by the Domain Decomposition Method

Abstract

In this thesis we look at the efficient numerical solution of two complex multi-physics problems which both have very important applications. The first problem is related to the numerical simulation of extrusion of aluminium. Extrusion of aluminium is an important industrial process where large pieces of aluminium is heated and squeezed into thin plates to be used in the design of a wide range of products. To optimize the production a quantitative knowledge of the fluid flow and temperature distribution in the aluminium and extrusion tools is needed. Because of the extreme physical conditions during the production process, it is hard to conduct any measurements or realistic experiments, and numerical simulations can therefore be a viable alternative.

The second problem we look at is concerned with the numerical approximation of blood flow in the human vascular system. More specifically, the blood is modeled as an incompressible Newtonian fluid while the displacement of the vessel wall is described by an elasticity equation. Simulation of the human artery system has recently been used for computer aided medical planning, allowing surgeons to predict the outcome of various medical procedures. For instance, many different designs for a bypass operation may be simulated on a computer before the surgeon makes a decision on which procedure to go for. The potential for testing several different procedures can reduce complications for the patients and lead to a larger number of successful operations. Until now the computer simulations have been restricted to simplified models where the vessel walls are fixed. However, it is known that the displacement of the wall can be as much as 10 % of the radius of the vessel, and that this influences the fluid flow significantly. Fluid-structure problems for the interaction between the blood flow and the vessel wall are therefore extensively studied by many research groups. These problems are particularly hard to solve numerically because the structure is very light compared to the fluid, which makes the fluid-structure coupling unstable. Only implicit coupling algorithms give satisfactory results, but they are usually computationally expensive. For the numerical simulations to be useful for the surgeons, the results must be available within a few hours or perhaps even within minutes. This requires both robust numerical algorithms for the fluid-structure coupling and efficient parallel implementations of the fluid and structure subproblems.