A Sharp Computational Method for the Simulation of the Solidification of Binary Alloys

Abstract

We present a numerical method for the simulation of binary alloys. We make use of the level-set method to capture the evolution of the solidification front and of an adaptive mesh refinement framework based on non-graded quadtree grids to efficiently capture the multiscale nature of the alloys' concentration profile. In addition, our approach is based on a sharp treatment of the boundary conditions at the solidification front. We apply this algorithm to the solidification of an Ni-Cu alloy and report results that agree quantitatively with theoretical analyses. We also apply this algorithm to show that solidification mechanism maps predicting growth regimes as a function of tip velocities and thermal gradients can be accurately computed with this method; these include the important transitions from planar to cellular to dendritic regimes.

Modeling binary alloy solidification by a random projection method

“…Hence the linear profile is preserved on the moving cell C 0 . The same property holds in the cell C 1 which coincides, at time t 1 , with [x 1/2 + vΔt, x 3/2 ] since the scheme (20) gives…”

“…There exists a large literature on this topic. Classical field concerns Stefan problems [8], see also [19], and more recently [20] for the Rubinstein's model. Note that these references propose finite differences approximation on cartesian grids.…”

“…Other solutions are to couple the level set method with (a) a particle tracking method (Lagrangian approach) known as particle level set method [25], or (b) a volume-of-fluid approach [26]. Finally, it is possible to develop a block adaptative meshing as it has been used for diphasic flow [27] or for the Rubinstein's problem in [20].…”

“…Compute the approximationsc n+1 I n andc n+1 I n +1 using (29) and the numerical flux (30) and (31). Compute the approximationsT n+1 I n andT n+1 I n +1 using (19) and (20) and the numerical fluxes (37)- (22). a.…”

“…Following [20], the velocity update at step 3c is obtained by mass conservation before sampling) and uses a discrete version the Rankine-Hugoniot condition of the concentration field (6). Note that in the case of the Stefan problem, that is when the concentration field is not taken into account, the velocity is updated using only the Stefan condition (5).…”