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

One University Avenue, Lowell, MA 01854 Dean_Wang@uml.edu

INTRODUCTION

The coarse-mesh nonlinear diffusion acceleration (CM-NDA) technique, or commonly known as coarse-mesh finite difference (CMFD), is one of the most widely used acceleration methods for numerical transport calculations. Although CM-NDA is very effective to reduce the iteration number of neutron transport sweep, it can sometimes become unstable, in particularly when the coarse mesh size becomes large. Over the last two decades a number of ad hoc techniques have been developed to stabilize the CM-NDA scheme [1, 2, 3].

In this paper, we present a new scheme, LR-NDA, which is based on CM-NDA with local refinement. Within the CM-NDA framework, the LR-NDA scheme incorporates solving an additional local boundary-value problem (BVP) of the neutron flux on each individual coarse mesh cell, yielding a finer flux profile than CM-NDA. Our assessments show that this new scheme can significantly improve the effectiveness of CM-NDA, and more importantly it is very stable even for extremely high optical thickness. A great feature of LR-NDA is that it is fully adaptive, which means that it can be easily applied to any local region of the problem domain, and local refinement can be performed using any numerical method (e.g., finite difference, finite element, or spectral element, etc.).

FORMULATION AND ALGORITHM OF LR-NDA

In this study, we focus on the neutron transport 𝑘-eigenvalue problem. The multigroup neutron transport equation with isotropic scattering can be expressed as

𝛀 ⋅ ∇𝜓! 𝑟,𝛀 + Σ!,!𝜓! 𝑟,𝛀 = !

!!Σ!,!!→!𝜙!! 𝑟

!!!!! + !!

!!""𝜈Σ!,!!𝜙!! 𝑟

!!!!! ,(1)

where 𝜓 = the angular flux, 𝜙 = 𝜓𝑑Ω!! , the scalar flux, 𝛀 = the neutron direction, 𝑟 = the spatial position, Σ!,!, Σ!,!!→!, Σ!,!! the total, scattering, and fission cross section, 𝜒 = the fission neutron spectrum, ν = the mean number of neutrons produced per fission, 𝑘!"" = the dominant eigenvalue,

𝑔,𝑔! = 1, 2,… ,𝐺, the neutron energy group index, and 𝐺 is the total number of groups. The above equation can be recast in a simple operator

form [3]: ℒΨ = !

!!𝑆 + !

!!""ℱ Φ, (2)

where ℒ = 𝛀 ⋅ ∇ + Σ!, the transport operator, 𝑆 = Σ!, the scattering operator, ℱ = 𝜒𝜈Σ!, the fission operator, Ψ = [𝜓!,𝜓!,… ,𝜓!], the angular flux vector, Φ = [𝜙!,𝜙!,… ,𝜙!], the scalar flux vector.

For general eigenvalue problems, the dominant eigenpair 𝑘!"",Ψ can be calculated using the power iteration method. However, it is well known that the convergence of the power iteration for the neutron transport equation is very slow, which is mainly determined by two parameters: the scattering ratio, 𝑐 = Σ!/Σ!, and the dominance ratio, 𝜌 = 𝜆!/𝜆! (𝜆! and 𝜆! are the first and second eigenvalues), respectively. When these two ratios are close to 1, the convergence will deteriorate greatly.

Over the past two decades, many techniques have been developed to accelerate the convergence of the neutron transport 𝑘-eigenvalue calculations. A recent study shows that the low-order NDA-based acceleration schemes are always superior to other techniques, e.g., Jacobian-Free Newton-Krylov (JFNK) and Nonlinear Krylov Acceleration (NKA) [5]. Although the fine-mesh NDA (FM-NDA) method is a more stable and effective acceleration scheme compared with CM-NDA, it requires much more computational cost, in particular for large problems. Therefore, CM-NDA is a more favorable acceleration method and has been adopted by many transport application codes. A problem with CM-NDA is that it will become ineffective and unstable when the optical thickness (Σ!Δ𝑥) increases above 1.0. In practice, some stabilization has to be used in combination with CM-NDA.

Most recently, we developed a new scheme, LR-NDA, which can achieve better acceleration and stability than CM-NDA. The key idea is that an additional local flux refinement is conducted on each coarse mesh cell to obtain a more accurate flux shape than CM-NDA, and this updated fine flux profile can effectively accelerate the neutron transport solution. The detailed LR-NDA

algorithm for the neutron transport 𝑘-eigenvlaue problem is illustrated in Fig. 1.

Fig. 1. LR-NDA algorithm flowchart.

