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Introduction Elements of Methodology Numerical Results Conclusion
Methodology for Decomposition into Transportand Diffusive Subdomains for the LD Method
Nicholas Stehle and Dmitriy Anistratov
Department of Nuclear EngineeringNorth Carolina State University
ICTT-22, Portland OR, 2011
Introduction Elements of Methodology Numerical Results Conclusion
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Background
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Background
Highly Diffusive Domains
A large class of radiative transfer problems contain highlydiffusive regions.
A highly diffusive region is characterized by a large opticalthickness and small absorption.
The diffusion (P1) approximation is valid in such domains.
It is possible to reduce computational costs by solving adiffusion problem in diffusive subdomains.
Introduction Elements of Methodology Numerical Results Conclusion
Background
Basic Approach
The basic idea is to apply domain decomposition methodsthat
split the problem domain into transport and diffusionsubregions,define interface conditions that couple subregions of twodifferent kinds,employ hybrid algorithms to solve the problem in eachsubdomain.
The goal is to approximate the original numerical transportsolution on a given spatial-angular grid with minimal error.Publications
A. Klar and N. Siedow (European J. Appl. Math, 1998)A. Klar, H. Neunzert & J. Struckmeier (TTSP, 2000)M. Yavuz & E. Larsen (TTSP, 1989)
Introduction Elements of Methodology Numerical Results Conclusion
Background
Current Work
We consider 1D slab geometry transport problems withisotropic scattering.
The methodology is developed and applied to the LinearDiscontinuous method.
Transport effects are evaluated by means of metrics basedon the quasidiffusion (Eddington) factors.
The second moment method is used as a basis for thedeveloped methodology.
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
P1 Approximation
Moment equations
dφ1
dx+ σaφ0 = q ,
23
dφ2
dx+
13
dφ0
dx+ σtφ1 = 0 ,
φn =
∫ 1
−1Pn(µ)ψ(x , µ)dµ ,
P1 equations
dφ1
dx+ σaφ0 = q ,
13
dφ0
dx+ σtφ1 = 0 .
M1(x)def= 2
∣∣∣∣dφ2
dx
/dφ0
dx
∣∣∣∣� 1 .
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
P1 Approximation
Moment equations
dφ1
dx+ σaφ0 = q ,
23
dφ2
dx+
13
dφ0
dx+ σtφ1 = 0 ,
φn =
∫ 1
−1Pn(µ)ψ(x , µ)dµ ,
P1 equations
dφ1
dx+ σaφ0 = q ,
13
dφ0
dx+ σtφ1 = 0 .
M1(x)def= 2
∣∣∣∣dφ2
dx
/dφ0
dx
∣∣∣∣� 1 .
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
P1 Approximation
Moment equations
dφ1
dx+ σaφ0 = q ,
23
dφ2
dx+
13
dφ0
dx+ σtφ1 = 0 ,
φn =
∫ 1
−1Pn(µ)ψ(x , µ)dµ ,
P1 equations
dφ1
dx+ σaφ0 = q ,
13
dφ0
dx+ σtφ1 = 0 .
M1(x)def= 2
∣∣∣∣dφ2
dx
/dφ0
dx
∣∣∣∣� 1 .
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
Metrics Based on QD (Eddington) Factor
Low-order quasidiffusion (QD) equations
dφ1
dx+ σaφ0 = q ,
dE [ψ]φ0
dx+ σtφ1 = 0 ,
E [ψ] =
∫ 1
−1µ2ψdµ
/∫ 1
−1ψdµ .
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
Metrics Based on QD (Eddington) Factor
Low-order quasidiffusion (QD) equations
dφ1
dx+ σaφ0 = q ,
Edφ0
dx+
dEdx
φ0 + σtφ1 = 0 ,
E [ψ] =
∫ 1
−1µ2ψdµ
/∫ 1
−1ψdµ .
If there are no transport effects, then
E(x) =13,
dE(x)
dx= 0 ,
M2(x)def=
∣∣∣∣E(x)− 13
∣∣∣∣ , M3(x)def=
∣∣∣∣dE(x)
dx
∣∣∣∣ .
