Boundary control of a Boussinesq equation for a crystal growth problem: aspects of numerical solution

Abstract

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.

Keywords:

• FV-discretization of the Boussinesq equation system
• Boussinesq approximation
• Optimization of partial differential equations
• FV-discretization of the primal and dual problem
• Iterative solution method of a KKT-system

Acknowledgements

This work is part of a cooperation with Professor Micheal Hinze from the Universität Hamburg. The topic of the work has been partially supported by the Mathematisches Forschungszentrum Matheon granted by the Deutsche Forschungsgemeinschaft. We have to thank Dr Klaus Böttcher and Professor Peter Rudolph from the Leibniz Institut für Kristallzüchtung Berlin for many helpful discussions about the physics of crystal growth. Finally, we thank the anonymous referees for the valuable comments which helped to improve the presentation of the material.

Additional information

Notes on contributors

Günter Bärwolff

Email: [email protected]