A weighted residual approach for solving a one-dimensional inverse solidification design problem involving a pure melt

Inverse solidification design studies are critical for assuring a desired solid-liquid interfacial motion that promotes certain morphological characteristics during a par-ticular solidification process. Knowledge of this motion is important since it is well known that the interfacial motion is directly related to the quality of the casting. This problem involves the determination of an unknown time-dependent boundary condition that produces a prescribed solid-liquid interfacial motion in a pure mate-rial. Though clearly conceived, this problem is considered mildly ill-posed since it has been shown that small changes in the input data can have a profound effect on its solution. Additionally, previous investigations have typically produced unstable numerical results that require some sort of stabilizing scheme in order to arrive at re-liable numerical results. The solution method offered in the present work uses a novel collocation method where the unknown temperatures are expanded in terms of global basis functions. All known auxiliary conditions are incorporated into the expansion.s forming a set of trial functions for each region. The method of weighted residuals, particularly orthogonal collocation, is used to resolve the expansion coefficients using a single matrix inversion. This is shown possible if the time variable is treated in an elliptic fashion. This novel adaption stabilizes the numerical solution in a natural manner since all future information is incorporated into the numerical scheme. This method has the potential to produce benchmark results. Numerical accuracy of the proposed method is demonstrated by comparison to an exact solution for a one-phase test case where the solid-liquid interface moves with a constant velocity. A two-phase investigation reveals that the time collocation method produces accurate numerical results to both direct and inverse problems.