Estimation of boundary condition of two-dimensional nonlinear PDE with application to continuous casting

Abstract

Solidification heat transfer process of billet is described by nonlinear partial differential equation (PDE). Due to the poor productive environment, the boundary condition of this nonlinear PDE is difficult to be fixed. Therefore, the identification of boundary condition of two-dimensional nonlinear PDE is considered. This paper transforms the identification of boundary condition into a PDE optimization problem. The Lipchitz continuous of the gradient of cost function is proved based on the dual equation. In order to solve this optimization problem, this paper presents a modified conjugate gradient algorithm, and the global convergence of which is analyzed. The results of the simulation experiment show that the modified conjugate gradient algorithm obviously reduces the iterative number and running time. Due to the ill-posedness of the identification of boundary condition, this paper combines regularization method with the modified conjugate gradient algorithm. The simulation experiment illustrates that regularization method can eliminate the ill-posedness of this problem. Finally, the experimental data of a steel plant illustrate the validity of this paper’s method.