Atomistic-to-continuum coupling methods for crystalline solids€¦ · Atomistic-to-continuum coupling methods for crystalline solids Christoph Ortner Warwick Mathematics Institute

Atomistic-to-continuum coupling methods for crystalline solids

Christoph Ortner

Warwick Mathematics Institute

NSF PIRE Summer School“New Frontiers in multiscale analysis and computing for materials”

June 21 – 29, 2012

Outline

1 Motivation and Perspective

2 Formulating a 1D Toy Model

3 Framework for Error Analysis

4 Full Atomistic Calculations

5 QCE Method

6 QNL Method: Construction and Consistency

7 QNL Method: Stability

8 B-QCE Method

9 A 2D Example

10 Conclusion





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

Molecular MechanicsContinuum Elastostatics: Reference domain Ω ⊂ Rd

Displacement u : Ω→ Rd

min Ec(u)−(f ,u)Ω :=

∫Ω

W (∇u) dx−∫

Ω

f · u dx

Molecular Statics: Refence domain ; index set: Λ ⊂ Zd

Displacement: u : Λ→ Rd

min Ea(u)−(f ,u)Λ :=∑`∈Λ

V(Du(`)

)−∑`∈Λ

f (`) · u(`)

Du(`) := u(`+ ρ)− u(`)ρ∈RV (Du(`)) = energy of atom `




∫Ω

W (∇u) dx−∫

Ω

f · u dx




V(Du(`)

)−∑`∈Λ

f (`) · u(`)





∫Ω

W (∇u) dx−∫

Ω

f · u dx




V(Du(`)

)−∑`∈Λ

f (`) · u(`)


Example displacement field

−25 −20 −15 −10 −5 0 5 10 15 20 25

−15

−10

−5

0

5

10

15

−60−40

−200

2040

60

−50−40

−30−20

−100

1020

3040

50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Displacement:x−component

Classical Interaction Potentials

Ea(u) =∑`∈Λ

V(Du(`)

)y := Λ + u

Pair potentials:Ea(y) =

∑(i,j)

φ(rij ) =∑

i

12

∑j 6=i

φ(rij )

rij = |yi − yj |,

J(x)

φ(r) = r−12 − 2r−6

Embedded atom method:

Ea(y) =∑

i

[12

∑j

φ2(rij ) + F(∑

j ρ(rij ))]

∠-Potentials, BOPs, GAPs, . . .Coulomb, Quantum mechanics, DFT

Global and Local Minimamin Ea(u)−(f ,u)Λ

Global Minimizers are Crystals:(hence crystal reference states)

Gardner & Radin (1974); 1D, LJ

Heitmann & Radin (1980); 2D, sticky

Theil (2006); 2D, LJ-like

E & Li (2008); 2D, bond-angle

Harris & Theil; 3D, LJ-like (but not really)

Defects are Local Minima: cracks, dislocations, voids, interstitials, . . .

















Coarse-GrainingApplications: defect energies, defect interaction, void growth, crack growth, . . .

Treat coarse graining as a numerical analysis problem.

“Early” Results:Blanc, Le Bris, Lions (2002); Blanc, Le Bris, Legoll (2005)Lin (2003, 2007); CO, Süli (2008); Dobson, Luskin (2008)

Computational Task

Reference domain:Λ := A · Z2 \ crack

Admissible displacements:U :=

u : Λ→ R2,u(`) ∼ 0

as |`| → ∞

,

ua ∈ arg min E a(U)

Since dimU =∞, ua is not directly computable.Instead, compute an approximation uh ∈ arg min Eh(Uh) such that

‖∇uh −∇ua‖L2 . (WORK)−β

(WORK := cost of one energy + gradient assembly)

Computational Task

Reference domain:Λ := A · Z2 \ crack

Admissible displacements:U :=

u : Λ→ R2,u(`) ∼ 0

as |`| → ∞

,

ua ∈ arg min E a(U)

Since dimU =∞, ua is not directly computable.Instead, compute an approximation uh ∈ arg min Eh(Uh) such that

‖∇uh −∇ua‖L2 . (WORK)−β

(WORK := cost of one energy + gradient assembly)

Method 1: Full Atomistic Calculation

;

diam(ΩN) ≈ N, UN :=

u ∈ Y∣∣u(`) = 0 for ` ∈ Λ \ ΩN

.

uaN ∈ arg min Ea(UN)

Approximation Parameter: N

Method 1: Numerical Result

Vacancy under macroscopic stretch and shear:

−10 −8 −6 −4 −2 0 2 4 6 8

−10

−8

−6

−4

−2

0

2

4

6

8

10

−100−80−60−40−20020406080100

−100

−50

0

50

100

−10

−9

−8

−7

−6

−5

−4

−3

xx

log(|u|)

(inspired by [Gavini; PRL 2008])

Method 1: Numerical ResultVacancy under macroscopic stretch and shear:

102 103 104

10−2

10−1

# DoFs

|uac!

u| 1

,2/|u

| 1,2

ATM

(# DoFs)−1/2

Can we beat this by coarse-graining?

Method 2: Galerkin Projection

;

Discretise Domain: ΩN = ∪Th, atomistic refinement near crackUh :=

uh ∈ UN

∣∣ p.w. affine w.r.t Th

uh ∈ arg min Ea(Uh)

Not useful: WORK[Ea(uh), δEa(uh)] ≈WORK[Ea(uN), δEa(uN)] .

Method 3: Quasicontinuum Method[Tadmor,Phillips,Ortiz; 1996]

;

QCE Method: use nonlinear elasticity in each blue triange

Ea(uh) ≈ Eqce(uh) :=∑`∈Λa

V(Duh(`)

)+

∫Ωc

W (∇uh(x)) dx

where W (F) = V (F · R) (Cauchy–Born stored energy density)

⇒WORK[Eqce(uh), δEqce(uh)] = O(#DoFs)


Approximation Parameters: N, K = diameter of Λa, h = mesh size

Method 3: Quasicontinuum Method[Tadmor,Phillips,Ortiz; 1996]

;



V(Duh(`)

)+

∫Ωc

W (∇uh(x)) dx




Approximation Parameters: N, K = diameter of Λa, h = mesh size

Methods 1, 3: Numerical Result

102 103 104

10−2

10−1

# DoFs

|uac!

u| 1

,2/|u| 1

,2

ATMQCE

(# DoFs)−1/2

N,K , Th balance the errors from the b.c, continuum approx. and FEM!

Methods 1, 3: Numerical Result

102 103 104

10−2

10−1

# DoFs

|uac!

u| 1

,2/|u| 1

,2

ATMQCE

(# DoFs)−1/2

N,K , Th balance the errors from the b.c, continuum approx. and FEM!

Failure of “Patch Test”Test Problem: Λ := A · Zd (no defect, no external forces)

min Eqce(uh) :=∑`∈Λa

V (Duh(`)) +

∫Ωc

W (∇uh) dx

QCE fails the patch test:δEa(0) = 0δEc(0) = 0,

δEqce(0) 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

Intuitive Explanation:interaction asymmetryacross interface

Failure of “Patch Test”Test Problem: Λ := A · Zd (no defect, no external forces)

min Eqce(uh) :=∑`∈Λa

V (Duh(`)) +

∫Ωc

W (∇uh) dx

QCE fails the patch test:δEa(0) = 0δEc(0) = 0,

δEqce(0) 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

Intuitive Explanation:interaction asymmetryacross interface

Approaches to A/C Coupling (a selection)1 Ancestors

Tewari (1973), Mullins (1982), Kohlhoff, Schmauder, Gumbsch (1989, 1991)QCE: Tadmor, Phillips, Ortiz (1995)

2 Force-based couplingDead-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, . . . )Force-based A/C: Dobson/Luskin (2008), Dobson/Luskin/CO (2010)Stress-based A/C: Makridakis/CO/Süli (2011); . . .

3 Quadrature approachesKnap/Ortiz (2003), Eidel/Stuchowski (2009), Gunzburger/Zhang (2010,2011)Luskin/Ortner (2009)

4 Blending methodsOverlapping domains: Belytschko & Xiao (2004), Klein & Zimmerman (2006), Parks,Gunzburger, Fish, Badia, Bochev, Lehoucq, et al. (2008)B-QCE: Van Koten, Luskin (2011); Luskin, CO, Van Koten (preprint)Force-blending: Lu, Ming (preprint); Li, Luskin, CO (preprint); . . .

5 Ghost Force Removal:Bond splitting: CO (2011); Li, Luskin (2012); Shapeev (2012); CO, Shapeev (2012)Geometry reconstruction: Shimokawa et al (2003), E, Lu, Yang (2006)CO (2012); CO, Zhang (preprint)Other ideas: Iyer, Gavini (2010)

A Curious Observation

If we can “spread” an error, then we control it in `2:

g(`) :=

K−1, ` = 1, . . . ,K

0, o.w.

⇒ ‖g‖`2 =

( K∑`=1

K−2)1/2

= K−1/2.

By constrast:

‖g‖`1 =K∑`=1

K−1 = 1.

Method 4: B-QCEBelytschko & Xiao (2004)

Idea: Spread the interface⇒ spread the error?

;

Eb(uh) :=∑`∈Λa

β(`)V (Duh(`)) +

∫Ωc

(1− β)W (∇uh) dx

β = “smooth” blending function [Luskin, VanKoten, CO (preprint)]

Approximation Parameters: N,K , Th, β

Method 4: B-QCEBelytschko & Xiao (2004)

Idea: Spread the interface⇒ spread the error?

;

Eb(uh) :=∑`∈Λa

β(`)V (Duh(`)) +

∫Ωc

(1− β)W (∇uh) dx


Approximation Parameters: N,K , Th, β

Methods 1, 3, 4: Numerical Result

102 103 104

10−2

10−1

# DoFs

|uac!

u| 1

,2/|u| 1

,2

ATMQCEB−QCE

(# DoFs)−1/2

Method 5: Quasi-nonlocal Coupling (QNL)

Can we remove the interface error altogether? Shimokawa et al (2003)

Eqnl(uh) =∑`∈Λa

V (Duh(`)) +∑`∈Λi

V`(Duh(`)) +

∫Ωc

W (∇uh) dx

Idea: Choose V` such that δEac(Fx) = 0 for all F.

(Quasi-nonlocal: idea is that interface atoms should interact through non-local laws with atomsin the atomistic region and through local laws with the continuum region.)

Approximation Parameters: N,K , Th

Method 5: Quasi-nonlocal Coupling (QNL)



V (Duh(`)) +∑`∈Λi

V`(Duh(`)) +

∫Ωc

W (∇uh) dx



Approximation Parameters: N,K , Th

Methods 1, 3, 4, 5: Numerical Result

102 103 104

10−2

10−1

# DoFs

|uac!

u| 1

,2/|u

| 1,2

ATMQCEB−QCEQNL

(# DoFs)−1/2

(# DoFs)−1

Perspectives 1: Rates for micro-cracks

103 104

10−2

10−1

100

# DoFs

|uac!

u| 1

,2/|u| 1

,2

ATMQCEB−QCEQNL

(# DoFs)−1/2

(# DoFs)−1

103 104

10−2

10−1

# DoFs

|uac!

u| 1

,!/|u| 1

,!

ATMQCEB−QCEQNL

Perspectives 1: Rates for micro-cracks

103 104

10−2

10−1

100

# DoFs

|uac!

u| 1

,2/|u| 1

,2

ATMQCEB−QCEQNL

(# DoFs)−1/2

(# DoFs)−1

103 104

10−2

10−1

# DoFs

|uac!

u| 1

,!/|u| 1

,!

ATMQCEB−QCEQNL

Perspectives 2: Electronic Structure

(Electron density in neighbourhood of a vacancy defect.)see, e.g., Gavini, V., Bhattacharya, K., Ortiz, M.;J. Mech. Phys. Solids 55, 2007.

