constructionandsharpconsistencyestimatesfor atomistic ......parameters c ijk at a dimerized si[100]...

Construction and sharp consistency estimates foratomistic/continuum coupling methods: a 2D model problem

Lei Zhangwith:

Christoph Ortner

Mathematical InstituteUniversity of Oxford

Point Defects in Crystal

[Wikipedia]

Defects in Crystals

4

When a small load is applied to a crystal it undergoes a deformation, so that the distances

between atoms change and attractive or repulsive forces arise which resist the external load. If

the original shape is restored when the load is removed, the deformation is known as elastic.

Stiffness describes the ratio of the load (stress) to deformation (strain) under elastic

conditions.

It is important to distinguish between stiffness and strength. Strength describes the resistance

to permanent (inelastic, plastic) deformation. It is expressed in terms of an applied stress

value beyond which deformation is irrecoverable. In order to understand the origin of the

strength of alloys it is important

to have a picture of how plastic

deformation takes place.

The easiest way to deform a

crystal plastically is by making

one densely populated plane of

atoms slide over the other, in a

manner similar to a stack of

cards. However, this requires

breaking and re-establishing a

huge number of interatomic

bonds simultaneously. An even

easier way to shear a crystal is

by propagating a dislocation

along the same plane, which is

similar to running a crease

(which can be thought of as a

line) across a carpet (Figure 2).

It is the ease of propagation of

dislocations across crystals that

determines their strength. You

will discuss the crystallographic

aspects of plastic deformation in

more detail in your A4 Material

Failure course.

Single crystals usually deform very easily under

low values of stress. Since they have a nearly

ideal structure, once the deformation begins at

one point, it can continue throughout the crystal.

Typical stress levels for plastic deformation are

in the range between 1 and 10 MPa, much too

low for most engineering purposes.

Most engineering alloys are used in

polycrystalline form (Figure 3). This means that

each piece of metal is made up of a great

number of single crystals, or grains, each having

Figure 2.

Figure 3. Korsunsky; lecture notes

(grain boundary)

R702 Topical Review

a) Time: 2.98 ps b) Time: 3.02 ps

c) Time: 3.15 ps d) Time: 3.47 ps

5.44 A700 A

(111)(112)

Figure 6. Four snapshots of the opening (111)[110] crack system simulated using LOTF. Atoms inred were treated quantum mechanically using the ab initio SIESTA package. The quantum regionfollows the crack tip as it moves from left to right. Note the pentagons and heptagons of the 2× 1Pandey reconstruction on the top surface. See the text for an explanation of the dynamical aspects.

atoms in the moving quantum region. In addition to showing smooth atomically flat surfacesopening, using such a sophisticated QM engine revealed that the surface reconstructions aredifferent on the upper and lower surfaces, with the upper surface showing the 2 × 1 Pandeyreconstruction, as indicated by the alternating pentagons and heptagons. The reason that onlyone surface shows this reconstruction can be understood by considering the dynamics of atomsnear the crack tip. The atoms, shown in black, that were part of a hexagon are forced towardseach other as the crack tip passes (see panels (a) and (b)), thus driving the formation of apentagon. In panel (d), the crack has moved and the next pentagon is about to form. Thecorresponding atoms on the lower surface (shown in grey) are forced away from each other,preventing pentagon formation. Further results on the crack propagation problem in siliconwill be published elsewhere.

6. Conclusion

At present, hybrid simulation techniques are not yet used very widely. This stems not onlyfrom the fact that the surge of interest in method development is very recent, but also fromthe technical difficulties in properly representing the boundary between the regions described

Csanyi etal;JPhys.:Condens.Matter(2005)

(crack tip in silicon) •M=Numbers of atoms in macroscopic material ~1023 atoms/cm3.

•Current computational limits for increasingly accurate potentials:

•Empirical potentials (EAM): ~1010 atoms

•Semi-empirical potentials (TB): ~106 atoms

•Quantum mechanics (DFT): ~104 atoms

•Nonconvex potential !(r) gives fracture, microstructure, etc.

•Can be modeled by continuum elasticity except near defects.

•AtC method: Use atomistic model near defects, remove degrees of

freedom elsewhere by continuum coarse-graining.

