The Sandwich Method for Determining Source Convergence in Monte Carlo Calculation

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The Sandwich Method for Determining SourceConvergence in Monte Carlo CalculationYoshitaka NAITO a & Jinan YANG aa NAIS Co., Inc. , 416 Tokai-mura, Naka-gun , Ibaraki , 319-1112Published online: 07 Feb 2012.

To cite this article: Yoshitaka NAITO & Jinan YANG (2004) The Sandwich Method for Determining Source Convergence inMonte Carlo Calculation, Journal of Nuclear Science and Technology, 41:5, 559-568

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The Sandwich Method for Determining Source Convergence

in Monte Carlo Calculation

Yoshitaka NAITO� and Jinan YANG

NAIS Co., Inc., 416 Tokai-mura, Naka-gun, Ibaraki 319-1112

(Received October 6, 2003 and accepted in revised form March 11, 2004)

In slow source convergence problems, it is often difficult to ascertain whether the source iteration has converged ornot. In order to solve this problem, a new ‘‘sandwich method’’ has been proposed. The essence of this method is that afinally converged eigenvalue keff is approached starting from two kinds of initial source guesses which give higher andlower neutron multiplication factors. It is especially important for evaluating nuclear criticality safety to know how tochoose a biasing source to obtain an upper limit for keff . In this paper, (1) an example is shown to explain the diffi-culties in ascertaining the source convergence, (2) a method is proposed to obtain the upper and lower limit curves forkeff by biasing the initial source distribution, (3) the sandwich method is applied to four benchmark problems proposedby the source convergence group of the OECD/NEA Working Party on Nuclear Criticality Safety.

Our calculation results show that the sandwich method is an effective means to confirm source convergence in suchslow convergence problems. Appendix is prepared to support the method theoretically.

KEYWORDS: slow source convergence, Sandwich method, Monte Carlo method, benchmark problems, initialsource, source iteration, keff value

I. Introduction

In slow convergence Monte Carlo problems, it is often dif-ficult to ascertain whether the source iteration has convergedor not. Although the source distribution appears to have es-sentially reached stability after a number of batches ofsource iteration, it can, at times, remain quite different fromthe ultimate one. Thus, the keff values found at such a pointneed to be further verified. So far, in Monte Carlo calcula-tions, there is no good way to know how many skip batchesare required to obtain the correct source distribution. In orderto solve this problem, we have proposed a new ‘‘sandwichmethod’’ for cases of slow source convergence. The essenceof this method is that both upper and lower keff ’s in eachsource iteration are obtained by controlling initial sourceguesses. A finally converged eigenvalue keff should fall be-tween these two (upper and lower) curves as shown in theappendix of this report.

In this report three items are described. (1) Using a prob-lem, as an example, difficulties in ascertaining the sourceconvergence are explained. (2) A method named ‘‘sandwichmethod’’ is proposed to obtain the upper and lower limitcurves for keff by biasing the initial source distribution. (3)The ‘‘sandwich method’’ is applied to four benchmark prob-lems1,3) proposed by the source convergence group of theOECD/NEA Working Party on Nuclear Criticality Safety.The calculation results show that the ‘‘sandwich method’’is an effective means to confirm source convergence in suchslow convergence problems. The detailed theoretical explan-ation of this method is presented in Appendix of this report.

II. Difficulties in Source Convergence Determina-tion

The benchmark problem No. 2 ‘‘Irradiated Pin Cell Ar-ray’’ is exhibited here to show difficulties encountered in de-termining the source convergence. The composition of theLWR spent fuel consists of low burn-up, more reactiveend regions separated by a long, less active, high burn-uppart. The calculation geometry is shown in Figs. 1(a) and(b). Such a geometry is similar to that of two loosely coupledreactors, for which it is difficult to determine source conver-gence using the Monte Carlo method. The Case2 3 of thebenchmark problem No. 2 is used, as an example, to givea detailed explanation regarding the difficulties.

In this case, the fuel pin is divided into nine uniform zonesalong the length with a slightly asymmetrical axial profile ofburned-up fuel composition, as shown in the lower part ofFig. 2. The source distribution and keff for such an asymmet-ric burn-up rod have been calculated with the Monte Carlocode MCNP.2) Figure 2 shows that, by starting from a uni-form initial source distribution, the source distribution con-verges steadily to a non-uniform profile with a strong peakat the reactive upper end and much lower source density inthe central and lower regions, even though there is only aslight difference in composition between the upper and low-er ends.