Perspective 3: QM-MM

[Kermode, Albaret, Sherman, Bernstein, Gumbsch, Payne, Csanyi, de Vita; Low speed fractureinstabilities in a brittle crystal, Nature, 2008]

References

Main Sources / References:E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag.A 73. 1996 (892 citations on Google scholar)

CO & M. Luskin: in preparationMatthew Dobson and Mitchell Luskin. An analysis of the effect of ghost force oscillation onquasicontinuum error. M2AN Math. Model. Numer. Anal., 43(3):591-604, 2009;arXiv:0811.4202

Christoph Ortner. A priori and a posteriori analysis of the quasinonlocal quasicontinuummethod in 1D. Math. Comp., 80(275):1265-1285, 2011; arXiv:0911.0671

Brian Van Koten and Mitchell Luskin, Analysis of energy-based blended quasicontinuummethods. SIAM Journal on Numerical Analysis, to appear. arXiv:1008.2138

M. Dobson, M. Luskin, and C. Ortner. Accuracy of quasicontinuum approximations nearinstabilities. J. Mech. Phys. Solids, 58(10):1741-1757, 2010; arXiv:0905.2914

A. V. Shapeev. Consistent energy-based atomistic/continuum coupling for two-bodypotentials in one and two dimensions. Multiscale Model. Simul., 9(3), 2011. arXiv:1010.0512

C. Ortner and A. Shapeev. Analysis of an Energy-based Atomistic/Continuum CouplingApproximation of a Vacancy in the 2D Triangular Lattice. to appear in Math. Comp. ArXive-prints, 1204.3705, 2012





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

2.1. MotivationModel problem: pair interactions with external potential

Generated by CamScanner from intsig.com

“Deformation” y : Z→ R, external dead load force f : Z→ R, then we write theenergy of a deformation as

E(y) =∑`∈Z

φ(y` − y`−1) +∑`∈Z

φ(y`+1 − y`−1)−∑`∈Z

f`y`

Boundary condition: y` ∼ A` as `→ ±∞φ is a Lennard-Jones type potential

J(x)

Too many infinities! Write energy-differences instead.

E(y) =∑`∈Z

[φ(y` − y`−1)− φ(A)

]+∑`∈Z

[φ(y`+1 − y`−1)− φ(2A)

]−∑`∈Z

f`(y` − A`)

=∑`∈Z

φ1(u` − u`−1) +∑`∈Z

φ2(u`+1 − u`−1)−∑`∈Z

fù`,

where u` := y` − A`. If u has compact support then E(u) is well-defined!




E(y) =∑`∈Z

φ(y` − y`−1) +∑`∈Z

φ(y`+1 − y`−1)−∑`∈Z

f`y`


J(x)


E(y) =∑`∈Z

[φ(y` − y`−1)− φ(A)

]+∑`∈Z

[φ(y`+1 − y`−1)− φ(2A)

]−∑`∈Z

f`(y` − A`)

=∑`∈Z

φ1(u` − u`−1) +∑`∈Z

φ2(u`+1 − u`−1)−∑`∈Z

fù`,





E(y) =∑`∈Z

φ(y` − y`−1) +∑`∈Z

φ(y`+1 − y`−1)−∑`∈Z

f`y`


J(x)


E(y) =∑`∈Z

[φ(y` − y`−1)− φ(A)

]+∑`∈Z

[φ(y`+1 − y`−1)− φ(2A)

]−∑`∈Z

f`(y` − A`)

=∑`∈Z

φ1(u` − u`−1) +∑`∈Z

φ2(u`+1 − u`−1)−∑`∈Z

fù`,


2.2. Displacement Space

Further simplification: assume f−` = −f` and hence u−` = −u`.Boundary conditions: u0 = 0 and u` ∼ 0 as `→∞.

Notation:Finite differences: u′` := u` − u`−1, u′′` := u`+1 − 2u` + u`−1

Continuous Interpolant:u(x) := u`−1 · (`− x) + u` · (x − `+ 1) for x ∈ (`− 1, `);then ∇u(x) = u′` for x ∈ (`− 1, `)

Interpret as displacements of Λ := 0, 1, 2, . . .

U :=

u : Λ→ R∣∣ u0 = 0, ‖u′‖`2 <∞

U0 :=

u ∈ U

∣∣ supp(u) is bounded

Lemma 2.11. U is a Hilbert space. 2. U0 is dense in U .

(U0 dense in U means that U encodes the boundary condition.)

2.3. Atomistic Energy


Internal energy: for u ∈ U0, define

Ea(u) := 12φ2(2u′1) +

∞∑`=1

φ1(u′`) + φ2(u′` + u′`+1)

.

We assume that φj ∈ C3(R) with bounded derivatives, φj (0) = 0.

Lemma 2.21. Ea : U0 → R is continuous. There exists a unique continuous extension to U .2. Ea ∈ C3(U).

External energy: for u ∈ U0, define

〈f ,u〉 := 12 f0u0 +

∞∑`=1

fù`.

We assume that 〈f ,u〉 ≤ C‖u′‖`2 for all u ∈ U0. Hence, there exists aunique continuous extension to U .

2.3. Atomistic Energy


Internal energy: for u ∈ U0, define

Ea(u) := 12φ2(2u′1) +

∞∑`=1

φ1(u′`) + φ2(u′` + u′`+1)

.

We assume that φj ∈ C3(R) with bounded derivatives, φj (0) = 0.

Lemma 2.21. Ea : U0 → R is continuous. There exists a unique continuous extension to U .2. Ea ∈ C3(U).

External energy: for u ∈ U0, define

〈f ,u〉 := 12 f0u0 +

∞∑`=1

fù`.

We assume that 〈f ,u〉 ≤ C‖u′‖`2 for all u ∈ U0. Hence, there exists aunique continuous extension to U .

2.4. Variational ProblemWe seek local minimizers:

ua ∈ arg minEa(u)− 〈f ,u〉

∣∣u ∈ U (M)

Quantities of interest: ∇u (defect geometry) and Ea(u) (defect energy)

Lemma 2.3

1 Necessary optimality conditions: If ua solves (M) then

〈δEa(ua), v〉 = 〈f , v〉 and 〈δ2Ea(ua)v , v〉 ≥ 0 ∀v ∈ U0.

2 Sufficient optimality conditions: Suppose that ua ∈ U satisfies

〈δEa(ua), v〉 = 〈f , v〉 and 〈δ2Ea(ua)v , v〉 ≥ c0‖v ′‖2`2 ∀v ∈ U0, (1)

for some c0 > 0, then ua solves (M).

(1) is also the correct physical notion of stability. We call a solution of (1) astrong minimizer.If 〈δ2Ea(0)v , v〉 ≥ c0‖v ′‖2

`2 , and ‖f‖U∗ suff. small then (1) has a locallyunique solution. (inverse function theorem)

2.4. Variational ProblemWe seek local minimizers:


∣∣u ∈ U (M)

Quantities of interest: ∇u (defect geometry) and Ea(u) (defect energy)

Lemma 2.3

1 Necessary optimality conditions: If ua solves (M) then

〈δEa(ua), v〉 = 〈f , v〉 and 〈δ2Ea(ua)v , v〉 ≥ 0 ∀v ∈ U0.

2 Sufficient optimality conditions: Suppose that ua ∈ U satisfies

〈δEa(ua), v〉 = 〈f , v〉 and 〈δ2Ea(ua)v , v〉 ≥ c0‖v ′‖2`2 ∀v ∈ U0, (1)

for some c0 > 0, then ua solves (M).

(1) is also the correct physical notion of stability. We call a solution of (1) astrong minimizer.If 〈δ2Ea(0)v , v〉 ≥ c0‖v ′‖2

`2 , and ‖f‖U∗ suff. small then (1) has a locallyunique solution. (inverse function theorem)

2.5. Computational Task


∣∣u ∈ U (M)

Task: Compute ua (defect geometry) towithin a prescribed tolerance.

Since dim(U) =∞, (M) cannot be solved directly.Construct

finite-dimensional approximation space Uhapproximate internal energy functional Eh : Uh → Rapproximate external forces 〈f ,uh〉h

and solveuh ∈ arg min

Eh(vh)− 〈f , vh〉h

∣∣ vh ∈ Uh

(Mh)Aim is to control the error∥∥∇ua −∇uh

∥∥L2 . (WORK)−β

where WORK is the cost of solving (Mh).(We will understand WORK = cost of energy and force assembly)





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

3.1. Abstract Error Analysis (1)

u ∈ arg minE(v) | v ∈ U, uh ∈ arg minEh(v) | vh ∈ Uh. (M, Mh)

U Hilbert space with norm ‖ · ‖; Uh ⊂ U ; E ,Eh ∈ C3(U)

Ingredients:Approximation operator: measures that best possible approximation of ufrom Uh.

Ih : U → Uh, s.t. ‖Ihu − u‖ ≤ cappx infvh∈U‖vh − u‖.

First-order optimality conditions:

〈δE(u), v〉 = 0 ∀v ∈ U , (2)〈δEh(uh), vh〉 = 0 ∀v ∈ Uh. (3)

Stability of the approximation:⟨δ2Eh(θh)vh, vh

⟩≥ c0‖vh‖2

for “suitable” θh ∈ Uh.


u ∈ arg minE(v) | v ∈ U, uh ∈ arg minEh(v) | vh ∈ Uh. (M, Mh)

U Hilbert space with norm ‖ · ‖; Uh ⊂ U ; E ,Eh ∈ C3(U)

Ingredients:Approximation operator: measures that best possible approximation of ufrom Uh.

Ih : U → Uh, s.t. ‖Ihu − u‖ ≤ cappx infvh∈U‖vh − u‖.

First-order optimality conditions:

〈δE(u), v〉 = 0 ∀v ∈ U , (2)〈δEh(uh), vh〉 = 0 ∀v ∈ Uh. (3)

Stability of the approximation:⟨δ2Eh(θh)vh, vh

⟩≥ c0‖vh‖2

for “suitable” θh ∈ Uh.


Proposition 3.1 (Elementary Error Estimate)

Let u solve (M) and uh solve (Mh) and let c0, c1 > 0 such that

〈δ2Eh(θh)vh, vh〉 ≥ c0‖vh‖2 ∀vh ∈ Uh, θh ∈ convuh, Ihu,〈δ2Eh(θ)vh,wh〉 ≤ c1‖vh‖ ‖wh‖ ∀vh,wh ∈ Uh, θ ∈ convu,uh.

Then,

‖u − uh‖ ≤ (1 + c1c0

)‖u − Ihu‖+ 1c0‖δE(u)− δEh(u)‖U∗h and

‖u − uh‖ ≥ 12cappx‖u − Ihu‖+ 1

2c1‖δE(u)− δEh(u)‖U∗h

Consistency: ‖δE(u)− δEh(u)‖U∗h is “controllable”

We call ‖δE(u)− δEh(u)‖U∗h the modelling error.


Proposition 3.2 (Error Estimate Based on Inverse Function Theorem)

Let u solve (M) and suppose that Eh ∈ C3(B(u, r)) for some r > 0 and thatthere exists c0 > 0 such that

〈δ2Eh(u)vh, vh〉 ≥ c0‖vh‖2 ∀vh ∈ Uh.

There exists ε > 0 such that, if ‖u − Ihu‖+ ‖δE(u)− δEh(u)‖U∗h ≤ ε, then thereexists a locally unique solution uh to (Mh) satisfying

‖u − uh‖ ≤(1 + 2c1

c0

)‖u − Ihu‖+ 2

c0‖δE(u)− δEh(u)‖U∗h ,

where c1 = ‖δ2Eh(u)‖.

CONSISTENCY + STABILITY⇒ CONVERGENCE

Let ‖T‖U∗h

:= supvh∈Uh,‖vh‖=1〈T , vh〉.