Atomistic Computation Scale Limits and the AtC Solution

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!(r)

Dobson & Luskin

(edge dislocation)

Fig. 7 shows the computed variation of the vacancy formation energy with sample size.This variation is reported for two sets of calculations: one where the atomic positions areheld fixed (unrelaxed) in their nominal position; and a second where the atomic positionsare relaxed. The vacancy formation energy is found to follow a power law close to n!0:5 inthe unrelaxed case and n!0:55 in the relaxed case, where n is the nominal number of atomsin the sample. This power-law behavior is an indication of the long-range nature of theunderlying physics. It is interesting to note that relaxation of the atomic positions reducesthe vacancy formation energy by 0.06 eV.Fig. 8 shows the radial displacement fields along h1 0 0i and h1 1 0i directions. The fields

have a long tail, another indication of long-range nature of the field of the vacancy. Themaximum displacement occurs in the h1 1 0i direction and amounts to 0.6% of the nearestatom distance. This value is less than predicted in previous calculations using Kohn–ShamDFT with periodic boundary conditions where the maximum displacement was estimatedto be 1–2% of the nearest atom distance (Mehl and Klein, 1991; Chetty et al., 1995; Turneret al., 1997). At this point there is no basis to decide whether the discrepancy is due to theorbital-free formulation or the use of periodic boundary conditions.

7.2. Di-vacancy

A di-vacancy consists of two vacancies at positions a1 and a2 within a crystal. We haveconducted calculations with a million-atom specimen subjected to Dirichlet boundaryconditions representing bulk values as before. The calculations use up to 2001representative atoms (slightly smaller when the vacancies are close to each other), have800,000 elements in the electronic-mesh and require 16–18 h on 64 700MHz processors.Figs. 9 and 10 display representative results. Fig. 9 shows the contours of the ground-stateelectron-density around a di-vacancy complex along h1 0 0i and h1 1 0i. Fig. 10 show the

ARTICLE IN PRESS

Fig. 5. (a) Contours of ground-state electron-density around the vacancy on (1 0 0) plane; (b) contours of ground-state electron-density around the vacancy on (1 1 1) plane.

V. Gavini et al. / J. Mech. Phys. Solids 55 (2007) 697–718710

Gavini; PhD Thesis

(vacancy in aluminium)

205

Figure 1. A fully atomistic simulation of a crack near a grain boundary is schematically shown in (a). In (b), mostof the 1.2 × 106 atoms in the simulation are not shown to reveal the important atomistic details of the disloca-tion loops emitting from the crack and impinging on the grain boundary (reproduced from [2], with permission,published by Taylor and Francis, www.tandf.co.uk).

of this paper to discuss all of them in detail, we will provide a brief literature review of thesetechniques.

The organization of the remaining sections is as follows. First, a brief review of atomisticmethods is provided in Section 2. This is considered relevant since the atomistic model isviewed as the benchmark “exact” description of material behaviour that the QC aims to repro-duce with reduced computational overhead. In Sections 3 and 4, the current state of the QC ispresented, based on the cumulative work presented in references [1, 6, 7, 8, 9]. In Section 5,a number of applications are presented. Section 6 discusses the current directions being takenwith the QC. Finally, we review related simulation techniques in Section 7.

As noted in the abstract, as an accompaniment to this paper, a website designed to serve as aclearinghouse for information on the QC method has been established at www.qcmethod.com.The site includes information on QC research, links to researchers, downloadable QC codeand documentation. The downloadable code is freely available and corresponds to the full QCimplementation discussed in Section 3.4.

2. Atomistic modeling

In the QC, the point-of-view which is adopted is that there is an underlying atomistic model ofthe material which is the “correct” description of the material behaviour. This could, in prin-ciple, be a quantum-mechanically based description such as density functional theory (DFT),but in practice the focus has been primarily on atomistic models based on semi-empiricalinteratomic potentials. A review of such methods can be found, for example, in [10]. Here,

Miller,Tadmor (2003)

(crack & grain bdry)

CT simulations using the fixed-parameter SW potential,two different TB schemes [22,23], and our hybrid scheme.To be able to perform several fully quantum-mechanicalsimulations we used a 64 Si atom cubic cell. As the cell isso small, here we compute the exact TB forces to be usedin the LOTF fit by direct diagonalization of the periodicsystem Hamiltonian. We note that the large differencebetween the results obtained by fitting the scheme on thetwo different quantum models is due to the differentpredictions of these models in accurate quantitative com-putations, reproduced by the present method. This em-phasizes the fact that the present scheme can at best beexpected to reproduce the results of the QM model that itis given [24], but can in no way improve its accuracy.

