) Pergamon

www.elsevier.com/locate/pnucene

Progress in Nuclear Energy, Vol. 39, No. 2, pp. 119-144, 2001 © 2001 Elsevier Sctence Ltd. All rights reserved

Printed in Great Britain 0149-1970/01/$ - see front matter

PII: S0149-1970(01)00007-5

3D R A D I A T I O N T R A N S P O R T B E N C H M A R K P R O B L E M S AND

RESULTS F O R S I M P L E G E O M E T R I E S W I T H VOID R E G I O N

KEISUKE KOBAYASHI* and NAOKI SUGIMURAt

Department of Nuclear Engineering, Kyoto University Yoshida, Sakyoku, Kyoto, Japan kobayasi@ip.media.kyoto-u.ac.j p

naoki_sugimura@gijutsu.neltd.co.j p

YASUNOBU NAGAYA

Department of Nuclear Energy System Japan Atomic Energy Research Institute

Tokai-mura, Naka-gun, Ibaraki, Japan nagaya@mike.tokai.j aeri.go.j p

Abstract - - Three-dimensional (3D) transport benchmark problems for simple geometries with void region were proposed at the OECD/NEA in order to check the accuracy of deterministic 3D transport programs. The exact total fluxes by the analytical method are given for the pure absorber cases, and Monte Carlo values are given for the 50% scattering cases as the reference values. The total fluxes of all contributions to the present benchmark problems calculated by the 3D transport programs, TORT, TORT with FNSUNCL3, PARTISN, PENTRAN, IDT, MCCG3D, EVENT and ARDRA are compared in figures. © 2001 Elsevier Science Ltd. All rights reserved.

Keywords: 3D Transport Calculations; Benchmark Problems; Benchmark Results; Void Problems

1 INTRODUCTION

In the OECD proceedings on 3D deterministic radiation transport computer programs edited by Sar- tori (1997), many 3D deterministic transport programs were presented. It is not simple, however, to determine their special features or their accuracy. One of the difficulties in multi-dimensional transport calculations concerns the accuracy of the flux distribution for systems which have void regions in a highly absorbing medium.

The method which is most widely used to solve the 3D transport equation is the discrete ordinates method. However, this method has the disadvantage of the ray effect and we must be cautions which S,, order and which quadrature set for the angular discretization should be used for such systems. On the other hand, the spherical harmonics method has the advantage of causing no ray effect, but

*Present status: Emeritus Professor of Kyoto University ?Present address: Nuclear Engineering Co., Tosabori 1-3-7, Nishiku, Osaka

119

120 K. Kobayashi et at

the equations are very complicated and it is difficult to derive the finite difference or discrete equations which satisfy the necessary boundary conditions at material interfaces as discussed by Kobayashi (1995). Then, the flux distribution by the spherical harmonics method shows some anomalies at the material interfaces of large cross section differences or at the material void interface. This was seen in the flux distribution of the 3D benchmark calculations with void region proposed by Takeda (1991), where appreciable discrepancies were observed in the flux distribution between programs based on the spherical harmonics method as shown by Kobayashi (1995).

Ackroyd and Pdyait (1989) investigated flux distributions for 2D void problems extensively, and their results show that these void problems are really difficult which suggests that 3D transport programs should also be checked for these problems.

The 3D benchmark void problems of simple geometries were proposed at the OECD/NEA in 1996 by Kobayashi (1997) which were simple extensions of the 2D void problems given by Ackroyd and Pdyait to 3D geometries. There are two kinds of one-group source problems. One is a system of a pure absorber with a void region so that the exact solution can be obtained by numerical integration. The other one has the same geometry as the pure absorber problem, however, the pure absorber is replaced by a material which has a scattering cross section of 50% of the total cross section intended as the case where ray effects are not too large. Preliminary results were presented at the Madrid conference by Kobayashi et al. (1999).

2 BENCHMARK PROBLEMS

The systems consist of three regions, source, void and shield regions, whose geometries are shown in Figs. 1 ,-, 8. An x - z or y - z plane geometry and a sketch of Problem 1 are shown in Figs. 1 and 2, respectively. An x - y or y - z plane geometry and a sketch of Problem 2 are shown in Figs. 3 and 4, respectively. Plane geometries and a sketch of Problem 3 are shown in Figs. 5 ,-~ 8, which is called the dog leg void duct problem. Reflective boundary conditions are used at the boundary planes x = 0, y = 0 and z = 0, and vacuum boundary conditions at all outer boundaries for all problems.

