A finite element formulation for macrosegregation during alloy solidification using a fractional step method and equal-order elements

A finite element formulation of the "minimal" solidification model is presented for the prediction of macrosegregation during two-dimensional (2D) columnar solidification of binary alloys. A fractional step method is extended to solve the thermosolutal convection that has a damping in the mushy zone during solidification. Using this method, the velocity and pressure are decoupled and interpolated by linear equal-order triangular elements, resulting in decoupled systems that can be solved simply and efficiently. For convection-diffusion equations of energy, solute and momentum, the consistent streamline upwind Petrov-Galerkin (SUPG) method and the second-order Crank-Nicolson scheme are used for the discretization and integration over the spatial domain, respectively. A solution procedure is designed to couple the resolutions of conservations of energy, solute and momentum, as well as the microsegregation model at an overall computational efficiency and accuracy. The formulation is first validated and then applied to predict macrosegregation during solidification of Pb-18 wt%Sn and Sn-10 wt%Pb alloys in a rectangular mold. The formation of macrosegregation is investigated, and comparisons with another finite element method (FEM) based code are made.