Distributed optimal control applied to Fluid Structure Interaction problems

In this paper we focus on an optimal control problem based on Lagrangian multipliers and adjoint variables applied to the steady state Fluid-structure interaction problem. FSI systems consist of one or more solid structures that deform by interacting with a surrounding fluid flow. We consider the Fluid-structure interaction problem written in variational monolithic form in order to balance automatically solid and liquid forces at the interface. The objective of the control problem is the minimization of a velocity matching functional and it is obtained with a distributed force control that deforms the shape of the solid domain. This distributed control for Fluid-structure interaction systems can be easily used to control the shape of the solid domain and the flow effects of the fluid-solid interactions. The optimality system is derived from the first order optimality conditions by taking the Fréchet derivatives of the Lagrangian with respect to all the variables involved. The optimal solution is then obtained through a steepest descent algorithm applied to the optimality system. To support this approach we report a few numerical tests where the distributed force deforms the solid structure in order to match the desired velocity profile. The results obtained show the feasibility and robustness of this approach and its possible use in many industrial relevant applications.