  The solidification and macro-segregation problem involving unsteady
multi-physics and multi-phase fields is typically a complex process with mass,
momentum, heat, and species transfers among solid, mushy, and liquid phase
regions. The quantitative prediction of phase change, chemical heterogeneities,
and multi-phase and multi-component flows plays critical roles in many natural
scenarios and industrial applications that involve many disciplines, like
material, energy, and even planet science. In view of this, some scholars and
research institutions have called for more contributors to join the benchmark
analysis of solidification and segregation problems. Our work proposes an
operator-splitting and matrix-based method to avoid non-linear systems. Also,
the combination of vectorization and forward equation-based matrix assembly
techniques enhances the implementability of extensions of 3D applications.
Lastly, the novel scheme is well validated through a bunch of 2D and 3D
benchmark cases. The numerical results also illustrate that this method can
ensure accurate prediction and adequately capture the physical details of
phenomena caused by the solutally and thermally driven flow, which include
channel segregation, the formation of freckles, edge effect, aspect ratio
effect, and 3D effect.
