Methodology for Decomposition into Transport and Diffusive ...

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


Outline

1 IntroductionBackground

2 Elements of MethodologyMetrics of Transport EffectsDomain Decomposition

3 Numerical ResultsTest ATest B

4 ConclusionSummary


Background

Outline




4 ConclusionSummary


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.


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)


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.


Metrics of Transport Effects

Outline




4 ConclusionSummary



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 .



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 .



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

∣∣∣∣ .


Domain Decomposition

Outline




4 ConclusionSummary



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 .



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) .



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.



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.


Test A

Outline




4 ConclusionSummary


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)


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


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


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


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 . ]


Test B

Outline




4 ConclusionSummary


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


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


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 . ]


Summary

Outline




4 ConclusionSummary


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.

Methodology for Decomposition into Transport and Diffusive ...

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