Multiscale Modeling and Hybrid Algorithms

Weinan E

Princeton University

Collaborators: Xiantao Li (PSU), Jianfeng Lu (Princeton),Pingbing Ming (CAS), Weiqing Ren (NYU) and Zhijian Yang(RIT).

Supported by ONR, DOE, AFOSR, ARO and NSF.

Hybrid, multi-physics algorithms have had a lot ofsuccesses

I Quantum mechanics-Molecular mechanics (QM-MM)methods (1975, Warshel and Levitt)

I Quasicontinuum method (1996, Tadmor, Ortiz and Phillips)

I Kinetic-hydrodynamic methods

I ......

There are still many fundamental issues

I Interface conditions

I Consistency

I Stability

Major difficulty:

I Microsopic models, such as molecular dynamics and electronicstructure models, are very poorly understood.

I Effect of noise or fluctuations?

I In some cases, connection between macro and micro models.

Coupled kinetic-hydrodynamic simulation

Dynamics of cracks

Error at the atomistic-continuum interface

Stability

Strategy:

I Use kinetic equation (e.g. Boltzmann) to resolve the shocks

I Use hydrodynamic (e.g. Euler) elsewhere.

t

x

t

x

Kinetic theory: f = f (x, v, t)

∂t f + v · ∇f =1

εC (f ).

ρ =

∫fdv , ρu =

∫f vdv , E =

∫f|v|22

dv .

∂t

ρρuE

+∇ · F = 0

F =

∫

R3

f

vv ⊗ v12 |v|2v

dv

No approximations yet!

When ε ¿ 1, the distribution function f is close to localequilibrium states, or the local Maxwellians,

M(x, v, t) =ρ(x, t)

(2πθ(x, t))3/2exp

(−(v − u(x, t))2

2θ(x, t)

)(1)

with θ being the absolute temperature.

Coupled scheme:

I Kinetic-based finite volume method over the whole region – toguarantee conservation.

I Local Maxwellian approximation inside hydrodynamic region –kinetic scheme.

I Appropriate interface condition.

Macro side: Kinetic scheme – finite volume scheme based onkinetic equations (similar in spirit to Lattice Boltzmann methods(LBM))

(ρ, ρu,E ) = (ρj , ρjuj ,Ej), x ∈ (xj−1/2, xj+1/2]

The finite volume scheme takes the form:

ρn+1j −ρn

j

∆t +

(F

(1)j+1/2

−F(1)j−1/2

)∆x = 0,

(ρu)n+1j −(ρu)nj

∆t +

(F

(2)j+1/2

−F(2)j−1/2

)∆x = 0,

En+1j −En

j

∆t +

(F

(3)j+1/2

−F(3)j−1/2

)∆x = 0

Fj+1/2 = (F(1)j+1/2,F

(2)j+1/2,F

(3)j+1/2)

T

Fj+1/2 = F+j+1/2 + F−j+1/2,

F±j+1/2 =

∫

R±f (x∓j+1/2, v , t)

vv2

12v3

dv .

To leading order, we have:

f (x , v , t) ∼ M(x − vt, v , tn)

F± =

ρuA±(S)± ρ

2√

πβB(S)

(p + ρu2)A±(S)± ρu

2√

πβB(S)

(pu + ρue)A±(S)± ρ

2√

πβ(

p

2ρ+ e)B(S)

A± =1 + erf(S)

2, B(S) = e−S2

, S =u√2RT

, p = ρRT .

Inside the kinetic region – use solutions of the kinetic equation:

Fk+1/2 =1

τ

∫ tn+τ

tn

dt∫

R+

f (x−k+1/2, v , t)

vv2

12v3

dv

+

∫

R−f (x+

k+1/2, v , t)

vv2

12v3

dv

.

(2)

At the interface – hybrid:

f (x , v , t) = M(x−k+1/2, v , t), v > 0

Fk+1/2 =1

τ

∫ tn+τ

tn

dt∫

R+

M(x−k+1/2, v , t)

vv2

12v3

dv

+

∫

R−f (x+

k+1/2, v , t)

vv2

12v3

dv

.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure: solid line – kinetic equation; ‘o’ – coupled kinetic/gas dynamicssimulation; ‘+’ – gas dynamics equation.

Coupled kinetic-hydrodynamic simulation

Dynamics of cracks

Error at the atomistic-continuum interface

Stability

Multiscale Computational models

Continuum

Atomistic

Atomistic and continuum models:

I models at the atomic scale: molecularstatics, molecular dynamics

I continuum models: elasticity,elastodynamics, etc.

