Optimal control of the free boundary in a two‐phase Stefan problem with flow driven by convection

Abstract

AbstractWe present an optimal control approach for the solidification process of a melt in a container. The process is described by a two phase Stefan problem including flow driven by convection and Lorentz forces. The free boundary (interface between the two phases) is modelled as a graph. We control the evolution of the free boundary using the temperature on the container wall and/or the Lorentz forces. The control goal consists in tracking a prescribed evolution of the free boundary. We achieve this goal by minimizing a appropriate cost functional. The resulting minimization problem is solved numerically by a steepest descent method with step size control, where the gradient of the cost functional is expressed in terms of the adjoint variables. Several numerical examples are presented which illustrate the performance of the method.