General hierarchical structure to solve transport phenomena with dissimilar time scales: Application in large-scale three-dimensional thermosolutal phase-field problems

A general hierarchical structure is developed for phase-field lattice-Boltzmann simulations with dissimilar time scales. The number of the grid levels can be artificially selected in a reasonable range, which can enhance the time marching step by two to three orders of magnitude in comparison with explicit methods. Constructed on a massively parallel platform, the mesh distribution is dynamically adjusted according to a gradient criterion. The developed high performance computing scheme is applied to simulate the coupled thermosolutal dendrite evolution. Numerical tests indicate that the computing efficiency can be further improved by two to three orders of magnitude, which makes numerical simulation of fully coupled thermosolutal dendrite growth viable for alloys with Lewis number $\sim {10}^4$. The domain size which equivalently consists of billions of uniform meshes is handled to simulate multidendrite evolution. Results show that the domain temperature becomes extremely uneven due to the release of latent heat, which causes a significant difference from isothermal solidification. A simple analytical model is proposed to predict the relation between growth velocity and Lewis number, and the growth morphologies of both equiaxed and directional multiple dendrites are discussed. The combination of the hierarchical mesh structure and the phase-field lattice-Boltzmann method provides an efficiency-driven approach to solve the coupled thermosolutal microstructure evolution.

Figure 1

Temperature distribution along the horizontal centerline for (a) 2D and (b) 3D cases, respectively. Both $T$ and $x$ are dimensionless.

Figure 2

Efficiency test for (a) 2D and (b) 3D cases by using the developed numerical algorithm to solve the thermosolutal PFLB equations. $n_p$ is the number of the parallel processes, and $n_l$ is the number of the grid levels employed during simulation.

Figure 3

Dimensionless average growth velocity $V_{\mathrm{ave}}$ of the 2D dendrite tip versus heat sink ${\dot{q}}_{\mathrm{LBM}}$. The insets show the thermal distribution (on the left side) and the dendrite contour with the patch-box architecture (on the right side).

Figure 4

3D dendrite morphology with ${\dot{q}}_{\mathrm{LBM}}$ being $5\times {10}^{-6}, 1\times {10}^{-5}, 5\times {10}^{-5}$, and $1\times {10}^{-4}$ from (a) to (d) [or from (e) to (h)] respectively. The first and second rows show the dendrite morphologies viewed from the [100] and [111] directions respectively.

Figure 5

Simulated single dendrite morphology including the phase field, solute field, temperature field, and hierarchical structure of the patch boxes during the AMR for (a) 2D and (b) 3D case. (c) Hierarchical grid structure according to the AMR for the 3D single dendrite.

Figure 6

Dimensionless temperature θ vs distance along the horizontal centerline of the 2D domain at the 300 000th, 600 000th, and 1 000 000th steps respectively. The inset is the dendrite contours at different time.

Figure 7

Tip growth velocity $V_{\mathrm{tip}}$ and dimensionless temperature θ vs time for the 3D thermosolutal single dendrite growth.

Figure 8

Average growth velocity $V_{\mathrm{ave}}$ and dimensionless temperature θ vs Le.

Figure 9

Evolution of 2D equiaxed multiple dendrites including (a)–(d) phase field, (e)–(h) temperature field, and (i)–(l) solute field. The column corresponds to the snapshots at the 40 000th, 400 000th, 800 000th, and 1 200 000th steps from left to right. Twenty randomly distributed seeds are initialized in the domain.

Figure 10

(a) Temperature fluctuations along the horizontal centerline of the computational domain corresponding to the temperature field cloud maps shown in Figs. 9–9. (b) The dimensionless maximum and minimum temperature at different time.

Figure 11

Evolution of 3D equiaxed multiple dendrites including (a)–(c) phase field, (d)–(f) temperature field, and (g)–(i) solute field. The column corresponds to the snapshots at the 10 000th, 50 000th, and 150 000th steps from left to right. Ten randomly distributed seeds are initialized in the domain.

Figure 12

Evolution of 2D directional multiple dendrites including (a)–(d) phase field, (e)–(h) temperature field, and (i)–(l) solute field. The column corresponds to the snapshots at the 100 000th, 800 000th, 1 800 000th, and 2 800 000th steps from left to right. Twenty randomly distributed seeds are initialized at the domain bottom.

Figure 13

(a) Temperature fluctuations along the vertical centerline of the computational domain corresponding to the temperature field cloud maps shown in Figs. 12–12. (b) The dimensionless maximum and minimum temperature at different time.

Figure 14

Evolution of 3D directional multiple dendrites including (a)–(c) phase field, (d)–(f) temperature field, and (g)–(i) solute field. The column corresponds to the snapshots at the 10 000th, 150 000th, and 300 000th steps from left to right. Ten randomly distributed seeds are initialized at the domain bottom.