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11-04820

Coarse Mesh Finite Difference in MCNP

Mitchell T. YoungForrest B. BrownBrian C. KiedrowskiWilliam R. Martin

MCNP Documentation



Mitchell T.H. Young Forrest Brown Brian KiedrowskiWilliam Martin

August 15, 2011


1 Introduction

2 CMFD Formulation

3 Implementation

4 Using CMFD in MCNP

5 Results

6 Conclusion


Introduction

Lee, et. al.[2] [3] have demonstrated the feasibility of applying aCoarse Mesh Finite Difference (CMFD) acceleration technique toaccelerate fission source distribution (FSD) convergence in montecarlo criticality calculations. Most of this work has been done in 1-and 2-D with multigroup monte carlo. In this work, a CMFD solverhas been implemented in MCNP to facilitate FSD acceleration in3-D with continuous-energy cross sections for more generalapplications. Promising results have been obtained for full-corereactor simulations.


CMFD Formulation

1 Introduction

2 CMFD Formulation

3 Implementation


5 Results

6 Conclusion


CMFD Formulation

Diffusion Equation

CMFD is based upon diffusion theory, represented (1-D) by theequation

−D∇φ(x) + Σa(x)φ(x) =1

kνΣf (x)φ(x). (1)

Fick’s law is used to represent neutron current,

J = −Ddφ

dx. (2)

Spatial discretization of the diffusion equation results in the FiniteDifference Method (FDM).


CMFD Formulation

Spatial Discretization

“Box scheme” used for discretization

Σr,i

DiΣr,i+1

Di+1

ϕ i

ϕ i+1

ϕ s ,Ji

hi

x i-1 x i x i+1

Solving for Js from both sides:

Js,l =−Di (φs − φi )

hi/2, and (3)

Js,r =−Di+1(φi+1 − φs)

hi+1/2. (4)

−Di (φs − φi )

hi/2=−Di+1(φi+1 − φs)

hi+1/2(5)

Solving for φs [1]:

φs =

Dihi

φi + Di+1

hi+1φi+1

Dihi

+ Di+1

hi+1

. (6)


CMFD Formulation

Spatial Discretization

Defining relative diffusivity, βi = Dihi

,

φs =βiφi + βi+1φi+1

βi + βi+1. (7)

Plug back into the current equation, (5) and after some goofyalgebra,

Ji = − 2βiβi+1

βi + βi+1(φi+1 − φi ). (8)

Coupling Coefficient

D̃i =2βiβi+1

βi + βi+1(9)

Our current equation for surface i is now

Ji = −D̃i (φi+1 − φi ) (10)


CMFD Formulation

CMFD Correction

The interface current from a higher-order solution is now preservedby applying following correction

Ji = −D̃i (φi+1 − φi ) + D̂i (φi + φi+1). (11)

Given the higher-order solution for φ and J, D̂ can be obtainedeasily,

D̂i =Ji + D̃i (φi+1 − φi )

φi+1 + φi. (12)


CMFD Formulation

Balance Equation

Multigroup neutron balance equation in 1D

Node i-1 Node i Node i+1

Ji-1J i

Fission source:

F ig = hiχ

ig

1

k

∑g∈G

νΣifgφi

g (13)

Scattering source:

S ig = hi

∑g ′ 6=g

Σisg ′gφi

g ′ (14)

J ig + Σi

rgφig = J i−1

g + F ig + S i

g (15)

φi−1g (−D̃ i−1

g − D̂ i−1g ) + φi

g (hi Σir,g + D̃ i−1

g + D̃ ig + D̂ i

g − D̂ i−1g ) + φi+1

g (−D̃ ig + D̂ i

g ) = S ig + F i

g

(16)


CMFD Formulation

Balance Equation

Extending balance equation to 3D

n

s

ew i

u

d

Current directionconventions:

west→ east

north→ south

top→ bottom

Most terms remain the same except:

Use volume for source terms,

four more current-based coupling terms, and

interface area must be included in node coupling.