Proof of Proposition 3.1: Let eh := Ihu − uh. Then

c0‖eh‖2 ≤

∫ 1

0

⟨δ

2Eh

((1− t)uh + tIhu

)eh︸︷︷︸

= ddt δEh(uh+teh)

, eh

⟩dt

=⟨δEh(Ihu)− δEh(uh), eh

⟩=⟨δEh(Ihu)− δEh(u), eh

⟩+⟨δEh(u)− δE(u), eh〉

≤(

c1‖u − Ihu‖ + ‖δEh(u)− δE(u)‖U∗h

)‖eh‖.

Dividing through by ‖eh‖ gives

‖u − uh‖ ≤‖u − Ihu‖ + ‖eh‖

≤(

1 +c1c0

)‖u − Ihu‖ + 1

c0|δEh(u)− δE(u)‖U∗

h.

Vice-versa,

cappx‖u − uh‖ ≥‖u − Ihu‖ by assumptions; and

c1‖u − uh‖ ≥‖δEh(u)− δEh(uh)‖U∗h‖δEh(u)− δE(u)‖U∗

h.

Combining these two lower bounds gives

‖u − uh‖ ≥ 12cappx

‖u − Ihu‖ + 12c1‖δEh(u)− δE(u)‖U∗

h.





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

4.1. Method 1: Full Atomistic Calculation (ATM)

;

diam(ΩN) ≈ N, UN :=

u ∈ Y∣∣u(ξ) = 0 for ξ ∈ Λ \ ΩN

.

uaN ∈ arg min Ea(UN)

4.2. ATM: 1D Formulation


UN := u ∈ U |u` = 0 for ` ≥ N

uN ∈ arg minEa(v)− 〈f , v〉 | v ∈ UN (∗)

WORK ∼ N


Decay Hypothesis (DH): ∃ r0 > 0 and α > 1/2 such that

|u(x)| . x1−α and |∇u(x)| . x−α for x ≥ r0.

4.2. ATM: 1D Formulation


UN := u ∈ U |u` = 0 for ` ≥ N

uN ∈ arg minEa(v)− 〈f , v〉 | v ∈ UN (∗)

WORK ∼ N




4.3. ATM: Error Estimate



Lemma 4.1Let N be even (for simplicity). Let IN : U → UN be defined by

INu(x) :=

u(x), 0 ≤ x ≤ N/2,

u(x)− x−N/2N/2 u(N), N/2 ≤ x ≤ N,

0, N ≤ x .

1. ‖∇u −∇INu‖L2 . N−1/2|u(N)|+ ‖∇u‖L2(N,∞) for all u ∈ U .

2. If u satisfies (DH), then ‖∇u −∇INu‖L2 . N1/2−α.

Theorem 4.2Let ua be a strong minimizer of (M) satisfying (DH). There exists N0 > 0 such that for allN > N0 there exists a locally unique solution ua

N of (∗) satisfying

‖∇ua −∇uaN‖L2 . N1/2−α.

4.3. ATM: Error Estimate



Lemma 4.1Let N be even (for simplicity). Let IN : U → UN be defined by

INu(x) :=

u(x), 0 ≤ x ≤ N/2,

u(x)− x−N/2N/2 u(N), N/2 ≤ x ≤ N,

0, N ≤ x .

1. ‖∇u −∇INu‖L2 . N−1/2|u(N)|+ ‖∇u‖L2(N,∞) for all u ∈ U .

2. If u satisfies (DH), then ‖∇u −∇INu‖L2 . N1/2−α.

Theorem 4.2Let ua be a strong minimizer of (M) satisfying (DH). There exists N0 > 0 such that for allN > N0 there exists a locally unique solution ua

N of (∗) satisfying

‖∇ua −∇uaN‖L2 . N1/2−α.

4.4. ATM: Numerical Result

Exact solution: ua = 0.1 · (1 + x2)(1−α)/2

101 102 103

10−3

10−2

#DoFs

!"u

a!

"uac !

L2

Convergence rates for ! = 1.25

ATMQCEQNLBQCE

#DoFs1 / 2!!

101 102 103

10−2

10−1

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs1 / 2!!

There is no modelling error, hence we expect from Proposition 3.1 that

‖∇ua −∇uaN‖L2 ≈ ‖∇ua −∇IN ua‖L2 ,

for some suitable approximation operator IN . Take the one defined in Lemma 4.1.

Proof of Lemma 4.1:

∇u −∇IN u =

0, 0 ≤ x ≤ N/2,

2N u(N), N/2 ≤ x ≤ N,∇u, N ≤ x.

⇒ ‖∇u −∇IN u‖L2 . N1/2 · N · |u(N)| + ‖∇u‖L2(N,∞).

This demonstrates that we need to control the far-field decay of u and∇u. Hence, we introduced the decayhypothesis (DH). If u satisfies (DH), then |u(N)| . N−α+1 and

‖∇u‖2L2(N,∞)

.

∫ ∞N

x−2α dx . N1/2−α.

Combining the results, we get the item 2.

Proof of Theorem 4.2: We apply Proposition 3.2 with E = Eh = Ea − 〈f , ·〉 and Uh = UN .From Lemma 4.1 we have

‖∇ua −∇IN ua‖L2 . N1/2−α,

which tends to zero as N →∞. Moreover, the energy is not approximated so there is no modelling error. Forthe same reason, the stability condition is trivial.

Hence, for N sufficiently large, the condition of Proposition 3.2 are satisfied.





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

5.1. Method 3: Quasicontinuum Method[Tadmor,Phillips,Ortiz; 1996]

;



V(Duh(`)

)+

∫Ωc

W (∇uh(x)) dx




5.2. QCE: Precise 1D FormulationRecall atomistic energy: Ea(u) = 1

2φ2(2u′1) +∑∞

`=1φ1(u′`) + φ2(u′` + u′`+1)

Rewrite Ea in terms of site energies:


Ea(u) =

∞∑`=0

Φa`(u), where

Φa`(u) := 1

2

φ1(u` − u`−1) + φ1(u`+1 − u`)

+φ2(u` − u`−2) + φ2(u`+2 − u`)

(for ` ≥ 2)

Cauchy–Born Approximation: If u is “smooth”⇔ u′′ small then

Ea(u) ≈ · · ·+∞∑`=1

φ1(u′`)+φ2(2u′`)

=

∫W (∇u) dx where W (F ) = φ1(F ) + φ2(F )

(we will return to this later)Combine into Quasicontinuum Energy: let K ∈ N,

Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx



2φ2(2u′1) +∑∞

`=1φ1(u′`) + φ2(u′` + u′`+1)



Ea(u) =

∞∑`=0

Φa`(u), where

Φa`(u) := 1

2

φ1(u` − u`−1) + φ1(u`+1 − u`)

+φ2(u` − u`−2) + φ2(u`+2 − u`)

(for ` ≥ 2)


Ea(u) ≈ · · ·+∞∑`=1

φ1(u′`)+φ2(2u′`)

=



Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx



2φ2(2u′1) +∑∞

`=1φ1(u′`) + φ2(u′` + u′`+1)



Ea(u) =

∞∑`=0

Φa`(u), where

Φa`(u) := 1

2

φ1(u` − u`−1) + φ1(u`+1 − u`)

+φ2(u` − u`−2) + φ2(u`+2 − u`)

(for ` ≥ 2)


Ea(u) ≈ · · ·+∞∑`=1

φ1(u′`)+φ2(2u′`)

=


(we will return to this later)

Combine into Quasicontinuum Energy: let K ∈ N,

Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx


5.2. QCE: Precise 1D Formulation



Ea(u) =

∞∑`=0

Φa`(u), where

Φa`(u) := 1

2

φ1(u` − u`−1) + φ1(u`+1 − u`)

+φ2(u` − u`−2) + φ2(u`+2 − u`)

(for ` ≥ 2)


Ea(u) ≈ · · ·+∞∑`=1

φ1(u′`)+φ2(2u′`)

=


(we will return to this later)

Combine into Quasicontinuum Energy: let K ∈ N,

Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx


5.2. QCE: Precise 1D Formulation



Ea(u) =

∞∑`=0

Φa`(u), where

Φa`(u) := 1

2

φ1(u` − u`−1) + φ1(u`+1 − u`)

+φ2(u` − u`−2) + φ2(u`+2 − u`)

(for ` ≥ 2)


Ea(u) ≈ · · ·+∞∑`=1

φ1(u′`)+φ2(2u′`)

=



Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx


5.3. QCE: Analysis

Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx

uqce ∈ arg minEqce(v)− 〈f , v〉

∣∣ v ∈ UN

(∗) Generated by CamScanner from intsig.com

Formally: If u is “smooth” in the elastic bulk, then Eqce(u) ≈ Ea(u).But to understand the error ∇ua −∇uqce, we need to understand the modellingerror (i.e., the forces)!

Lemma 5.1‖δEa(u)− δEqce(u)‖U∗N ≥ C|φ′2(u′K + u′K +1)|

Theorem 5.2Let ua solve (M) and let uqce solve (∗). If ua satisfies (DH) and K ≥ r0, then

‖∇ua −∇uqce‖L2 ≥ C1|φ′2(0)| − C2K−α.

For a much finer analysis of the effect of the ghost forces see [Dobson & Luskin; 2009]

5.3. QCE: Analysis

Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx


∣∣ v ∈ UN





‖∇ua −∇uqce‖L2 ≥ C1|φ′2(0)| − C2K−α.


5.3. QCE: Analysis

Eqce(u) :=

K∑`=0

Φa`(u) +

∫ N

K +1/2

W (∇u) dx


∣∣ v ∈ UN





‖∇ua −∇uqce‖L2 ≥ C1|φ′2(0)| − C2K−α.


5.4. QCE: Numerical Result


101 102 103

10−3

10−2

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs1 / 2!!

101 102 103

10−2

10−1

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs1 / 2!!

5.5. QCE: Numerical Result – Ghost Forces

−30 −20 −10 0 10 20 30−0.02

0

0.02

0.04

0.06

0.08

0.1

x

∇ u

qce (x

)

We ignore FE coarsening (this will only detract from the main message). We expect from Proposition 3.1 that

‖∇ua −∇uqce‖L2 ≈ ‖∇ua −∇IN ua‖L2 + ‖δEqce(ua)− δEa(ua)‖U∗N.

Proof of Lemma 5.1: Let G(u) := Eqce(u)− Ea(u), then

G(u) =

∫ ∞K +1/2

W (∇u) dx −∞∑

`=K +1

Φa`(u)

=

∞∑`=K +1

12φ1(u′`) + 1

2φ1(u′`+1) + 12φ2(2u′`) + 1

2φ2(2u′`+1)

− 12φ1(u′`)− 1

2φ1(u′`+1)− 12φ2(u′`−1 + u′`)− 1

2φ2(u′`+1 + u′`+2)

=

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1)− 12φ2(u′`−1 + u′`)− 1

2φ2(u′`+1 + u′`+2).

We compute the first variation 〈δG(u), v〉 and group by v ′`. We only need the coefficient of v ′K , and just denotethe remaining ones by R`:

〈δG(u), v〉 =

∞∑`=K +1

φ′2(2u′`)v ′` + φ

′2(2u′`+1)v ′`+1

− 12φ′2(u′`−1 + u′`) · (v ′`−1 + v ′`)− 1

2φ′2(u′`+1 + u′`+2) · (v ′`+1 + v ′`+2)

= − 1

2φ′2(u′K + u′K +1) · v ′K +

∞∑`=K +1

R`v′`.

On the previous page, we have shown that

〈δG(u), v〉 = − 12φ′2(u′K + u′K +1) · v ′K +

∑∞`=K +1

R`v′`,

for some (unimportant values of R`). We can now construct a test function v that will extract the term before thesum: There exists a unique v ∈ UN such that

v ′K = s/√

2 and v ′K−1 = −s/√

2,

where s = −sign(φ′2(u′K + u′K +1). Testing with v = v we obtain

〈δG(u), v〉 = 1√8|φ′2(u′K + u′K +1)|.

