Figure 7From: Optimal control of buoyancy-driven liquid steel stirring modeled with single-phase Navier–Stokes equationsOptimization of nozzles’ positions. Results obtained for levels 2 (top) and 3 (bottom), \(J_{1}\) (left), and \(J_{2}\) (right). The reduction of the functionals is depicted in accordance with the legend, i.e., the size of the circles represents \(1-R_{i}\) with the corresponding reduction \(R_{i}\) from equation (8). The red crosses are the computed weighted centers, see equation (9). The black plus signs indicate the default positions of the nozzles as given in Table 1Back to article page