Discretization of an unsteady spatial model describing the solidification process of an ingot in a mold of a continuous casting machine

Abstract

Continuous casting machines (CCM) provide high quality metal. Optimization of technological processes, at present, cannot be based only on an empirical approach based on a generalization of production experience. This is due to the fact that each caster is a complex technical system with its own specifics and features. The improvement of continuous casting technology is based, first of all, on the creation of mathematical models describing technological processes taking into account many technological and structural factors. When modeling crystallization processes, most researchers, in order to simplify the solution of the problem, consider stationary plane-parallel models. Such models cannot describe the studied phenomenon well in principle and therefore have a limited scope. Therefore, the development of methods for solving the unsteady spatial model of continuous casting is of great scientific interest and expands the possibilities of solving very complex boundary value problems with phase transitions. The finite element method (FEM) is one of the most universal numerical methods for solving differential and integral equations and their systems. FEM leads to a system of algebraic equations. The article describes the use of FEM for numerical modeling of the solidification process of an ingot in a continuous casting mold with the aim of conducting computational experiments. The discretization process of the system of differential equations and the problem solution domain is presented. In addition, an algorithm of finite-difference approximation of non-stationary members of the equations is described. To sample the region, three-dimensional finite elements in the form of a parallelepiped with nodes located at the vertices of the elements are used. The solution is in the space of piecewise linear functions.