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 2/16
CMFD • Very effective to accelerate neutron transport iteration, but • Degrades and even fails when the problem thickness becomes large. • Stabilization needed. 3/16
Stabilization Techniques Ø Multiple transport sweeps Ø Underrelaxation 𝐸 $%&/(∗ = 𝜄! 𝐸 $%&/( + (1 − 𝜄)! • ! 𝑬 underrelaxation: ! 𝐸 $1&/( 89: • Flux update with underrelaxation: 𝜚 $%& = 𝜚 $%&/( 1 + 𝜄 3 4567 < 89:/= − 1 ; & Ø Artificial Diffusion: 𝐸 = >? @ + 𝜄Δ & • pCMFD (Cho et al., 2003): It is algebraically “equivalent” to 𝜄 = B • odCMFD (Larsen, 2003; Zhu et al., 2016): 𝜄 = 𝜄(Σ D Δ) 4/16
lpCMFD 1D: 𝒚 − 𝒚 𝒋1 ⁄ 𝟐 𝟑 𝜺𝝔 𝒋 (𝒚) = 𝜺𝚾 𝒋1 ⁄ 𝟐 𝟑 + (𝜺𝚾 𝒋% ⁄ 𝟐 𝟑 − 𝜺𝚾 𝒋1 ⁄ M 𝟐 𝟑 𝒚 𝒋% ⁄ 𝟐 𝟑 − 𝒚 𝒋1 ⁄ 𝟐 𝟑 & ( = 1 & ( + Φ I $%& − L $%& − L $% ⁄ $% ⁄ & ( 𝜀Φ I1 ⁄ Φ I1& 𝜚 I1& 𝜚 I 2 𝜀Φ GH = & ( = 1 & ( + Φ I%& $%& − L $%& − L $% ⁄ $% ⁄ & ( 𝜀Φ I% ⁄ Φ I 𝜚 I 𝜚 I%& 2 2D: 5/16
Comparison of Flux Correction 6/16
Fourier Analysis C=0.6 C=0.8 C=0.9 C=0.99 7/16
1D Iron-Water Test Problem Transport: S 10 Gauss-Legendre, DD Fine Mesh: 0.1 cm Coarse Mesh: 1 cm 8/16
2D Fixed-Source Problem (Wang and Xiao 2018) Transport: S 12 Gauss-Legendre, DD Fine Mesh: 0.1 cm Coarse Mesh: 1 cm 9/16
2D K-Eigenvalue Problem (Wang and Xiao 2018) Transport: S 12 Gauss-Legendre, DD Fine Mesh: 0.1 cm Coarse Mesh: 1 cm 10/16
An Extension: LR-NDA (Wang 2016; Xiao, 2017, 2018) Fine Mesh: Coarse Mesh: Local Mesh: Local Refinement BVP: 𝛂 V − 𝟐 𝟐 𝟑 𝝔 𝒎𝒑𝒅𝒃𝒎 𝒎%𝟐 + (𝜯 𝒖 − 𝜯 𝒕 )𝝔 𝒎𝒑𝒅𝒃𝒎 𝒎% ⁄ 𝛂 + ! 𝒎%𝟐 𝑬 𝑮𝑵 = 𝑹 𝟒𝜯 𝒖 2 ( Φ $%& Φ $%& $%& = 1 $% ⁄ & ( BCs: 𝜚 GH b + b ) 𝜚 c,e L L 𝜚 $% ⁄ 𝜚 $% ⁄ & ( & ( % 1 11/16
LR-NDA 2D K-eigenvalue Problem S12 Solution Accelerated with LR-NDA 3 3 1.0E+00 3 x 3 2.5 1.0E-01 5 x 5 Normalized Scalar Flux 2 Keff Relative Error 1.0E-02 1.5 1.0E-03 1.0E-04 1 1.0E-05 0.5 1.0E-06 0 60 1.0E-07 40 1.0E-08 20 1 10 100 50 45 Y 40 Transport Sweep # 35 30 0 25 20 15 10 5 0 X 12/16
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. 13/16
Acknowledgements • US DOE and NRC for support 14/16
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) 15/16
Thank You! More info: http://faculty.uml.edu/Dean_Wang 16/16
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