Hand-shake region

Figure 7 Schematic illustration of the hand-shake treatment near the FE/MD interface in the CLS method. A one-to-one atom-to-node mapping is considered in the FE zone up to a distance equal to twice the classical potential cutoff. After that, the mesh expands. Elements that are partially contained in the MD zone are displayed as shaded and contribute less than their entire energy to the Hamiltonian. Similarly, on the MD side, bonds crossing the interface (dashed lines) contribute less than their full weight, while bonds completely enclosed in the MD area (solid lines) have weight equal to one. Bi and B2 are the boundaries of the hand-shake region.

In the ODD method, the atomistic and continuum regions are coupled through a standard hand-shake region, and a single energy functional is considered for the complete domain. As in most methods, the continuum is... 

One of the issues in dynamical multiscale coupling is the tailoring of the time step to the different subdomains. If the same time step is used in both the atomistic and the continuum regions, computations will be wasted in the continuum model. However, if in the hand-shake region the size of the FEM elements is reduced to coincide with the individual atoms, it is difficult to tailor the time step. Therefore, the authors of the ODD method chose to use a uniform mesh for the continuum domain, so that a much larger time step could be used in the continuum model than in the atomistic one. A description of such a multiple-time-step algorithm is provided in the paper. 

Figure 9 Coupling between the atomistic and continuum domains in the BED method In contrast to the ODD case, no hand-shake region is considered, and the coupling of the two regions occurs through an interface.

SO that atoms remain near the corresponding mesh point on either side of the interface at all times. The way the hand-shake Hamiltonian is constructed is based on the idea that the interactions at the interface can be approximated to first order by an average of the two descriptions. This means that bonds completely contained in the MD region, or elements in the FE one, contribute with a full weight to the Hamiltonian, while bonds or elements only partially contained in their natural region contribute with a reduced weight to the overall Hamiltonian. More specifically, when considering as semiempirical potential the SW one, the hand-shake Hamiltonian becomes... 

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