École Nationale Supérieure de
Techniques Avancées

Résumé

Reaction-diffusion system (RDS) is a mathematical model describing a range of phenomena with the coexistence of reaction and diffusion. In this research, RDS is applied for diffusion controlled carburization of Fe-Cr alloys. Firstly, a micro-macro modeling strategy is used to build up two mathematical models in this system: (1RDS) for one single reaction and (2RDS) for two coupled reactions. Particularly, alloy’s grain-boundary microstructure and nucleation reaction mechanism are considered. Next, by constructing a series of iterative problems, (1RDS) is proved to be well-posed. And when reaction rate constant k →+∞(so-called fast reaction limit), the solution of (1RDS) converges to the solution of a specific Stefan problem (P1∞). The same strategy is also tried for a simplified (2RDS). Because of the complexity, only some clues are given. In the third section, finite difference numerical method is used to simulate the evolution of solutions. Because of the stiffness of reaction rate terms, implicit/implicit-explicit (IMEX) backwards differential formula (BDF) is chosen. In algorithms, coupled nonlinear equations are solved by two optimization methods: Multidirectional search (MDS) for low-dimension convex functions; Anderson mixing/Basinhopping strategy for large-scale nonlinear functions with several local optimums. In the last part, simulations are realized in Python and numerical results are compared with experiments. Some special behaviors, such as inter/intragranular diffusion interface, phase transition between two carbides, and differences between finite/infinite reaction rate, are observed in numerical solutions.