CMFD Formulation

Balance Equation

3D Balance Equation for node i , group g

φwg Ax(−D̃w

g − D̂wg ) + φe

gAx(−D̃eg + D̂e

g ) + φigAx(D̃

wg + D̃e

g + D̂eg − D̂w

g )+

φngAy (−D̃n

g − D̂ng ) + φs

gAy (−D̃sg + D̂s

g ) + φigAy (D̃

ng + D̃s

g + D̂sg − D̂n

g )+

φugAz(−D̃u

g − D̂ug ) + φd

gAz(−D̃dg + D̂d

g ) + φigAz(D̃

ug + D̃d

g + D̂dg − D̂u

g )+

ViφigΣ

ir,g = S i

g + F ig (17)

F ig = Viχ

ig1

k

Xg∈G

νΣifgφi

g (18)

S ig = Vi

Xg′ 6=g

Σisg′gφi

g′ (19)


CMFD Formulation

Balance Equation

The previous balance equation is formed for each mesh region,forming a large system of linear equations.

Power method is applied directly to the system

Mφ = (λF + S) φ (20)

to find the eigenvalue, λ = 1/k.

The F and S matrices contain the fission and scatteringsources.

The migration matrix, M contains the LHS of Eq. (17).

φ is a vector caintaining the flux in each node/group.


Implementation

1 Introduction

2 CMFD Formulation

3 Implementation


5 Results

6 Conclusion


Implementation

Obtaining Constants

To solve the CMFD equations we need

fluxes φ,partial currents J,diffusion coefficients D,removal cross sections Σr ,scattering cross sections Σsg ′g ,fission neutron production cross section νΣf .

Scalar flux is obtained with a normal FMESH tally.

Partial currents are tallied using a modified FMESH.

Basic cross sections are obtained using tally multipliers for theinteration(s) of interest and forming the ratio

Σ =reaction tally

flux tally. (21)

Diffusion coefficients are calculated as 1/3Σt .


Implementation

Obtaining Constants

For general multigroup calculations, a full scattering matrixΣS [g ′, g ] would be needed.

A two-group calculation is used instead, such that Σs12 canbe calculated from the group 2 balance equation,

Σs12φ1 = Σa2φ2 + J+net,2 − J−net,2. (22)

Typically, Σrg ≡ Σtg − Σsgg . In the absence of Σsgg

Σr1 = Σa1 + Σs12 andΣr2 = Σa2.


Implementation

Solving the System

Nodes arranged in a “natural ordering” scheme.The Migration matrix assumes the form below.

Matrices and flux vector sorted with group-majorscheme.

Energy group is major sort index.Nodes sorted within each energy group.

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

y

x

φ =

26666666666666666666666666666666666666666666666666664

.

.

.26666666664

.

.

.

φi−1

φi

φi+1

.

.

.

37777777775g−126666666664

.

.

.

φi−1

φi

φi+1

.

.

.

37777777775g26666666664

.

.

.

φi−1

φi

φi+1

.

.

.

37777777775g+1

.

.

.

37777777777777777777777777777777777777777777777777775


Implementation

Solving the System

Outer iterations are used to update fission and scatteringsources.

Gauss-Seidel method is used to solve fixed-source,single-group system.

Four outer bands (n,s,u,d coupling) of M subtracted to LHS.Previous estimate of flux is used for these couplings.Several inner iterations are required to achieve convergencewithin outer iterations.The remaining RHS (tridiagonal matrix) represents a series ofstrips along x-axis.Each strip is solved directly using LU decomposition andforward-backward substitution.


Implementation

Solving the System


Implementation

Resampling the Fission Source

New fission source distribution (FSD) must now be given toMCNP for the next cycle.

Existing fission bank points are used.

Based on the FSD obtained from CMFD, points in someregions are drawn more preferentially than others.

The resultant fission bank is made up of points from the oldfission bank, but with the corrected FSD.


Using CMFD in MCNP

1 Introduction

2 CMFD Formulation

3 Implementation


5 Results

6 Conclusion


Using CMFD in MCNP

Modifications to the Input Deck

FMESH tallies must be added for the following:

Flux (no multiplier)Total cross sectionAbsorption cross section (capture + fission)Fission production (νΣf )Partial current

All meshes must have the same geometric properties.