Moreover, taking averages of keff over 10 cycles AVðnÞ¼0:1� SumðI¼ðn�1Þ�10þ1 to n�10) of keff ðIÞ, e.g., onebatch of 10 keff cycles with 10,000 particles per cycle, thedifference of adjacent keff ’s �keff¼ðAVðnþ1Þ�AVðnÞÞ=ðAVðnþ1ÞþAVðnÞÞ=2 are scanned as a function of the startcycle of each batch, which is defined as ðn�1Þ�10þ1. Therelative keff difference, �keff , obtained for Case2 3 is shownin Fig. 3. From Fig. 3, we note that the keff difference curvefalls within �1:0E-3 to 1.0E-03 area, i.e., the source distri-�Corresponding author, Tel. +81-29-270-5000, Fax. +81-29-270-

5001, E-mail: [email protected]

Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 41, No. 5, p. 559–568 (May 2004)

559

ORIGINAL PAPER

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bution appears to have essentially reached stability after 500active cycles. However, further examination of Fig. 4 showsthat keff after 500 active cycles of source iteration remains

far from the final keff after 2,000 active cycles. In otherwords, with further source iteration, keff still increases slight-ly for every active cycle. Therefore, in such a case, it is verydifficult to ascertain whether source convergence has beenattained or not.

In order to solve this problem, a method is necessary toestablish an upper limit for keff . If we could find both upperand lower limits for keff in each source iteration, then thefinal keff would fall between these two curves. The sourcedistribution then can be assumed to have converged whenthe difference between the upper and lower keff ’s becomesinsignificant.

III. A Method for Obtaining the Upper and LowerLimit Curves for keff

The Boltzmann equation for K-calculation is expressed by

� ¼ ð1=KÞD�1F� � ð1=KÞT�; ð1Þ

where D is the destruction operator, F the fission operator,� the neutron flux and K the neutron multiplication factor.

The neutron flux after the m-th iteration is expressed by

Light water

Pellet R=0.412 cm

Cladding Zircalloy−4

R=0.475 cm

Reflective

Reflective

Ref

lect

ive

Ref

lect

ive

1.33 cm

1.33cm

(a)

Vacuum

Water 30cmEnd plug 1.75cm

Fuel region−1 5cm

Vacuum

Fuel region−2 5cmFuel region−3 10cm

Fuel region−4 20cm

Fuel region−5 285.7cm



End plug 1.75cm

Fuel region−8 5cmFuel region−9 5cm

Water 30cm

(b)

Fig. 1 (a) Horizontal cross section of the problem geometry forbenchmark 2

(b) Vertical cross section of the problem geometry for bench-mark 2 (regions not to scale)

0.0E+00

5.0E−03

1.0E−02

1.5E−02

2.0E−02

2.5E−02

3.0E−02

3.5E−02

4.0E−02

0 50 100 150 200 250 300 350 400

Axial direction (cm)

Sou

rce

dist

ribut

ion

cycle 100

cycle 500

cycle 1000

Burnup

GWD/MTU 21 24 2430 30 3040 4055

Fig. 2 Cycle number effect on the source distribution

−8.E−03

−6.E−03

−4.E−03

−2.E−03

0.E+00

2.E−03

4.E−03

6.E−03

10 100 1000

Start cycles.

delta

kef

f

Fig. 3 Relative keff difference obtained from each batch of 10 keffcycles for Case2 3

560 Y. NAITO and J. YANG

JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

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�ðmÞ ¼ NðmÞXn

að0Þn ð�n=�0Þm’n;

NðmÞ ¼ �m0

Ym�1

t¼0

ð1=KðmÞÞ;ð2Þ

where �n is the eigenvalue of the eigenfunction ’n of theoperator T , and satisfies,

�0 � �1 � � � � � �N � 0:

When m is sufficiently large, Eq. (2) is approximatelyexpressed as

�ðmÞ ¼ NðmÞ að0Þ0 ’0 þ að0Þ1 ð�1=�0Þm’1

� �; ð3Þ

where að0Þn is the expansion coefficient of ’n for initialneutron flux �ð0Þ.

As shown in Appendix, under the above assumption, thatis, when neutron flux after the m-th iteration is approximatedby Eq. (3), the neutron multiplication factor KðmÞ satisfies thefollowing relation, if the first cycle neutron multiplicationfactor Kð0Þ is higher than the finally converged one �0,

�0 � Kðmþ1Þ � KðmÞ � Kð0Þ: ð4Þ

On the other hand, if Kð0Þ is less than �0, the followingrelation is satisfied,

�0 � Kðmþ1Þ � KðmÞ � Kð0Þ: ð5Þ

The relations (4) and (5) mean that if the source iterationstarts with a neutron multiplication factor higher (lower)than the final one, the neutron multiplication factor graduallydecreases (increases) with source iteration cycles and ap-proaches the eigenvalue corresponding to the fundamentalmode. This supports the ‘‘sandwich method’’. In the caseswhere the above assumption is not satisfied, that is, highermode neutron fluxes affect strongly to keff , the trend of mul-tiplication factors vs. iteration cycles dose not necessarilyshow monotonous decreases or increases but oscillatory.

