Coupling of continuum and particle models - DAMTP · Coupling of continuum and particle models Milana Gataric ... molecular statics and dynamics full particle models continuum models

Introduction Domain Decomposition Methods Particle-to-continuum coupling

Coupling of continuum and particle models

Milana Gataric

Supervisors: Fehmi Cirak and Carola-Bibiane Schönlieb

University of Cambridge

January 2012


Motivation

Motivation

Reducing the computational cost of simulations inmolecular statics and dynamics

full particle modelscontinuum models

U s e c o u p l i n g m o d e l smaterial domain is decomposed into particle andcontinuum subdomainsparticle models are used only on small, strategicallychosen subdomains where some irregularities areexpected, and most of the body is modeled by continuummodels (finite element method)


Motivation

Motivation





Motivation

Motivation


full particle models

continuum models



Motivation

Motivation





Motivation

Motivation



U s e c o u p l i n g m o d e l s

material domain is decomposed into particle andcontinuum subdomainsparticle models are used only on small, strategicallychosen subdomains where some irregularities areexpected, and most of the body is modeled by continuummodels (finite element method)


Motivation

Motivation



U s e c o u p l i n g m o d e l smaterial domain is decomposed into particle andcontinuum subdomains

particle models are used only on small, strategicallychosen subdomains where some irregularities areexpected, and most of the body is modeled by continuummodels (finite element method)


Motivation

Motivation





Approach

Approach

Couple models by using the ideas of classical domaindecomposition methods for solving partial differentialequations in divide and conquer manner

continuum-to-continuum particle-to-continuum


Approach

Approach


continuum-to-continuum

particle-to-continuum


Approach

Approach


continuum-to-continuum particle-to-continuum


Alternating Schwarz Method


problem:

Lu ≡ −∇ · (a(x)∇u) = f (x), in Ω ⊂ <n

u = 0, on ∂ΩLw1 = f , in Ω1

w1 = w2, on Γ1

w1 = 0, on ∂Ω ∩ Ω1Lw2 = f , in Ω2

w2 = w1, on Γ2

w2 = 0, on ∂Ω ∩ Ω2

u(x) = wi(x) on Ωi for i = 1,2




problem:

Lu ≡ −∇ · (a(x)∇u) = f (x), in Ω ⊂ <n

u = 0, on ∂Ω

Lw1 = f , in Ω1

w1 = w2, on Γ1

w1 = 0, on ∂Ω ∩ Ω1Lw2 = f , in Ω2

w2 = w1, on Γ2

w2 = 0, on ∂Ω ∩ Ω2





problem:

Lu ≡ −∇ · (a(x)∇u) = f (x), in Ω ⊂ <n

u = 0, on ∂Ω

Lw1 = f , in Ω1

w1 = w2, on Γ1

w1 = 0, on ∂Ω ∩ Ω1Lw2 = f , in Ω2

w2 = w1, on Γ2

w2 = 0, on ∂Ω ∩ Ω2





problem:

Lu ≡ −∇ · (a(x)∇u) = f (x), in Ω ⊂ <n

u = 0, on ∂ΩLw1 = f , in Ω1

w1 = w2, on Γ1

w1 = 0, on ∂Ω ∩ Ω1Lw2 = f , in Ω2

w2 = w1, on Γ2

w2 = 0, on ∂Ω ∩ Ω2





problem:

Lu ≡ −∇ · (a(x)∇u) = f (x), in Ω ⊂ <n

u = 0, on ∂ΩLw1 = f , in Ω1

w1 = w2, on Γ1

w1 = 0, on ∂Ω ∩ Ω1Lw2 = f , in Ω2

w2 = w1, on Γ2

w2 = 0, on ∂Ω ∩ Ω2





Lu(k+1)1 = f , in Ω1 Lu(k+1)

2 = f , in Ω2

u(k+1)1 = u(k)|Γ1 , on Γ1 u(k+1)

2 = u(k+1)1 |Γ2 , on Γ2

u(k+1)1 = 0, on ∂Ω ∩ Ω1 u(k+1)

2 = 0, on ∂Ω ∩ Ω2

the (k + 1)-th iteration is definedby:

u(k+1) =

u(k+1)

2 on Ω2

u(k+1)1 on Ω \ Ω2

The method converges: ‖u − u(k)‖ ≤ ρk‖u − u(0)‖, ρ ∈ (0,1)




Lu(k+1)1 = f , in Ω1 Lu(k+1)

2 = f , in Ω2

u(k+1)1 = u(k)|Γ1 , on Γ1 u(k+1)

2 = u(k+1)1 |Γ2 , on Γ2

u(k+1)1 = 0, on ∂Ω ∩ Ω1 u(k+1)

2 = 0, on ∂Ω ∩ Ω2

the (k + 1)-th iteration is definedby:

u(k+1) =

u(k+1)