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
Metrics Based on QD (Eddington) Factor
Low-order quasidiffusion (QD) equations
dφ1
dx+ σaφ0 = q ,
Edφ0
dx+
dEdx
φ0 + σtφ1 = 0 ,
E [ψ] =
∫ 1
−1µ2ψdµ
/∫ 1
−1ψdµ .
If there are no transport effects, then
E(x) =13,
dE(x)
dx= 0 ,
M2(x)def=
∣∣∣∣E(x)− 13
∣∣∣∣ , M3(x)def=
∣∣∣∣dE(x)
dx
∣∣∣∣ .
Introduction Elements of Methodology Numerical Results Conclusion
Metrics of Transport Effects
Metrics Based on QD (Eddington) Factor
Low-order quasidiffusion (QD) equations
dφ1
dx+ σaφ0 = q ,
Edφ0
dx+
dEdx
φ0 + σtφ1 = 0 ,
E [ψ] =
∫ 1
−1µ2ψdµ
/∫ 1
−1ψdµ .
If there are no transport effects, then
E(x) =13,
dE(x)
dx= 0 ,
M2(x)def=
∣∣∣∣E(x)− 13
∣∣∣∣ , M3(x)def=
∣∣∣∣dE(x)
dx
∣∣∣∣ .
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Second Moment (SM) Method
A transport sweep (high-order equation)
µ∂ψ(s+1/2)
∂x+ σtψ
(s+1/2) =12σsφ
(s) +12
q .
Low-order SM equations
dJ(s+1)
dx+ σaφ
(s+1) = q ,
13
dφ(s+1)
dx+ σtJ(s+1) =
ddx
F (s+1/2) .
The closure term
F (s+1/2) =
∫ 1
−1
(13− µ2
)ψ(s+1/2)dµ .
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Algorithm
Transport Subdomain
Lψ(s+1/2) =12σsφ
(s) +12
q ,
F (s+1/2)=
∫ 1
−1
(13−µ2)ψ(s+1/2)dµ.
Diffusion Subdomain
F (s+1/2) = 0 .
Whole problem domain
dJ(s+1)
dx+ σaφ
(s+1) = q ,
13
dφ(s+1)
dx+ σtJ(s+1) =
ddx
F (s+1/2) .
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Algorithm
Transport Subdomain
Lψ(s+1/2) =12σsφ
(s) +12
q ,
F (s+1/2)=
∫ 1
−1
(13−µ2)ψ(s+1/2)dµ.
Diffusion Subdomain
F (s+1/2) = 0 .
Whole problem domain
dJ(s+1)
dx+ σaφ
(s+1) = q ,
13
dφ(s+1)
dx+ σtJ(s+1) =
ddx
F (s+1/2) .
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Algorithm
Transport Subdomain
Lψ(s+1/2) =12σsφ
(s) +12
q ,
F (s+1/2)=
∫ 1
−1
(13−µ2)ψ(s+1/2)dµ.
Diffusion Subdomain
F (s+1/2) = 0 .
Whole problem domain
dJ(s+1)
dx+ σaφ
(s+1) = q ,
13
dφ(s+1)
dx+ σtJ(s+1) =
ddx
F (s+1/2) .
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Interface Conditions for the High-Order Equation
We define the angular flux coming into a transportsubregion from a neighboring diffusion subregion for thehigh-order (transport) equation.
P1 approximation
ψ(x , µ) =12φ+
32µJ .
Asymptotic interface conditions in discrete form
ψ(x , µ) =
[12φ− µ3
2
(1
3σt
)dφdx
]h.
No interface conditions are needed for the low-orderequations.
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Interface Conditions for the High-Order Equation
We define the angular flux coming into a transportsubregion from a neighboring diffusion subregion for thehigh-order (transport) equation.
P1 approximation
ψ(x , µ) =12φ+
32µJ .
Asymptotic interface conditions in discrete form
ψ(x , µ) =
[12φ− µ3
2
(1
3σt
)dφdx
]h.
