Donald Harrison

Finite Element Analysis of a Coupled Fluid-Solid Problem: Mathematical and Computational Approach

This dissertation addresses issues related to modeling, mathematical analysis, and numerical solution for the behavior of a composite medium consisting of an elastic solid enclosing a Newtonian fluid.

Since the Navier-Stokes equations for fluid flow are nonlinear, problems involving fluids enclosed in elastic solid are difficult and only the simplest of problems can be solved with analytical solutions. Current computational methods commonly use iteration with separate treatment for the solid and the fluid. This dissertation presents a method used to solve for a global velocity field in the fluid and solid in once coupled linear system. Assuming small elastic solid displacements, thick solid walls, and linearized flow to reduce the complexity of the Navier-Stokes equations, this dissertation develops a new mathematical model for a coupled fluid/solid problem. The existence and uniqueness of a solution to this problem is established.

The finite element mixed method is used for spatial discretization and a finite difference scheme is applied to discretize the problem in the time variable. Error estimates for a finite element approximation and for a global discretization are derived. The discretization involves a global treatment of the velocity and pressure which leads to a linear system with block structured matrix. A FORTRAN code for a computer implementation of the method is developed and used to solve a test problem. Results of numerical experiments are provided.