NUMERICAL ANALYSIS OF AXISYMMETRIC FLOWS AND METHODS FOR FLUID-STRUCTURE INTERACTION ARISING IN BLOOD FLOW SIMULATION

In this thesis we propose and analyze the numerical methods for the approximation of axisymmetric flows as well as algorithms suitable for the solution of fluid-structure interaction problems. Our investigation is aimed at, but are not restricted to, the simulation of the blood flow dynamics. The first part of this work deals with an axisymmetric fluid model based on three-dimensional incompressible Stokes or Navier–Stokes equations which are solved on a two-dimensional half-section of the domain under consideration. In particular we show optimal a priori error estimates for P1isoP2/P1 axisymmetric finite elements for the steady Stokes equations under the assumption that the domain and the data are axisymmetric and that the data have no angular component. Our analysis is carried out in the framework of weighted Sobolev spaces and takes advantage of a suitably defined Cl´ement type projection operator. We then introduce an axisymmetric formulation of the Navier–Stokes equations in moving domains and, starting from existing results in three-dimensions, we set up an Arbitrary Lagrangian–Eulerian (ALE) formulation and prove some stability results. In the second part, we deal with algorithms for the solution of fluid-structure interaction problems. We introduce the problem in a generic form where the fluid is described by means of incompressible Navier–Stokes equations and the structure by a viscoelastic model. We account for large deformations of the structure and we show how existing algorithms may be improved to reduce the computational time. In particular we show how to use transpiration boundary conditions to approximate the fluid-structure problem in a fixed point strategy. Moreover, in a quasi-Newton strategy we reduce the cost by replacing the Jacobian with inexact Jacobians stemming from reduced physical models for the problem at hand. To speed up the convergence of the Newton algorithm, we also define a dynamic preconditioner and an acceleration scheme which have been successfully tested in haemodynamics simulations in two and three dimensions.