Boundary control for non-isothermal Navier-Stokes flows by using domain decomposition and extended flow methods

Optimal boundary control problems associated with the Navier-Stokes equations coupled with the energy equation have a wide and important range of applications. Velocity, pressure and temperature describe most systems and the control through their boundary conditions is the most common in their optimal design. Despite the fact that this field has been extensively studied, determining the optimal boundary control for a complex system of equations is still a difficult and time consuming task. In this work, we study a class of stationary optimal temperature and optimal flow control problems and their implementation. In order to implement the boundary control the boundary control problem is transformed into an extended distributed problem. This approach gives robusteness to the boundary control algorithm which can be solved by standard distributed control techniques over the extended part of the domain. Optimal control computations with distributed and boundary controls are presented by using a new multigrid approach for reliable and robust optimization. In fact the multigrid solver is based on a local Vanka-type solver for the Navier-Stokes, temperature and the adjoint system of equations. The solution is achieved by solving and relaxing element by element the optimal control problem which is formulated by using an embedded domain approach. The class of Vanka-type smoothers is characterized, in each smoothing step, by the solution of small local linear systems of equations in a Gauss-Seidel manner. In this work these solvers are applied to simple optimality systems and its adjoint system. The optimal solution is achieved by solving and relaxing element by element the optimal control problem. By using this technique only local subsystems are stored and solved which allows this method to be very effective in large problems where other solvers cannot satisfy memory and cpu requirements. Also the adjoint and the original system can be solved exactly over small domains and t