Dendritic Solidification of Binary Mixtures of Metals under the action of Magnetic Field: Modeling, Mathematical and Numerical Analysis

In order to understand the behavior of materials in the presence of impurities during the solidification process, it was required to develop appropriate methodologies for an analysis and an effective control of the topological changes of the microstructures (e.g., the formation of dendrites) during the different phases of transformation. The objective of this thesis is to build a relevant model of solidification of binary alloys under the action of magnetic fields, to analyze the obtained systems, from a theoretical and a numerical point of view, and finally, to develop an optimal control method to control the dynamics of the solidification front by the action of magnetic fields. Initially, we have described the physics of the problem and the fundamental laws necessary for modeling, then we built a new model of phase field, which takes into account the influence of the action of magnetic field on the movement of the solidification front. The model thus developed is characterized by the coupling of three systems: one of magnetohydrodynamic type, a second of Boettingger Warren-convection type (representing the evolution of the solidification front and the concentration of the binary mixture) and a third representing the evolution of the temperature. The equations of the complete system describing the model, in a domain Ω subset of R^{n}, n ≤ 3, are time-dependent, nonlinear, coupled and anisotropic. In a second part, we have performed the theoretical analysis of the model in the two-dimensional, isothermal and isotropic case. We have obtained results of existence, regularity, stability and uniqueness of the solution, under certain conditions on nonlinear operators of the system. Finally, we have developed a nonlinear optimal control method: the magnetic field (which acts multiplicatively) plays the role of the control, and the observation is the desired state of the dynamics of the front. We have proved the existence of an optimal solution and obtained the sensitivity of the operator solution and the optimality conditions by introducing an adjoint problem. The theoretical part of the thesis is supplemented by an important numerical work. The analysis and numerical simulations have been conducted on the complete two-dimensional nonlinear (isotropic and anisotropic) problem. We used, for discretization, the method of lines which consists to consider separately the spatial and temporal discretization. The spatial discretization is performed by using a mixed finite elements scheme and the resolution of the obtained algebraic differential system is performed by using the DASSL solver. The discretization of the domain is performed by unstructured triangular meshes. In the realistic case, they correspond to a non-uniform mesh that is very fine in area of the dendrite and at the interface. We have obtained error estimates for the different state variables of the model and analyzed the robustness and stability of the approximation schemes. This numerical code has been validated on various examples, and gives excellent results. Then we have used the code to treat a realistic problem, namely the dendritic solidification of a binary alloy Nickel-Copper, and to analyze the influence of magnetic fields on the development of dendrites. The results show the effectiveness of the approach to reproduce the experimental observations.