2 on Ω2

u(k+1)1 on Ω \ Ω2

The method converges: ‖u − u(k)‖ ≤ ρk‖u − u(0)‖, ρ ∈ (0,1)


Matrix form of the alternating Schwarz Method

Matrix form of the alternating Schwarz method

after the finite element discretization of PDE:

Au = f

A1 = RT1 AR1,

A2 = RT2 AR2





Au = f

A1 = RT1 AR1,

A2 = RT2 AR2





Au = f

A1 = RT1 AR1,

A2 = RT2 AR2




the alteranting Schwarz gives

A1w(k+ 1

2 )

1 = RT1 (f− Au(k)),

u(k+ 12 ) = u(k) + R1w

(k+ 12 )

1 ,

A2w (k+1)2 = RT

2 (f− Au(k+ 12 )),

u(k+1) = u(k+ 12 ) + R2w (k+1)

2 .

introduce the projection operators as

Pi ≡ RTi A−1

i RiA, for i = 1,2

the error equation is

e(k+1) = (I − P2)(I − P1)e(k)





A1w(k+ 1

2 )

1 = RT1 (f− Au(k)),

u(k+ 12 ) = u(k) + R1w

(k+ 12 )

1 ,

A2w (k+1)2 = RT

2 (f− Au(k+ 12 )),

u(k+1) = u(k+ 12 ) + R2w (k+1)

2 .


Pi ≡ RTi A−1

i RiA, for i = 1,2


e(k+1) = (I − P2)(I − P1)e(k)





A1w(k+ 1

2 )

1 = RT1 (f− Au(k)),

u(k+ 12 ) = u(k) + R1w

(k+ 12 )

1 ,

A2w (k+1)2 = RT

2 (f− Au(k+ 12 )),

u(k+1) = u(k+ 12 ) + R2w (k+1)

2 .


Pi ≡ RTi A−1

i RiA, for i = 1,2


e(k+1) = (I − P2)(I − P1)e(k)


So far, we have discussed domain decomposition method incontinuum-to-continuum context.

continuum-to-continuum


Let us now use domain decomposition ideas inparticle-to-continuum coupling.

particle-to-continuum


Models

Global particle (atomistic) model

in physics known as atom-spring modelatmistic lattice (example in the one-dimensional case):

global atomisic model: K ag ua

g = fag

the upscaling by h→ 0 gives the corresponding continuummodel, which in the one-dimensional case looks as

− ddx

(kc

dudx

)= f ,

for a density function u and kc = k1 + 4k2


Models

Global particle (atomistic) modelin physics known as atom-spring model

atmistic lattice (example in the one-dimensional case):


g = fag


− ddx

(kc

dudx

)= f ,



Models

Global particle (atomistic) modelin physics known as atom-spring modelatmistic lattice

(example in the one-dimensional case):


g = fag


− ddx

(kc

dudx

)= f ,



Models

Global particle (atomistic) modelin physics known as atom-spring modelatmistic lattice (example in the one-dimensional case):


g = fag


− ddx

(kc

dudx

)= f ,



Models



g = fag


− ddx

(kc

dudx

)= f ,



Models



g = fag

the upscaling by h→ 0 gives the corresponding continuummodel

, which in the one-dimensional case looks as

− ddx

(kc

dudx

)= f ,



Models



g = fag


− ddx

(kc

dudx

)= f ,



Models

Global finite element model

finite element grid in one dimensional case:

global finite element model: K feg ufe

g = ffeg

Coupled modelin the one dimensional case, the overlapping decmpositionof domain:

restrict global to local models, and then couple them:

K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Models

Global finite element modelfinite element grid

in one dimensional case:


g = ffeg



K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Models

Global finite element modelfinite element grid in one dimensional case:


g = ffeg



K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Models



g = ffeg



K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Models



g = ffeg

Coupled model

in the one dimensional case, the overlapping decmpositionof domain:


K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Models



g = ffeg



K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Models



g = ffeg



K ag ua

g = fag

K feg ufe

g = ffeg

K aua = fa

K feufe = ffe


Alternating Schwarz method (particle-to-continuum)


global to local:

K a = RT1 K a

g R1 K fe = RT2 K fe

g R2

the (k + 1)th iteration:

K aw(k+ 1

2 )

1 = RT1 (fa

g − K ag u(k)),

u(k+ 12 ) = u(k) + R1w

(k+ 12 )

1 ,

K few (k+1)2 = RT

2 (ffeg − K fe

g u(k+ 12 )),

u(k+1) = u(k+ 12 ) + R2w (k+1)

2 .




global to local:

K a = RT1 K a

g R1 K fe = RT2 K fe

g R2

the (k + 1)th iteration:

K aw(k+ 1

2 )

1 = RT1 (fa

g − K ag u(k)),

u(k+ 12 ) = u(k) + R1w

(k+ 12 )

1 ,

K few (k+1)2 = RT

2 (ffeg − K fe

g u(k+ 12 )),

u(k+1) = u(k+ 12 ) + R2w (k+1)

2 .



