## 6 Conclusions  
This paper presents a new methodology for the rapid optimization of electrode resistance furnaces, the aim of which is to increase the intensity of bath mixing and, thus, improve its homogeneity. Omitting the hydrodynamic model from the optimization cycle allowed us to shorten its duration by two orders of magnitude. Owing to this, it was possible to optimize the electrode furnace using local algorithms, as part of the presented research. The proposed surrogate criterion was based solely on Joule heat analysis due to the negligible influence of convection forced by Lorentz forces (only about 0.0015 N/kg). Based on the accelerated optimization loop, various variants of the furnace design solutions were determined, depending on the adopted definitions of the criteria and the optimization parameter space. Hydrodynamic simulations were performed on the basis of these solutions, and the target criterion value was obtained, i.e., the average bath flow velocity. All variants obtained using the proposed optimization methodology showed a significant increase in the mixing intensity of the processed bath compared with solutions based on geometric relationships given in the literature (at least a 631% improvement). The best variant, determined for the substitute criterion, maximized the lowering of the Joule heat  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Variant 0** & **Variant 1** & **Variant 2** & **Variant 3** \\ \hline Vertical position of JHC [m] & \(-\)0.0662 & \(-\)0.0996 & \(-\)0.0950 & \(-\)0.0918 \\ \hline Distance of JHC from axis [m] & 0.0000 & 0.0850 & 0.0011 & 0.15421 \\ \hline Average velocity [m/s] & 0.0008 & 0.00782 & 0.00845 & 0.00593 \\ \hline Maximum velocity [m/s] & 0.01067 & 0.04000 & 0.04185 & 0.04635 \\ \hline Total Lorentz force [N/kg] & 0.00136 & 0.00175 & 0.00173 & 0.00148 \\ \hline \end{tabular}
\end{table}
Table 3: Summary of electrode geometry variants.  
generation center while forcing its location closer to the system axis. This increased the mixing intensity by 956% compared to the base variant.  
Using a surrogate criterion drastically reduced the criterion determination time (from 6 h to 3 min on a computer with an AMD EPYC 7302, 16 Core, 3.00 GHz processor), allowing for the methodology to be extended toward global optimization algorithms (e.g., PSO, GA). This could increase the chances of finding a global optimum and further improve the bath mixing intensity.  
The developed methodology was applied to a two-electrode furnace as an example. However, based on the diagram shown in Figure 1, it can be seen that the location of the heat release center in the three-electrode furnace also depends on the geometry of the electrodes, especially the position of their lower ends. As shown in the article, controlling the position of the Joule heat release center allows for increasing the intensity of bath mixing. Regardless of whether this heat is generated as a result of a single-phase current flow between two electrodes or a three-phase current flow between three electrodes, the convection-driving mechanism remains the same. Therefore, it seems that the proposed surrogate criterion can also be used in the optimization of three-electrode furnaces. This is a general, conceptual analysis, but it encourages further research involving three-electrode furnaces.  
Conceptualization, R.Z. and S.G.; methodology, R.Z. and S.G.; software, optimization algorithms, J.W.; post-processing, P.C.; analysis, T.S. and P.M.; writing, R.Z. and S.G.; visualization, R.Z. and P.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Polish Ministry of Science and Higher Education, grant number DWD/6/0460/2022.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.