Figure 1

Stress-strain curves for five simulations with varying grain size. To analyze the strength, the stress in the sample was calculated as $\sigma =dF/d{ϵ}_{yy}$, where ${ϵ}_{yy}$ is the applied uniaxial strain.Reuse & Permissions

Figure 2

(Color online) Three time slides capturing deformation in the $50\times 50{\text{nm}}^2$ sample. Average grain size is 15 nm. Dislocations A and C are absorbed by the surface or grain boundary, respectively. Dislocation B is ejected from the grain boundary (GB) and annihilated at the surface. Small arrows without labels point to a dislocation trajectory.Reuse & Permissions

Figure 3

Two frames showing a portion of a sample with a rotating grain. Initial time is $t=30$ and final time is $t=40$. Dashed lines $A$ and $A^{\prime }$ are parallel. The dotted line shows the original shape of the grain. It is superimposed onto the deformed grain on the right.Reuse & Permissions

Figure 4

Three voids A, B, and C are displayed in four consecutive times 25, 35, 45, and 50. Voids A and B are nucleated and continue to grow as simulation proceeds. In the second half [slides (c) and (d)], a third void C is nucleated, while the A and B voids are almost annihilated. It is interesting to point out the dynamic nature of the process evidenced by the voids “gliding” along grain boundaries.Reuse & Permissions

Figure 5

Hall-Petch-type plot with yield stress plotted against the inverse square root of grain size. Solid points represent MPFC simulation results. Unfilled circles and diamonds are experimental measurements [39]. The data are rescaled about the highest stress and the corresponding grain size. The dashed line is a guide for the eyes.Reuse & Permissions

Figure 6

The convergence history of different multigrid cycles for a typical run of the MPFC model during a solidification experiment.Reuse & Permissions

Figure 7

Iterations to convergence vs number of grid points for successive over-relaxation (SOR) and multigrid (MG) methods. Both methods are solving model (3). The top and bottom lines represent the theoretical complexity for the SOR and multigrid approaches, respectively. The SOR and multigrid are further compared during a solidification simulation, where the iterative method increases the number of operations by a factor of 1.3 with increasing sample size, while multigrid method increases the same by a factor of 1.1. The performance of the multigrid approach improved very slightly when simulating a deformation in a perfect crystal. In this case, the number of operations increased 1.07 times with the number of computational nodes.Reuse & Permissions