M A S
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Atomistic-to-Continuum Coupling
Huan Wu, University of Warwick
Mathematics Institute
Introduction
Atomistic-to-Continuum Coupling
Crystal defects play an important role in material behaviour. The following picture by F oll
demonstrates an example of different types of defects.
An example: 2D GR-AC model of a anti-plane problem
Take the 1D case for an example, (u a )00 will be of order O(1) in a neighbourhood of defects,
therefore the Cauchy–Born model produces an O(1) error. To improve the accuracy, a/c
coupling methods decompose the reference lattice Z into an atomistic region A that contains
the defect core and is described by the exact atomistic model, and a continuum region C
where (u a )0 varies slowly and the Cauchy–Born model is used.
0.04
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
10
0
−10
−10
−15
0
−5
10
5
15
Figure : Decomposition of the atomistic chain into an atomistic region A and a continuum region C.
Figure : This a visualization of the solution u a ch using P2 finite element method on a coarse mesh.
Figure : a) Interstitial impurity atom, b) Edge dislocation, c) Self interstitial atom, d) Vacancy, e) Precipitate of
impurity atoms, f) Vacancy type dislocation loop, g) Interstitial type dislocation loop, h) Substitutional impurity atom
Hence a general form of a coupling method is as follows
Z
X
X
Eac(u) =
V (D(u)) +
Vxi (Du(x)) +
W (∇u)dξ,
x∈A
Atomistic-to-continuum (a/c) coupling methods are a class of computational multiscale
schemes that combine the accuracy of atomistic models and the efficiency of continuum
models. In short, these schemes admit the atomistic model in a neighbourhood of defect cores
and approximate using a continuum elasticity model at far fields. These methods contain
various approximation errors, including pure modelling errors and computational errors.
Error estimates
Ωc
x∈I
where A, I and C are the atomistic, interface and continuum regions respectively. The picture
below is an example of a 2D GR-AC coupling model.
Let u a be a strongly stable atomistic solution and let uhac be the coupling solution obtained by
the finite element scheme. After consistency and stability analysis we can find the following
error estimates:
coarsening error
modelling error
k∇u a − ∇uhackUh∗
}|
{ z far-field
z
}|
{ z
}| error {
. k∇2u a kL2(I) + k∇3u a kL2(Ch ) + khk ∇k+1u a kL2(Ch ) + k∇u a kL2(R2\ΩN ) .
k∇u a − ∇uhackUh∗
}|
{
z
}|
{ z
2 a
3 a
2 a 2
k k+1 a
. k∇ u kL2(I) + k∇ u kL2(Ch ) + k∇ u kL4(Ch ) + kh ∇ u kL2(Ch ) +
coarsening error
modelling error
The Atomistic Model in 1D and 2D
(1D QNL)
z far-field
}| error {
+ k∇u a kL2(R2\ΩN ) .
In 1D, we consider a finite atomistic chain, denoted by L := AZ. A displacement of the chain is
represented by a map u : Z → R. In my Msc thesis I studied the next-nearest-neighbour
interaction using Lennard-Jones potential.
(2D GR-AC)
It is clear that the modelling error and the far-field error cannot be improved by the finite
element setting. Thus we aim to make the coarsening error small enough such that it is
dominated by other terms. It can be shown that for the 1D QNL model, P2 finite element
method is sufficient to achieve the optimal convergence rate. In the 2D GR-AC model, P2 finite
element method is also sufficient.
Finite Element Coarsening
Numerical Results of 1D QNL
Figure : Illustration of the next-nearest-neighbour interaction
In 2D, we consider a nearest interaction model on a triangular lattice denoted by
1 cos(π/3)
L := AZ2, with A =
.
0 sin(π/3)
Convergence rates for slowly decaying solution
−1
−1
Convergence rates for rapidly decaying solution
10
10
N1/2−α
T
1/2−α
NT
−2
10
−2
In terms of numerics, we use Pk finite element coarsening. We apply the exact atomistic
calculation on the atoms in the atomistic region and the interface region, and project E a onto a
coarsened mesh in the continuum region.
Figure : A triangular lattice with nearest-neighbour interactions
The atomistic energy is therefore represented in terms of energy sites:
X
a
E (u) :=
V (Du(x)),
x∈L
where V is a suitable potential and Du is the discrete difference of u.
Figure : A coarse mesh in 2D. The red dots are the connanical atomistic sites in the atomistic and the interface
regions.
E c (u) =
W (∇u)dx,
Rn
where W is the Cauchy–Born strain energy density function that approximates the atomistic
potential V . Note that the discrete displacement map u is identified with its piecewise linear
interpolant.
MASDOC CDT, University of Warwick
N−1/2−α
−1/2−α
NT
T
−3
10
−1−α
−4
NT
10
−1−α
ATM
QNL−P1
QNL−P2
−4
−5
10
NT
ATM
QNL−P1
QNL−P2
10
0
10
1
2
10
10
3
10
0
10
1
2
10
10
N
T
T
We assume the regularity of the atomistic solution u a to be
8
6
|∇j u a (x)| . |x|−α+1−j for x > r∗, j = 0, 1, 2, 3.
4
The results for α = 0.8 (slow decay) are shown on the left and the result for α = 1.3 (rapid
decay) are shown on the right. The x-axis NT is the number of degrees of freedom. We
observe precisely the rates predicted in the previous analysis.
0
To approximate the atomistic energy, we consider the continuum elasticity model with an
energy functional of the form
Z
−3
10
N
2
Cauchy–Born Continuum Model
error
Figure : Illustration of the constuction of a quasi-non-local method in 1D with a coarse mesh in the continuum
region. The triangles represent the nodes of a coarsened finite element mesh.
error
10
−2
Acknowledgement
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Supervisors: Christoph Ortner, Andreas Dedner
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Mail: masdoc.info@warwick.ac.uk
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M A S
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WWW: http://www2.warwick.ac.uk/fac/sci/masdoc/