It is important to find an initial source distribution whichgives a higher or lower Kð0Þ. The region where neutrons gen-

erate the most progenies will be hereafter referred to as themost important region. Concentrating the source in this re-gion will create a higher eigenvalue for the first cycle. Tofind such a region, calculations are separately performedwith locally uniform initial source distributions in suitablydefined regions. The keff ’s obtained from calculations ofone active cycle are compared. The region whose sourcegives the highest value of keff after a one-cycle calculationwill be the most important region. If the initial source is as-sumed to be concentrated only in that region, the estimatedeigenvalue has a higher value and gradually decreases withsource iterations but never becomes lower than the con-verged keff value. In other words, an upper limit curve forkeff is obtained.

On the other hand, in many cases, a lower limit curve canbe obtained by assuming a flat, uniform initial source guess.As shown in Appendix, keff value after a one-cycle calcula-tion is proportional to source weighted average importance.Uniform source weight corresponds to volume weight. Theimportance weighted by the finally converged source is usu-ally larger than the volume weighted one. That is, keff after aone-cycle calculation with uniform source is, usually, lowerthan that of finally converged keff value. With this initialsource, a lower limit curve for keff is obtained. The depend-ence of an initial keff , K

ð0Þ, on initial source distribution isdiscussed in Appendix.

The above results are valid for the case of a deterministicsource iteration method. There may be some difference be-tween deterministic methods and statistical methods suchas Monte-Carlo methods. However, the fundamental trendsof the neutron multiplication factor in source iteration calcu-lations are similar. In this report, this point has been illustrat-ed with sample problems.

IV. Benchmark Calculations3)

To verify the efficiency of the sandwich method, it hasbeen applied to the benchmark problems listed below.K-calculations are performed with two types of initial

1.036

1.038

1.04

1.042

1.044

1.046

1.048

1.05

1.052

1.054

10 100 1000

Start cycles

Kef

f

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

SD

(%)

Keff(500 cycles)

Final keff(2000 cycles)

SD(%)

Fig. 4 keff and standard deviation of each batch foe Case2 3

The Sandwich Method for Determining Source Convergence in Monte Carlo Calculation 561

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source, i.e., a uniform source and an important region sourceas defined above. keff values for source iteration cycles willgradually increase for the uniform source and will graduallydecrease for the important region source. When the differ-ence between the keff values obtained from the two sourceconditions becomes less than a chosen limiting value, thesource iteration is assumed to be converged. Thus, iterationsbefore that convergence point are to be skipped. Subsequent-ly, keff calculations will continue until the standard deviationbecomes sufficiently small.

1. OECD/NEA Source Convergence Benchmark 1:Checkerboard Storage of AssembliesThis model comprises a 24�3 light water reactor (LWR)

fuel storage rack with fuel array stored in alternate locations.The fuel array consists of �5% enriched-by-weight fuel el-ements located within fully water-flooded steel storage rackssurrounded by close-fitting full concrete reflection on threesides, water on the remaining side, and water on the topand bottom. Each fuel assembly is made up of a 15�15 lat-tice of Zr-clad UO2 rods. The calculation geometry is shownin Figs. 5(a)–(c).

In order to obtain the most important region for this prob-lem, K-calculations have been carried out using one activecycle with 100,000 particles. The initial source is then sam-pled uniformly for each fuel assembly. The resulting keff ’sare shown in Table 1. From the table, we may conclude thatthe fuel assembly in position ð1; 3Þ is the most importantregion.

Next, criticality calculations are performed with two dif-ferent initial source guesses. The uniform source is sampleduniformly within all fuel assemblies. On the other hand, theimportant region source is sampled uniformly only withinthe fuel assembly in position ð1; 3Þ. The initial guess forkeff is set to 1.0, while the number of particles per cycle isset to 20,000. The computed keff results obtained from2,000 active cycles and 0 skip cycles are shown in Fig. 6.

It can be seen that the source convergence is very slowand the keff -curves are waved. This comes from a highermode effect with statistical variation. However, both curves,upper and lower ones, seem not to step over the finally con-verged keff , which lies between the curves. Both curves arenot smooth near the starting cycles. This comes from thebad initial guess for keff .