This completes the proof of the lemma.

Proof of Theorem 5.2: This is an immediate consequence of Lemma 5.1 and (DH).

Remark on Ghost Forces: A simple calculation shows that δG(0) has the following represenation:

〈δG(0), v〉 = − 12φ′2(0)v ′K + 1

2φ′2(0)v ′K +2.

This is the weak form of the “ghost forces” (spurious forces that appear even under homogeneous deformation).





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

6.1. Method 5: Quasi-nonlocal Coupling (QNL)



V (Duh(`)) +∑`∈Λi

V`(Duh(`)) +

∫Ωc

W (∇uh) dx



6.2. QNL: The Bond Splitting Idea

***BOARD***

Eqnl(u) := 12φ2(2u′1) +

∞∑`=1

φ1(u′`)

+K∑`=1

φ2(u′` + u′`+1) +∞∑

`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).

Rewrite as a QNL Energy:


Eqnl(u) =K−1∑`=0

Φa`(u) +

K +1∑`=K

Φi`(u) +

∫ ∞K +3/2

W (∇u) dx (4)

Φi`(u) := 1

2

φ1(u′`) + φ1(u′`+1) + φ2(u′`−1 + u′`) + φ2(2u′`+1)

.


Eqnl(u) := 12φ2(2u′1) +

∞∑`=1

φ1(u′`)

+K∑`=1

φ2(u′` + u′`+1) +∞∑

`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).



Eqnl(u) =K−1∑`=0

Φa`(u) +

K +1∑`=K

Φi`(u) +

∫ ∞K +3/2

W (∇u) dx (4)

Φi`(u) := 1

2

φ1(u′`) + φ1(u′`+1) + φ2(u′`−1 + u′`) + φ2(2u′`+1)

.


Eqnl(u) := 12φ2(2u′1) +

∞∑`=1

φ1(u′`)

+K∑`=1

φ2(u′` + u′`+1) +∞∑

`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).



Eqnl(u) =K−1∑`=0

Φa`(u) +

K +1∑`=K

Φi`(u) +

∫ ∞K +3/2

W (∇u) dx (4)

Φi`(u) := 1

2

φ1(u′`) + φ1(u′`+1) + φ2(u′`−1 + u′`) + φ2(2u′`+1)

.

6.3. QNL: Consistency ErrorEa(u) = 1

2φ2(2u′1) +

∞∑`=1

φ1(u′`) +

∞∑`=1

φ2(u′` + u′`+1)

Eqnl(u) = 12φ2(2u′1) +

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).

By construction, δEqnl(uF ) = 0, i.e., QNL has no ghost force.Can we now control the modelling error?

Lemma 6.11. Let u ∈ U , and Λc

K := K + 1,K + 2, . . . , then

‖δEa(u)− δEqnl(u)‖U∗ . ‖u′′‖`2(ΛcK )

2. If u satisfies (DH) and K ≥ r0, then

‖δEa(u)− δEqnl(u)‖U∗ . K−α−1/2


|u`| . `1−α, |u′`| . `−α and |u′′` | . `−α−1 for ` ≥ r0.


2φ2(2u′1) +

∞∑`=1

φ1(u′`) +

∞∑`=1

φ2(u′` + u′`+1)

Eqnl(u) = 12φ2(2u′1) +

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).



K := K + 1,K + 2, . . . , then





|u`| . `1−α, |u′`| . `−α and |u′′` | . `−α−1 for ` ≥ r0.


2φ2(2u′1) +

∞∑`=1

φ1(u′`) +

∞∑`=1

φ2(u′` + u′`+1)

Eqnl(u) = 12φ2(2u′1) +

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).



K := K + 1,K + 2, . . . , then





|u`| . `1−α, |u′`| . `−α and |u′′` | . `−α−1 for ` ≥ r0.


2φ2(2u′1) +

∞∑`=1

φ1(u′`) +

∞∑`=1

φ2(u′` + u′`+1)

Eqnl(u) = 12φ2(2u′1) +

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).



K := K + 1,K + 2, . . . , then





|u`| . `1−α, |u′`| . `−α and |u′′` | . `−α−1 for ` ≥ r0.

6.4. QNL: FE Coarsening

We want to turn the QNL method into an efficient scheme. We coarsen thecontinuum region [K ,N] using finite elements.


Notation:Nodes: Nh := n0, n1, . . . , nJ, where n0 = 0, n1 = 1, . . . , nK +2 = K + 2 and nJ = N.]Elements: Th := [nj−1, nj ] | j = 1, . . . , JMesh size: hj := nj − nj−1, h(x) := hj for x ∈ (nj−1, nj )P1 FE Displacement space:

Uh :=

uh ∈ UN | uh is piecewise affine w.r.t. Th

Eqnl(uh) =

K−1∑`=0

Φa`(uh) +

K +1∑`=K

Φi`(u) +

∫ N

K +3/2

W (∇uh) dx

=

K−1∑`=0

Φa`(uh)︸︷︷︸

WORK=O(K )

+

K +1∑`=K

Φi`(u)︸︷︷︸

WORK=O(1)

+ 12 W (u′h,K +2) +

J∑j=K +3

hjW (∇uh|(nj−1,nj ))︸︷︷︸WORK=O(J−K )





Uh :=


Eqnl(uh) =

K−1∑`=0

Φa`(uh) +

K +1∑`=K

Φi`(u) +

∫ N

K +3/2

W (∇uh) dx

=

K−1∑`=0

Φa`(uh)︸︷︷︸

WORK=O(K )

+

K +1∑`=K

Φi`(u)︸︷︷︸

WORK=O(1)

+ 12 W (u′h,K +2) +

J∑j=K +3






Uh :=


Eqnl(uh) =

K−1∑`=0

Φa`(uh) +

K +1∑`=K

Φi`(u) +

∫ N

K +3/2

W (∇uh) dx

=

K−1∑`=0

Φa`(uh)︸︷︷︸

WORK=O(K )

+

K +1∑`=K

Φi`(u)︸︷︷︸

WORK=O(1)

+ 12 W (u′h,K +2) +

J∑j=K +3


6.5. QNL: Coarsening ErrorWe now consider the coarsened QNL method: (〈f , vh〉h is a trapezoidal rule approximation to 〈·, ·〉)

uqnl ∈ arg minEqnl(vh)− 〈f , vh〉h

∣∣ vh ∈ Uh,

From the abstract error analysis we expect (we cheated here)

‖∇ua −∇uqnl‖L2 . ‖∇ua −∇Ihua‖L2 + ‖δEqnl(ua)− δEa(ua)‖U∗h

+ ‖〈f , ·〉 − 〈f , ·〉h‖U∗h.

Approximation operator: Ih : U → Uh (nodal interpolant of INu)Ihu(nj ) := (INu)(nj ) for j = 0, . . . , J.

Standard interpolation error estimates + the analysis of IN give

‖∇u −∇Ihu‖L2 . ‖h∇2u‖L2(K ,N) + N1/2−α

for all u ∈ C1,1 with u(`) = u`.

Lemma 6.2 (Optimising the FE mesh)Suppose that u satisfies (DH) and K ≥ r0. Then we can choose Th and N such that J . K and

‖∇u −∇Ihu‖L2 . K−α−1/2

Lemma 6.3 (Error in External Forces)Suppose that f satisfies (DH) with α ; α− 2, then with the mesh from Lemma 6.2

‖〈f , ·〉 − 〈f , ·〉h‖U∗h. log(K )K−α−3/2



∣∣ vh ∈ Uh,



+ ‖〈f , ·〉 − 〈f , ·〉h‖U∗h.



‖∇u −∇Ihu‖L2 . ‖h∇2u‖L2(K ,N) + N1/2−α



‖∇u −∇Ihu‖L2 . K−α−1/2


‖〈f , ·〉 − 〈f , ·〉h‖U∗h. log(K )K−α−3/2



∣∣ vh ∈ Uh,



+ ‖〈f , ·〉 − 〈f , ·〉h‖U∗h.



‖∇u −∇Ihu‖L2 . ‖h∇2u‖L2(K ,N) + N1/2−α



‖∇u −∇Ihu‖L2 . K−α−1/2


‖〈f , ·〉 − 〈f , ·〉h‖U∗h. log(K )K−α−3/2



∣∣ vh ∈ Uh,



+ ‖〈f , ·〉 − 〈f , ·〉h‖U∗h.



‖∇u −∇Ihu‖L2 . ‖h∇2u‖L2(K ,N) + N1/2−α



‖∇u −∇Ihu‖L2 . K−α−1/2


‖〈f , ·〉 − 〈f , ·〉h‖U∗h. log(K )K−α−3/2



∣∣ vh ∈ Uh,



+ ‖〈f , ·〉 − 〈f , ·〉h‖U∗h.



‖∇u −∇Ihu‖L2 . ‖h∇2u‖L2(K ,N) + N1/2−α



‖∇u −∇Ihu‖L2 . K−α−1/2


‖〈f , ·〉 − 〈f , ·〉h‖U∗h. log(K )K−α−3/2

6.6. QNL: Summary of Consistency Analysis

Coarsened QNL Method:


∣∣ vh ∈ Uh, (∗)

Given K , we can choose N and construct the mesh Th such that J . K and:Modelling error: ‖δEqnl(ua)− δEa(ua)‖U∗h . K−α−1/2

Coarsening error: ‖∇ua −∇Ihua‖L2 . K−α−1/2

Note in particular that the modelling error is no worse than the coarseningerror!

To apply the abstract theorems we still need a stability result. If we canget this (LECTURE 4), then we would obtain

‖∇ua −∇uqnl‖L2 . K−α−1/2 ≈WORK−α−1/2.

6.6. QNL: Summary of Consistency Analysis

Coarsened QNL Method:


∣∣ vh ∈ Uh, (∗)

Given K , we can choose N and construct the mesh Th such that J . K and:Modelling error: ‖δEqnl(ua)− δEa(ua)‖U∗h . K−α−1/2


Note in particular that the modelling error is no worse than the coarseningerror!

To apply the abstract theorems we still need a stability result. If we canget this (LECTURE 4), then we would obtain

‖∇ua −∇uqnl‖L2 . K−α−1/2 ≈WORK−α−1/2.

6.7. QNL: Numerical Result


101 102 103

10−3

10−2

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs! 1 / 2!!

#DoFs1 / 2!!

101 102 103

10−2

10−1

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs! 1 / 2!!

#DoFs1 / 2!!

6.8. QNL: Outlook

The bond-splitting idea can be generalized to 2D (Shapeev; 2011), seeLecture 5, to some extent also to 3D (Shapeev; preprint), but is restrictedto pair interactions.Other constructions exist that remove the ghost force; most prominentlygeometric reconstruction:

Shimokawa et al (2003): described basic idea; explicit construction, but it is only“consistent” for short-range pair interactionsE, Lu and Yang (2006): generalized approach; general conditions and explicitconstruction for “flat” a/c interfacesCO, Zhang (preprint): generalized approach; explicit construction and analysis for 2Dnearest neighbour interactions (general interface geometry). . . many unresolved cases left . . .

Consistency of more general QNL-type methods is fairly well understoodeven in 2D/3D but some gaps remain: CO (2012); CO, Shapeev(preprint); . . .

Stability is a different story: see lecture 4.

Derivation of the bond-splitting idea: In the derivatin of the Cauchy–Born approximation used to constructthe QCE method, we used that

φ2(u′` + u′`+1) = φ2(2u′`) + O(|u′′` |).

