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An applied functional and numerical analysis of a 3-D fluid-structure interactive PDE
We will present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. In Chapter 1, the wellposedness of this PDE model is established by means of constructing for it a nonstandard semigroup generator representation; this representation is essentially accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain O being coupled to a fourth order plate equation, possibly with rotational inertia parameter p >0, which evolves on a flat portion Ω of the boundary of O. The coupling on Ω is implemented via the Dirichlet trace of the Stokes system fluid variable – and so the no-slip condition is necessarily not in play – and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on this elastic portion of the boundary. We note here that inasmuch as the Stokes fluid velocity does not vanish on Ω, the pressure variable cannot be eliminated by the classic Leray projector; instead, the pressure is identified as the solution of a certain elliptic boundary value problem. Eventually, wellposedness of this fluid-structure dynamics is attained through a certain nonstandard variational (“inf-sup”) formulation. Chapter 1 also includes two abstract results. The first qualitative result shows that zero is in the resolvent set of the operator which generates the C0-semigroup in the wellposedness argument. The second establishes the backward uniqueness property for the fluid-structure system.^ Subsequently, in Chapter 2 we show how our constructive proof of wellposedness naturally gives rise to a certain mixed finite element method for numerically approximating solutions of this fluid-structure dynamics. This method is demonstrated for a certain test problem in the p = 0 case. In addition, error estimates for the rate of convergence of the numerical method are provided and a test problem is solved to demonstrate the efficacy of the numerical code.^
Clark, Thomas J, "An applied functional and numerical analysis of a 3-D fluid-structure interactive PDE" (2014). ETD collection for University of Nebraska - Lincoln. AAI3618565.