Essentially, LR-NDA is a multi-level approach, i.e., a local refinement level is added on top of the original two-level Transport/CM-NDA algorithm. When the CM-NDA calculation is done, a local refinement is carried out on each coarse mesh cell by solving a nonlinear neutron diffusion equation, which has exactly the same form as the CM-NDA formulation:

∇ ⋅ − !!!!

∇ + 𝐷! Φ + Σ! − Σ! Φ = !!!""! 𝐹Φ, (3)

where Φ = the local flux,

𝐷! =!!!!!,!! !

!!!∇!!,!

!!,!, the drift coefficient.

For the local refinement calculation, an example discretization using the finite difference method is shown in Fig. 2. The unknown scalar flux is defined at the interfaces of the fine cells within a coarse mesh cell, Φ!!!/!, 𝑖 = 1, 2,… ,𝑃 − 1, where 𝑃 denotes the cell index. Φ! , and Φ! , are the coarse mesh left and right edge boundary values, which will be defined later on. The drift coefficient at each cell center is calculated from the neutron transport results.

Fig. 2. LR-NDA mesh structure.

In LR-NDA, there are three levels of mesh structures. The transport equation is solved on the fine mesh, and the CM-NDA is solved on the coarse mesh. Local refinement is carried out on the local mesh. Note that other numerical discretization methods can be used for solving the local diffusion equation too.

The boundary conditions (BCs) for the local BVP are obtained by weighting the transport flux values at the coarse mesh edges with the coarse mesh flux ratio between the CM-NDA and transport results:

Φ!"#$ =!!

!!"!,!

!!"!,!

!+ !!"

!,!

!!"!,!

!Φ!"#$!,! , (4)

where Φ!"#$ = the flux BCs, e.g., Φ! and Φ! in Fig. 2, Φ!"#$!,! = the transport flux at the coarse mesh

boundary, Φ!"!,! = the CM-NDA piecewise constant flux,

Φ!"!,! = the transport average flux on the coarse mesh.

It was found in our study that the boundary conditions play a crucial role in the performance of the LR-NDA scheme. The above treatment of BCs is superior to other treatments.

After the local refinement calculation is done for each coarse mesh cell, the resulting fine scalar flux is sent back to the neutron transport sweep for the next iteration calculation.

NUMERICAL RESULTS

The test problem considered in this study is a 25-cm slab with reflective boundaries. The monoenergetic neutron transport 𝑘-eigenvalue equation was discretized using the finite difference method with the S10 Gauss-Legendre quadrature set, and solved with the sweeping

method. The model specifications are summarized in Table 1.

Table 1. Test problem specifications.

Σ! 𝑐𝑚!!

Σ! 𝑐𝑚!!

𝜈Σ! 𝑐𝑚!!

Δ𝑥!" 𝑐𝑚

Δ𝑥!" 𝑐𝑚

0.1 - 40 𝑐Σ! 1 − 𝑐 Σ! 0.1 1.0 𝑐 = Σ!/Σ!, the scattering ratio

The reflective boundary conditions are used for both left and right surfaces of the slab. The problem domain is uniformly divided into 25 coarse mesh cells with the size Δ𝑥!" = 1.0 cm, and each coarse mesh cell contains 10 uniform fine mesh cells with the fine mesh Δ𝑥!" =0.1 cm. The S10 transport equation was discretized and solved on the fine mesh, while CM-NDA were discretized and solved on the coarse mesh. Local refinement was carried out on the local mesh, which in this problem has the same number of uniform cells as the fine mesh in each coarse mesh. For comparison, the FM-NDA results were also given, which were solved on the same fine mesh as the S10 transport solution. It should be noted that no stabilization was used for the implementation of all the three NDA schemes in this study.

In order to investigate the convergence sensitivity of the three acceleration schemes to different optical thickness values (𝜏 = Σ!Δ𝑥!"), the total cross section (Σ!) was varied from 0.1 to 40 cm-1. For the results reported in this paper, the scattering ratio, 𝑐 = Σ!/Σ! , was set at 0.8. The 𝜈Σ! values were always equal to 0.2Σ!. The convergence rate of Keff as a function of the neutron transport sweep number is plotted in Figs. 3a through 3f for the coarse mesh optical thickness at 0.1, 1.0, 1.5, 2, 15, and 40, respectively.

Fig. 3a. Keff convergence for τ = 0.1.

Fig. 3b. Keff convergence for τ = 1.0.

Fig. 3c. Keff convergence for τ = 1.5.

Fig. 3d. Keff convergence for τ = 2.0.

