Numerical Methods and Analysis for Fluid-Structure Interaction Problems

Fluid-structure interaction (FSI) problems help to describe many physical phenomena in our world, including topics such as aerodynamics, hydrodynamics, geophysical flows, engineering, and biomedical applications. The problems may be mathematically modeled using a system of Navier-Stokes and elastodynamics partial differential equations (PDEs). The fluid of interest is viscous, incompressible, and Newtonian, and the solid is elastic and thick, meaning the solid may be described in the same dimension as the fluid. When the densities of the fluid and solid are comparable, an added-mass effect is introduced where the force from the fluid acts as a mass on the fluid-structure interface. This added complexity makes finding solutions to this two-way coupled problem especially difficult to numerically solve.

In order to solve these FSI problems, three novel algorithms are presented and analyzed. More specifically, the solvers developed throughout this dissertation are based on partitioning, using both loosely and strongly coupled techniques. The proposed schemes are then analyzed for stability and convergence and their performance is investigated in numerical examples. One of the main novelties of each of the three schemes is based on the implementation of generalized Robin-like boundary conditions, which are imposed on each subproblem in order to account for matching velocities and stresses at the fluid-structure interface. The adaptation of our method compared to classical Robin-Robin boundary conditions allows us to use energy estimates to prove our algorithm applied to a moving domain problem is unconditionally stable.

The time-discretization varies between methods and spatial discretization is done using the finite element method (FEM). When considering spatial discretization, it is important to note that a saddle-point problem arises due to the fluid’s incompressibility. Hence, certain conditions of the finite element spaces are met, which is addressed in our choices of finite elements used in the numerical examples.