No interface conditions are needed for the low-orderequations.
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Interface Conditions for the High-Order Equation
We define the angular flux coming into a transportsubregion from a neighboring diffusion subregion for thehigh-order (transport) equation.
P1 approximation
ψ(x , µ) =12φ+
32µJ .
Asymptotic interface conditions in discrete form
ψ(x , µ) =
[12φ− µ3
2
(1
3σt
)dφdx
]h.
No interface conditions are needed for the low-orderequations.
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Interface Conditions for the High-Order Equation
We define the angular flux coming into a transportsubregion from a neighboring diffusion subregion for thehigh-order (transport) equation.
P1 approximation
ψ(x , µ) =12φ+
32µJ .
Asymptotic interface conditions in discrete form
ψ(x , µ) =
[12φ− µ3
2
(1
3σt
)dφdx
]h.
No interface conditions are needed for the low-orderequations.
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Linear Discontinuous Method
LD equations
µm(ψm,i+1/2 − ψm,i−1/2) + σt ,i∆xiψm,i =∆xi
2(σs,iφi + qi) ,
µmθi(ψm,i+1/2+ψm,i−1/2−2ψm,i)+σt ,i∆xi ψ̂m,i =∆xi
2(σs,i φ̂i +q̂i) ,
ψm,i∓1/2 =
{ψm,i − ψ̂i , µm < 0 ,ψm,i + ψ̂i , µm > 0 .
Asymptotic expansion of the LD solution in the interior of adiffusive domain leads to the following interface condition:
ψm,i+1/2 =
12
(φi +φ̂i
)− µm
2σt,i ∆xi
(φi+1/2−φi−1/2
), µm > 0,
12
(φi+1−φ̂i+1
)− µm
2σt,i+1∆xi+1
(φi+3/2−φi+1/2
), µm < 0.
Introduction Elements of Methodology Numerical Results Conclusion
Domain Decomposition
Linear Discontinuous Method
LD equations
µm(ψm,i+1/2 − ψm,i−1/2) + σt ,i∆xiψm,i =∆xi
2(σs,iφi + qi) ,
µmθi(ψm,i+1/2+ψm,i−1/2−2ψm,i)+σt ,i∆xi ψ̂m,i =∆xi
2(σs,i φ̂i +q̂i) ,
ψm,i∓1/2 =
{ψm,i − ψ̂i , µm < 0 ,ψm,i + ψ̂i , µm > 0 .
Asymptotic expansion of the LD solution in the interior of adiffusive domain leads to the following interface condition:
ψm,i+1/2 =
12
(φi +φ̂i
)− µm
2σt,i ∆xi
(φi+1/2−φi−1/2
), µm > 0,
12
(φi+1−φ̂i+1
)− µm
2σt,i+1∆xi+1
(φi+3/2−φi+1/2
), µm < 0.
Introduction Elements of Methodology Numerical Results Conclusion
Test A
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Test A
Definition of the Test
Region 0≤x≤1 1≤x≤11σt 2 100σs 0 100q 0 0
∆x 0.1 1BC’s ψ|x=0=1 ψ|x=11=0
Double S4 Gauss-Legendrequadrature setPointwise convergence criterionwith its parameter = 10−15
0 2 4 6 8 1 0 1 20 . 0
0 . 1
0 . 2
0 . 3
S c a l a r F l u x ( φ)
Introduction Elements of Methodology Numerical Results Conclusion
Test A
Metrics
0 2 4 6 8 1 0 1 21 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 51 0 - 41 0 - 31 0 - 21 0 - 11 0 0
Metric
M 1 , i M 2 , i M 3 , i
Converged
0 2 4 6 8 1 0 1 21 0 - 1 71 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 51 0 - 41 0 - 31 0 - 21 0 - 11 0 01 0 1
Metric
x
M 1 , i M 2 , i M 3 , i
Estimated
Introduction Elements of Methodology Numerical Results Conclusion
Test A
Metrics
0 2 4 6 8 1 0 1 21 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 51 0 - 41 0 - 31 0 - 21 0 - 11 0 0
