A Coarse-Mesh Nonlinear Diffusion Acceleration Scheme with Local - - PowerPoint PPT Presentation

A Coarse-Mesh Nonlinear Diffusion Acceleration Scheme with Local Refinement for Neutron Transport Calculations

2016 ANS Winter Meeting, Las Vegas, NV

Dean Wang, Sicong Xiao, and Ryan Magruder

University of Massachusetts Lowell

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:

• 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

25-cm slab with reflective boundaries

Numerical results – 1D problem

Numerical results – 1D problem

Numerical results – 2D problem

1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-02 1.0E+00 1 10 100 Keff Reltative Error Transport Sweep #

Coarse Mesh Optical Thickness: 15

CM-NDA FM-NDA LR-NDA

1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-02 1.0E+00 1 10 100 Keff Reltative Error Transport Sweep #

Coarse Mesh Optical Thickness: 2.0

CM-NDA FM-NDA LR-NDA

Numerical results – local adaptivity

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 Reltative Error Transport Sweep #

LR-NDA Local Adaptivity

3 3x3 5x5

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

1cm 1cm

Summary

• LR-NDA incorporates a local refinement

solution on the coarse mesh structure based

• n 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

Stabilize

Modified CM-NDA Algorithm

100 101 102 103

Transport Sweep #

10-10 10-8 10-6 10-4 10-2

Flux Residual Acceleration Schemes for S12 on 2D Optical Thickness = 5

FM-NDA LR-NDA Modified CM-NDA CM-NDA with underrelaxation of 0.3

100 101 102 103

Transport Sweep #

10-10 10-8 10-6 10-4 10-2 100

Flux Residual Acceleration Schemes for S12 on 2D Optical Thickness = 1

FM-NDA LR-NDA Modified CM-NDA CM-NDA

100 101 102 103

Transport Sweep #

10-10 10-8 10-6 10-4 10-2

Flux Residual Acceleration Schemes for S12 on 2D Optical Thickness = 10

FM-NDA LR-NDA Modified CM-NDA CM-NDA with underrelaxation of 0.1

100 101 102 103

Transport Sweep #

10-10 10-8 10-6 10-4 10-2 100

Flux Residual Acceleration Schemes for S12 on 2D Optical Thickness = 15

FM-NDA LR-NDA Modified CM-NDA CM-NDA with underrelaxation of 0.1

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.