The error equation:

e(k+1) = (I − P fe)(I − Pa)e(k) + P fed, where d ≡ ufeg − ua

g

The bound on the error (M. Parks and P. Boachev):

‖e(k+1)‖ ≤ σk+1κ(‖e(0)‖ − ‖P

fed‖1−σ

)+ κ‖P

fed‖1−σ

Numerical results (1-dimensional case):

δ ‖u − uag‖ Nit σ t

0 1.5161× 10−07 55 0.9177 0.20611 9.7438× 10−08 33 0.8480 0.12972 7.3157× 10−08 25 0.7840 0.06114 6.5386× 10−08 20 0.7247 0.05515 5.9744× 10−08 17 0.6697 0.0358



The error equation:


g


‖e(k+1)‖ ≤ σk+1κ(‖e(0)‖ − ‖P

fed‖1−σ

)+ κ‖P

fed‖1−σ



0 1.5161× 10−07 55 0.9177 0.20611 9.7438× 10−08 33 0.8480 0.12972 7.3157× 10−08 25 0.7840 0.06114 6.5386× 10−08 20 0.7247 0.05515 5.9744× 10−08 17 0.6697 0.0358



The error equation:


g


‖e(k+1)‖ ≤ σk+1κ(‖e(0)‖ − ‖P

fed‖1−σ

)+ κ‖P

fed‖1−σ



0 1.5161× 10−07 55 0.9177 0.20611 9.7438× 10−08 33 0.8480 0.12972 7.3157× 10−08 25 0.7840 0.06114 6.5386× 10−08 20 0.7247 0.05515 5.9744× 10−08 17 0.6697 0.0358


Non-matching finite element grid and paricle lattice


make finite element grid to be coarser than particle latticein the one dimensional case:

the alternating Schwarz method:

u(k+ 12 ) = u(k) + R1(K a)−1RT

1 (fag − K a

g u(k)) (1)

u(k+1) = u(k+ 12 ) + T2R2(K fe)−1RT

2 T1(ffeg − K fe

g u(k+ 12 )) (2)

grid transfer operators T1 and T2 can be written as

T1 =12

T T2

the error equation: e(k+1) = (I − P fe)(I − Pa)e(k) + P fed




make finite element grid to be coarser than particle lattice

in the one dimensional case:


u(k+ 12 ) = u(k) + R1(K a)−1RT

1 (fag − K a

g u(k)) (1)

u(k+1) = u(k+ 12 ) + T2R2(K fe)−1RT

2 T1(ffeg − K fe

g u(k+ 12 )) (2)


T1 =12

T T2







u(k+ 12 ) = u(k) + R1(K a)−1RT

1 (fag − K a

g u(k)) (1)

u(k+1) = u(k+ 12 ) + T2R2(K fe)−1RT

2 T1(ffeg − K fe

g u(k+ 12 )) (2)


T1 =12

T T2







u(k+ 12 ) = u(k) + R1(K a)−1RT

1 (fag − K a

g u(k)) (1)

u(k+1) = u(k+ 12 ) + T2R2(K fe)−1RT

2 T1(ffeg − K fe

g u(k+ 12 )) (2)


T1 =12

T T2







u(k+ 12 ) = u(k) + R1(K a)−1RT

1 (fag − K a

g u(k)) (1)

u(k+1) = u(k+ 12 ) + T2R2(K fe)−1RT

2 T1(ffeg − K fe

g u(k+ 12 )) (2)


T1 =12

T T2







u(k+ 12 ) = u(k) + R1(K a)−1RT

1 (fag − K a

g u(k)) (1)

u(k+1) = u(k+ 12 ) + T2R2(K fe)−1RT

2 T1(ffeg − K fe

g u(k+ 12 )) (2)


T1 =12

T T2




Numerical results for 1-dimensional coupled model

the non-matching case:

δ ‖u − uag‖ t σ

1 3.5091× 10−07 0.1297 12 1.9706× 10−07 0.0611 13 1.4318× 10−07 0.0551 14 1.0996× 10−07 0.0358 1

the matching case:


1 1.5161× 10−07 0.3124 0.91772 9.7438× 10−08 0.2065 0.84803 7.3157× 10−08 0.1757 0.7840



Numerical results for 1-dimensional coupled model

the non-matching case:


1 3.5091× 10−07 0.1297 12 1.9706× 10−07 0.0611 13 1.4318× 10−07 0.0551 14 1.0996× 10−07 0.0358 1

the matching case:


1 1.5161× 10−07 0.3124 0.91772 9.7438× 10−08 0.2065 0.84803 7.3157× 10−08 0.1757 0.7840


Thank you.

Documents

Coupling of continuum and particle models - DAMTP · Coupling of continuum and particle models Milana Gataric ... molecular statics and dynamics full particle models continuum models