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In this paper, we study a nonlinear parabolic type model with an inequality constraint on the solution describing the phase liquid-solid heat transfer during the solidification process in continuous casting. We consider a semi-discretization with respect to time (implicit Euler method in time) of the studied thermal problem. We have a sequence of stationary constrained problems. Reformulating the problem into a variational inequality problem and thanks to the approximation (linearization) of the nonlinear boundary condition (radiation condition) and appropriate assumptions, we establish a result of existence and uniqueness of the solution of the stationary problem. We also consider a multivalued formulation of the same problem allowing the analysis of the behavior of the iterative relaxation algorithm used for the numerical solution of the discretized problem. Finally, numerical simulations are displayed.

A variational inequality approach for the numerical simulation of steel solidification of two-phase continuous casting process