In a unidirectional solidification design problem, the solidification velocity and liquid-side interfacial temperature gradient are of principle interest due to their effect on the morphology of the cast structure. The design challenge is the prediction of the temporal conditions at the boundaries such that the solidification velocity and liquid-side temperature gradient at the solid-liquid interface follow a prescribed scenario. This design problem requires the resolution of two inverse heat conduction problems: one in an expanding solid domain and the second in a shrinking liquid domain. Resolution of the solid domain results in a transient boundary condition that yields the prescribed solidification velocity, while resolution of the liquid domain results in a transient boundary condition that yields the prescribed interfacial temperature gradient.
An innovative and robust solution technique is proposed and demonstrated for resolution of the liquid-side temperature gradient design problem during unidirectional solidification. The technique, termed the Function Decomposition Method (FDM), is an innovative combination of function decomposition for the superposition of direct solutions, continuous least squares, and the weighted residual method which transforms the mildly ill-posed inverse heat conduction problem in the shrinking liquid region into a set of well-posed direct problems. To demonstrate its application, a set of test cases are presented which provide illustrative results highlighting the flexibility of the methodology.