2. OECD/NEA Source Convergence Benchmark 2:Pin-cell Array with Irradiated FuelIn order to obtain the most important region for Case2 3

of the benchmark 2, we have investigated the problem byfirst assuming that each pin-cell includes three parts: an up-per end region (upper 40 cm), a middle region and a lowerend region (lower 40 cm). K-calculations were performed

728

40 40

40

30

81

Fuel assembly location

Concrete

Water channel & water reflector

position (1,1)

position (12,2)All dimensions in cm

(a)

position (23,3)

Fuel assembly location Water channel location

27

27

15x15 latticewater moderatedcentrally located

pitch 1.4fuel radius 0.44clad radius 0.49

steel wallthickness 0.5

(b)

Concrete

Water reflector 30

360

Water reflector 30

Concrete

(c)

Fig. 5 (a) Horizontal cross section of the problem geometry forbenchmark 1

(b) Fuel assembly and water channel for benchmark 1(c) Vertical cross section of the problem geometry for

benchmark 1

Table 1 keff ’s obtained from one active cycle with 100,000 particles for benchmark 1

Position1 2 3 4 5 6 7 8 9 10 11 12ði:jÞ

3 0.87784 0.87249 0.87252 0.87252 0.87252 0.872522 0.87294 0.87320 0.87320 0.87320 0.87320 0.873201 0.86429 0.86034 0.86034 0.86034 0.86034 0.86034

Position13 14 15 16 17 18 19 20 21 22 23 24ði:jÞ

3 0.87252 0.87252 0.87252 0.87252 0.87252 0.871772 0.87320 0.87320 0.87320 0.87320 0.87324 0.871941 0.86034 0.86034 0.86034 0.86034 0.86034 0.86004



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with specific initial source guesses sampled uniformly ineach such region. The keff ’s obtained from one active cyclewith 100,000 particles are shown in Table 2. From the table,we can conclude that the upper end region is the most impor-tant region.

Moreover, considering that the upper end region includesfour burn-up regions with different atomic densities shownin Fig. 1(b), we must further investigate the problem to ob-tain the most important region among these four burn-up re-gions. K-calculations were performed with specific initialsource sampled uniformly in each region. The keff resultsare shown in Table 3. From the table, we now see that re-gion 3 is the most important region.

Criticality calculations for Case2 3 have been performedusing the sandwich method with two different assumptionsfor the most important region. First, the important regionsource is confined to uniformly sampling at the 40 cm upperend region, while the uniform source is sampled uniformlyover the entire volume of the pin pellet. The initial guessfor keff is set to 1.0; the number of particles per cycle isset to 20,000. The computed keff results obtained from

1,000 active cycles and 0 skip cycles are shown in Fig. 7.Next, we consider limiting the important region source to

the most reactive region 3 in order to compare the trend ofconvergence with different important regions. The resultsare also shown in Fig. 7. From this figure, we see that thesetwo important region cases converge to essentially the samevalue of keff .

3. OECD/NEA Source Convergence Benchmark 3:Three Thick One-dimensional Slabs4)

This benchmark problem is composed of a one-dimen-sional infinite slab geometry as shown in Fig. 8. A slab ofwater separates two fissile units. The thickness of unit 1 isfixed at 20 cm. The dimensions of thickness of unit 2 andof the water layer for the cases studied are given in Table 4.

0.874

0.876

0.878

0.88

0.882

0.884

0.886

0.888

0 500 1000 1500 2000

Keff cycles

Kef

f

Uniform source

Biasing source at location(1,3)

Fig. 6 keff vs. the keff cycles using sandwich method with two in-itial source distributions for benchmark 1

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

0 200 400 600 800 1000

Keff cycles

Kef

f

Uniform sourceBiasing source in upper end regionBiasing source in region 3

Fig. 7 Comparison of keff ’s vs. the keff cycles using differentimportant regions for Case2 3

Table 2 keff ’s obtained from one active cycle with 100,000 parti-cles for Case2 3 in benchmark 2

Defined region keff

Upper end region 1.07294Middle region 0.98192Lower end region 1.06705

Table 3 keff ’s obtained from one active cycle for the upper fourregions for Case2 3

Region No. keff

1 0.915812 1.075903 1.130784 1.08257

Table 4 Thickness of unit 1, unit 2 and water layer for the casesstudied in benchmark 3

Case No.Unit 1 Unit 2 Water(cm) (cm) (cm)

3 2 20 18 303 6 20 18 203 10 20 18 103 13 20 18 53 14 20 18 23 16 20 18 0

Unit 1 Water Unit 2

Reflective

Reflective

20 cm variable variable

Vac

uum

Vac

uum

Fig. 8 The problem geometry for benchmark 3


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In order to obtain the most important region for the abovecases, we first extend our consideration to the thickness ofthe water layer. Since the water layer is fairly thick for Cases3 2, 3 6, 3 10, and 3 13, the configuration can be consideredas a loosely coupled asymmetrical fuel system. Therefore,the thicker fissile unit 1 can be treated as the most importantregion.