Metric
M 1 , i M 2 , i M 3 , i
ε = 10−6
0 2 4 6 8 1 0 1 21 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 51 0 - 41 0 - 31 0 - 21 0 - 11 0 0
Metric
M 1 , i M 2 , i M 3 , i
ε = 10−9
Introduction Elements of Methodology Numerical Results Conclusion
Test A
Metrics
0 2 4 6 8 1 0 1 21 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 51 0 - 41 0 - 31 0 - 21 0 - 11 0 0
Metric
M 1 , i M 2 , i M 3 , i
ε = 10−6
3 ≤ x ≤ 10
0 2 4 6 8 1 0 1 21 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 51 0 - 41 0 - 31 0 - 21 0 - 11 0 0
Metric
M 1 , i M 2 , i M 3 , i
ε = 10−9
4 ≤ x ≤ 9
Introduction Elements of Methodology Numerical Results Conclusion
Test A
Numerical Solution with Domain Decomposition
0 2 4 6 8 1 0 1 2
1 0 - 1 6
1 0 - 1 5
1 0 - 1 4
1 0 - 1 3
1 0 - 1 2
1 0 - 1 1
1 0 - 1 0
1 0 - 9
1 0 - 8
1 0 - 7
1 0 - 6
Relat
ive Er
ror
x
ε = 10−6 [ P 1 ] ε = 10−6 [ A s y . ] ε = 10−9 [ P 1 ] ε = 10−9 [ A s y . ]
Introduction Elements of Methodology Numerical Results Conclusion
Test B
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Test B
Definition of the Test
Region 0≤x≤10 10≤x≤30σt 1 10σs 0.5 9.9q 0.5 0.01
∆x 1 0.5BC’s ψ|x=0=0 ψ|x=30=0
Double S4 Gauss-Legendrequadrature setPointwise convergence criterionwith its parameter = 10−15
0 5 1 0 1 5 2 0 2 5 3 0
0 . 0
0 . 5
1 . 0
1 . 5
2 . 0
Scala
r Flux
x
Introduction Elements of Methodology Numerical Results Conclusion
Test B
Metrics
0 5 1 0 1 5 2 0 2 5 3 01 0 - 9
1 0 - 8
1 0 - 7
1 0 - 6
1 0 - 5
1 0 - 4
1 0 - 3
1 0 - 2
1 0 - 1
1 0 0
1 0 1
Metric
M 1 , i M 2 , i M 3 , i
Converged
0 5 1 0 1 5 2 0 2 5 3 01 0 - 1 0
1 0 - 9
1 0 - 8
1 0 - 7
1 0 - 6
1 0 - 5
1 0 - 4
1 0 - 3
1 0 - 2
1 0 - 1
1 0 0
1 0 1
Metric
x
M 1 , i M 2 , i M 3 , i
Estimated
Introduction Elements of Methodology Numerical Results Conclusion
Test B
Numerical Solution with Domain Decomposition
0 5 1 0 1 5 2 0 2 5 3 01 0 - 1 71 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 81 0 - 71 0 - 61 0 - 5
Relat
ive Di
fferen
ce
x
ε = 10−4 [ P 1 ] ε = 10−4 [ A s y . ] ε = 10−5 [ P 1 ] ε = 10−5 [ A s y . ] ε = 10−6 [ P 1 ] ε = 10−6 [ A s y . ]
Introduction Elements of Methodology Numerical Results Conclusion
Summary
Outline
1 IntroductionBackground
2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition
3 Numerical ResultsTest ATest B
4 ConclusionSummary
Introduction Elements of Methodology Numerical Results Conclusion
Summary
We developed a new spatial decomposition method andapplied it to a particular transport discretization in 1D.The numerical results showed that this method has highaccuracy in approximating the transport solution computedon a given grid.The accuracy of the solution depends on the definition of adiffusion subdomain.The metrics of transport effects can be used to determineaccurately diffusion subdomains.The proposed methodology can be applied to differenttransport methods and extended to multidimensionalgeometries and multigroup transport problems.