Main ideas:

I Atomistic models to resolve local defects

I Continuum models away from the defects

I Matching at the interface

Existing Coupling Methods

Some existing methods:

I quasicontinuum method (1996)

I finite temperature quasicontinuum (2005)

I coarse-grained molecular dynamics (1998,2005)

I bridging scale decomposition (2003)

I heterogeneous multiscale methods (2003)

I bridging domain method (2004)

I many others ...

The continuum model

Elastodynamics:ρu = ∇ · σ.

I u: displacement.

I σ = σ(∇u): stress.

Formulation of Conservation Laws:

wt + fx + gy = 0.

w =

ε11

ε12

ε21

ε22

ρ0v1

ρ0v2

, f =

−v1

0−v2

0−σ11

−σ21

, g =

0−v1

0−v2

−σ21

−σ22

,

The atomistic model

Reference position and displacement:

ui = ri − Ri .

Molecular dynamics

mui = −∇ui V .

Embedded atom potential:

V =1

2

∑φ(rij) +

∑E (ρi ), ρi =

∑

j

ρ(rij).

Coupling of the two models

j j+1

k

k+1

t= nσ .

t= nσ .

t= nσ .

t= Σ f ij

Continuum Region Atomistic Region

Coupling method:

I In the continuum region:constitutive relation

I at the atomistic/continuuminterface: boundary conditions.

Finite volume formulation

Average velocity:

ρ0vJ+ 12,K+ 1

2=

1

|AJK |∑

i∈AJK

mivi ,

Momentum balance:

ρ0d

dtvJ+ 1

2,K+ 1

2=

∑

∂AJK

t.

Computing the traction at the cell edges:

I In the continuum region: approximated by a continuum solver(Roe scheme) t = σ · n.

I At the atomistic/continuum interface: computed based on theatomic position t =

∑i∈AJKj /∈AJK

fij .

Overall computational procedure

∆ x δ x

∆ tδ t

Interface condition I: Linearizing the atomic interaction

Partition of the system:

u = (uA,uC ).

I uA: displacement in the atomistic region.

I uC : displacement in the continuum region.

Mechanical equilibrium:

∇uCV (uA,uC ) = 0, uC = RCA(uA).

Linearization:

V ≈ V (uA,RCA(uA)) +1

2

(uC − RCA(uA)

)TDCC

(uC − RCA(uA)

).

Interface condition II: boundary condition for molecularstatics

Harmonic approximation:

DCAuA + DCCuC = 0, uC = BCAuA.

Boundary map:

BCA = −D−1CCDCA = GCAG−1

AA .

Lattice Green’s function:

−∑

j

Di−jGj ,k = δik .

Di ,j ,Gj ,k are elements of matrix D,GDi−j = Di ,j due to translational invariance

Interface condition III: approximate solution at theinterface

Hamilton’s equation:

muA = −∇uAV (uA,BCAuA)− DACuC ,

muC = −DCCuC − DCAuA.

Initial condition is interpolated from the continuum solution:

uC (0) = uC , vC (0) = vC .

Interface condition IV: boundary condition for MD

Decomposition of the displacement at the interface:u

C= u(1)

C+ u(2)

C.

u(1)C

= D−1/2CC sin(D

1/2CC t)(v

C− BCAv

A) + cos(D

1/2CC t)(u

C− BCAu

A)

u(2)C

=

∫ t

0α(τ)u

A(t − τ)dτ.

Displacement in the atomistic region:

muA = −∇uAV

(uA,uC

)−∫ t

0θ(τ)vA(t − τ)dτ − θ(t)u(0)

Boundary conditions at finite temperature

Partition of the system:

u = (uA,uC ), v = (vA, vC )

The generalized Langevin equation:

muA = −∇uAV −

∫ t

0Θ(τ)uA(t − τ)dτ + R(t).

R(t) : stationary Gaussian process.

⟨R(t)R(s)T

⟩= kBTΘ(t − s).

1D shock propagation

−40 −30 −20 −10 0 10 20 30 40

0

0.005

0.01

0.015

0.02

0.025

Str

ain

Before the shock leaves the atomistic region

−40 −30 −20 −10 0 10 20 30 40

0

0.005

0.01

0.015

0.02

0.025

Str

ain

After the shock leaves the atomistic region

Solution in the atomistic regionSolution at the continuum region

Solution in the atomistic regionSolution at the continuum region

−40 −30 −20 −10 0 10 20 30 40−0.005

0

0.005

0.01

0.015

0.02

0.025Before the shock enters the atomistic region

−40 −30 −20 −10 0 10 20 30 40−0.005

0

0.005

0.01

0.015

0.02

0.025After the shock enters the atomistic region

Solution in the atomistic regionSolution at the continuum region

Solution in the atomistic regionSolution at the continuum region

Crack propagation: Preparation

Displacement from the anisotropic elasticity solution (Sihand Liebowitz 1968):

u(x , y) = KI

[p1s2s2−s1

φ(z) + p2s1s1−s2

ψ(z).]