But this still led to the ghost force issues. If we recall that the cause of this was a break-down of symmetry offorces, then this might motivate us to consider the following approximation instead:

φ2(u′` + u′`+1) ≈ 12φ2(2u′`) + 1

2φ2(2u′`+1)

Define the two functionals ψa`(u) := φ2(u′` + u′`+1) and ψc

`(u) := 12φ2(2u′`) + 1

2φ2(2u′`+1), then we obtain

〈δψa(u), v〉 =φ′2(u′` + u′`+1) · (v ′` + v ′`+1)

〈δψc`(u), v〉 =φ

′2(2u′`)v ′` + φ

′2(2u′`+1)u′`+1.

In particular, for any homogeneous displacement u = Fx, F ∈ R,

〈δψa(Fx), v〉 = φ′2(0) · (v ′` + v ′`+1) = 〈δψa(Fx), v〉.

This means that these two different bond energies generate the same forces under (locally) homogeneousdisplacements. In particular, this implies that the following energy will have no ghost forces:

Eqnl(u) :=

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1).

That is, δEqnl(Fx) = 0 for all F .

Rewriting bond-splitting as a QNL method: We ignore φ1 which are treated in the same way by the atomisticand continuum model. The modifications of the calculation near the origin are straightforward and will also beignored (replaced with . . . ) For second-neighbour bonds we have:

Eqnl(u) = · · · +K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1)

= · · · +K−1∑`=2

12

φ2(u′`−1 + u′`) + φ2(u′`+1 + u′`+2)

+ 1

2φ2(u′K−1 + u′K ) + 12φ2(u′K + u′K +1) + 1

2φ2(2u′K +1)

+

∞∑`=K +2

φ2(2u′`).

From here, one can easily see that Eqnl can be written in the form (4).

Proof of Lemma 6.1, 1: Define G := Ea − Eqnl, then

G(u) =

∞∑`=K +1

φ2(u′` + u′`+1)− 1

2φ2(2u′`)− 12φ2(2u′`+1)

,

and hence

〈δG(u), v〉 =

∞∑`=K +1

φ′2(u′` + u′`+1) · (v ′` + v ′`+1)− φ′2(2u′`) · v ′` − φ

′2(2u′`+1) · v ′`+1

=

∞∑`=K +1

(φ′2(u′` + u′`+1)− φ′2(2u′`)

)v ′` +

(φ′2(u′` + u′`+1)− φ′2(2u′`+1)

)v ′`+1

.

Now notice that we have ∣∣φ′2(u′` + u′`+1)− φ′2(2u′`)∣∣ . |u′`+1 − u′`| = |u′′` |,

and analogously for the second similar term. Using this estimate and applying a Cauchy–Schwarz inequality,∣∣〈δG(u), v〉∣∣ . ∞∑

`=K +1

|u′′` |(|v ′`| + |v

′`+1|). ‖u′′‖

`2K ,K +1,... ‖v′‖`2 .

Proof of Lemma 6.1, 2: Straightforward application of (DH):

‖u′′‖`2 .

(∫ ∞K

x−2α−2 dx)≈ K−1/2−α

.

Sketch of the coarsening anaysis: This is standard technology in numerical analysis, hence we will be veryformal.We construct a finite element mesh Th in the continuum region [K ,N]. This mesh should be chosen so that theinterpolation error ‖ua − Ihua‖L2 is minimized for a given number of degrees of freedom. For convenience wedrop the superscript.First, we need to understand the interpolation error. Suppose that u is a C1,1 interpolant of u with|∇2u(x)| . x−α−1. (One can always construct such an interpolant using piecewise cubic polynomials. ) Thenthe interpolation error is

‖∇u −∇Ihu‖L2(K ,N)≤ ‖∇u −∇u‖L2(K ,N)

+ ‖∇u −∇Ihu‖L2(K ,N). ‖h∇2u‖L2(K ,N)

.

That is the error behaves like ‖hx−2α−2‖L2(K ,N). The number of degrees of freedom can be seen to be

approximately equal to

J ≈

∫ N

K

1h(x)

dx.

Minimizing the error subject to keeping J approximately fixed,

min

∫ N

K

h2x−2α−2 dx subject to

∫ N

K

1h

dx = Const,

Yields the Euler–Lagrange equation, for some Lagrange multiplier λ,

hx−2α−2 = λ1h2, and hence we choose h(x) ≈

(xK

)− 23 (α+1)

.

From this formula, we can easily confirm that J . K and that ‖h∇2u‖L2(K ,N). K−1/2−α.





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

7.1. ReviewQNL method (bond-splitting formulation):

Eqnl(u) :=

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1)

.

Efficient formulation with FE coarsening possible:uqnl ∈ argminEqnl(vh)− 〈f , vh〉h | vh ∈ Uh.

Coarsening error and modelling error estimates: (decay hypothesis!)

‖∇ua −∇Ihua‖L2 + ‖δEa(ua)− δEqnl(ua)‖U∗h . J−1/2−α,

where J = #DoFs ≈WORK and α > 1/2.

From Proposition 3.1 we expect

‖∇ua −∇uqnl‖L2 ≈ J−1/2−α,

but we can make this rigorous only if we can prove

STABILITY: ⟨δ2Eqnl(ua)v , v

⟩≥ c′0‖∇v‖2

L2 ∀v ∈ U0.

for some c′0 > 0


Eqnl(u) :=

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1)

.






‖∇ua −∇uqnl‖L2 ≈ J−1/2−α,



⟩≥ c′0‖∇v‖2

L2 ∀v ∈ U0.

for some c′0 > 0


Eqnl(u) :=

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1)

.






‖∇ua −∇uqnl‖L2 ≈ J−1/2−α,



⟩≥ c′0‖∇v‖2

L2 ∀v ∈ U0.

for some c′0 > 0


Eqnl(u) :=

∞∑`=1

φ1(u′`) +

K∑`=1

φ2(u′` + u′`+1) +

∞∑`=K +1

12φ2(2u′`) + 1

2φ2(2u′`+1)

.






‖∇ua −∇uqnl‖L2 ≈ J−1/2−α,



⟩≥ c′0‖∇v‖2

L2 ∀v ∈ U0.

for some c′0 > 0

7.2. Key Difficulty

In classical numerical analysis one has the same issue to face, however, it isusually possible to estimate the error in the hessian in an operator norm

‖δ2E(u)− δ2Eh(Ihu)‖ . εh.

This immediately gives the estimate

〈δ2Eh(Ihu)vh, vh〉 ≥ 〈δ2E(u)vh, vh〉 − εh‖∇vh‖2 ≥ (c0 − ε)‖∇vh‖2.

That is, stability of E and control of εh imply stability of Eh.

In the QNL (or any other a/c coupling) this is not true: for simplicity we consideronly u = 0:

Proposition 7.1

Let u ∈ U , then ‖δ2Ea(u)− δ2Eqnl(u)‖ & |φ′′2 (0)|

This means that the best we can hope to prove is

inf δ2Ea(u) > 0 ??⇒ inf δ2Eqnl(u) > 0

(What about⇐?)

7.2. Key Difficulty







Proposition 7.1




(What about⇐?)

7.2. Key Difficulty







Proposition 7.1




(What about⇐?)

7.3. Stability of homogeneous statesWe first look at the case of homogeneous deformations; for simplicity of notation u = 0.Here we can prove a sharp stability result by simply computing the stability constantsexplicitly:

Proposition 7.2

inf‖v ′‖`2 =1〈δ2Eqnl(0)v , v〉 ≥ inf‖v ′‖

`2 =1〈δ2Ea(0)v , v〉.

The key idea in the proof was to write

|v ′` + v ′`+1|2 = 2|v ′`|2 + 2|v ′`+1|2 − |v ′′` |2

This yields the following represenations of the hessians

〈δ2Ea(0)v , v〉 = A∞∑`=1

|v ′`|2 − φ′′2 (0)

∞∑`=1

|v ′′` |2,

〈δ2Eqnl(0)v , v〉 = A∞∑`=1

|v ′`|2 − φ′′2 (0)

K∑`=1

|v ′′` |2,

where A = φ′′1 (0) + 4φ′′2 (0) = W ′′(0).


Proposition 7.2


`2 =1〈δ2Ea(0)v , v〉.


|v ′` + v ′`+1|2 = 2|v ′`|2 + 2|v ′`+1|2 − |v ′′` |2


〈δ2Ea(0)v , v〉 = A∞∑`=1

|v ′`|2 − φ′′2 (0)

∞∑`=1

|v ′′` |2,

〈δ2Eqnl(0)v , v〉 = A∞∑`=1

|v ′`|2 − φ′′2 (0)

K∑`=1

|v ′′` |2,

where A = φ′′1 (0) + 4φ′′2 (0) = W ′′(0).


Proposition 7.2


`2 =1〈δ2Ea(0)v , v〉.


|v ′` + v ′`+1|2 = 2|v ′`|2 + 2|v ′`+1|2 − |v ′′` |2


〈δ2Ea(0)v , v〉 = A∞∑`=1

|v ′`|2 − φ′′2 (0)

∞∑`=1

|v ′′` |2,

〈δ2Eqnl(0)v , v〉 = A∞∑`=1

|v ′`|2 − φ′′2 (0)

K∑`=1

|v ′′` |2,

where A = φ′′1 (0) + 4φ′′2 (0) = W ′′(0).

7.4. Stability of nonlinear deformations

Ingredients we have available:Assume ua is a strong minimizer: 〈δ2Ea(ua)v , v〉 ≥ c0‖v ′‖2

`2

QNL is stable in reference state:

Lemma 7.3If there exists u ∈ U with 〈δ2Ea(u)v , v〉 ≥ c0‖v ′‖2

`2 , then

〈δ2Ea(0)v , v , 〉 ≥ c0‖v ′‖2`2 and hence, 〈δ2Eqnl(0)v , v , 〉 ≥ c0‖v ′‖2

`2

Can we therefore hope that δ2Eqnl(ua) is also stable?

Theorem 7.4Let u ∈ U be stable: 〈δ2Ea(u)v , v〉 ≥ c0‖v ′‖2

`2 . Then there exists K0 such that,for all K ≥ K0, ⟨

δ2Eqnl(u)v , v⟩≥ 1

2 c0‖v ′‖2`2 .



`2



`2 , then


`2





2 c0‖v ′‖2`2 .



`2



`2 , then


`2





2 c0‖v ′‖2`2 .

7.5. QNL: Error Estimate

We can now apply the general results to obtain a fully rigorous convergencerate. We have proven so far: if ua is a strong minimizer satisfying (DH) and Kis sufficiently large then

Modelling error: ‖δEqnl(ua)− δEa(ua)‖U∗h. K−α−1/2


Stability:⟨δ2Eqnl(ua)v , v

⟩≥ 1

2 c0‖v ′‖2`2 .

Theorem 7.5 (QNL Error Estimate)

Let ua be a strong solution of (M) satisfying (DH). There exists K0 > 0 suchthat, for all K ≥ K0 we can choose the mesh Th and N in such a way that thereexists a locally unique solution uqnl ∈ arg minEqnl(vh)− 〈f , vh〉h | vh ∈ Uh,which satisfies

‖∇ua −∇uqnl‖L2 . J−α−1/2.

7.5. QNL: Error Estimate

We can now apply the general results to obtain a fully rigorous convergencerate. We have proven so far: if ua is a strong minimizer satisfying (DH) and Kis sufficiently large then

Modelling error: ‖δEqnl(ua)− δEa(ua)‖U∗h. K−α−1/2


Stability:⟨δ2Eqnl(ua)v , v

⟩≥ 1

2 c0‖v ′‖2`2 .

Theorem 7.5 (QNL Error Estimate)

Let ua be a strong solution of (M) satisfying (DH). There exists K0 > 0 suchthat, for all K ≥ K0 we can choose the mesh Th and N in such a way that thereexists a locally unique solution uqnl ∈ arg minEqnl(vh)− 〈f , vh〉h | vh ∈ Uh,which satisfies

‖∇ua −∇uqnl‖L2 . J−α−1/2.