On the other hand, since the water layer thickness is only2 cm for Case 3 14 and 0 cm for Case 3 16, the configurationcan’t be considered as a loosely coupled system. To obtainthe most important region for cases 3 14 and 3 16, units 1and 2 are split into 8 sub-regions as shown in Fig. 9. K-cal-culations were performed with initial source distributionsampled uniformly in each sub-region. The keff ’s obtainedfrom one active cycle with 100,000 particles are shown inTable 5. From the table, we can conclude that region 4 isthe most important region for both Case 3 14 and Case 3 16.

Meanwhile, units 1 and 2 can also be split into 8 sub-re-gions for Cases 3 2, 3 6, 3 10 and 3 13 to obtain a more pre-cisely defined important region. K-calculations were thenperformed for Case 3 6. From Table 5, we can see that re-gion 3 is the most important region for Case 3 6.

Criticality calculations for the above cases were per-formed using the sandwich method with two initial sourcedistributions. The uniform source is sampled uniformly overunit 1 and unit 2. For the important region source, i.e. bias-ing source, uniform sampling is limited to unit 1 for cases3 2, 3 6, 3 10 and 3 13, while the important region sourceis confined to uniform sampling of the most reactive re-gion 4 for cases 3 14 and 3 16. The initial guess for keff isset to 1.0, while the number of particles per cycle is set to20,000. The computed keff results obtained from 1,000 activecycles and 0 skip cycles are shown in Figs. 10–15.

For cases 3 2, 3 14 and 3 16, comparison of keff ’s vs. thekeff cycles using different biasing sources are shown in thosefigures as well.

4. OECD/NEA Source Convergence Benchmark 4:Array of Interacting SpheresIn this benchmark a lattice of 5�5�1 separated, highly

Water

Reflective

Reflective

Vac

uum

Vac

uum region1

region2

region3

region4

region5

region6

region7

region8

Fig. 9 Calculation configuration for Cases 3 14 and 3 16

Table 5 keff ’s obtained from one active cycle with 100,000 parti-cles for Cases 3 6, 3 14, and 3 16 in benchmark 3

Region No. Case 3 6 Case 3 14 Case 3 16

1 0.61012 0.62463 0.627372 0.97777 1.01671 1.023663 1.05041 1.16588 1.185904 0.88889 1.19076 1.234115 0.86106 1.17857 1.226976 1.00372 1.13601 1.161857 0.92045 0.97437 0.984028 0.57963 0.60078 0.60510

0.91

0.915

0.92

0.925

0.93

0 200 400 600 800 1000

Keff cycles

Kef

f

Biasing source in unit 1

Uniform source

Fig. 10 keff vs. the keff cycles using sandwich method with twoinitial source distributions for Case 3 2

0.91

0.915

0.92

0.925

0.93

0.935

0.94

0.945

0 200 400 600 800 1000

Keff cycles

Kef

f


Uniform source

Biasing source in region 3

Fig. 11 Comparison of keff ’s vs. the keff cycles using differentbiasing source for Case3 6

0.945

0.95

0.955

0.96

0.965

0 200 400 600 800 1000

Keff cycles

Kef

f


Uniform source

Fig. 12 keff vs. the keff cycles using sandwich method with twoinitial source distributions for Case3 10



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enriched uranium metal spheres is considered. The separat-ing material is air. The center-to-center distance betweenspheres is 80 cm. The radius of the central sphere is 10 cm,while the radius of all the other spheres is 8.71 cm.Figure 16 describes the overall geometry. This benchmarkis an adaptation from Kadotani et al. (Proc. ICNC91,Oxford, 1991).

Since the overall geometry is symmetric, we may investi-gate the spheres included in a one-eighth section of the arrayto obtain the most important region. Meanwhile, we split thecentral sphere into two parts: an inner 5 cm radius sphere andan outer 5 cm thick spherical shell. The central sphere is inposition ð3; 3Þ. Accordingly, the central point of the sphereis labeled position ð3; 3; 0Þ; the inner 5 cm radius sphere islabeled position ð3; 3; 1Þ; and the outer 5 cm thick sphericalshell is labeled position ð3; 3; 2Þ.

K-calculations were performed with initial source distri-butions sampled uniformly in each region. The keff ’s ob-tained from one active cycle with 100,000 particles areshown in Table 6. From the table, we may conclude that po-sition ð3; 3; 1Þ is the most important region, and the centralpoint ð3; 3; 0Þ is the most important point. The smallerspheres are clearly less important.