K cI =

√πν2

1√(b11b22/2d0)(

√b22/b11+(2b12+b66)/(2b11))

.

I K cI is chosen based on the Griffith criterion.

I s1, s2 are roots of characteristic equation reduced from thestress function.

I bij : compliance parameters of corresponding plane stress state.

I p1 = a11s21 + a12 − a16s1, p2 = a11s

22 + a12 − a16s2.

I aij : compliance parameters of corresponding plane strain state.

Relaxation of the crack KI = K cI

Relaxation of the crack KI = 1.2K cI

Crack front waves in the continuum region

Mesh size ∆x = ∆y = 6√

2a0.Time step ∆t = 8δt.

Scattering of the elastic wave off the crack tip

crack under mode I loading.

Velocity and displacement at the cross sections

Displacement and velocity ahead of the crack tip.

Pulse

Dynamic loading condition.

Coupled kinetic-hydrodynamic simulation

Dynamics of cracks

Error at the atomistic-continuum interface

Stability

Quasicontinuum method

I An adaptive mesh and model refinement procedureI Based on linear finite elementsI Representative atoms define the triangulationI Near defects, the mesh becomes fully atomisticI Local (continuum) and nonlocal (atomistic) regions

Consistency

I Consistency in the bulk: For simple systems, the two modelsshould produce consistent results.

I Consistency at the local-nonlocal interface.

Consistency between atomistic and continuum models

Nonlinear elasticity theory:

E (u) =

∫

ΩW (∇u)dx

W (·) = stored energy density.In linear elasticity, W = a quadratic function of ∇u.

E (y1, · · · , yN) =∑

i ,j

V2(yi , yj) +∑

i ,j ,k

V3(yi , yj , yk) + · · ·

Question: Can we relate W to the atomistic model?

The Cauchy-Born rule

Given A, a 3× 3 matrix, W (A) =?

Deform the crystal uniformly: yj = xj + Axj = (I + A)xj

W (A) = energy density of deformed unit cell, computed accordingto the given atomistic or electronic structure model.

Validity of Cauchy-Born rule: ConsistencyOne dimension model: xk = kε.Assume: yk = xk + u(xk) and u is a smooth function.

V =1

2

∑

i 6=k

V0(yi − yk) (3)

≈ 1

2

∑

i

∑

k 6=i

V0(1 +du

dx(xi ))kε

=∑

i

W

(du

dx(xi )

)ε ≈

∫W

(du

dx(x)

)dx ,

where

W (A) =1

2ε

∑

k

V0((1 + A)kε)

X. Blanc, C. Le Bris and P. L. Lions (2002) considered generalcase, including some QM models.

Validity of Cauchy-Born rule: Counterexample

Example: Lennard-Jones potential, next nearest neighborinteraction

I Triangular lattice, Cauchy-Born rule is valid

I Square lattice, Cauchy-Born gives negative shear modulus(unstable), can’t speak of elasticity theory.

Validity of Cauchy-Born rule: Stability

I Continuum level (Born criteria) – Elastic stiffness tensor ispositive definite

I Atomic level (Lindemann criteria) – Phonon spectra(dispersion relation for the lattice waves) remain “positivedefinite”

I Electronic level – Disperson relation for the charge-densitywaves and spin waves

Phonon stability condition

Acoustic branch: dynamics of the Bravais lattice

Optical branch: relative motion of the internal degrees of freedom

1st Brillouin zone: Voronoi cell of the origin of the dual lattice

Consistency in the bulk

Under these conditions, the solutions to the Cauchy-Borncontinuum model and the atomistic model are close.

I E and Ming (2007), molecular mechanics models.

I E and Lu (2008), several classes of quantum mechanicsmodels.

These conditions are sharp!!Violation of the stability conditions signals onset of plasticdeformation or structural (or electronic) phase transformation.Related work of Ju Li, S. Yip et al. (Λ-criterion),R. Elliott et al.

Consistency at the loca-nonlocal interfaceThe issue of “ghost force” (e.g. in quasicontinuum methods)

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15Ghost forces with 5th nearest neighbor interaction

Original QCQuasi−nonlocalLinear construction

I Left side: using continuum model based on the Cauchy-Bornrule (effectively a nearest neighbor model).

I Right side: Using full atomistic model, next nearest neighborinteraction.

Explicit solution for quadratic potential (Ming and Yang, 09)

0 5 10 15 20 25 30 35−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

N

D+(y

qc−

x)

1. The deformation gradient has O(1) error at the interface.

2. The influence of the ghost force decays exponential fast awayfrom the interface.

3. Away from an interfacial region of width O(ε| log(ε)|), theerror in the deformation gradient is of O(ε) (see also recentwork of Dobson and Luskin).

