Computational Physics

442 The finite element method for partial differential equations

MD

FEM

Figure 13.8. Strip modelled partly by finite elements and partly by molecular

dynamics.

Three remarks are in place. In the original formulation[14], the MD region is
three-dimensional, whereas the FEM region is only two-dimensional. The transition
is made by averaging the MD points over thez-direction which is taken perpen-
dicular to the FEM grid. Here we shall consider the strictly two-dimensional case
for simplicity. The second remark concerns the treatment of the FEM masses. As
we have seen above, the mass matrix couples the kinetic degrees of freedom at
the vertices of the FEM triangles. However, in the handshake region, we strictly
want to assign the mass of a real atom to the point. For this reason, we use the
lumped massapproximation in the finite element description. In this formulation,
we assign one-third of the mass of each triangle to each of its vertices. This means
that the mass matrix has become diagonal, so that the numerical integration of
the equations of motion has become much simpler as the solution of an implicit
equation at each time step is avoided. The FEM mass is derived from the MD equi-
librium by requiring that the same amount of mass is present per unit area in both
descriptions.
The final remark is that the boundaries of the system in the MD and FEM descrip-
tion do not fit onto each other. In the FEM description, the triangles are taken
uniform, but a MD system with a boundary will have a slightly smaller distance
between the outermost layers than in the bulk, as a result of the fact that the next
nearest-neighbour interactions pull the outermost particles slightly more towards
the interior of the system. This deviation is minor so we do not correct for it.