On the numerical sensitivity of cellular automata grain structure predictions to large thermal gradients and cooling rates

Abstract

Cellular automata (CA) models of as-solidified grain structure, originally developed and applied to casting, have become a common means of predicting grain structure resulting from Additive Manufacturing (AM) processes. The majority of these models are based on the decentered octahedron approach, which attempts to correct for the effect of grid anisotropy on the prediction of competitive solidification of dendritic grains. However, AM solidification occurs under cooling rates ($\dot{T}$) and thermal gradients ($G$) that are orders of magnitude larger than those encountered in casting, and no systematic investigation on the effect of the CA model cell size ($\Delta x$) and time step ($\Delta t$) on AM microstructure predictions has been performed. In this study, such an investigation is first performed via simulation of individual grains of various crystallographic orientations with a fixed, unidirectional $G$, showing that CA prediction of the steady-state undercooling matched the expected values based on the interfacial response function at small $G$ and deviated from the expected values at large $G$. Simulation of competitive growth of multiple grains showed a weakening of the predicted texture as $G$ and $\Delta x$ became large. Simulation of solidification under AM conditions, where $G$ and $\dot{T}$ vary spatially across the melt pools, showed that not only does grain selection weaken and deviate from expectations at large $\Delta x$, but grains with crystallographic $〈100〉$ aligned with the grid directions are more adversely affected by the temperature field discontinuities than grains with other crystallographic orientations. Despite the fact that the exact grain competition results depended on $\Delta t$, the overall texture development was notably less sensitive to $\Delta t$ than $\Delta x$, provided that a reasonable value of $\Delta t$ is selected based on the ratio of $\Delta x$ to the maximum local solidification velocity in the simulation domain. Finally, from the directional solidification and AM simulation results, an analysis of computational cost compared to simulation resolution is performed based on an equation derived to quantify the relatively inaccuracy in grain selection based on the model and temperature field inputs. From this analysis, it is concluded that there is a need for algorithmic improvements to improve CA grain competition accuracy for large $G$ processing conditions as sufficiently small $\Delta x$ to resolve the necessary competition is intractable for many AM processing conditions.