Criticality calculations for the above cases are performedusing the sandwich method with two initial source distribu-

Table 6 keff ’s obtained from one active cycle with 100,000particles for benchmark 4

Location keff

Position ð1; 1Þ 0.90366Position ð1; 2Þ 0.90759Position ð1; 3Þ 0.90754Position ð2; 2Þ 0.91252Position ð2; 3Þ 0.91318Position ð3; 3Þ 1.01032Position ð3; 3; 0Þ 1.49746Position ð3; 3; 1Þ 1.40099Position ð3; 3; 2Þ 0.95590

1.01

1.014

1.018

1.022

1.026

1.03

0 200 400 600 800 1000

Keff cycles

Kef

f


Uniform source

Fig. 13 keff vs. the keff cycles using sandwich method with twoinitial source distributions for Case3 13

1.07

1.075

1.08

1.085

1.09

1.095

1.1

1.105

1.11

0 200 400 600 800 1000Keff cycles

Kef

f

Biasing source in region 4

Uniform source



1.09

1.095

1.1

1.105

1.11

1.115

1.12

1.125

1.13

0 200 400 600 800 1000

Keff cycles

Kef

f

Biasing source in region 4Uniform sourceBiasing source in unit 1


5 x 5 x 1 spheres

Position(5,1)

Position(1,1)

80 cm

80 cm

Umetal R=8.71cm

Umetal R=10.0cm

Air

X

Y

Fig. 16 Calculation geometry for the benchmark 4


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tions. The uniform source is sampled uniformly over all the25 spheres. The important region initial source, i.e. biasingsource, is sampled uniformly over the inner 5 cm radiussphere ð3; 3; 1Þ. The initial guess for keff is set to 1.0, whilenumber of particles per cycle is set to 20,000. Finally the im-portant region source is confined at the most important pointð3; 3; 0Þ in order to compare the trend of convergence be-tween the important region and the important point. FromFig. 17, we see that there is no significant difference be-tween the keff results obtained from the important regionand the important point. Therefore, we may conclude thatboth the important region and the important point wouldbe adequate for the problem when the sandwich method isemployed.

V. Concluding Remarks

According to deterministic calculation method, a multipli-cation factor keff at each source iteration is higher (lower)than the finally converged one when iteration calculationstarts with initial source confined in a higher (lower) impor-tance region. Two curves of keff , upper and lower limitcurves, are obtained by performing source iterations twotimes and finally converged keff is in between these twocurves. Based on this characteristics of source iteration fordeterministic calculation, the ‘‘sandwich method’’ has beendeveloped and applied to ascertain convergence for slowconvergence problems in Monte Carlo calculations.

To evaluate the effectiveness of this method in MonteCarlo calculation, the sandwich method has been appliedto the four OECD/NEA benchmark problems on source con-vergence. An upper limit for keff is obtained by assuming theinitial source to be confined within the most important re-gion. The most important region can be determined by com-paring keff results from first-cycle K-calculations. As forbenchmark No. 1, the location ð1; 3Þ, i.e., top left-hand fuelassembly, is the most important region; for benchmarkNo. 2, region 3 is the most important region; for benchmarkNo. 3, region 3 is the most important region for Case3 6,while, region 4 is the most important region for Case3 14and Case3 16; and for benchmark No. 4, location ð3; 3; 1Þ,

i.e., the inner 5 cm radius sphere, is the most importantregion.

In benchmark No. 4, if the biasing source is assumed to beconfined at the central 10 cm radius sphere ð3; 3Þ, the keff ob-tained from the first cycle lies below the converged value.This means that the importance of the central sphere is lessthan the average importance of the whole system. However,the importance of the central point ð3; 3; 0Þ and the impor-tance of the central region ð3; 3; 1Þ are higher than the aver-age importance. So, if the initial biasing source is assumed tobe confined at any regions or points whose importance ishigher than the average one, a higher keff can be obtainedfrom the first cycle calculation.

Every lower limit keff curves of 4 benchmark problems areobtained by assuming the initial source to be uniform. Thisshows that a uniform initial source usually leads to a lowerkeff in each iteration cycle.

The calculation results obtained for the above benchmarkproblems show that the ‘‘Sandwich Method’’ is an effectivemeans for criticality safety evaluation.

References

1) M. Takano, H. Okuno, OECD/NEA Burnup Credit CriticalityBenchmark—Result of Phase IIA, NEA/NSC/DOC (96) 01,JAERI Research 96-003, (1996).

2) J. Briesmeister, Ed., MCNP—A General Monte CarloN-Particle Transport Code, Version 4B, LA-12625-M, Rev. 2,Los Alamos National Lab. (1997).

3) R. N. Blomquist, A. Nouri, ‘‘The OECD/NEA source conver-gence benchmark program,’’ Trans. Am. Nucl. Soc., 87, (2002).

4) T. Yamamoto, T. Nakamura, Y. Miyoshi, ‘‘Fission sourceconvergence of Monte Carlo criticality calculations in weaklycoupled fissile arrays,’’ J. Nucl. Sci. Technol., 37, 41–52 (2002).

