Multi-scale Modeling of Alloy Solidification and Phase-field Model

Abstract

In this chapter we formulate and study robust control problems for a twodimensional, non-linear, time-dependent and solutal phase-field model of the Warren–Boettinger-type (TDWB), which describes the isothermal solidification of a binary alloy. This model contains two unknowns, the relative concentration and a non-conserved structural order parameter (which is said to be the phase-field variable) coming from thermodynamics. The phase-field theory is a direct consequence of the Cahn–Hilliard and Ginzburg–Landau type classical field theoretic approaches to phase boundaries.

The order parameter describes the phase of the underlying substance: the order parameter is close to 1 if the system is in a liquid phase and is close to 0 if it is in a solid phase.

It is well known that thermal fluctuations and material impurities affect considerably the solidification microstructure dynamics. These effects are modeled by variants of Warren–Boettinger model containing additive noise due to thermal fluctuations and by modification of some operators to take into account impurities.

The objective is the prediction and stabilization of microstructure dynamics by taking into account the influence of fluctuations and data noises. First, a variant of TDWB model (MTDWB) is introduced and analyzed. Second, we introduce the perturbation problem of the non-linear governing coupled system of the MTDWB equations (the deviation from the desired target). The existence and uniqueness of the solution of the perturbation are proved as well as their stability under mild assumptions. Afterwards, some robust control problems are formulated for the cases when the control is in the initial condition of the concentration field and when the worst disturbance is the noise due to thermal fluctuations or is in the initial condition of the phasefield parameter. We show the existence of an optimal solution, and we also find the necessary conditions for a saddle point optimality.

This work is a generalization of recent research developed by Belmiloudi in [40].