Mechanics of the current tally require a “halo” of ghost cellsto obtain incoming current at the domain boundary.

The meshes must be uniform in each direction.


Using CMFD in MCNP


Example FMESH definition:

C nu-Sigma_fFMESH24:n GEOM=xyz

ORIGIN=-203.49 -203.49 -211.8IMESH=203.49 IINTS=19 $ 17 active mesh regions $JMESH=203.49 JINTS=19 $ 17 active mesh regions $KMESH=201.800 KINTS=22 $ 20 active mesh regions $EMESH=1.e-6 20.0 EINTS=1 1 $ 2-group structure $

FM24 -1.0 0 -6 -7


Using CMFD in MCNP


Table: Magic numbers and FM cards for each FMESH tally. Interactionnumbers assume continuous energy.

Interaction FMESH Number FM Card[4]

Partial current 1a NoneFlux 4 NoneTotal 14 FM -1.0 0 -1Absorption 34 FM -1.0 0 -2:-6Fission 24 FM -1.0 0 -6 -7a Any number ending in a 1 will result in a partial cur-

rent tally.


Using CMFD in MCNP


IDUM Array

The MCNP IDUM array is used to provide several inputs to theCMFD module:

1 KCODE cycle at which to implement CMFD,

2 number of inner iterations per outer iteration, and

3 the maximum number of outer iterations allowed.

IDUM [cycle] [inner] [outer]


Results

1 Introduction

2 CMFD Formulation

3 Implementation


5 Results

6 Conclusion


Results

Boxy Kord Smith Challenge

Problem

Blue = Water

Green = Core/Fuel

Red = Reactor vessel


Results


5.45

5.5

5.55

5.6

5.65

5.7

5.75

5.8

5.85

5.9

5.95

0 10 20 30 40 50 60 70 80 90 100

Sou

rce

Ent

ropy

Cycle

CMFD Cycle 10No CMFD


Results


Before CMFD

1e-08 2e-08 3e-08 4e-08 5e-08 6e-08 7e-08 8e-08 9e-08 1e-07 1.1e-07

X

Y

’plane_f2’

0 2 4 6 8 10 12 14 16 18 0

2

4

6

8

10

12

14

16

18

After CMFD

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X

Y

’plane_fs2’

0 2 4 6 8 10 12 14 16 18 0

2

4

6

8

10

12

14

16

18


Conclusion

1 Introduction

2 CMFD Formulation

3 Implementation


5 Results

6 Conclusion


Conclusion

Pure FDM shown to greatly accelerate convergence of FSD,though not perfectly accurate.

Still some issues with CMFD correction.

FDM solution tends to be close enough to significantly aidconvergence.

Discrete FSD sampling method results in over-concentratedFSD.


Conclusion

Future Work

Investigate other FSD re-sampling methods to moreaccurately reproduce CMFD solution.

Debug CMFD correction.

Improve solver stability and error-handling capabilities.

Thoroughly examine applicability of method to other problemstypes.


Conclusion

References

H. G. Joo.

Solution of one-dimensional, one-group neutron diffusion equation.NERS 561 Lecture Notes. University of Michigan., 2011.

Min-Jae Lee, Han Gyu Joo, Deokjung Lee, and Kord Smith.

Investigation of cmfd accelerated monte carlo eigenvalue calculationi with simplified low dimensionalmultigroup formulation.PHYSOR, 2010.

Min-Jae Lee, Han Gyu Joo, Deokjung Lee, and Kord Smith.

Multigroup monte carlo reactor calculation with coarse mesh finite difference formulation for real variancereduction.Joint International Conference on Supercomputing in Nuclear Applications and Monte Carlo, 2010.

X-5 Monte Carlo Team.

MCNP A General Monte Carlo N-Particle Transport Code, Version 5 Volume II: Users Guide.Los Alamos National Laboratory, LA-CP-03-0245, February 2008.


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