To further elucidate how the scheme adapts to the localenvironment, we report in Fig. 3 the time evolution of theparameters Cijk at a dimerized Si[100] surface at roomtemperature. In the bulk these angle parameters remain atall times close to their !1=3 equilibrium value (left).However, on the reconstructed surface, the equilibriumangle on the lower side of the buckled dimers (inset) islowered to almost 90". The corresponding parameters‘‘learn’’ this by switching to a value close to zero, andflip back and forth between zero and !1=3 as the buck-ling direction varies with time (right).

Moving to a problem where a fully quantum approachwould be practically unfeasible, we simulate the glidingmotion at 900 K of an opposing pair of 30" partialdislocations in Si, using a 4536 atom unit cell. We flagfor QM treatment all atoms within 7.0 A of the danglingbonds (undercoordinated atoms) formed during disloca-tion motion, corresponding to two disjoint QM regions of100–120 atoms, which follow the diffusing kink of eachpartial dislocation. On a single 1200 MHz Pentium IIIprocessor, the hybrid simulation takes #3–4 CPU minutesper MD step using a cluster radius of 7 A. For comparison,a MD step takes 11 h using a state-of-the-art linearscaling tight binding code [25], while a conventional

direct diagonalization would be approximately an orderof magnitude slower. Improving on the accuracy of theclassical potentials is crucial to simulate this extendeddefect. LOTF simulations on this system reveal a bondtopology for the free energy minimum at 900 K whichdiffers from that obtained in the same conditions (and at0 K [26]) from the SW model, with a square of atomsformed adjacent to an antiphase defect (red atom inFig. 4), and a different defect migration pathway. Whilethe TB model used [23] may still not capture everyrelevant feature of the phenomenon under study, theseresults indicate the need to enforce electronic-structureprecision. Indeed, in dislocation dynamics the inaccuratecanonical phase space sampling provided by classicalpotentials is regarded as the main source of discrepancybetween theory and experiment [26].

0.7 0.8 0.9 1−6

−5.5

−5

−4.5

−4

1000/T [K]

log 10

D [c

m2 /s

ec]

Stillinger−Weber

Kwon et al.

Lenosky et al.

FIG. 2 (color). Arrhenius plot of the silicon vacancy diffu-sivity obtained by the present scheme fitted ‘‘on the fly’’ on dif-ferent TB models (red), compared with the results of fully TBsimulations (black) and fixed-parameter SW potential (blue).

0

1

2

3

4

5−0.5

0

0.5

Time [ps]

Ang

le P

aram

eter

s, C

Surface Angles

Bulk Angles

θ = 90∼

FIG. 3 (color). Time evolution of angle parameters near aSi[100] reconstructed surface. A representative ‘‘surfaceangle’’ is marked in the inset with a black arc. The ‘‘bulk angleparameters’’ are all for angles centered on atoms of the thirdatomic layer.

FIG. 4 (color). Equilibrium structure of the Si 30" partialdislocation left kink from a 900 K hybrid simulation (only theglide plane atoms are shown). One undercoordinated atom (red)and its neighbors (blue) are flagged for QM treatment. The grayatoms further away complete the set of atoms for whichpotential optimization is performed. The parameters corre-sponding to interactions between the yellow atoms in theexternal region are not updated. Empty circles represent thestable 0 K structure [26].

VOLUME 93, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S week ending22 OCTOBER 2004

175503-3 175503-3

Csanyi et al; PRL 93; 2004

(dislocation in silicon)

Scales in Materials Modelling

17Jun2002

8:14

AR

AR162-15-COLOR.tex

AR162-15-COLOR.SGM

LaTeX2e(2002/01/18)

P1:GDL

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Annu. Rev. Mater. Res. 2002.32:377-400. Downloaded from arjournals.annualreviews.orgby RADCLIFFE SCIENCE LIBRARY on 09/07/05. For personal use only.