7.6. QNL: Numerical Result


101 102 103

10−3

10−2

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs! 1 / 2!!

#DoFs1 / 2!!

101 102 103

10−2

10−1

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs! 1 / 2!!

#DoFs1 / 2!!

Atomistic Stability: The atomistic hessian δ2Ea(0) has the representation

⟨Ea(0)v , v

⟩= 2φ′′2 (2u′1)|v ′1|

2 +

∞∑`=1

φ′′1 (0)|v ′`|

2 +

∞∑`=1

φ′′2 (0)|v ′` + v ′`+1|

2.

The key step is to rewrite the interacting bonds in terms of non-interacting bonds and a strain gradient correction(parallelogram law):

|v ′` + v ′`+1|2 = 2|v ′`|

2 + 2|v ′`+1|2 − |v`+1 − v ′`|

2 = 2|v ′`|2 + 2|v ′`+1|

2 − |v ′′` |2.

This gives us ⟨Ea(0)v , v

⟩=

∞∑`=1

(φ′′1 (0) + 4φ′′2 (0))|v ′`|2 −

∞∑`=1

φ′′2 (0)|v ′′` |

2.

We should now distinguish two cases: φ′′2 (0) ≤ 0 and φ′′2 (0) > 0. We only consider the φ′′2 (0) ≤ 0 and leavethe other case as an exercise. In this case, we obtain⟨

Ea(0)v , v⟩≥ A‖v ′‖2

`2 , where A = φ′′1 (0) + 4φ′′2 (0).

To see that this is sharp, let

v (n)

`:=

`/√

n, ` = 0, . . . , n,1, ` ≥ n ,

then ‖(v (n))′‖2`2 = 1 but ‖(v (n))′′‖

`2 = 1/n. Hence, 〈δ2Ea(0)v (n), v (n)〉 → A as n→∞.

QNL Stability: The QNL hessian δ2Eqnl(0) has the representation

⟨Eqnl(0)v , v

⟩= 2φ′′2 (2u′1)|v ′1|

2+

∞∑`=1

φ′′1 (0)|v ′`|

2+

K∑`=1

φ′′2 (0)|v ′`+v ′`+1|

2+

∞∑`=K +1

2φ′′2 (0)|v ′`|

2+2φ′′2 (0)|v ′`+1|2.

Using again the parallelogram law,

|v ′` + v ′`+1|2 = 2|v ′`|

2 + 2|v ′`+1|2 − |v ′′` |

2,

we obtain ⟨Ea(0)v , v

⟩=

∞∑`=1

(φ′′1 (0) + 4φ′′2 (0))|v ′`|2 −

K∑`=1

φ′′2 (0)|v ′′` |

2.

Clearly, if φ′′2 (0) ≤ 0 then⟨Ea(0)v , v

⟩≥ A‖v ′‖2

`2 and by testing with v that is supported in the continuumregion, we obtain that this is sharp. Thus, we have even shown thatinf‖v′‖

`2 =1〈δ2Eqnl(0)v , v〉 = inf‖v′‖

`2 =1〈δ2Ea(0)v , v〉.

Remark 1: That case φ′′2 > 0 is a little more involved. In this case we only obtain thatinf‖v′‖

`2 =1〈δ2Eqnl(0)v , v〉 ≥ inf‖v′‖

`2 =1〈δ2Ea(0)v , v〉 as stated in Proposition 7.2.

Remark 2: In the case φ′′2 (0) < 0 we have δ2Ea(0) ≥ δ2Eqnl(0), while in the case φ′′2 (0) > 0 we have the

opposite inequality δ2Eqnl(0) ≥ δ2Ea(0).

Sketch of the proof of Theorem 7.4: Using the parallelogram identity, we rewrite the QNL-hessian as

〈δ2Eqnl(u)v , v〉 =

∞∑`=1

A`|v ′`|2 +

K∑`=1

B`|v ′′` |2.

The strategy of the proof is to “split” the test function into two components: morally v ∼ w + z where w issupported in the atomistic region, and z is supported in [K/2,∞], and show that up to a controllable error wecan split the hessian as well. Precisely, we will construct w, z ∈ U such that |w ′|2 + |z′|2 = |v ′|2, w ′` = 0 for`geqK , z′` = 0 for ` ≤ K/2 and

〈δ2Eqnl(u)v , v〉 = 〈δ2Eqnl(u)w,w〉 + 〈δ2Eqnl(u)z, z〉 + O(

K−1‖v ′‖2`2

). (5)

Suppose that we can prove (5). Since w is supported in the atomistic region only, it doesn’t “see” the QNLcoupling at all and we get

〈δ2Eqnl(u)w,w〉 = 〈δ2Ea(u)w,w〉 ≥ c0‖w ′‖2`2 .

Secondly, since z′ = 0 in [0,K/2], we can apply (DH) to obtain

〈δ2Eqnl(u)z, z〉 = 〈δ2Eqnl(0)z, z〉 + O(

K−α‖z′‖2`2

)Now we can apply the sharp homogeneous stability result to obtain

〈δ2Eqnl(u)z, z〉 ≥ c0‖z′‖2`2 + O

(K−α‖z′‖2

`2

).

Inserting these estimates into (5) we have arrive at

〈δ2Eqnl(u)v , v〉 ≥ 〈δ2Eqnl(u)w,w〉 + 〈δ2Eqnl(u)z, z〉 − CK−1‖v ′‖2`2

≥ c0

(‖w ′‖2

`2 + ‖z′‖2`2 b)− C(K−α + K−1)‖v ′‖2

`2 =[

c0 − C(K−α + K−1)]‖v ′‖2

`2 .

For K sufficiently large, we obtain the stated result. If (DH) does not hold, then we simply need to replae K−α

with ‖u′‖ΛcK/2→ 0 as K →∞. We are only left to prove (5).

Proof of (5): Supppose, wlog that K is even. Let χ, ψ be Lipschiz functions, such that

χ(x) =

1, x ≤ K/2,0, ` ≥ K , ψ(x) =

0, x ≤ K/2,1, ` ≥ K , χ

2+ψ2 ≡ 1, ‖∇χ‖L∞+‖∇ψ‖L∞ . K−1.

One can always find such functions, e.g., by defining χ(x) = sin(πx), ψ(x) = cos(πx), and then rescalingχ(x) = χ((x − K/2)/(K/2)) and ψ analogously.In the following we argue very formally. It is best to think of v , χ, ψ,w, z as continuous objects and accept thatall errors made are controllable (O(K−1)). Upon “discretising” the argument one then gets a rigorous proof.We define

v ′ = (χ2 + ψ2)v ′ = χw ′ + ψz′,

where w ′ = χv ′ and z′ = ψv ′. In addition, we require that w0 = z0 = 0 so that w, z ∈ U . We can nowcalculate

|v ′′|2 = (χw ′ + ψz′)′v ′′ = (χw ′)′v ′′ + ψz′)′v ′′.

Take one of the two groups and modify further:

(χw ′)′v ′′ = χ′w ′v ′′ + w ′′(χv ′′) = χ

′w ′v ′′ − w ′′χ′v ′ + w ′′(χv ′)′

Since we have ‖χ′‖∞ = O(K−1) and ‖v ′′‖`2 . ‖v ′‖

`2 and ‖w ′‖`2 . ‖v ′‖

`2 , we obtain

(χw ′)′v ′′ = |w ′′|2 + O(K−1|v ′|2).

Thus, we end up to|v ′′|2 = |w ′′|2 + |z′′|2 + O(K−1|v ′|2)

From this it is now easy to get (5).





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

8.1. Method 4: B-QCE

Belytschko & Xiao (2004)Idea: Spread the interface⇒ spread the error?

;

Eb(uh) :=∑`∈Λa

β(`)V (Duh(`)) +

∫Ωc

(1− β)W (∇uh) dx


8.2. B-QCE: 1D FormulationThe Cauchy–Born site energy: for u ∈ U ,∫ ∞

K +1/2

W (∇u) dx =

∞∑`=K +1

Φc`(u), where Φc

`(u) := 12

(W (u′`) + W (u′`+1)

)

Blending Function: β` ∈ [0, 1] for ` ∈ ΛIn practise choose 0 < K < L < N, take (K , L) as blending region:β = 0 in [0,K ] and β = 1 in [L,∞].


B-QCE Energy functional:

Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)


K +1/2

W (∇u) dx =

∞∑`=K +1

Φc`(u), where Φc

`(u) := 12

(W (u′`) + W (u′`+1)

)Blending Function: β` ∈ [0, 1] for ` ∈ ΛIn practise choose 0 < K < L < N, take (K , L) as blending region:β = 0 in [0,K ] and β = 1 in [L,∞].



Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)


K +1/2

W (∇u) dx =

∞∑`=K +1

Φc`(u), where Φc

`(u) := 12

(W (u′`) + W (u′`+1)

)Blending Function: β` ∈ [0, 1] for ` ∈ ΛIn practise choose 0 < K < L < N, take (K , L) as blending region:β = 0 in [0,K ] and β = 1 in [L,∞].



Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)

8.3. B-QCE: Modelling Error

B-QCE Energy Functional:

Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)

B-QCE Method: with FE coarsening of the continuum region [L,N]

ubqce ∈ arg minEbqce(vh)− 〈f , vh〉h

∣∣ vh ∈ Uh

Again, we need to control the coarsening error and the consistency error:

Coarsening error: ‖∇ua −∇Ihua‖L2 . L−1/2−α, while J . L (same as QNL)

Lemma 8.1 (B-QCE Consistency Error)

Let u ∈ U , and suppose that β` = 0 for ` ≤ K and β` = 1 for ` ≥ L; then‖δEbqce(u)− δEa(u)‖U∗ . ‖β′′‖`2 + ‖u′′‖`2(Λc

K−1)

8.3. B-QCE: Modelling Error

B-QCE Energy Functional:

Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)



∣∣ vh ∈ Uh

Again, we need to control the coarsening error and the consistency error:

Coarsening error: ‖∇ua −∇Ihua‖L2 . L−1/2−α, while J . L (same as QNL)

Lemma 8.1 (B-QCE Consistency Error)

Let u ∈ U , and suppose that β` = 0 for ` ≤ K and β` = 1 for ` ≥ L; then‖δEbqce(u)− δEa(u)‖U∗ . ‖β′′‖`2 + ‖u′′‖`2(Λc

K−1)

8.4. B-QCE: Optimising the blending functionB-QCE Energy Functional:

Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)

Coarsening Error: ‖∇ua −∇Ihua‖L2 . L−1/2−α, while J . LModelling Error: ‖δEbqce(u)− δEa(u)‖U∗ . ‖β′′‖`2 + ‖u′′‖`2(Λc

K−1)

How should we choose β so as to minimize the error? (or at least minimize theerror estimate?) The function that minimizes ‖∇2β‖L2 is a cubic polynomial!

Lemma 8.2 (Optimising the Blending Function)

Suppose that β` = β(`) where β is the unique cubic polynomial withβ(K ) = β′(K ) = β′(L) = 0 and β(L) = 1.

Suppose also that u satisfies (DH) and K ≥ r0 + 1; then‖δEbqce(u)− δEa(u)‖U∗ . (L− K )−3/2 + K−1/2−α.

Key Property: Increasing the blending width improves accuracy! I.e., B-QCE is aconsistent method!

8.4. B-QCE: Optimising the blending functionB-QCE Energy Functional:

Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)

Coarsening Error: ‖∇ua −∇Ihua‖L2 . L−1/2−α, while J . LModelling Error: ‖δEbqce(u)− δEa(u)‖U∗ . ‖β′′‖`2 + ‖u′′‖`2(Λc

K−1)

How should we choose β so as to minimize the error? (or at least minimize theerror estimate?) The function that minimizes ‖∇2β‖L2 is a cubic polynomial!

