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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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Motivation
Motivation
Reducing the computational cost of simulations inmolecular statics and dynamics
full particle models
continuum 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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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 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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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 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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
Alternating Schwarz Method
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
u(x) = wi(x) on Ωi for i = 1,2
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
Alternating Schwarz Method
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
u(x) = wi(x) on Ωi for i = 1,2
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
Alternating Schwarz Method
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz Method
Alternating Schwarz Method
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Matrix form of the alternating Schwarz Method
Matrix form of the alternating Schwarz method
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Matrix form of the alternating Schwarz Method
Matrix form of the alternating Schwarz method
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Matrix form of the alternating Schwarz Method
Matrix form of the alternating Schwarz method
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)
Introduction Domain Decomposition Methods Particle-to-continuum coupling
So far, we have discussed domain decomposition method incontinuum-to-continuum context.
continuum-to-continuum
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Let us now use domain decomposition ideas inparticle-to-continuum coupling.
particle-to-continuum
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global particle (atomistic) modelin physics known as atom-spring model
atmistic 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global particle (atomistic) modelin 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global particle (atomistic) modelin 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global particle (atomistic) modelin 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global particle (atomistic) modelin 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global particle (atomistic) modelin 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global finite element modelfinite 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global finite element modelfinite 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global finite element modelfinite 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global finite element modelfinite element grid in one dimensional case:
global finite element model: K feg ufe
g = ffeg
Coupled model
in 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global finite element modelfinite 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Models
Global finite element modelfinite 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz method (particle-to-continuum)
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 .
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz method (particle-to-continuum)
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 .
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz method (particle-to-continuum)
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz method (particle-to-continuum)
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Alternating Schwarz method (particle-to-continuum)
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
Non-matching finite element grid and paricle lattice
make finite element grid to be coarser than particle lattice
in 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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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:
δ ‖u − uag‖ t σ
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Non-matching finite element grid and paricle lattice
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:
δ ‖u − uag‖ t σ
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
Introduction Domain Decomposition Methods Particle-to-continuum coupling
Thank you.