An optimization problem for a Boussinesq equation system will be formulated. We are looking for a temperature profile or an appropriate velocity on the boundary of the considered region of the thermal coupled flow problem to induce a forced convection, which implies a velocity field close to a prescribed one. For such tracking type optimization problems with tracking type minimization functionals, the evaluation of the first-order necessary optimality condition leads to an optimality system consisting of the forward (primal) and adjoint (dual) mathematical model. Besides the derivation of the optimality system we discuss aspects of numerical solution, e.g. the spatial and time discretization and the iteration method for the solution of the resulting coupled nonlinear primal and dual problem in this paper. The optimization concept will be applied to a crystal growth flow and results of two-dimensional and three-dimensional model problems will be presented.
['Günter Bärwolff']
Boundary control of a Boussinesq equation for a crystal growth problem: aspects of numerical solution