St Stab abiliz ilizin ing g CMFD D wit ith Lin Linear ar - - PowerPoint PPT Presentation

St Stab abiliz ilizin ing g CMFD D wit ith Lin Linear ar Prolongation: : lpC lpCMFD

Dean Wang, Sicong Xiao (University of Massachusetts Lowell) Yulin Xu, Thomas Downar (University of Michigan) Emily Shemon (Argonne National Laboratory) Yulong Xing (Ohio State University)

PHYSOR 2018, April 22 – 26, Cancun, Mexico

Outline

• CMFD
• Current stabilization methods
• lpCMFD
• Summary and Remarks

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CMFD

• Very effective to accelerate

neutron transport iteration, but

• Degrades and even fails when

the problem thickness becomes large.

• Stabilization needed.

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Stabilization Techniques

ØMultiple transport sweeps ØUnderrelaxation

• !

𝑬 underrelaxation: ! 𝐸$%&/(∗ = 𝜄! 𝐸$%&/( + (1 − 𝜄)! 𝐸$1&/(

• Flux update with underrelaxation: 𝜚$%& = 𝜚$%&/( 1 + 𝜄

34567

89:

; <89:/= − 1

ØArtificial Diffusion: 𝐸 =

& >?@ + 𝜄Δ

• pCMFD (Cho et al., 2003): It is algebraically “equivalent” to 𝜄 =

& B

• odCMFD (Larsen, 2003; Zhu et al., 2016): 𝜄 = 𝜄(ΣDΔ)

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lpCMFD

𝜀ΦGH = 𝜀ΦI1 ⁄

& ( = 1

2 ΦI1&

$%& − L

𝜚I1&

$% ⁄ & ( + ΦI $%& − L

𝜚I

$% ⁄ & (

𝜀ΦI% ⁄

& ( = 1

2 ΦI

$%& − L

𝜚I

$% ⁄ & ( + ΦI%& $%& − L

𝜚I%&

$% ⁄ & (

M 𝜺𝝔𝒋(𝒚) = 𝜺𝚾𝒋1 ⁄

𝟐 𝟑 +

𝒚 − 𝒚𝒋1 ⁄

𝟐 𝟑

𝒚𝒋% ⁄

𝟐 𝟑 − 𝒚𝒋1 ⁄ 𝟐 𝟑

(𝜺𝚾𝒋% ⁄

𝟐 𝟑 − 𝜺𝚾𝒋1 ⁄ 𝟐 𝟑

2D: 1D:

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Comparison of Flux Correction

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Fourier Analysis

C=0.6 C=0.8 C=0.9 C=0.99

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1D Iron-Water Test Problem

Transport: S10 Gauss-Legendre, DD Fine Mesh: 0.1 cm Coarse Mesh: 1 cm 8/16

2D Fixed-Source Problem

(Wang and Xiao 2018)

Transport: S12 Gauss-Legendre, DD Fine Mesh: 0.1 cm Coarse Mesh: 1 cm 9/16

2D K-Eigenvalue Problem

(Wang and Xiao 2018)

Transport: S12 Gauss-Legendre, DD Fine Mesh: 0.1 cm Coarse Mesh: 1 cm 10/16

An Extension: LR-NDA

(Wang 2016; Xiao, 2017, 2018) Local Refinement BVP: BCs:

𝛂 V − 𝟐 𝟒𝜯𝒖 𝛂 + ! 𝑬𝑮𝑵

𝒎% ⁄ 𝟐 𝟑 𝝔𝒎𝒑𝒅𝒃𝒎 𝒎%𝟐 + (𝜯𝒖 − 𝜯𝒕)𝝔𝒎𝒑𝒅𝒃𝒎 𝒎%𝟐

= 𝑹

b 𝜚GH

$%& = 1

2 ( Φ$%& L 𝜚$% ⁄

& ( %

+ b Φ$%& L 𝜚$% ⁄

& ( 1

) 𝜚c,e

$% ⁄ & (

Fine Mesh: Coarse Mesh: Local Mesh:

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LR-NDA

1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1 10 100 Keff Relative Error Transport Sweep # 3 3 x 3 5 x 5

0.5 60 1 1.5 2 Normalized Scalar Flux 2.5 40 3 Y 2D K-eigenvalue Problem S12 Solution Accelerated with LR-NDA 20 X 50 45 40 35 30 25 20 15 10 5

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Summary and Remarks

• The new lpCMFD scheme employs a linear prolongation for

flux update to replace the standard flux ratio based approach.

• lpCMFD is a stable and effective scheme, which performs

better than CMFD and other stabilization techniques.

• It can be easily implemented in any codes with CMFD.
• LR-NDA can be viewed as an extension of lpCMFD since it

solves a local refinement BVP to obtain a finer flux than linear interpolation.

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Acknowledgements

• US DOE and NRC for support

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References

1. Wang and Xiao, “A Linear Prolongation Approach to Stabilizing CMFD,” Nucl. Sci. Eng., 190, 1, 45 (2018). 2. Xiao et al., “A Local Adaptive Coarse-Mesh Nonlinear Diffusion Acceleration Scheme for Neutron Transport Calculations,” Nucl. Sci. Eng., 189, 3, 272 (2018). 3. Xiao et al., “Convergence Study of LR-NDA Using Fourier Analysis," Trans. AM. Nucl. Soc., 116, 2017. 4. Wang et al., “A Coarse-Mesh Nonlinear Diffusion Acceleration Scheme with Local Refinement for Neutron Transport Calculations," Trans. AM. Nucl. Soc., 115, 2016. 5. Zhu et al., “An Optimally Diffusive Coarse Mesh Finite Difference Method to Accelerate Neutron Transport Calculations,” Ann. Nucl. Energy, 95, 116 (2016) 6. Cho et al., “Partial Current- Based CMFD Acceleration of the 2D/1D Fusion Method for 3D Whole-Core Transport Calculations,” Trans. Am. Nucl. Soc., 88, 594 (2003). 7. Larsen, “Infinite Medium Solutions of the Trans- port Equation, SN Discretization Schemes, and the Diffusion Approximation,” Transp. Theory Stat. Phys., 32, 633 (2003)

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Thank You!

More info: http://faculty.uml.edu/Dean_Wang

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