Lemma 8.2 (Optimising the Blending Function)

Suppose that β` = β(`) where β is the unique cubic polynomial withβ(K ) = β′(K ) = β′(L) = 0 and β(L) = 1.

Suppose also that u satisfies (DH) and K ≥ r0 + 1; then‖δEbqce(u)− δEa(u)‖U∗ . (L− K )−3/2 + K−1/2−α.

Key Property: Increasing the blending width improves accuracy! I.e., B-QCE is aconsistent method!

8.5. B-QCE: B-QCE Error EstimateB-QCE Energy Functional:

Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)



∣∣ vh ∈ Uh

(∗)

Summarizing the consistency estimates: If u satisfies (DH) then we can chooseK , L,N, β and Th such that J . K and

‖∇u −∇Ihu‖L2 + ‖δEbqce(u)− δEa(u)‖U∗h. J−1/2−min(α,1)

Theorem 8.3 (B-QCE Error Estimate)

Let ua be a solution of (M), satisfying (DH). There exists K0 > 0 such that, for all K ≥ K0

we can choose K , L,N, β and Th in such a way that there exists a locally uniquesolution ubqce of (∗), which satisfies

‖∇ua −∇ubqce‖L2 . J−1/2−min(α,1)

Remarks: 1. We have again skipped the stability estimate!2. For slowly decaying fields (α < 1) the B-QCE is comparable to QNL; for fastdecaying fields (α > 1) QNL converges more rapidly.


Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)



∣∣ vh ∈ Uh

(∗)









Ebqce(u) :=

∞∑`=0

(1− β`)Φa

`(u) + β`Φc`(u)



∣∣ vh ∈ Uh

(∗)








8.6. B-QCE: Outlook

Main Conclusion from this section: even though B-QCE does not removethe ghost force, its complexity is close to that of QNL (qualitatively identical forslowly decaying solutions):

‖∇ua −∇uaN‖L2 .WORK1/2−α,

‖∇ua −∇uqce‖L2 &1,

‖∇ua −∇uqnl‖L2 .WORK−1/2−α,

‖∇ua −∇ubqce‖L2 .WORK−1/2−min(α,1)

The method is straightforward to generalize to 2D/3D and even to complexcrystals.The consistency analysis can be extended to 2D/3D, many-bodyinteractions and complex crystalsThe stability is well-understood in 1D, but completely open in 2D/3D

8.6. B-QCE: Outlook

Main Conclusion from this section: even though B-QCE does not removethe ghost force, its complexity is close to that of QNL (qualitatively identical forslowly decaying solutions):

‖∇ua −∇uaN‖L2 .WORK1/2−α,

‖∇ua −∇uqce‖L2 &1,

‖∇ua −∇uqnl‖L2 .WORK−1/2−α,

‖∇ua −∇ubqce‖L2 .WORK−1/2−min(α,1)

The method is straightforward to generalize to 2D/3D and even to complexcrystals.The consistency analysis can be extended to 2D/3D, many-bodyinteractions and complex crystalsThe stability is well-understood in 1D, but completely open in 2D/3D

8.7. BQCE: Numerical Result


101 102 103

10−3

10−2

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs! 1 / 2!!

#DoFs1 / 2!!

#DoFs! 3 / 2

101 102 103

10−2

10−1

#DoFs

!"u

a!

"uac !

L2


ATMQCEQNLBQCE

#DoFs! 1 / 2!!

#DoFs1 / 2!!





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

9.1. 2D: Atomistic Model

Discrete domain: Λ ⊂ Z2

Vacancy: Λ = Z2 \ 0Micro-cavity: Λ = Z2 \ B(0, r)Micro-crack: Λ = Z2 \ (s, 0) | j = −S, ,S. . .

u : Z2 → R2 then u : R2 → R2

by canonical P1 interpolation

Bonds: B := (`, `′) | `, `′ ∈ Λ, |`− `′| ≤ rcutDbu := u(`′)− u(`) for b = (`, `′)

Displacement space:

U :=

u : Λ→ R2∣∣∇u ∈ L2

Atomistic Energy:

Ea(u) :=∑b∈B

φb(Dbu),

where φb ∈ C3(R2) and φb(0) = 0.(φb(s) = φ(rb + s)− φ(rb))

9.2. 2D: Generalising the 1D Bond Splitting***BOARD***

Separate bonds into two sets: B = Ba ∪ Bc

For b ∈ Bc make the approximation

φb(Dbu) ≈ −∫

b

φb(∇bu)db,

where ∇bu = ∇u · r for b = (`, `+ r).This results in the energy functional

Eecc(u) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇bu)db

Coarse-graining:Polygonal domains Ω,Ωa;Ωc := (Ω \ Ωa)

Th triangulation of Ωc,Uh associated FE space

Ba := b ∈ B | length(b ∩ Ωa) > 0

uecc ∈ arg minEecc(vh)

∣∣ vh ∈ Uh

(∗)

Wc

Wa

Key Issue: Can we make (∗) efficient as in 1D?

9.2. 2D: Generalising the 1D Bond Splitting



φb(Dbu) ≈ −∫

b

φb(∇bu)db,


Eecc(u) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇bu)db





∣∣ vh ∈ Uh

(∗)

Wc

Wa


9.2. 2D: Generalising the 1D Bond Splitting



φb(Dbu) ≈ −∫

b

φb(∇bu)db,


Eecc(u) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇bu)db





∣∣ vh ∈ Uh

(∗)

Wc

Wa


9.3. 2D: Consistency

The ECC-Method is analyzed in the form

Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇buh)db

The abstract error analysis needs to be slightly modified, but roughly still applies inthis setting, so we expect

‖∇ua −∇uecc‖L2 ≈ ‖∇ua −∇Ihua‖L2 + ‖δEa(ua)− δEecc(ua)‖U∗h

We focus again on consistency:

Lemma 9.1 (Consistency of the ECC-Method)

1. Let u ∈ U and let u be a C1,1-conforming FE interpolant, then

‖δEa(u)− δEecc(u)‖U∗h. ‖∇2u‖L2(Ωc),

where Ωc is a “slightly enlarged continuum region”.2. If u satisfies the decay hypothesis |∇j u(x)| . |x |1−α−j , j = 0, 1, 2, for |x | ≥ r0, withα > 1, then we can construct the finite element mesh in such a way that

‖∇ua −∇Ihua‖L2 + ‖δEa(ua)− δEecc(ua)‖U∗h. (#DoFs)−α/2.



Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇buh)db







where Ωc is a “slightly enlarged continuum region”.

2. If u satisfies the decay hypothesis |∇j u(x)| . |x |1−α−j , j = 0, 1, 2, for |x | ≥ r0, withα > 1, then we can construct the finite element mesh in such a way that




Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇buh)db







where Ωc is a “slightly enlarged continuum region”.2. If u satisfies the decay hypothesis |∇j u(x)| . |x |1−α−j , j = 0, 1, 2, for |x | ≥ r0, withα > 1, then we can construct the finite element mesh in such a way that


9.4. 2D: The Bond Density Formula

Lemma 9.2 (Shapeev (2011))

Let r ∈ Z2, let T be a triangle with cornersbelonging to Z2, and let

χT (x) :=

1, x ∈ int(T ),

1/2, x ∈ bdry(T ),0, x ∈ R2 \ T .

Then,∑`∈Z2

−∫ `+r

`

χT db = |T |

Corollary 9.3

Eecc(uh) =∑b∈Ba

φb(Dbuh)−∑b∈Ba

−∫

bχΩcφb(∇buh)db +

∫Ωc

W (∇uh) dx ,

where W (F ) =∑

r φr (Fr) is the Cauchy–Born stored energy density.

Conclusion: WORK . #DoFs!




χT (x) :=

1, x ∈ int(T ),

1/2, x ∈ bdry(T ),0, x ∈ R2 \ T .

Then,∑`∈Z2

−∫ `+r

`

χT db = |T |

Corollary 9.3

Eecc(uh) =∑b∈Ba


−∫


∫Ωc

W (∇uh) dx ,

where W (F ) =∑






χT (x) :=

1, x ∈ int(T ),

1/2, x ∈ bdry(T ),0, x ∈ R2 \ T .

Then,∑`∈Z2

−∫ `+r

`

χT db = |T |

Corollary 9.3

Eecc(uh) =∑b∈Ba


−∫


∫Ωc

W (∇uh) dx ,

where W (F ) =∑



9.5. 2D: OutlookOne could again state a convergence theorem:“Let ua be an atomistic solution satisfying the decay hypothesis with α > 1, as well as[STABILITY CONDITION]. Then there exists K0 > 0 such that, for K ≥ K0 we can choose Thand N in such a way that there exists a locally unique uecc ∈ arg minEecc(vh) | vh ∈ Uhsuch that ‖∇ua −∇uecc‖L2 . (#DoFs)−α/2."

In some simple cases, one can prove stability of ECC, but the results are not nearly as sharpas in 1D. Stability of ECC can be considered an open question.

The generalisation of the bond-splitting

Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇buh)db

to 3D is straightforward, but the bond density formula∑`∈Z3

−∫ `+r

`

χT db = |T |

is FALSE! It is difficult (but still computationally efficient) to compute the direction-dependentvolumes vr [T ] :=

∑`∈Z3 −

∫ `+r`

χT db. [Shapeev; preprint]

Given the elegant construction of the ECC method, the consistency analysis was relativelystraightforward. Can one construct similarly simple/elegant methods in 2D/3D for EAMinteractions?

Ea(u) =∑`,r

φ(Dr u`) +∑`

F(∑

rρ(

Dr u`))

??




Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇buh)db


−∫ `+r

`

χT db = |T |


∑`∈Z3 −

∫ `+r`



Ea(u) =∑`,r

φ(Dr u`) +∑`

F(∑

rρ(

Dr u`))

??




Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−∫

b

φb(∇buh)db


−∫ `+r

`

χT db = |T |


∑`∈Z3 −

∫ `+r`



Ea(u) =∑`,r

φ(Dr u`) +∑`

F(∑

rρ(

Dr u`))

??

9.6. QNL: OutlookWhile bond-splitting + bond-density formula are a very elegant formulation of a/ccoupling, they seem at the moment to end in a cul-de-sac. To generalize to many-bodyinteractions, we may need to return to the original QNL idea:


V (Duh(`)) +∑`∈Λi

V`(Duh(`)) +

∫Ωc

W (∇uh) dx

Geometric reconstruction idea to constructions of V :Shimokawa et al (2003): described basic idea; explicit construction, but it is only“consistent” for short-range interactionsE, Lu and Yang (2006): generalized approach; general construction and necessaryconditions for “consistency”; provided parameters for “flat” a/c interfacesCO, Zhang (preprint): generalized approach; explicit construction and analysis for 2Dnearest neighbour interactions (general interface geometry). . . many unresolved cases left:Problem: does there exist a “simple” “consistent” a/c coupling for EAM potentials?

Consistency (coarsening + modelling errors) of these general QNL-type methodsis well understood even in 2D/3D (but some gaps remain here as well): CO(2012); CO, Zhang (preprint); CO, Shapeev (preprint); . . .

Stability: none of the results of Lecture 4 have been generalized so far

To make the bond-integral formulation efficient, we begin by rewriting it in terms of the contributions fromelements. Let χTT∈Th be a partition of unity of Ωc, then

Eqnl(uh) =∑b∈Ba

φb(Dbuh) +∑b∈Bc

−

∫φb(∇buh)db

=∑b∈Ba

φb(Dbuh) +∑b∈Bc

−

∫ ∑T∈Th

χTφb(∇buh)db

=∑b∈Ba

φb(Dbuh) +∑T∈Th

∑ρ∈Z2

φρ(∇ρuh|T )∑b∈Bc

b=(`,`+r)

−

∫ ∑T∈Th

χT db

=:∑b∈Ba

φb(Dbuh) +∑T∈Th

∑ρ∈Z2

φρ(∇ρuh|T )volρ[T ].

