The Joule heat generated by current flow between electrodes in a resistance furnace not only melts and heats the charge but also induces mixing of the molten material. Increased mixing promotes improved chemical and temperature uniformity within the bath. This paper presents a novel approach to effectively optimizing electrode geometry in resistance furnaces. The method relies on a surrogate criterion derived exclusively from an electromagnetic submodel, which governs the process hydrodynamics. This criterion is based on the location of the Joule heat generation center in the bath. Its idea is to lower this center as much as possible while keeping it close to the vertical bath axis. Owing to this, the best conditions for the development of natural convection were obtained. The developed methodology was demonstrated through an application to a two-electrode furnace. The results showed that the influence of forced MHD convection is negligible in this furnace (with a Lorentz force of only about 0.0015 N/kg). The validation of the optimized geometry, derived using solely the electromagnetic submodel, was carried out using a full process model, including time-consuming hydrodynamic calculations. The proposed optimization methodology enabled a 10-fold increase in the average mixing velocity (from 0.0008 to 0.0084 m/s). The main significance of the presented study is the introduction of a surrogate criterion that allows for a multiple reduction in the time of numerical optimization of the mixing intensity in electrode resistance furnaces in comparison to the standard solution based on the flow velocity criterion determined from the hydrodynamic model.

Melting reactive metals and alloys, such as titanium, poses a significant challenge due to the risk of crucible damage and contamination of the molten material. Full levitation melting presents a promising solution; however, its application has largely been limited to small laboratory samples. This paper introduces a methodology for modeling (in a 2D axisymmetric domain) and optimizing a new large-scale levitation melting system and demonstrates its application to pure titanium. The system features a torus-shaped load positioned within a gutter-shaped coil. Numerical experiments using this approach confirm the feasibility of stable levitation for a substantial mass (2.6 kg) within a newly designed electromagnetic levitation system.

This paper addresses new optimization approaches for resistance furnaces with electrodes for processing molten slag. The thermal and hydrodynamic behavior of the melt is strongly influenced by the Joule heat distribution generated by the electric current flowing between the electrodes, making electrode geometry a key design variable affecting the final process conditions. The aim of this study is to combine two approaches: a surrogate criterion based on process physics and an additional high-level surrogate criterion generated by a Gaussian process. The solutions were validated using a full model of the slag melting process in a dual-electrode furnace, incorporating electromagnetic and hydrodynamic phenomena to evaluate the target criterion—slag mixing intensity. The results confirm the usefulness of the physics-based surrogate criterion, while also revealing limitations of the Gaussian-process layer under nonlinear geometric constraints. Further development of feasibility-aware acquisition strategies and validation on additional benchmark cases is required to establish its broader applicability.