The cross sections and source strength S are shown in Table 1. The cross section in a void region is assumed to be not zero but 10-4cm - l so that 3D transport programs based on the second order differential form can be used.

Tab le 1 One-group cross sections and source strength S

Problem i Problem ii S St ,Us

Region (n cm-3s -1) (cm -1) (am -1) 1 1 0.1 0 0.05 2 0 10 -4 0 0.5 x 10 -4 3 0 0.1 0 0.05

In problems l-i, 2-i and 3-i, the systems consist of a pure absorber, and in problems 1-ii, 2-ii and 3-ii, the systems have a scattering cross section of 50% of the total cross section, namely, E , = 0.5Zt. It is expected that the total flux distributions at mesh points shown in Tables 2 --~ 4 be calculated. These mesh points are chosen so that the programs which give the fluxes at the mesh center can be used. It is expected that mesh width, CPU time, the required memory size, and name of computer used should be given.

3D radtatwn transport benchmark problems 121

2.1 Problem 1

E 0

v

0

. e 4

e~

Z Vacuum

Reg. 3

Reg. 2 1 Void

50 Reflective

• ~ x , y

0 10 100 (cm) Reg. 1 Source Fig . 1. x - z or y - z plane of Problem 1, Shield with square void

• t • . . . . . . . . . . .

z . . . . . . . . . . . . . . . . . . .

0 10 50 100 (era)

100

F ig . 2. Sketch of Problem 1, Shield with square void

2.2 Problem 2

Vacuum

~. :~ Reg. 3 ~ i too

Reg 1

0 1 0 6 0 ( c m ) 10 60 (c.)

R e f l e c t i v e

F ig . 3. x - y or y - z plane of Problem 2, Shield with void duct

F ig . 4. Sketch of Problem 2, Shield with void duct

1 2 2

2.3 Problem 3

K. Kobayashi et al.

"B Y Vacuum

~ eg. 3 ~ Z

_ ~ Vacuum

| ~[ Reg. li ~ Void

~ Sou~ce ~ | Reg. 1 I ] [ /Source ] ] Reg. 3

X ~ ~ [ i ~eg . 2 Void] Y 0 10 30 40 60(cm) 0 10 50 60 100(cm)

Ref lec t ive Reflective

Fig. 5. x - y plane of Problem 3, Shield with dog leg void duct

Fig. 6. y - z plane of Problem 3, Shield with dog leg void duct

. . . . . . . . . . . . . . . " ~ LO0

~ .e,l ~ 1 ~ ~ . . . . . . . . . . . . . . . 0

Reg. 1 Source eq ^_

o / / I ~

0 10 30 40 60 (cm) Reflective o 10 3o 4o 60 (c~)

Fig. 7. x - z plane of Problem 3, Shield with dog leg void duct

Fig. 8. Sketch of Problem 3, Shield with dog leg void duct

3.1

3D radiatJon transport benchmark problems

3 REFERENCE SOLUTIONS

Exact Total Flux for Pure Absorber Problems

123

In the case of no scattering, fluxes can be obtained simply by numerical integration, namely, the neutron total flux at the observation position r in a pure absorber can be calculated by

f ¢(r) = ~ fv. d r ' e x p ( - Ir - r'12 ' (1)

where ¢(r), S(rl), S~ and V~ are the total flux, external source, absorption cross section and source region, respectively.

We assume that the source S(r) is constant in space and use the relation

l = r p - r, dr ~ = dl = 12dld~t = 12dl sin 0 d~ d0,

where 0 and ~ are the polar and azimuthal angles, respectively.

(2)

Y

y

Source re~tion b X

Fig. 9. Observation point is outside the source region

Fig. 10. Observation point is inside the source region

If the observation position r is at the outside of the source region as shown in Fig. 9, the total flux of Eq.(1) can be expressed as

¢(r) = ~1 ~,[" d / e x p ( - fz o" S~(l')dl'l 2- fifo S~(l')dl')S(l)

= s_ f f 12dl exp(-E°'l' - Soo/o) exp(- fZ*o S.,dl') 4~r ~ J~o l 2

S f J d/2 exp( -E~, / , - Z~o/o)(1 - exp(--Z~,(/b --l~))), (3)

