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

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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.