Mathematics, Statistics, and Computer Science
Fluid-structure interaction, 3D linearized Navier-Stokes, Kirchhoff plate, finite element method, Babuška-Brezzi theorem
We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain O coupled to a fourth order plate equation, possibly with rotational inertia parameter ρ>0. This plate PDE 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 Ω. We note that as the Stokes fluid velocity does not vanish on Ω, the pressure variable cannot be eliminated by the classic Leray projector; instead, it is identified as the solution of an elliptic boundary value problem. Eventually, wellposedness of the system is attained through a nonstandard variational (``inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.
Source Publication Title
Evolution Equations and Control Theory
American Institute of Mathematical Sciences
Avalos, G., & Clark, T. (2014). Mixed Variational Formulation for the Wellposedness and Numerical Approximation of a PDE Model Arising in a 3-D Fluid-Structure Interaction. Evolution Equations and Control Theory, 3 (4), 557. https://doi.org/10.3934/eect.2014.3.557