Rountree et al,Annu.Rev.Mater.Res.2002

Current computational limits:

Empirical potentials (EAM): ≈ 1010 atoms

Semi-empirical potentials (TB): ≈ 106 atoms

‘Almost’ ab initio (KS-DFT): ≈ 104 atoms

Schrodinger Equation: a few electrons

Density of atoms in typical materials:≈ 1020 atoms/mm3

Goal: QM MM FEM

Applications of Coarse-Grainingcomplicated defects, defect interaction, crack growth, void growth, . . .Application: Cu on Au [Iglesias/Leiva, Acta Mater. ’06]

B Langwallner (Oxford University) The Quasicontinuum Method[Iglesia & Leiva, 2006], [Miller et al, 1998],[QC software tutorial, Miller & Tadmor]

Application: Crack at Grain Boundary [Miller et al.,’98]

≈ 8m atoms, 15000 repatoms

ductile case:a lot of dislocationsemitted

brittle case:crack happens“instantly”

B Langwallner (Oxford University) The Quasicontinuum Method

Application: Shear in Twinned Aluminium

[Miller/Tadmor,QC Softwaretutorial]

B Langwallner (Oxford University) The Quasicontinuum Method

Atomistic Mechanics (0T statics)Atomistic body: N atoms at positions y = (yn)N

n=1 ∈ Rd×N

Total energy of configuration y :

minE tota (y) := Ea(y) + Pa(y)

Ea = interaction potential, Pa = potential of external frcs

Crystallisation:[Theil, Theil/Harris, . . . ]

The Quasicontinuum Idea

(a) (b)

Molecular statics problem: ya ∈ argmin E tota (Y )

Coarse grained problem: yh ∈ argmin E tota (Yh)

where mesh Th resolves the defect, and Yh = Y ∩ P1(Th).

Cauchy–Born ApproximationAtomistic Stored Energy:

Ea(y) =∑

x∈L

V(y(x + r)− y(x); r ∈ R

)Cauchy–Born Stored Energy:

Ec(y) =

∫ΩW (∇y) dV, where W (F) = V (Fr ; r ∈ R).

“Theorem:” [Similar to result by E/Ming; 2007]

Let ya ∈ argmin E tota be “sufficiently smooth globally”, then

there exists yc ∈ argmin E totc such that

‖∇ya −∇yc‖L2 . C(‖∇3ya‖L2 + ‖∇2ya‖2L4

)If there are no defects, then the Cauchy–Born model is a highly accuratecontinuum approximation.

Atomistic/Continuum Coupling: First Attempt

(a) (b) (c)

Ea(yh) ≈ Eac(yh) :=∑

x∈La

ωxVx +

∫Ωc

W (∇yh) dx

Fails the patch test:δEa(yF) = 0

andδEc(yF) = 0,

butδEqce(yF) 6= 0 !

−10

−50

5

10

−10

−50

5

10−0.02

−0.01

0

0.01

0.02

Error: x−Component

−10−5

05

10

−10

−5

0

5

10

−0.01

0

0.01

Error: y−Component

−8

−6

−4

−2

0

2

4

6

8

x 10−3

Alternative Approaches

1 Energy-based coupling: interface correction

2 Force-based coupling:FeAt: Kohlhoff, Schmauder, Gumbsch (1989, 1991)Dead-load GF removal: Shenoy, Miller, Rodney, Tadmor, Phillips, Ortiz (1999)AtC: Parks, Gunzburger, Fish, Badia, Bochev, Lehoucq, et al. (2007, . . . )CADD: Shilkrot & Curtin & Miller (2002, . . . )Stress-based coupling: Makridakis/Ortner/Süli (2011). . .

3 Blending methods:Belytschko & Xiao (2004)Klein & Zimmerman (2006)Parks, Gunzburger, Fish, Badia, Bochev, Lehoucq, et al. (2008). . .

A Priori Error Analysis

Framework: Let ya ∈ argmin E tota , yac ∈ argmin E tot

ac , then. . . (standard ideas) . . .

