Adaptive Mesh Refinement Computation of Solidification Microstructures Using Dynamic Data Structures

Abstract

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular, we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater than 2 17 × 2 17 elements. We examine the convergence properties of our algorithm. We also present two-dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive.

Three-Dimensional Lattice Boltzmann Modeling of Dendritic Solidification under Forced and Natural Convection

“…(0, 0, 0) for i = 0 (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) for i = [1][2][3][4][5][6] (±1, ±1, ±1) for i = 7-14 (1) where c = ∆x/∆t is the lattice speed, ∆x is lattice spacing, and ∆t is time step size. It is assumed that the temperature is constant and the undercooling does not change throughout the simulation.…”

“…The variety of numerical models of dendrite growth at the microscale falls into one of the three dominant categories: those based on the Phase-Field (PF) method [1][2][3][4][5]; models based on the Level Set (LS) method [6][7][8][9], and models that perform a Direct…”

Atomistic and continuum modeling of dendritic solidification

“…While such algorithms have been extremely helpful to extend simulations to a small velocity regime [158][159][160]186], they do not fundamentally eliminate the stiffness introduced by a diffuse interface. For most solidification conditions (save perhaps very rapid rates), surmounting this difficulty requires a formulation of the phase-field equations in such a way that they reduce to the correct set of sharp-interface equations for a mesoscale interface thickness.…”

“…For typical equiaxed grains in castings, p is in the range of 0.01-0.1. The adaptive meshing [159,160] and other hybrid [186] algorithms developed recently for dendritic growth cope efficiently with the disparity between r and '. They do not, however, eliminate the several orders of magnitude disparity between r and d 0 that follows from the smallness of p and s in Eq.…”

Solidification in a Supercomputer: From Crystal Nuclei to Dendrite Assemblages

“…1,2 Although the cellular automaton method has been widely used for polycrystal solidification simulations, 15,16,82 and has been employed for large-scale solidification simulations, 83,84 multiple-dendrite-growth simulation by the phase-field method is crucial for accurate prediction of the solidification microstructure. An adaptive mesh refinement technique, in which fine meshes are used only around the interface, [85][86][87][88][89] can reduce the computational cost. However, its applicability to polycrystal solidification with a large interface area fraction is not flexible.…”