Applying the bond-density formula it is now easy to get the stated result stated in Corollary 9.3

Proof of the Bond-density formula; Lemma 9.2:

If we set Λ = Z2, then the bond-density formula states volρ[T ] = |T |. To establish this, one verifies that forelements Ti with disjoint interior, and for integer translations a ∈ Z2,

volρ[T1 ∪ T2] = volρ[T1] + volρ[T2], volρ[a + T ] = volρ[T ], volρ[−T ] = volρ[T ].

Since we can write mT = ∪m2i=1Ti (take one of the corners of T to be the origin!) we obtain

volρ[mT ] =

m2∑i=1

volρ[Ti ] = m2volρ[T ].

Moreover, by a simple counting argument, one quickly oberserves that

volρ[mT ] = |mT | + O(m) = m2|T | + O(m),

there the O(m) stands for a “counting error” that scales like the boundary length of mT . Hence, we can concludethat

m2vol[T ] = m2|T | + O(m),

and letting m →∞, we arrive at the result.

Proof of Lemma 9.1: The idea of the proof is the same as in 1D, but some additional technicalities arise, thatwe will largely gloss over. We analyze the consistency in the bond-integral formulation

Eecc(uh) =∑b∈Ba

φb(Dbu) +∑b∈Bc

−

∫b

φb(∇buh)db

Let

G(u) := Φecc(u)− Φa(u) =∑b∈Bc

−

∫b

φb(Dbu)db− φb(Dbu)db

then

〈δG(u), v〉 =∑b∈Bc

−

∫b

φ′b(∇bu) · ∇bvdb− φ′b(Dbu) · Dbv

=∑b∈Bc

−

∫b

φ′b(∇bu)− φ′b(Dbu)

· ∇bvdb

.

(∑b∈Bc

w−1|b| −

∫b

∣∣φ′b(∇bu)− φ′b(Dbu)∣∣2db)1/2

·(∑

b∈Bc

w(|b|)−

∫b

|∇bv |2db)1/2

,

where the weights are chosen so that∑

ρ∈Z2\0w(|ρ|) <∞. One can now apply the bond-density formula

(or less sharp variants of the result) to prove that(∑b∈Bc

w(|b|)−

∫b

|∇bv |2db)1/2

. ‖∇v‖L2 .

It remains to estimate the errors in the bond-forces: Let u be a C1,1-conforming FE interpolant, then (note thatDb u = Dbu) ∣∣φ′b(∇bu)− φ′b(Dbu)

∣∣ . ∣∣φ′b(∇bu)− φ′b(∇b u)∣∣ +∣∣φ′b(∇b u)− φ′b(Db u)

∣∣Since ‖∇u −∇u‖L∞(b) . ‖∇2u‖L∞(b) we can estiamte

−

∫b

∣∣φ′b(∇bu)− φ′b(∇b u)∣∣2db . ‖∇bu −∇b u‖2

L∞(b) . ‖∇2u‖2L∞(b).

Moreover, using

−

∫b

∣∣φ′b(∇bu)− φ′b(∇b u)∣∣2db . ‖∇bu −∇b u‖2

L∞(b) . ‖∇2u‖2L∞(b).

Similarly, using also Db y = −∫

b∇udb, we obtain

−

∫b

∣∣φ′b(∇b u)− φ′b(Db u)∣∣2db . ‖∇b u − −

∫b∇b udb‖2

L∞ . ‖∇2u‖2L∞(b).

Thus, we obtain that

〈δG(u), v〉 ≤(∑

b∈Bc

C(|b|)‖∇u‖2L∞(b)

)1/2

· ‖∇v‖L2 .

The remainder of the proof is technical but uses only classical ideas of FE analysis, in particular inverse

estimates and local norm-equivalence relations.





5 QCE Method



8 B-QCE Method

9 A 2D Example

10 Conclusion

10.1. One-Page Summary

A/C methods are numerical approximation schemes of atomistic models ofcrystalline solids. They can be analyzed in the traditional setting of numericalanalysis:

CONSISTENCY + STABILITY ⇔ CONVERGENCE

〈δ2Eac(ua)v , v〉 ≥ c0‖v ′‖2`2 .

‖∇ua −∇uac‖L2 ≈ ‖∇ua −∇Ihua‖L2 + ‖δEa(ua)− δEac(ua)‖U∗h

The new aspect is to understand the effect of the a/c interface treatment onmodelling error and stability.

Modelling Error:QCE-Method: ‖δEa(ua)− δEqce(ua)‖U∗

h& 1

QNL-Method: ‖δEa(ua)− δEqnl(ua)‖U∗h. ‖(ua)′′‖`2(Λc

K).

B-QCE-Method: ‖δEa(ua)− δEbqce(ua)‖U∗h. ‖(ua)′′‖`2(Λc

K) + ‖β′′‖`2 .

Complexity: rewrite coarsening and modelling errors in terms of allapproximation parameters (N,K , h, β), balance errors, and obtain error estimatesin terms of WORK. ; (quasi-) optimal numerical algorithms




〈δ2Eac(ua)v , v〉 ≥ c0‖v ′‖2`2 .




h& 1


K).


K) + ‖β′′‖`2 .





〈δ2Eac(ua)v , v〉 ≥ c0‖v ′‖2`2 .




h& 1


K).


K) + ‖β′′‖`2 .


10.2. Brief History Lesson (1)

Ancestors: Tewari (1973), Mullins (1982), Kohlhoff, Schmauder,Gumbsch (1989, 1991)Starting point: Quasicontinuum method by Tadmor, Phillips, Ortiz (1995)(892 citations on Google scholar)

Force-based methodsQC GF Removal: Shenoy, Miller, Rodney, Tadmor, Phillips, Ortiz (1999);www.qcmethod.comForce-based QC: Dobson, Luskin (2008)(but similar variants were proposed throughout [1999, 2008])Dobson, Luskin and Ortner (2010): QCF unstable in W 1,p, p <∞.Lu, Ming (preprint): force blending, H2-stability for O(N) blendingLi, Luskin, CO (preprint): “conditional” H1-stability of for o(N1/5) blending

But the simplest method is still “pure” QCF: unknown whether it is stableor not in 2D/3D; KEY QUESTION: in which norm?

Λ = Λa ∪ Λc, Fqcf` (u) :=

F a` (u), ` ∈ Λa,F c` (u), ` ∈ Λc

Fbqcf` (u) := (1− β(`))F a

` (u) + β(`)F c` (u).


Ancestors: Tewari (1973), Mullins (1982), Kohlhoff, Schmauder,Gumbsch (1989, 1991)Starting point: Quasicontinuum method by Tadmor, Phillips, Ortiz (1995)(892 citations on Google scholar)Force-based methods

QC GF Removal: Shenoy, Miller, Rodney, Tadmor, Phillips, Ortiz (1999);www.qcmethod.comForce-based QC: Dobson, Luskin (2008)(but similar variants were proposed throughout [1999, 2008])Dobson, Luskin and Ortner (2010): QCF unstable in W 1,p, p <∞.Lu, Ming (preprint): force blending, H2-stability for O(N) blendingLi, Luskin, CO (preprint): “conditional” H1-stability of for o(N1/5) blending



F a` (u), ` ∈ Λa,F c` (u), ` ∈ Λc

Fbqcf` (u) := (1− β(`))F a

` (u) + β(`)F c` (u).


Ancestors: Tewari (1973), Mullins (1982), Kohlhoff, Schmauder,Gumbsch (1989, 1991)Starting point: Quasicontinuum method by Tadmor, Phillips, Ortiz (1995)(892 citations on Google scholar)Force-based methods

QC GF Removal: Shenoy, Miller, Rodney, Tadmor, Phillips, Ortiz (1999);www.qcmethod.comForce-based QC: Dobson, Luskin (2008)(but similar variants were proposed throughout [1999, 2008])Dobson, Luskin and Ortner (2010): QCF unstable in W 1,p, p <∞.Lu, Ming (preprint): force blending, H2-stability for O(N) blendingLi, Luskin, CO (preprint): “conditional” H1-stability of for o(N1/5) blending



F a` (u), ` ∈ Λa,F c` (u), ` ∈ Λc

Fbqcf` (u) := (1− β(`))F a

` (u) + β(`)F c` (u).


Ancestors: Tewari (1973), Mullins (1982), Kohlhoff, Schmauder, Gumbsch(1989, 1991)Starting point: Quasicontinuum method by Tadmor, Phillips, Ortiz (1995)(892 citations on Google scholar)“Consistent” Energy-based Methods

Shimokawa et al (2003): described basic idea; explicit construction, but it is only“consistent” for short-range interactions and there are problems with cornersE, Lu and Yang (2006): generalized approach; introduce free parameters andnecessary conditions for “consistency”; provided parameters for “flat” a/c interfacesA. Shapeev (2011): simple explicit construction for 2D pair interactions; partialgeneralisation to 3D (preprint)CO, Zhang (preprint): generalized E/Lu/Yang approach; explicit construction andoptimal consistency estimates for 2D nearest neighbour interactions for generalinterface geometry. . . many unresolved cases left: Problem: find a “simple” and “consistent” a/c couplingfor EAM potentials? . . . for general many-body interactions?

Consistency of these schemes is maturing (CO (2012), CO/Shapeev (preprint),CO/Zhang (preprint)), though some interesting gaps remain; Stability:completely open in 2D/3D


V (Duh(`)) +∑`∈Λi

V`(Duh(`)) +

∫Ωc

W (∇uh) dx


Ancestors: Tewari (1973), Mullins (1982), Kohlhoff, Schmauder,Gumbsch (1989, 1991)Starting point: Quasicontinuum method by Tadmor, Phillips, Ortiz (1995)(892 citations on Google scholar)Energy Blending:

Belytschko and Xiao (2004). . . many variants throughout [2004, 2012] . . .Vankoten and Luskin (2011): clear formulation of the algorithm and first rigorousanalysis

Eb(uh) :=∑`∈Λa

β(`)V (Duh(`)) +

∫Ωc

(1− β)W (∇uh) dx

The only known practical energy-based a/c scheme with controllableconsistency error for general a/c interface geometries in 1D/2D/3D ?Consistency is well-understood, but stability is again completely open.

10.5. More Open Problems

Construction of “consistent” QNL-type methodsStability of practical a/c couplings in 2D/3DOther classes of a/c couplings: overlapping domains (. . . ), other types ofa/c decomposition such as u = uc + ψ (Park), field-based QC (Gavini &Iyer), summation rules, . . .Optimise approximation parameters, e.g., using a posteriori adaptiontechniques, higher order elementshigher order continuum modelsapproximation of bifurcation diagrams and critical phenomena (formationand motion of defects)prediction of defect nucleationapproximation of transition states. . . . . . . . .

Electronic Structure: Jianfeng Lu

Temperature: Mitch Luskin

10.5. More Open Problems

Construction of “consistent” QNL-type methodsStability of practical a/c couplings in 2D/3DOther classes of a/c couplings: overlapping domains (. . . ), other types ofa/c decomposition such as u = uc + ψ (Park), field-based QC (Gavini &Iyer), summation rules, . . .Optimise approximation parameters, e.g., using a posteriori adaptiontechniques, higher order elementshigher order continuum modelsapproximation of bifurcation diagrams and critical phenomena (formationand motion of defects)prediction of defect nucleationapproximation of transition states. . . . . . . . .

Electronic Structure: Jianfeng Lu

Temperature: Mitch Luskin

Documents

Atomistic-to-continuum coupling methods for crystalline solids€¦ · Atomistic-to-continuum coupling methods for crystalline solids Christoph Ortner Warwick Mathematics Institute