∥∥∇[ya − yac]∥∥

L2 ≈CONSISTENCYSTABILITY =

‖δEa(ya)− δEac(ya)‖H−1inf‖∇u‖L2=1〈δ2Eac(ya)u, u〉

3 Steps:1 CONSISTENCY: 〈δEac(y)− δEa(y), u〉 .

∥∥∇2ya∥∥

L2(Ωc)

2 STABILITY: 〈δ2Eac(y)u, u〉 ≥ Cstab‖∇u‖2L2

3 REGULARITY: bounds on ∇2ya, note that r−a for defects

ConsistencyNumerical Analysis Literature on Energy-Based Coupling:

1D, NN, variational analysis: [Blanc/LeBris/Legoll, 2005]1D, Error estimates for QCE and QNL: [Dobson/Luskin, 2009], [Ming/Yang, 2010],

[Ortner, 2010], [Ortner/Wang, 2011]1D, Sharp Stability Analysis, Linear Regime: [Dobson/Luskin/Ortner, 2010]1D, finite range: [Li/Luskin, preprint]1D, EAM potentials: [Li/Luskin, 2011]1D, blending methods: [Luskin/van Koten, preprint]2D, pair interactions, defects: [Ortner/Shapeev, preprint]2D, first order consistency for general finite-range interactions: [Ortner, preprint]2D, multi-body interactions: [Ortner/Zhang, preprint]

Analysis for Force-based Coupling:Dobson & Luskin (2008); Ming (2009); Dobson & Luskin & Ortner (2009, 2010, 2010);Makridakis & Ortner & Süli (2010, preprint); Dobson & Ortner & Shapeev (preprint), Lu & Ming(manuscript)

Analysis for Multi-Lattices:Dobson &Elliott & Luskin & Tadmor (2007); Abdulle & Lin & Shapeev (preprint)

Patch Test Consistency implies First-orderConsistency?

Suppose the interface potentials Vx are fitted numerically, s.t. Eac becomespatch test consistent:

δEac(yF) = 0 ∀F ∈ Rd×d

Does this automatically imply that Eac is first-order consistent?

“Theorem:” [First-order Consistency]

Suppose δEac passes the patch test, V finite range multi-body po-tential + technical conditions +• d = 1; or• d = 2, Ωa connected; or• d = 3, flat interface, homogeneous Vx

then ⟨δEac(y)− δEa(y), uh

⟩. ‖h∇2y‖L2(Ωc)‖∇uh‖L2

If Eac is also stable then this would imply ‖∇ya −∇yac‖L2 . ‖h∇2ya‖L2(Ωc)

Consistency in 1d

no ghost forces = consistent in a negative Sobolev space.A/C method Eac has no ghost forces if

Fac(yF; x) = −∂Eac(y)

∂y(x)= 0 ∀F > 0 ∀x ∈ L .

“Theorem:” [sharp estimate in 1d]

Assume that the A/C energy Eac has no ghost forces, then there existsa constant C depending only on M2(µ) and M3(µ), such that

supu∈RL

‖u′‖`p′=1

〈δEac(y)−δEa(y), u〉 ≤ C(‖y ′′‖`p(I)+‖y ′′′‖`p(C)+‖y ′′‖2

`2p(C)

).

Consistency in 1dno ghost forces = consistent in a negative Sobolev space.

1 A/C stress function 〈δEac(y), u〉 =∑

x∈LΣac(y ; x)u′(x), where

Fac(y ; x) = Σac(y ; x + 1)− Σac(y ; x). Similarly Σa(y ; x), Σc(y ; x).2 2nd order error for Σc(y ; x) (by symmetry, V(b)=V(-b))∣∣Σa(y ; x)− Σc(y ; x)

∣∣ ≤ C(‖y ′′′‖`∞(x+R) + ‖y ′′‖2

`∞(x−R)

).

3 Σac(y ; x) = Σa(y ; x) for x ∈ A; Σac(y ; x) = Σc(y ; x) for x ∈ C,4 Fac(yF; x) = 0⇒ Σac(yF; x) = Σa(yF; x) = Σc(yF; x),5 Σac and Σa are Lipschitz continous when V is smooth∣∣Σac(y ; x)− Σa(y ; x)

∣∣ ≤ ∣∣Σac(y ; x)− Σac(yF; x)∣∣+∣∣Σa(yF; x)− Σa(y ; x)

∣∣≤ C‖F− y ′‖`∞(I±)

⇒∣∣Σac(y ; x)− Σa(y ; x)

∣∣ ≤ C‖y ′′‖`∞((x−R)∩I)

6 use 〈δEac(y)− δEa(y), u〉 =∑

x∈L(Σac(y ; x)− Σa(y ; x))u′(x) to complete the

proof.

Construction of General A/C Schemes

Eac(yh) =∑

x∈La

Vx +∑

x∈Li

Vx +∑

x∈Lc

V cx

Construct V s.t. δEac(yF) = 0 for all F ∈ Rd×d .

