Efficient Simulation Schemes for Large-Scale Phase-Field Modelling of Polycrystalline Growth During Alloy Solidification

Abstract

Ameliorating the computing efficiency is always of importance in phase-field simulations of material microstructure formation and evolution. Borrowing from the nonlinear preconditioning treatment of diffuse interface models [1], the usual quantitative phase-field model for a binary alloy [2] has been transformed to make it easier to compute accurately (see Fig. 1 the transform procedure). The transformation yields a new variable whose value changes linearly across the interface. The dependences of simulated results of the nonlinearly preconditioned phase-field formula on the interface grid size and the discretization time step have been examined in detail through numerical experiments, including the growth velocity, the radius and the solute concentration of a steady tip. The results show that the new evolution equations are able to be solved on a computational mesh with interface grids 2–4 times coarser than those used in the conventional method, as show in Fig. 2. In combination with the front-tracking method to capture the crystallographic orientation of each crystal, the orientation gradient energy is incorporated into the nonlinearly preconditioned phase-field model, which enables simulations of grain boundary behaviors. The algorithm of the distributed parallel finite element method on an adaptive mesh is applied to further raise the computing efficiency. Simulations of multi-dendrites growth of Al-4 wt.% Cu alloy in undercooled melt cooling down continuously are performed. The results demonstrate that the proposed fast simulation approaches allow quantitative simulations of a large number of dendrites growth on the scale of centimeters or millimeters, respectively in two (Fig. 3) or three dimensions (Fig. 4), just using an ordinary workstation instead of clusters or supercomputers [3]. The proposed fast simulation schemes make it possible to perform full-scale simulations of the in situ and real-time observation experiments [4, 5, 6, 7]. In addition, the nonlinear preconditioning transformation can be adopted to other phase-field models used for simulations of the solid-state grains growth, the precipitation of a second phase, the crack propagation, etc. Open image in new window Fig. 1. The transformation of the quantitative phase-field model for dilute binary alloy solidification proposed by Karma [2] using a nonlinear hyperbolic tangent function \( \varphi (\varvec{r},t) = \tanh [\psi (\varvec{r},t)/\sqrt 2 ] \). Open image in new window Fig. 2. Variation of the steady-state tip velocity and number of grid size with the interface (minimum) grid size for a free dendrite growth from undercooled melt in two (a, b) and three (c, d) dimensions, which demonstrates the available large grid size that can be employed at the interface to obtain the converged results. The statistical histogram show that as the interface grid size decreases, the computing amounts can be reduced orders of magnitude. Open image in new window Fig. 3. The 2D phase-field simulated thousands of dendritic crystals growth in a centimeter-scale domain during continuously cooling-down of Al-Cu alloy melt. (a) The solute concentration, (b) the magnified local view of the boxed dendrites in (a) and the corresponding mesh. The cooling rate is 0.8 K/s. The domain size is of 1.058 × 1.058 cm2 (10000W0 × 10000W0). The grain number with different orientations is 2000. The simulations are implemented in a tower workstation with 2 Intel® Xeon E5-2697 v3 (2.6 GHz) CPUs and 256 GB of memory. 48 cores are employed for parallel computing and 5 days are taken to finish the simulation. Open image in new window Fig. 4. The evolution of dendrite morphology during solidification in 3-D simulation with cooling rate R = 0.8 K/s. The different colors represent different crystallographic orientations. The domain size is of 1.058 × 1.058 × 1.058 mm3 and the grain number is 125. The configuration of computation is identical to the 2-D simulation, but takes 30 h to finish.