1.0E-16

1.0E-13

1.0E-10

1.0E-07

1.0E-04

1.0E-01

1 10 100

KeffRe

lta)v

eError

TransportSweep#

CoarseMeshOp)calThickness:0.1

CM-NDAFM-NDALR-NDA

1.0E-16

1.0E-13

1.0E-10

1.0E-07

1.0E-04

1.0E-01

1 10 100

KeffRe

lta)v

eError

TransportSweep#

CoarseMeshOp)calThickness:1.0

CM-NDAFM-NDALR-NDA

1.0E-16

1.0E-13

1.0E-10

1.0E-07

1.0E-04

1.0E-01

1 10 100

KeffRe

lta)v

eError

TransportSweep#

CoarseMeshOp)calThickness:1.5

CM-NDAFM-NDALR-NDA

1.0E-16

1.0E-13

1.0E-10

1.0E-07

1.0E-04

1.0E-01

1 10 100

KeffRe

lta)v

eError

TransportSweep#

CoarseMeshOp)calThickness:2.0

CM-NDAFM-NDALR-NDA

Fig. 3e. Keff convergence for τ = 15.

Fig. 3f. Keff convergence for τ = 40.

The following observations can be drawn from the 1D results: • CM-NDA is only effective for the coarse mesh

optical thickness less than 1. It becomes unstable and fails to converge when the thickness is larger than 2.

• Overall both LR-NDA and FM-NDA perform better than CM-NDA. The acceleration performance of LR-NDA is very comparable with FM-NDA before the optical thickness reaches 10.

• When the optical thickness is larger than 15, FM-NDA starts to loose effectiveness and becomes unstable at 20. It is interesting to notice that CM-NDA and FM-NDA will be more stable if the exact reflective boundary conditions can be preserved during each iteration. However, it is not the case for the sweeping method.

• LR-NDA is always stable and effective even for very large optical thickness and the scattering ratio close to 1. For example, we successfully tested it with a very challenging problem, which has an alternating optically thick (Σ!Δ𝑥!" = 1000) and thin (0.001) configuration.

CONCLUSIONS

This paper presents our latest work on the development and assessment of the nonlinear LR-NDA acceleration scheme for neutron transport calculations. LR-NDA incorporates a local refinement solution on the coarse mesh structure based on the Transport/CM-NDA algorithm, i.e., the new scheme calculates a fine flux profile on the coarse mesh rather than a piecewise constant. Such local refinement has greatly improved effectiveness and stability of CM-NDA. It should be pointed out that computational cost incurred by local refinement is insignificant because of its compactness and efficient parallel implementation.

The assessment of the developed LR-NDA was performed with the 1D transport 𝑘-eigenvalue problem. The results show that LR-NDA is superior to both FM-NDA and CM-NDA in terms of effectiveness and stability for accelerating the neutron transport calculations. Our testing also shows that LR-NDA is unconditionally stable in fact. Although the assessment was conducted on the 1D problem, it would be very straightforward to apply the new scheme for the 2D/3D neutron transport problems.

A final remark is that LR-NDA is a truly local adaptive method, which means that the scheme can be easily implemented for any region of the problem domain. In addition, the mesh structure for local refinement can be varied pin-by-pin or assembly-by-assembly. This novel feature would be very useful for large-scale 2D/3D neutron transport problems.

ACKNOWLEDGEMENTS

The research was supported by the Department of Energy Nuclear Energy University programs.

REFERENCES

1. M. Jarrett et al., “Stabilization Methods for CMFD Acceleration,” Proc. M&C, American Nuclear Society, Nashville, TN, April 19-23, (2015).

2. K. P. Keady and E. W. Larsen, “Stabilization of Sn K-Eigenvalue Iterations Using CMFD Acceleration,” Proc. M&C, American Nuclear Society, Nashville, TN, April 19-23, (2015).

3. N. Z., 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).

4. H. Park et al., “Nonlinear Acceleration of Transport Criticality Problems,” Nucl. Sci. Eng., 172, (2012).

5. J. Willert, et al., “A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem,” J. Comput. Phys., 274, (2014).

1.0E-14

1.0E-11

1.0E-08

1.0E-05

1.0E-02

1 10 100

KeffRe

lta)v

eError

TransportSweep#

CoarseMeshOp)calThickness:15

CM-NDAFM-NDALR-NDA

1.0E-16

1.0E-13

1.0E-10

1.0E-07

1.0E-04

1.0E-01

1 10 100

KeffRe

lta)v

eError

TransportSweep#

CoarseMeshOp)calThickness:40

CM-NDAFM-NDALR-NDA