General Construction: [Shimokawa et al, 2004; E/Lu/Yang, 2006]

Vx = V(gx ,r ; r ∈ R

)gx ,r =

∑s∈RxCx ,r ,sgs

→ Find Cx ,r ,s s.t. δEac(yF) = 0 ∀F→ geometric conditions only!

Explicit constructions for flat interfaces:[E/Lu/Yang, 2006]Explicit constructions for 2D generalinterface: [Ortner/Zhang; preprint]In general: compute Cx,r,s numerically inpreprocessing

2d, NN, multibody po-tential, triagular lattice


Eac(yh) =∑

x∈La

Vx +∑

x∈Li

Vx +∑

x∈Lc

V cx


General Construction:

Vx = V(gx ,r ; r ∈ R

)gx ,r =

∑s∈RxCx ,r ,sgs

→ Find Cx ,r ,s s.t. δEac(yF) = 0 ∀F→ geometric conditions only!

1. Local Energy Consistency V (yF) = V (yF)

⇒ r =∑

s∈RxCx,r,ss. (a)

2. Patch Test Consistency

0 =〈δEac(yF), u〉

=∑x∈L

∑r∈R

VF,r∑s∈R

Cx,r,sDsu

=∑x∈L

∑r∈R

∑s∈R

(Cx−as ,r,sVF,r − Cx,r,sDr VF,r )u(x)

⇒∑r∈R

∑s∈R

(Cx−s,r,sVF,r−Cx,r,sVF,r ) = 0. (b)

Solve (a) + (b) + B.C. in La and Lc to obtain Cx ,r ,s for x ∈ Li.unknowns: |LI ||R|2, eqns: 2|LI ||R|.


Eac(yh) =∑

x∈La

Vx +∑

x∈Li

Vx +∑

x∈Lc

V cx


General Construction:Flat interface

Interface with corner

Cx ,r ,r for NN interaction, multibody potential, one-sided construction.1. works for general interface in 2d

2. preprocessing for longer interaction range

Consisency of The Schemes

Eac(yh) =∑

x∈La

Vx +∑

x∈Li

Vx +∑

x∈Lc

V cx


“Theorem:” [Ortner/Zhang]

There exists a constant C that depends only on M2(V ) and M3(V ) suchthat∥∥δEac(y)−δEa(y)

∥∥U −1,p ≤ C

(‖∇3y‖`p(Ωc) +‖∇2y‖2`2p(Ωc) +‖∇2y‖`p(ΩI)

).

Stress Function in 2DDefine atomistic and continuum stress,

〈δEa(y), u〉 =∑

T∈T

|T |Σa(y ;T ) : ∇Tu

〈δEc(y), u〉 =∑

T∈T

|T |Σc(y ;T ) : ∇Tu

〈δEa(y), u〉

=∑x∈L

6∑j=1

Vx,j · Dj u(x)

=∑

T

|T |( 12|T |

6∑j=1

VxT,j ,j ⊗ aj)

: ∇T u

Since Dj u(x) = 12 (∇Tx,j u · aj +∇Tx,j−1u · aj ).

index conventions:


〈δEa(y), u〉 =∑

T∈T

|T |Σa(y ;T ) : ∇Tu

〈δEc(y), u〉 =∑

T∈T

|T |Σc(y ;T ) : ∇Tu

〈δEa(y), u〉

=∑x∈L

6∑j=1

Vx,j · Dju(x) (Dju(x) = −Dj+3u(x + aj ))

=∑

T∈T

|T | 12|T |

6∑j=1

VxT,j ,j ⊗ aj︸︷︷︸=Σ1

a(y ;T )

: ∇T u

=∑

T∈T

|T | 12|T |

6∑j=1

12(VxT,j ,j − VxT,j+3,j+3)⊗ aj︸︷︷︸

=Σ2a(y ;T )

: ∇T u

using Dju(x) = −Dj+3u(x + aj ) to obtain last equality.

index conventions:


〈δEa(y), u〉 =∑

T∈T

|T |Σa(y ;T ) : ∇Tu

〈δEc(y), u〉 =∑

T∈T

|T |Σc(y ;T ) : ∇TuSimilarly,⟨

δEc(y), u⟩

=∑

T∈T

|T |∂W (∇T y) : ∇T u(T )

