Estimation of the boundary condition of a 3D heat transfer equation using a modified hybrid conjugate gradient algorithm

Billet production is for the most part completed in a continuous casting machine. To produce a billet of suitable quality, the secondary cooling control system needs to provide an appropriate value of water flow rate. At present, most secondary cooling control methods are based on a heat transfer model for the billet, the accuracy of which can directly impact the cooling effect and further affect its quality. In practice, it is difficult to determine the heat flux at the billet’s boundaries due to the complexity of continuous casting. Therefore, this study focuses on identifying the boundary conditions for the 3D heat transfer model of a billet with a linear coefficient. First, we transform the identification of boundary conditions into an optimization problem, prove the Lipschitz continuity of the cost function of the optimization problem, and obtain the Lipschitz constant. Second, based on the Lipschitz continuity of the cost function, we present a modified hybrid conjugate gradient algorithm (MHCGA) to solve the optimization problem and then prove the global convergence of this MHCGA. Compared with other methods, the results of the simulation experiments clearly show that this MHCGA can reduce running time and iteration number. Third, to eliminate the ill-posedness of the inverse problem for identifying the boundary condition, we combined a regularization method with MHCGA. Simulation experiments confirmed that this method can effectively estimate the boundary conditions. Finally, the experimental data from a steel plant verified the validity of our method, and the prediction results of the shell thickness were confirmed by the nail shooting method.