A penalty-projection algorithm for incompressible fluid-structure interaction

This paper deals with boundary optimal control problems for the temperature and Navier-Stokes equations, where we propose to transform a boundary optimal control problem into a distributed problem through the lifting function approach of nonhomogeneous boundary conditions. The lifting function approach defines controls in the function spaces which are naturally associated to the volume variables, without stronger regularity requirements. For strong and robust optimization the state-adjoint system must be solved in a coupled way. To this purpose, we propose the use of domain decomposition Vanka-type solvers. With this type of solvers the problem is split into small blocks of finite element subdomains with a small number of degrees of freedom, so that an optimal solution is computed by solving the fully coupled state- adjoint system on each subdomain. We present a numerical study of a class of optimal control problems where temperature is the observed quantity and the control quantity corresponds to the boundary values of the fluid temperature in a portion of the boundary. The control region for the observed quantity is a part of the domain where it is interesting to match a desired temperature value. In a multi-physics framework the desired temperature matching can be achieved by the contribution of different physical mechanisms, involving not only boundary temperature control but also boundary velocity, the overall effect of which is studied. We consider a classical finite element method for the numerical discretization of such problems and we illustrate the results of some test cases for the state, adjoint and control solution, in order to show that candidate boundary controls can be computed in an effective manner.