Optimal control of buoyancy-driven liquid steel stirring modeled with single-phase Navier–Stokes equations

Journal of Mathematics in Industry

Table 4 Optimization of gas flow rates. Minimum objectives \(M_{J, \lambda }\) and corresponding frequencies \(\omega _{i}\), \(i\in \{1,2\}\). The last column is the number of required optimization iterations. \(\Delta t=0.05\), mesh 3

Objective J	λ	\(M_{J,\lambda }\)	Reduction \(M_{J,\lambda }\)	\(\omega _{1}\)	\(\omega _{2}\)	No. it
J1	0.00001	−53.84143	0.90192	0.13656	0.025	41
J2	0.00001	−1.43724	0.32240	0.15859	0.02785	33
J3(1)	0.00001	0.40925	0.97046	0.03342	0.03312	28
J3(100)	0.00001	129.15223	0.96660	0.15117	0.025	29
J1	0.0001	−53.98697	0.89091	0.11986	0.025	38
J2	0.0001	−1.10799	0.28574	0.18136	0.03019	41
J3(1)	0.0001	0.71228	0.50805	0.29053	0.04984	31
J3(100)	0.0001	129.54287	0.96569	0.15123	0.02522	43
J1	0.001	−49.8165	0.87434	0.12039	0.02576	36
J2	0.001	2.05743	0.12980	0.25298	0.23902	30
J3(1)	0.001	3.3579	0.13667	0.14809	0.14976	31
J3(100)	0.001	133.52497	0.95737	0.15854	0.0272	35
J1	0.01	−15.51777	0.66315	0.14386	0.13765	28
J2	0.01	28.32234	0.01441	0.14907	0.15026	33
J3(1)	0.01	29.38429	0.01813	0.15032	0.15016	32
J3(100)	0.01	162.50446	0.82260	0.14795	0.15085	26
J1	0.1	247.01347	0.14722	0.15016	0.14946	26
J2	0.1	288.20955	0.00147	0.14975	0.15028	28
J3(1)	0.1	289.08829	0.00261	0.15028	0.15024	34
J3(100)	0.1	422.61609	0.31780	0.15033	0.1497	28