Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature Page: 4 of 35

2

Micron
Resonator
CG''' ,

FIG. 1: Schematic diagram of a coarse-grained simulation of
a NEMS silicon microresonator.4-6 The CG region comprises
most of the volume, but the MD region contains most of the
simulated degrees of freedom. Note that the CG mesh is
refined to the atomic scale where it joins with the MD lattice.
with conventional continuum elastic theory and finite el-
ements, and too large to be modeled by conventional
atomistics. Even in single crystals, sub-micron dynami-
cal regions bounded by surfaces or interfaces are affected
by Angstrom- and nano-scale physics which causes de-
viations from continuum elastic theory;12 dynamical re-
gions larger than 0.1 micron cubed exceed the current
limit of about one billion atoms for atomistic simulation
of solids on a supercomputer.13 The atomistic effects are
compounded in materials with local defects or cracks that
couple to long-range strain fields.14 The situation is not
entirely intractable, however, because the most impor-
tant atomistic effects are often localized to small regions
of the system: surfaces, defects, regions of large deflection
or internal strain, and regions of localized heating per-
haps due to friction. The challenge is to develop a robust
model for such an inhomogeneous system which captures
the important atomistic effects without the prohibitive
computational cost of a brute force atomistic simulation
for the entire system. In this article we focus on the link
between the micron scale and the nanoscale and develop
a model, coarse-grained molecular dynamics (CGMD),15
which bridges the disparate scales seamlessly.
The motivation to use atomistic models at the finest
resolution is motivated in some cases by the fact that the
inherent length scale of the process of interest is the in-
teratomic spacing and in other cases by the ability to de-
rive interatomic potentials from quantum mechanics and
hence built a model from first principles. Yet another mo-
tivation is that the processes of interest may be thermally
activated, and molecular dynamics provides a means to
simulate the thermal effects directly. Entropic and ther-
mal effects are often paramount in soft matter systems,
and in hard matter thermal activation is important in de-

feet diffusion, the motion of dislocations in metals with
high Peierls barriers and many nucleation phenomena.
Temperature is important in other ways, too, such as in
causing phase transitions. The population of phonons
increases with temperature, causing thermal expansion,
changes (typically softening) in the elastic constants and
dislocation drag at high strain rates. These are but a few
well-known examples of the important role temperature
plays, and thus in our development of multiscale models
we search for methodologies capable of handling non-zero
temperatures.
The variation of the strain field/atomic displacements
in inhomogeneous solid systems suggests the use of dif-
ferent computational methodologies for different regions,
as we mentioned above. The challenge is to meld
them into a seamless, monolithic simulation. The first
such proposal implemented a coupling between molecu-
lar dynamics16,17 (MD) and a finite element model,19
(FEM) implementation of continuum elastic theory us-
ing stress consistency as the boundary condition at the
interface.20 More recently, a dynamical instability in the
original formulation has been eliminated through the use
of a mean force boundary condition together with uni-
form symplectic time evolution.21 In both of these formu-
lations, the same constitutive relation is used regardless
of the size of the cells in the FEM mesh, leading to a
discontinuity at the atomic limit.
At its heart the FEM description of such a system relies
on the ability to improve the accuracy of the simulation
by going to a finer mesh.19 A mesh of varying coarseness
is chosen, adaptively or by fiat, such that no single region
contributes disproportionately to the error. These errors
typically result from large strain gradients which violate
the discrete expression for the integral of the elastic en-
ergy density of a continuous medium. This approxima-
tion may be improved by mesh refinement. There is a
limit, however. As the mesh size approaches the atomic
scale, the constitutive equations have significant errors
because the expression for the elastic energy does not
represent localized bonds and the standard distributed
mass expression for the kinetic energy does not account
for the fact that essentially all of the mass is localized
in the nuclei, at least four order of magnitude smaller
than the interatomic spacing. At this point, the physics
of the governing equations is wrong, and further mesh
refinement does not help.
The approach of Refs. 20 and 21 to improve this situa-
tion replaces the FEM equations of motion on regions of
the mesh that are atomic-sized with MD equations of mo-
tion and implements a hand shaking between the MD and
FEM regions. Although this technique is remarkably suc-
cessful, the union is not perfectly seamless. In the FEM
cells approaching the atomic limit, the energy density
varies smoothly within each cell, whereas on the other
side of the interface, the MD energy is effectively local-
ized to interatomic bonds. The short-wavelength modes
of the system are able to probe this discrepancy, leading
to errors that grow with the wavenumber.

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Rudd, R E & Broughton, J Q. Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature, article, May 30, 2005; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc877423/m1/4/ocr/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.

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