Numerical investigation of a binary solidification design problem

This exposition presents the development and application of a methodology for control of unidirectional solidification of a binary alloy. In particular, it is desired to produce a casting that has a uniform cast structure throughout its entire length. Furthermore, the methodology allows the specification, a priori, of the cast structure with respect to both scale, i.e., fine or coarse, and mor- phology, i.e., dentritic or cellular. This specification is in the form of a map that relates solidification characteristics, i.e., scale and morphology, to the solidification velocity and liquid-side interfacial temperature gradient. Thus design is accomplished by controlling these two parameters during the solidification process. With this in mind, the goal of what is termed the binary solidification design problem is the prediction of a set of boundary temperatures and heat fluxes which when applied will result in the desired interfacial motion and temperature gradient and therefore cast structure. Mathematical models for problems of this type lead to what are termed ill-posed systems in that they may not exhibit existence, uniqueness, or continuous dependence on boundary data. The resolution of this class of problems requires advanced techniques to overcome the instabilities encountered due to their ill-posed nature.

The methodology developed herein employs the classical weight residual approach in a innovative manner. Normally, in the solution of a parabolic partial differential equation, such as the heat equation, a spatial series expansion with time varying coefficients is utilized along with a minimiza- tion technique to reduce the partial differential equation to a set of first order ordinary differential equations. This set can be solved using any number of numerical technique, i.e, Runge-Kutta, to obtain the temporal variation of the coefficientsThese types of time stepping techniques eventually lead to the onset of instability when employed for the resolution of ill-posed problems due to the build-up of round-off errors. In this exposition, time stepping is replaced by the further expansion of the time varying expansion coefficients into a series unto itself. Minimization in both space and time is simultaneously applied leading to a set of algebraic equations whose solution leads to the resolution of the entire space-time domain. This treatment of time in an elliptic fashion stabilizes the solution of the ill-posed problem and forms the basis of what is termed the Global Time Method, GTM.

The results obtained for the control boundaries indicate that the control measures required to accomplish the design solidification are not only physically realistic but relatively undemanding to implement. Furthermore, under the design solidification conditions, it was observed that once formed, a mushy zone of constant thickness was maintained throughout the transient. This observation gave rise to a quasi-steady state analysis of the mushy zone which lead to the development of a set of design tools.