=∑

T∈T

|T |1

2|T |

6∑j=1

∂j V (∇T ya)⊗ aj︸︷︷︸=Σ1

c(y ;T )=∂W (∇T y)

: ∇T u

=∑

T∈T

|T |1

2|T |

6∑j=1

12(

VT ,j + VTT,j ,j)⊗ aj︸︷︷︸

=Σ2c(y ;T )

: ∇T u

index conventions:


〈δEa(y), u〉 =∑

T∈T

|T |Σa(y ;T ) : ∇Tu

〈δEc(y), u〉 =∑

T∈T

|T |Σc(y ;T ) : ∇Tu

stress functions are not uniqueΣ1

a(y ;T )− ∂W (∇T y) = O(|D2y |)Σ2

a(y ;T )−Σ2c(y ;T ) = O(|D2y |2 + |D3y |)⇒

“Theorem:” [Consistency for CB energy]∥∥δEa(y)− δEc(y)∥∥

U−1,p ≤ C(‖∇3y‖`p + ‖∇2y‖2

`2p

).

index conventions:

Discrete Divergence Free Tensor FieldThe stress function is not unque, up to a divergence free tensor field, namely, σsuch that ∑

T∈T

σ : ∇Tu = 0

Characterization of div-free field, [Arnold/Falk, Polthier/Preuß]Lemma:v piecewise constant vector on T , v divergence free, i.e.,∑

T∈T

v · ∇Tu = 0

iff ∃ a function w ∈ N1(T ), such that v = J∇w , where J isthe counter-clockwise rotation by π/2.

N1(T ) is Crouzeix–Raviart finite element space.

For basis ζf ∈ N1(T ) associated with f = T1 ∩ T2,∑T∈T

J∇ζf · ∇T u = Jn1 · ∇T1u + Jn2 · ∇T2u = u1 − u2 + u2 − u1 = 0

Characterization of stress functionIf an A/C energy Eac satisfies patch test consistency,

0 = 〈δEac(yF), u〉 =∑

T∈T

|T |Σac(yF;T ) : ∇Tu

Lemma:∃ a function ψ(F,T ) ∈ N1(T )2, such that

Σac(yF;T ) = ∂W (F) + J∇ψ(F;T )

For general deformation y , deformation gradient average forpatch ωf = (T1 ∪ T2), Ff (y) = −

∫ωf∇y dx ,

Corrector function: ψ(y ; ·) =∑

f∈F ψ(Ff (y);mf

)ζf

Define the ’modified’ stress function,

Σac(y ;T ) := Σac(y ;T )− J∇ψ(y ;T ), for T ∈ T .

Σac(yF; T ) = ∂W (F) = Σa(yF; T )

Σac(y ; T ) = Σa(y ; T ), T ∈ ΩA and Σac(y ; T ) = Σc(y ; T ),T ∈ ΩC

Numerical ExperimentTest Problem: single vacancy in the triangular lattice,

−300 −200 −100 0 100 200 300

−200

−100

0

100

200

atom

ic s

paci

ngs

Vacancy defect in 269,100 atom cell

−20 −10 0 10 20−15

−10

−5

0

5

10

15

103 104 105

10−6

10−4

10−2

100

# DoFs

|Eac − Ea|/|Ea − E0|

QCE

B-QCE

GR-AC(DoF)−2

(DoF)−1

103 104 105

10−3

10−2

10−1

100

# DoFs

|yac − ya|1 ,2/ |ya − id|1 ,2

B-QCE

GR-AC

(DoF)−1

(DoF)−1/2

QCE

103 104 105

10−4

10−3

10−2

10−1

100

# DoFs

|yac − ya|1 ,∞/ |ya − x|1 ,∞

QCE

B-QCE

(DoF)−1

GR-AC

Error in Energy H1 Error W 1,∞ Error

Outlook on A/C MethodsSummary

Ghost force removal ⇒ Patch test consistency ⇒ consistencyConstruction of practical energy-based a/c methodsSharp consistency error estimates through stress based formulation

Open ProblemsConsistency in 3D: proof or counterexampleStabilityA/C coupling at surfacesImplementation, benchmarks, applications

Major Open ProblemsA/C methods for multi-latticesA/C methods for Coulomb interactionA/C methods for electronic structure models(done only for insulators)

constructionandsharpconsistencyestimatesfor atomistic ......parameters c ijk at a dimerized si[100]...

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