Removing the ghost force

Ghost force may induce numerical artifacts (e.g. plasticdeformation) at the interface.

I Force-based approach (Tadmor et al., Miller, Dobson andLuskin)

I Quasi-nonlocal atoms (Jacobson et al.)

I Geometrically consistent scheme (E, Lu and Yang)

Classical numerical analysis viewpoint:

I Truncation error = O(ε) in a weak sense

I Stability conditions (similar to the ones discussed above)

Uniform O(ε) accuracy for smooth solutions.

Ghost force for the coupled OF-DFT/EAM method(Choly, Lu, E, Kaxiras)

-0.03

-0.02

-0.01

0

0.01

-0.03

-0.02

-0.01

0

0.01

−3

∆ρ (A−3

) ∆ρ (A−3

)

1 nm

0.14A−3

A

(d)

0.22

(c)

(b)(a)

Loss of fluctuations

Example: Coupled KMC-continuum models of epitaxial crystalgrowth (Schulze, Smereka and E):

I Around the step-edges, use KMC, since fluctuations areimportant

I On the terraces, use continuum (e.g. diffusion) models

Mean position and variance of step edge

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 200000 400000 600000 800000 1e+06

σe

D/F

+++++++++

+

33333333

3

3

+++++++++

+

+++++++++

+

Figure 6: The time and space averaged surface adatom density for the KMC simulations(diamonds) and the hybrid scheme (crosses) using cell-widths M = 20, 25, 40.

0

0.005

0.01

0.015

0.02

0 200000 400000 600000 800000 1e+06

σρ

D/F

++++++++++

33

33333

33

3

++++++++++ ++++++

+++

+

Figure 7: The standard deviation (in time) of the surface averaged adatom density as afunction of the ratio D/F . The solid curve (diamonds) are from the KMC simulations andthe remaining curves are for the hybrid scheme with cell-widths M = 20, 25, 40 sites percell.

21

Other examples: See work of Garcia, Bell and Donev, et al.

Coupled kinetic-hydrodynamic simulation

Dynamics of cracks

Error at the atomistic-continuum interface

Stability

Stability of coupled continuum/MD methods

Work of Weiqing Ren (NYU)

I Example of fluids

I General strategy: Domain decomposition (with overlap)

I Coupling:

III

z=−az=−b

z=az=b

z=1

z=−1u=0

u=0

I

II

In continuum region (I, III): ρut − µuzz = 0

In particle region (II): md2xj

dt2 = f j

Four coupling schemes:

1. velocity(MD)-velocity(C),

2. velocity(MD)-flux(C),

3. flux(MD)-velocity(C),

4. flux(MD)-flux(C)

Particular features as a domain decomposition method

I The MD (molecular dynamics) domain is very small.

I Statistical error cannot be avoided

Numerical solutions for equilibrium states

0 0.5 1 1.5 2

x 104

0

0.02

0.04

t / T

||u|| 2

0 0.5 1 1.5 2

x 104

0

0.02

0.04

t / T

||u|| 2

0 0.5 1 1.5 2

x 104

0

0.02

0.04

t / T

||u|| 2

0 0.5 1 1.5 2

x 104

0

0.2

0.4

t / T

||u|| 2

Upper panel: velocity-velocity Upper panel: velocity-fluxLower panel: flux-velocity Lower panel: flux-flux

0

1

2

u

−100 −50 0 50 1000

1

2

z

u

0 2000 4000 6000 80000

0.1

0.2

t / Tc

e u

Steady-state calculation: Tc = ∞Amplification factors g for the four schemes:

I velocity-velocity, flux-velocity

un(z) =n∑

i=1

gn−iξi1− z

1− a, 〈‖un‖2〉 ≤

(1

3(1− g2)

)1/2

σv

ξi : Statistical errors in velocity BC; σv =⟨ξ2i

⟩

g = a(1−b)b(1−a) for velocity-velocity; g = a

a−1 for flux-velocity

I velocity-flux, flux-flux

un(z) =n∑

i=1

gn−iξ(z − 1)

g = b−1b for velocity-flux; (g > 1 → Diverge)

g = 1 for flux-flux → 〈‖un‖2〉 ≤ 3−1/2(1− a)n1/2στ

Stability: Finite Tc

0 200 400 600 800 10000

0.5

1

Tc

k vv ,

k fv

10 30 50 70 900.8

0.9

1

1.1

Tc

k vf ,

k ff

Figure: The amplification factor g versus Tc/∆t for the four schemes:VV (squares), FV (diamonds), VF (triangles) and FF (circles).

Remarks:

I There are many different variants of atomistic/continuumcoupling schemes.

I Errors and artifacts are difficult to understand.

I What I have described are examples of efforts to try to putthings on a solid foundation.