Optimal control of the free boundary in a two-phase Stefan problem

Abstract

We 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 with the free boundary (interface between the two phases) modelled as a graph. We control the evolution of the free boundary using the temperature on the container wall. 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. The novelty of the approach presented consists in using a sharp interface model for the control of the free boundary. This guarantees direct access to the free boundary as optimization variable in terms of its parametrization as a graph. Moreover at any stage of the optimization process the physical laws constituted by the mathematical model are conserved, and the heat flux into the free boundary is variable and thus does not need to be specified a priori in the optimization process. This guarantees more flexibility than in the optimization approaches to two phase Stefan problems taken so far.