# Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature Page: 7 of 35

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III. SHAPE FUNCTIONS

In addition to the general framework we have presented

for CGMD, a specific choice of the weighting functions

is required for calculations. They result from the intro-

duction of a set of shape functions {Nj(x)}j7l" on the

mesh from which the interpolated fields are constructed.

The shape functions have the following properties:

i. Nj x) -- k,

Nnoae

ii. N(x) 1,

j=1

iii. C0 continuity.

The first property states that the functions are normal-

ized and local on the mesh nodes, xk. The second states

that the functions form a partition of unity, so the cen-

ter of mass mode is represented. The third states that

the functions are continuous but their derivatives need

not be. This continuity guarantees that the elastic en-

ergy, proportional to an integral of the square of the

strain (Ou)sym, is well-behaved in the continuum limit.

The interpolated displacement field is then defined by

u(x) E uj Nj (x). Often there are additional consid-

erations, such as the need to refine the mesh onto a par-

ticular crystal lattice at the MD/CG interface.38

Given any set of atomic displacements we can find the

displacement field represented on the CG mesh which

best fits these data in the least squares sense:

2

X2 = u - uj Ng, , (9)

where NN, Nj(x o). This X2 error is minimized by

u E fj, uN, [cf. (1)] where

f NNk~) Nk,. (10)

This equation defines the weighting function fg- of (1)

in terms of the interpolating function Nj(x). We note

that recently this relationship introduced in CGMD has

been generalized for use in the bridging scale, and other

L2 projection techniques.27

The formulation we have described is appropriate to

retain the low-lying acoustic phonon modes in the coarse-

grained system. In some cases it is desirable to retain the

long-wavelength optical phonons, as well. For example,

in the study of III-V epitaxial quantum dots, internal re-

laxation of the zinc blende structure in the strained dots

leads to important changes in the optical spectra of the

dots.39 If it is important to model the optical phonons

or to capture the internal relaxation in a crystal lattice

with a multiple-atom basis, each interpolation function

carries a band index, a, in addition to the nodal index,j: Na)(x). Then the basis requirements are somewhat

different. The functions should be local and normalized

within each band. They should independently (band-by-

band) form a partition of unity. And each of the functions

should be C0 continuous. For example, denote the dis-

placement associated with the k 0 phonon for band a

as u. , normalized such that2 )

pCunit cellNbasis

(11)

where Nbasis is the number of atoms in the Wigner-Seitz

unit cell. Then the shape functions can be defined asNa)(Xw) = u a)N.(xw)

(12)

where Nj (x) is a conventional shape function, such as lin-

ear interpolation. Note that in the case of a monatomic

unit cell this shape-function basis is a linear combination

of the shape functions we have discussed above. Then

u." is the same for all lattice sites, and the orthonor-

mal vectors corresponding to the three acoustic-mode

phonons span three-dimensional space. We do not dis-

cuss an example of a polyatomic CGMD including optical

phonons explicitly, but the CGMD formalism continues

to work in this case. It should be emphasized, however,

that even in polyatomic materials this band-index ex-

tension may not be needed to capture the mechanical

response of interest.

IV. THE CGMD HAMILTONIAN

We now turn to the calculation of the CGMD energy.

A. Harmonic Lattices

The CG energy (3) may be computed in closed form

using analytic techniques in the case of a harmonic lat-

tice. The expression was originally presented in Ref. 15.

We take the form of the atomistic Hamiltonian to be

2

HMD YEoh + -y u - DN1u,,, (13)

where Ecoh is the cohesive energy of atom p and DN is

the dynamical matrix. It acts as a tensor on the com-

ponents of the displacement vector at each site. We re-

express the CG energy (3) using a parametric derivative

of the log of the constrained partition function (4),E(uk, Uk)

O, log Z(uk, nk; 0),

(14)

and we introduce the Fourier transform representation

of the delta function (a form of Lagrange multiplier) to

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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/m1/7/: accessed May 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.