Reducing drag of the obstacle in the channel by boundary control: Theory and numerics
Fluid structure interaction comprising of an elastic body immersed in the moving fluid is considered. The fluid is modeled by an incompressible Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions. The goal is to reduce a drag of the obstacle by changing the flow profile on the inlet. This leads to a boundary control problem with a minimization of a hydro-elastic pressure on the interface between the solid and the fluid. The latter is expressed in a form of shape functional. The interface is "free" and it is itself an unknown variable. The problem is reformulated as a quasilinear PDE-control with a free boundary. A distinct feature of the model is that the boundary conditions imposed on the fluid domain are mixed [change from Dirichlet to Neumann]. This feature is well known to cause singularities in elliptic solutions. Handling of these requires a careful analysis of local singularities which depend on the geometry of the domain. The final result provides an existence of optimal control [with volume constraints] which minimizes the drag of the obstacle, under the assumption of sufficiently small strains. The obtained results are illustrated by numerical simulations which confirm and interpret the theoretical findings.
Lasiecka, I., Szulc, K., & Zochowski, A. (2019). Reducing drag of the obstacle in the channel by boundary control: Theory and numerics. IFAC-PapersOnLine, 52, 168-173. https://doi.org/10.1016/j.ifacol.2019.08.030