Appendix

Initial Source Distribution Dependence of Neutron Mul-tiplication Factor in Each Source Iteration Cycle

The Boltzmann equation applied to Monte CarloK-calculation is simply expressed by

D��1

KF� ¼ 0; ðA1Þ

where D is the destruction operator, F the fission operator, �the neutron flux and K the neutron multiplication factor,which is obtained from

K ¼

Z’�0F�d�Z

’�0D�d�

ðA2Þ

where ’�0 is the adjoint neutron flux for a fundamental mode

proportional to neutron importance.Equation (A1) can be rewritten in the following form:

� ¼1

KD�1F� �

1

KT�: ðA3Þ

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

0 200 400 600 800 1000

Keff cycles

Kef

f

Uniform sourceBiasing source at the center pointBiasing source at location (3 3 1)Biasing source at location (3 3)

Fig. 17 Comparison of keff ’s vs. the keff cycles using differentbiasing source for benchmark 4



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In the iteration method, the neutron flux after the m-thiteration can be expressed by

�ðmÞ ¼1

Kðm�1Þ T�ðm�1Þ

¼1

Kðm�1Þ �1

Kðm�2Þ T2�ðm�2Þ ¼ � � �

¼Ym�1

t¼0

1

KðtÞ

!Tm�ð0Þ: ðA4Þ

The initial neutron flux �ð0Þ is expanded in terms of eigen-functions ’n corresponding to the eigenvalues �n of theoperator T ,

�ð0Þ ¼Xn

að0Þn ’n

�0 � �1 � � � � � �N � 0: ðA5Þ

Then the neutron flux after the m-th iteration is expressed by

�ðmÞ ¼ NðmÞXn

að0Þn

�n

�0

� �m

’n; ðA6Þ

where

NðmÞ ¼ �m0

Ym�1

t¼0

1

KðtÞ

� �: ðA7Þ

When m is sufficiently large, Eq. (A6) becomes

�ðmÞ ¼ NðmÞ að0Þ0 ’0 þ að0Þ1

�1

�0

� �m

’1

� �: ðA8Þ

Because the higher mode að0Þ1 ð�1=�0Þm’1 decreases with in-creasing m, the neutron flux �ðmÞ and the neutron multiplica-tion factor KðmÞ approach the fundamental mode ’0 and thecorresponding eigenvalue �0, respectively.

The neutron flux �ðmÞ is expanded with the same eigen-functions as in Eq. (A5)

�ðmÞ ¼Xn

aðmÞn ’n ¼ aðmÞ0 ’0 þXn6¼0

aðmÞn ’n: ðA9Þ

Equations (A8) and (A9) lead to

aðmÞ0 ¼ NðmÞað0Þ0 and aðmÞ1 ¼ NðmÞað0Þ1 ð�1=�0Þm: ðA10Þ

The multiplication factor KðmÞ is obtained as

KðmÞ ¼Z

’�0D

�1F�ðmÞd� ¼ aðmÞ0 �0; ðA11Þ

where ’�0 is the adjoint function of ’0.

At each iteration step, the amount of neutron sourcegeneration or destruction is normalized to a value such as,for example,Z

D�ðmÞd� ¼Z

DaðmÞ0 ’0d� þXn¼1

ZDaðmÞn ’nd�

¼ aðmÞ0

ZD’0d� þ

Xn¼1

aðmÞn

ZD’nd� ¼

ZD’0d�:

ðA12ÞHereafter, neutron flux terms of higher than 2nd order are

assumed to be negligibly small. Then, Eq. (A12) is rewrittenas follows:

aðmÞ0 þ aðmÞ1

ZD’1d�

� ZD’0d�

� �¼ 1:0: ðA13Þ

That is,

aðmÞ0 ¼ 1� aðmÞ1

ZD’1d�

� ZD’0d�

� �� 1� AðmÞ;

ðA14Þ

where

AðmÞ ¼ NðmÞað0Þ1 ð�1=�0ÞðmÞ

�Z

D’1d�

� ZD’0d�

� �:

ðA15Þ

Under the above assumptions, the following relations areobtained.

In the case of Kð0Þ��0, from Eq. (A11), Kð0Þ¼að0Þ0 �0��0,that is,

að0Þ0 ¼ 1� Að0Þ � 1:0: ðA16Þ

This means Að0Þ must be negative and the sign ofað0Þ1 ð

RD’1d�=D’0d�Þ is also negative. That is, from

Eq. (A15)

AðmÞ � 0; ðA17Þ

and

Aðmþ1Þ=AðmÞ ¼ ð�0=KðmÞÞð�1=�0Þ: ðA18Þ

From Eqs. (A11) and (A14)

KðmÞ ¼ aðmÞ0 �0 ¼ ð1� AðmÞÞ�0

¼ ð1þ jAðmÞjÞ�0 � �0:ðA19Þ

In Eq. (A18), the values of �0=KðmÞ and �1=�0 being less

than 1.0, the following relation is obtained

jAðmþ1Þj � jAðmÞj; ðA20Þ

and then from Eq. (A19)

�0 � Kðmþ1Þ � KðmÞ � Kð0Þ: ðA21Þ

Equation (A21) shows that if an initial multiplicationfactor is larger than the final one, �0, multiplication factorsmonotonously decrease with source iteration cycles andnever become less than the final one.

