Optimal Control of Stefan Problems with Constraints

Free and moving boundary problems are ubiquitous in technical applications. They can serve as a model for free surfaces of liquids, or phase boundaries in phase transition phenomena etc. Free and moving boundary problems are characterized by the property that the domain of definition of the underlying partial differential equation is a priori unknown. By contrast, the domain is determined along with the solution of the equation. This project is concerned with the optimal control of moving boundary problems involving time-dependent partial differential equations on two- and three-dimensional domains. We focus on problems of Stefan type which model phase transitions, for instance in solidification problems. In many industrial solidification processes, including casting and crystal growth, it is desirable to influence the evolution and shape of the free boundary, which affect the product quality or the duration of the production cycle. Mathematically, this can be expressed in terms of an objective functional, and the problem can be cast as an optimal control problem for a moving boundary problem. The objective function may depend on the location, area, or curvature of the free boundary, as well as on the state (temperature) and control (heat fluxes and sources) variables. For instance, the objective function may express the desired evolution of the moving boundary, its desired final location, or it may penalize large boundary curvature. Often, inequality constraints are required in order to avoid excessive boundary heat fluxes, to guide the moving boundary in a band around its desired track, or to prevent the moving boundary from entering a specified area. In the proposed project, we shall investigate numerical techniques for free boundary optimal control problems of Stefan type, with constraints on the location of the free boundary. In contrast to previous approaches, we shall utilize the structure of the optimality system, incorporate second-order information and employ level-set techniques to describe the evolution of the free surface.

The main results of the project concern new optimality conditions as well as techniques for their numerical solution pertaining to optimization problems involving material models undergoing a phase change. The classical Stefan problem as well as solidification phenomena in crystal growth applications are important examples for the problem class under consideration. The novelty of the approach pursued in this project is its ability to handle both closed and non-closed liquid-solid interfaces. This is achieved by employing a level-set description of the interface in contrast to, e.g., a description by the graph of a function. Within the project, we have developed novel first-order optimality conditions. The techniques leading to these conditions can be used for other optimization problems as well as long as they involve a level-set description of the moving interface. The results of the project may thus have a potentially substantial impact on the treatment of a broad class of optimization problems with free boundaries and moving interfaces. In order to demonstrate the practicality of the proposed method, techniques were also developed for the numerical solution of the optimality system.