Adaptive mesh simulations of polycrystalline materials using a Cartesian representation of an amplitude expansion of the phase-field-crystal model

This paper introduces improvements to an amplitude expansion of the phase-field-crystal model. An auxiliary field describing local grain rotation is introduced and used to enable the adaptive mesh to be coarsened in all grains, regardless of their orientation. Only a Cartesian representation of the amplitude equations is used.

Figure 1

Beats in misaligned grains. The images show (a) real component of the first complex amplitude $\text{Re}\left(A_1\right)$, and (b) the reconstructed atomic density field $\psi$ in three seeds, rotated for (clockwise from bottom left) $\theta =0$, $\pi /24$, and $\pi /6$.

Figure 2

Microstructure evolution in time. Comparison of the AMR model with and without local rotation. When using a local rotation, the mesh coarsens in all seeds, regardless of their orientation, and remains dense on grain boundaries where dislocations are formed. Images show the average amplitude field (${\sum }_j\left|A_j\right|/3$) at different times.

Figure 3

Beats and mesh refinement in rotated and nonrotated grains. The grains are rotated by $\theta =\pi /12$ and 0. From top to bottom the image shows (a) the average absolute amplitude (${\sum }_j\left|A_j\right|/3$), (b) the real part of locally rotated complex amplitude $\left[\text{Re}\left(A_1^{\vartheta }\right)\right]$, (c) the phase angle of the complex amplitude $A_1$ $\left[\mathrm{\Phi }\left(A_1\right)\right]$, (d) local mesh refinement ($\delta x$), (e) reconstructed atomic density function ($\psi$), and (f) the fields $\delta x$, ${\sum }_j\left|A_j\right|/3$ in cross section.

Figure 4

Comparison of simulation results with different models at $t=360$. All columns show PFC model with complex-amplitudes extension. The first column shows results when using a regular grid, the second column shows results when using adaptive mesh refinement techniques, and the third column shows results when using local rotation with the help of an auxiliary rotation field. From top to bottom: reconstructed atomic density field $\psi$, mesh density $\delta x$, average amplitude ${\sum }_j\left|A_j\right|/3$, real part of the first complex amplitude $\text{Re}\left(A_1\right)$, and local grain rotation in degrees $\theta$ are shown.

Figure 5

Selected variables in excerpt and cross section at $t=560$. From top to bottom: (a) absolute value of the first complex amplitude $|A_1|$, (b) local mesh refinement $\delta x$, (c) reconstructed density field $\psi$, and (d) local rotation $\theta$. Values of $\theta$, $|A_1|$, and ${\sum }_j\left|A_j\right|/3$ in cross section on the marked line at $y=183\pi /2\approx 287$ are shown in the last image (e). The local rotation field correctly tracks the grain rotation and shows large changes only on the grain boundaries.

Figure 6

Amplitude $|A_1|$ in cross section cut along the lines $x=201\pi \approx 631$ and $y=203\pi \approx 638$ at times $t=160$ and 560. Results obtained with AMR using a local rotation on an adaptive grid match the results obtained with simulations on a uniform grid. Exact values at the intersected computation nodes of the adaptive grid are shown in combination with interpolated values from the same grid in comparison with values obtained on a uniform grid.

Figure 7

Number of computational nodes as a function of time for the summations shown in Figs. 2 and 4. After the liquid freezes, the number of nodes stops increasing. When using the scheme with a local rotation, the mesh coarsens in all nodes and therefore consists of a much smaller total number of nodes.

Figure 8

Accuracy of the simulations under conditions where local rotation might be incorrectly calculated or where large deformations exist. (a) Microstructure of a simulation run where our algorithm determined local rotation. (b) Microstructure of a simulation run where local rotation was fixed to a chosen function $\mathrm{\Theta }\left(\mathbf{x}\right)=\mathrm{\Theta }\left(x\right)$. The microstructure in both simulation runs matches closely, with minor differences observed only around dislocations. (c) Local rotation field $\mathrm{\Theta }$ as determined by our algorithm. (d) Deformation remainder $\mathrm{\Delta }\epsilon$, shown in logarithmic scale. (e) Local rotation $\mathrm{\Theta }$ as determined by our algorithm, the chosen function $\mathrm{\Theta }\left(x\right)$, and deformation remainder $\mathrm{\Delta }\epsilon$ in cross section. $\mathrm{\Delta }\epsilon$ remains low inside grains, in the area with pure elastic deformation at $x\approx 280$ shows a small increase coupled with some inaccuracy in the calculation of a local rotation field $\mathrm{\Theta }$ and rapidly increases in the vicinity of the dislocation at $x\approx 120$.