Nonconforming finite element methods for fluid-structure interaction problems

Accurately simulating the interaction between a fluid and a structure remains a challenging problem in computational mathematics. One difficult aspect of this process is to efficiently couple the geometry of each domain as well as the systems of equations which model the physical properties of each media. The primary objective of this dissertation is to systematically develop and analyze Sophisticated computational techniques which employ finite element methods for solving fluid-structure interaction problems that arise in science and engineering applications.

First, a one-dimensional problem is presented in order to introduce the approximation techniques we will extend to higher dimensions. Additionally, many two-dimensional fluid-structure interactions can be reduced to one dimension under certain assumptions about the geometry of the subdomains along with inflow and outflow boundary conditions. In this context, we will establish consistency and stability properties for our discretization methods. Secondly, we will develop a nonconforming finite element methodology using a three-field formulation to solve a coupled physical system comprised of two adjacent domains, one containing a viscous, incompressible fluid and the other an elastic structure. Our method will employ an arbitrary Lagrangian-Eulerian strategy to formulate the governing equations for the fluid coupled with a linear elasticity model describing the deformation of the solid in order to simulate a full unsteady physical phenomenon. This formulation is analogous to the one used in the one-dimensional problem, although more complicated constraints are required due to the geometry of the problem. Again, consistency and stability properties are established for this technique. Finally, computational results which establish consistency and convergence of our numerical methods for the one-dimensional problem are presented. These results include verification that our discretization technique is first order in time and that when a nonconforming technique is applied, i.e., when different polynomial degrees of approximation are used in the fluid and structure domains, the solution obtained is no worse than those computed using a conforming method. Additionally, convergence of the method under refinement of the computational grid, known as the h-method, is explored.

Thus, our numerical experiments provide confidence in the discretization techniques established in the previous chapters as well as insight on how to appropriately construct finite element code for the two-dimensional problem.