A Coarse-Mesh Nonlinear Diffusion Acceleration Scheme with Local Refinement for Neutron Transport Calculations Dean Wang, Sicong Xiao, and Ryan Magruder University of Massachusetts Lowell 2016 ANS Winter Meeting, Las Vegas, NV
Background • CMFD is one of the most widely used acceleration methods for numerical neutron transport solutions – Very effective to reduce the iteration number of neutron transport sweep, but – Degrades and even fails when the problem thickness becomes large • Current ad hoc fixups can improve its stability at high thickness but not much effectiveness, e.g., underrelaxation or artificial diffusion
A new scheme: LR-NDA • We developed a new stabilization method for CMFD. • This method employs a local refinement calculation on coarse mesh cells where the thickness is high. • It can greatly improve the effectiveness of CMFD.
LR-NDA Local Boundary Value Problem: BCs:
Numerical results – 1D problem 25-cm slab with reflective boundaries • Mesh – Fine Mesh: 0.1 cm – Coarse Mesh: 1.0 cm • Monoenergetic neutron transport 𝑙 -eigenvalue problem with isotropic scattering – Diamond difference method – S10 Gauss-Legendre quadrature • Nonlinear diffusion acceleration schemes – FM-NDA – CM-NDA – LR-NDA
Numerical results – 1D problem
Numerical results – 1D problem
Numerical results – 2D problem Coarse Mesh Optical Thickness: 2.0 1.0E+00 CM-NDA 1.0E-02 Keff Reltative Error FM-NDA LR-NDA 1.0E-04 1.0E-06 1.0E-08 1.0E-10 Coarse Mesh Optical Thickness: 15 1 10 100 Transport Sweep # 1.0E+00 CM-NDA FM-NDA 1.0E-02 Keff Reltative Error LR-NDA 1.0E-04 1.0E-06 1.0E-08 1.0E-10 1 10 100 Transport Sweep #
Numerical results – local adaptivity 1cm 1cm 2D K-eigenvalue Problem S12 Solution Accelerated with LR-NDA LR-NDA Local Adaptivity 3 1.0E+00 2.5 3 1.0E-01 3x3 Normalized Scalar Flux 2 5x5 1.0E-02 Keff Reltative Error 1.5 1.0E-03 1 1.0E-04 0.5 1.0E-05 0 1.0E-06 60 1.0E-07 40 1.0E-08 20 1 10 100 50 45 Y 40 35 30 0 25 20 Transport Sweep # 15 10 5 0 X
Summary • LR-NDA incorporates a local refinement solution on the coarse mesh structure based on the CMFD algorithm. • Very effective for small and high thickness • It is a truly local adaptive method since it can be easily implemented for any region of the problem domain.
General Remarks • Consistency: The nonlinear diffusion accelerated Sn solution should converge to the unaccelerated Sn solution – The drift coefficient should be calculated in a consistent way with the diffusion discretization: • It is interesting to notice that both CM-NDA and FM- NDA become more stable if the reflective boundary conditions are exactly imposed during each transport calculation. However, it is not the case for the sweeping method.
What is going on now • LR-NDA – Convergence and stability analysis – More benchmarks: C5G7, etc. • A new prolongation method for CMFD • Coarse-mesh diffusion synthetic methods: CM-DSA • Stay tuned http://faculty.uml.edu/Dean_Wang/research.htm
Acknowledgement • This work is funded by the DOE NEUP program • We are collaborating with – Yulong Xing, UC Riverside – Thomas Downar and Yulin Xu, Umichagan – Emily Shemon, ANL
Thank You! 14
Modified CM-NDA Algorithm Stabilize
Acceleration Schemes for S12 on 2D Acceleration Schemes for S12 on 2D Optical Thickness = 1 Optical Thickness = 5 10 0 FM-NDA FM-NDA LR-NDA LR-NDA Modified CM-NDA Modified CM-NDA 10 -2 CM-NDA CM-NDA with underrelaxation of 0.3 10 -2 10 -4 Flux Residual 10 -4 Flux Residual 10 -6 10 -6 10 -8 10 -8 10 -10 10 -10 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 Transport Sweep # Transport Sweep # Acceleration Schemes for S12 on 2D Acceleration Schemes for S12 on 2D Optical Thickness = 10 Optical Thickness = 15 FM-NDA FM-NDA LR-NDA 10 0 LR-NDA 10 -2 Modified CM-NDA Modified CM-NDA CM-NDA with underrelaxation of 0.1 CM-NDA with underrelaxation of 0.1 10 -2 10 -4 Flux Residual Flux Residual 10 -4 10 -6 10 -6 10 -8 10 -8 10 -10 10 -10 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 Transport Sweep # Transport Sweep #
Remarks • The new prolongation method can effectively stabilize CM-NDA (CMFD) • Advantages of this modified CM-NDA method: – Does NOT require any relaxation parameter – Very stable and robust even for very high optical thickness – Can be easily implemented with CMFD in any code.
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