Numerical methods for solving optimal control for Stefan problems

Abstract

The article describes the mathematical model and a numerical method for the calculation and optimization of temperature fields with regard to phase transformations and the nonlinear material properties. It proposes a finite-difference method and a computer program that will effectively implement the computer simulation and optimization of thermal processes during melting and crystallization of the product. Direct Stefan problem was solved on the basis of one of the options through “enthalpic” method. The solution of the dual problem is found by smoothing the concentrated heat capacity and other parameters and characteristics of a feature such as a delta function. The article deals with a number of examples of optimization problems under various restrictions: minimizing energy consumption for melting the material, finding the maximum (minimum) temperature field, as well as two-sided estimate gradient of the solution at a given point in the area. In the above case, the functions of control are the source of the bulk power density, the values of which are located in a strip of arbitrary width. The results can be used in the practice of research and design in the field of metallurgy, electrical appliances, сryogenic etc. Refs 12. Figs 7. Tables 3.