Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region

The quasicontinuum method is a way of reducing the number of degrees of freedom in an atomistic simulation by removing the majority of the atoms in regions of slowly varying strain fields. Due to the different ways the energy of the atoms is calculated in the coarse-grained regions and the regions where all the atoms are present, unphysical forces called “ghost forces” arise at the interfaces. Corrections may be used to almost remove the ghost forces, but the correction forces are nonconservative, ruining energy conservation in dynamic simulations. We show that it is possible to formulate the quasicontinuum method without these problems by introducing a buffer layer between the two regions of space. The method is applicable to short-ranged potentials in the face-centered cubic, body-centered cubic, and hexagonal close-packed crystal structures.

Figure 1

Coupling of atomic and continuum model with one vacancy. To the left is a fully atomistic model, to the right a quasicontinuum model. In the quasicontinuum model open circles show nonlocal atoms and solid circles show local atoms. In the small gradient fields far from the vacancy, each local atom in the quasicontinuum model represents several nonlocal atoms in the fully atomistic model.

Figure 2

Representative region ${\omega }_{\beta }^e$ of local atom $\beta$ in element $e$ is obtained by using the relation between the Delaunay triangulation (solid lines) and the Voronoi diagram (broken lines). When a corner of the Voronoi polyhedron of the element $e^{\prime }$ is outside the element, the center of gravity (solid square) is used to define ${\omega }_{{\beta }^{\prime }}^{e^{\prime }}$.

Figure 3

Concept of a quasi-nonlocal atom. (a) Quasi-nonlocal atoms (double circles) are located between nonlocal and local atomic regions. (b) The quasi-nonlocal atom $\gamma$ images neighbor atoms (gray circles) by using first neighbor atomic positions (big open and double circles) except for nonlocal atoms (small open circles) within the interactive field.

Figure 4

(a) In a 2D hexagonal lattice, the quasi-nonlocal atom 1 can extrapolate the positions of its next-nearest neighbors (such as atom 14) and its third-nearest neighbors (such as atom 9) from common nearest neighbors (atoms $5+6$ and atom 3, respectively). But it cannot extrapolate the position of fourth-nearest neighbors (such as atom 23) from common nearest neighbors. (b) If a neighbor such as atom 9 is a nonlocal atom, its position is not extrapolated.

Figure 5

First-nearest neighbors to the black atom $\gamma$ and superposed atomic configurations on (111), (110), and (0001) planes for (a) fcc, (b) bcc, and (c) hcp structures.

Figure 6

Atomic configuration of each model. (a) Full atomistic model, (b) old style QC model without correction of ghost forces, (c) and (d) improved QC model with new buffer layers. Open, solid, and double circles represent nonlocal, local, and quasi-nonlocal atoms, respectively.

Figure 7

Atomic force in perfect fcc structure in the initial configuration. No ghost force appears in models $C$ and $D$ with quasi-nonlocal atoms.

Figure 8

Displacement in the $z$ direction of each atom from its initial configuration in the equilibrium state (perfect fcc structure).

Figure 9

Atomic configurations around the vacancy in an equilibrium state. (a) Absolute value of atomic displacement in an equilibrium state from initial configuration of model $A$, $\mid {\mathbf{r}}^{\mathrm{A}}-{\mathbf{r}}_0^{\mathrm{A}}\mid$. (b) The error in the atomic displacement in the equilibrium states of models $B$ and $C$ compared to the fully atomistic model $A$: $\mid {\mathbf{r}}^{\mathrm{B}}-{\mathbf{r}}^{\mathrm{A}}\mid$, and $\mid {\mathbf{r}}^{\mathrm{C}}-{\mathbf{r}}^{\mathrm{A}}\mid$. Orientation of the atomic plane is {111}. Darker colors correspond to larger atomic displacements. $\mathrm{Max}\mid {\mathbf{r}}^{\mathrm{B}}-{\mathbf{r}}^{\mathrm{A}}\mid \simeq 0.04\mathrm{\AA }$ and $\max \mid {\mathbf{r}}^{\mathrm{C}}-{\mathbf{r}}^{\mathrm{A}}\mid \simeq 0.0001\mathrm{\AA }$.

Figure 10

A one-third part of the atomic configuration of a $\Sigma 5$ grain boundary of a full atomic model (upper part), a QC model with ghost force (middle part), and a QC model with quasi-nonlocal atoms (lower part). Open circles, solid circles, and double circles show nonlocal atoms, local atoms, and quasi-nonlocal atoms, respectively. Periodic boundary conditions are adopted in the $x$ and $z$ directions.

Figure 11

Displacement from the initial configuration in an equilibrium state in the $y$ direction of the full atomic model and the QC models. $N$, $Q$, and $L$ are the nonlocal, quasi-nonlocal, and local atomic regions.