The paper presents a multi-scale mathematical model dedicated to a comprehensive simulation of resistance heating combined with the melting and controlled cooling of steel samples. Experiments in order to verify the formulated numerical model were performed using a Gleeble 3800 thermo-mechanical simulator. The model for the macro scale was based upon the solution of Fourier-Kirchhoff equation as regards predicting the distribution of temperature fields within the volume of the sample. The macro scale solution is complemented by a functional model generating voluminal heat sources, resulting from the electric current flowing through the sample. The model for the micro-scale, concerning the grain growth simulation, is based upon the probabilistic Monte Carlo algorithm, and on the minimization of the system energy. The model takes into account the forming mushy zone, where grains degrade at the melting stage – it is a unique feature of the micro-solution. The solution domains are coupled by the interpolation of node temperatures of the finite element mesh (the macro model) onto the Monte Carlo cells (micro model). The paper is complemented with examples of resistance heating results and macro- and micro-structural tests, along with test computations concerning the estimation of the range of zones with diverse dynamics of grain growth.

The paper presents the results of work leading to the construction of a spatial hybrid model based on finite element (FE) and Monte Carlo (MC) methods allowing the computer simulation of physical phenomena accompanying the steel sample testing at temperatures that are characteristic for soft-reduction process. The proposed solution includes local density variations at the level of mechanical solution (the incompressibility condition was replaced with the condition of mass conservation), and at the same time simulates the grain growth in a comprehensive resistance heating process combined with a local remelting followed by free/controlled cooling of the sample tested. Simulation of grain growth in the entire computing domain would not be possible without the support of GPU processors. There was a 59-fold increase in the computing speed on the GPU compared to single-threaded computing on the CPU. The study was complemented by examples of experimental and computer simulation results, showing the correctness of the adopted model assumptions.