Transient Anisotropy-Undercooling Coupling in Dendritic Solidification: Grid-Qualified Phase-Field Ensembles and Finite Element Cross Verification.

Interfacial anisotropy and undercooling control dendritic growth, but numerical artifacts often distort their apparent effects. To address this, we reimplemented the nondimensional Kobayashi model using isotropic nine-point operators, full-interface sub-grid tip tracking, and a grid-independent seed. Our balanced 4 × 4 study explored undercooling (Δ = 0.4-1.0) and sixfold anisotropy (δ = 0-0.025) across 160 simulations at Δx = 0.01333 and Δt = 2.5 × 10-5. Because extended runs continuously decelerated, we reported common-time late-window kinetics (t = 0.054-0.090). Undercooling accounted for 88.7% of the growth rate variance, whereas anisotropy explained 6.9% and their interaction 4.4%. Relative to isotropic growth (δ = 0), maximum anisotropy (δ = 0.025) increased rates by 8.9-141.1% depending on Δ. The marginal gain from δ = 0.02 to 0.025 was only 0.5-2.8%, indicating diminishing sensitivity. Additionally, the δ × Δ interaction explained 17.6-28.7% of morphological shape variance. Numerical reliability was confirmed via a spatial convergence order of 2.14 and a fine-grid Grid Convergence Index (GCI95) of 2.45%. Furthermore, MATLAB-COMSOL cross-verification reproduced solid-fraction histories within a 2.42-3.08% mean relative error. By strictly limiting main claims to positive-stiffness amplitudes, this study delivers a rigorously verified transient numerical interpretation rather than material-specific physical validation.