A Mixed Finite Element Method for the Coupling of Linear Elasticity and Stokes Flow
The complex interaction between fluids and structures require the coupling the laws concerning structure mechanics and fluid dynamics and are of vital importance to many scientific and engineering fields. We propose a method for modeling the coupling of a linearly elastic solid and slow fluid flow modeled by Stokes equations. The model equations are expressed in terms of displacement, velocity and stress. With these primary variables, we use a single mixed finite element space based on the Hellinger-Reissner variational principle for linear elasticity to discretize the resulting system spatially. This results in more accurate approximations for stress than those obtained by using standard finite element methods followed by post processing and simple imposition of the interface conditions. Although the solid and fluid subproblems using this mixed form have been studied separately, to our knowledge, a fluid-structure interaction (FSI) problem has not been studied using these models. In this dissertation, we first discuss the linear elasticity and fluid flow models separately. This includes an improvement to the proved convergence rate of a nonconforming mixed finite element method originally discovered by Arnold and Winther [MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, 13 ,3, March 2003] for the steady state problem with certain boundary conditions. Also a description of the special attention required to obtain approximations to the initial conditions for the fluid model is required to make use of a second order in time discretization. A fully discrete implicit in time model is presented for the complete FSI problem. A priori error estimates are derived for all primary variables. These error estimate show a unfavorable effect caused by the presence of an interface. However, this effect may be not be noticeable when compared with the approximation properties of the mixed finite element space chosen for spatial discretization of the problem. Numerical results using conforming and nonconforming finite element spaces are provided for the interaction problem.
Bean, Maranda Lee, "A Mixed Finite Element Method for the Coupling of Linear Elasticity and Stokes Flow" (2018). ETD Collection for University of Texas, El Paso. AAI13421820.