On the other hand, if the initial multiplication factor is lessthan final one, AðmÞ is positive and the following relation issatisfied

�0 � Kðmþ1Þ � KðmÞ � Kð0Þ: ðA22Þ

The above two relations (A21) and (A22) support the ba-sic idea of the ‘‘Sandwich Method’’.

The above results are obtained under the assumption thathigher order neutron fluxes are by far smaller than the 0thand 1st order ones. In many cases, this assumption is correct.In particular, after many iteration cycles, higher order neu-tron fluxes being small as shown in Eq. (A6) or (A8), this as-sumption must be satisfied. In cases where the summation ofhigher order neutron fluxes is so large that the sign of AðmÞ

changes for some iteration cycles, the trend of multiplication


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factors vs. iteration cycles will not show a monotonousdecrease or increase, but an oscillatory behavior.

The dependence of the neutron multiplication factor after1st source iteration cycle on the initial source distribution isexplained using Fig. A1. The figure shows four types of ini-tial source distribution and neutron importance distribution’�0 which is assumed to be known. Here Sð0Þ1 is an initial

source distribution which is nonzero in a region where fis-sion neutron importance is higher than the average impor-tance. The source expands with source iteration cycles andapproaches Sð0Þ2 , which is a source distribution correspondingto the eigenfunction of the fundamental mode ’0. S

ð0Þ3 is a

flat source distribution, and Sð0Þ4 an initial source distributionwhich is nonzero where neutron importance is lower than the

average value. These source distributions expand withsource iteration cycles and approach Sð0Þ2 .

Initial neutron multiplication factors Kð0Þ’s correspondingto initial source guesses are respectively obtained by

Kð0Þ1

Z’�0D�

ð0Þd� ¼Z

’�0S

ð0Þ1 d� ¼ h’�

01iSð0Þ1 ��

Kð0Þ2

Z’�0D�

ð0Þd� ¼Z

’�0S

ð0Þ2 d� ¼ h’�

02iZ

Sð0Þ2 d�

Kð0Þ3

Z’�0D�

ð0Þd� ¼Z

’�0S

ð0Þ3 d� ¼ h’�

03iSð0Þ3

Zd�

Kð0Þ4

Z’�0D�

ð0Þd� ¼Z

’�0S

ð0Þ4 d� ¼ h’�

04iSð0Þ4 ��0; ðA23Þ

where the quantities of the sources Sð0Þ1 ��,RSð0Þ2 d�, Sð0Þ3

Rd�

and Sð0Þ4 ��0 are normalized to unity. Thus, Kð0Þi is proportion-

al to the source-weighted average importance h’�0ii.

In the case of initial source Sð0Þ1 , the neutron multiplicationfactor Kð0Þ

1 decreases with iteration cycles and approachesKð0Þ2 , which is the same as the final neutron multiplication

factor �0 corresponding to the fundamental mode. In the caseof Sð0Þ4 , the neutron multiplication factor Kð0Þ

4 increases withiteration cycles to Kð0Þ

2 . The source weighted average impor-tance h’�

02i is usually larger than the volume-weighted aver-age one h’�

03i as shown in Fig. A1, so Kð0Þ2 is larger than Kð0Þ

3 .Under the above assumptions, the initial neutron multiplica-tion factors have the following relationship

Kð0Þ1 > Kð0Þ

2 ¼ �0 > Kð0Þ3 > Kð0Þ

4 : ðA24Þ

In the case of Sð0Þ1 , the neutron multiplication factor Kð0Þ1

decreases with iteration and approaches Kð0Þ2 , which is the

same as �0. In the cases of Sð0Þ3 and Sð0Þ4 , the neutron multipli-

cation factor increases to Kð0Þ2 .

(0)3S

S

τ∆τ′∆τ

∗0ϕ

(0)2S

(0)4S

(0)1S

*0ϕ

< (1)>*0ϕ

< (2)>*0ϕ

< (3)>*0ϕ

< (4)>*0ϕ

Fig. A1 Average importants weighted by typical initial sourcedistributions



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Documents

The Sandwich Method for Determining Source Convergence in Monte Carlo Calculation