4~ 'Za l

where the external source is assumed to be in the region whose absorption cross section is S~1, lo and l~ are the total length of the path whose absorption cross sections are S~0 and Z~I, respectively, and lb - l~ is the path length in the source region. If the observation position r is in the inside of the source region as shown in Fig. 10, the total flux of Eq.(1) becomes

1 ft o S~(l')dl')S(l) = ~ dl2 lZdl e xp ( - sa ' l ) ¢(r) = ~ d / ex p ( - 12 4~r 12

_ S f d~(1 - exp(-Z~l/b)) , (4) 4~rZ~l J

124 K. Kobayashi et al

w h e r e lb is t h e l eng th of t h e p a t h f rom t h e obse rva t i on pos i t i on r to t h e b o u n d a r y of t h e source region.

T h e t o t a l f luxes g iven by Eqs . (3) a n d (4) a r e c a l cu l a t ed us ing t h e t r a p e z o i d a l rule in 0 and ~ var iab les ,

w h e r e t h e n u m b e r of m e s h po in t s used for b o t h 0 and ~ var iab les is 20,000. C o n v e r g e n c e w i t h r e spec t to

t h e n u m b e r of m e s h po in t s is checked by c o m p a r i n g t h e fluxes w i t h 10,000 m e s h po in t s a n d con f i rming

t h a t t h e r e is no d i f fe rence b e t w e e n t h e m . T h e f luxes thus o b t a i n e d are s h o w n in Tab les 2, 3 a n d 4 for

each p rob l em.

Z m i j a r e v i c a n d Sanchez (2001) also c a l c u l a t e d the fluxes f r o m Eq . (1 ) us ing t h r e e - d i m e n s i o n a l n u m e r i c a l

q u a d r a t u r e , a n d the i r f luxes a re e x a c t l y t h e s a m e as t h e p re sen t ones for all six d ig i t s g iven in Tab les

2, 3 a n d 4 e x c e p t smal l d i f ferences a t m e s h po in t s (45,95,35) a n d (55,95,35) of P r o b l e m 3.

T a b l e 2 T o t a l flux for P r o b l e m 1

Case i (No Scattering) Case ii (50% Scat ter ing) Case Coordinates Analyt ica l M e t h o d Monte Carlo M e t h o d by G M V P

1A

1B

1C

(cm) Tota l F lux Tota l F lux FSD a Tota l F lux FSD ( x , y , z ) (cm-2s -1) (cm-2s -1) l a ( % ) (cm-2s -1) l a ( % )

5 ,5 ,5 5 15, 5 5 25, 5 5 35, 5 5 45, 5 5 55, 5 5 65,5 5 75,5 5 85,5 5 95, 5

5.95659 x 10 - ° 5.95332 x 10 - ° 0.308 8.29260 x 10 - ° 0.021 1.37185 x 10 -o 1.37116 x 10 -o 0.053 1.87028 x 10 -o 0.005 5.00871 x 10 -1 5.00789 x 10 -1 0.032 7.13986 x 10 -1 0.003 2.52429 x 10 -1 2.52407 x 10 -1 0.027 3.84685 x 10 -1 0.004 1.50260 x 10 -1 1.50251 x 10 -1 0.025 2.53984 x 10 -1 0.006 5.95286 x 10 -2 5.95254 x 10 -2 0.023 1.37220 x 10 -1 0.073 1.53283 x 10 -2 1.53274 x 10 -2 0.022 4.65913 x 10 -2 0.117 4.17689 x 10 -3 4.17666 x 10 -3 0.022 1.58766 x 10 -2 0.197 1.18533 x 10 -3 1.18527 x 10 -3 0.021 5.47036 x 10 -3 0.343 3.46846 x 10 -4 3.46829 x 10 -4 0.021 1.85082 x 10 -3 0.619

5 ,5 ,5 15, 15, 15 25, 25, 25 35, 35, 35 45, 45, 45 55, 55, 55 65, 65, 65 75, 75, 75 85, 85, 85 95, 95, 95

5.95659 x 10 - ° 5.95332 x 10 - ° 0.308 8.29260 x 10 - ° 0.021 4.70754 x i0 -t 4.70489 x 10 -1 0.040 6.63233 x 10 -1 0.004 1.69968 x 10 - l 1.69911 x 10 -1 0.025 2.68828 x 10 - I 0.003 8.68334 x 10 -2 8.68104 x 10 -2 0.021 1.56683 x 10 -1 0.005 5.25132 x 10 -2 5.25011 x 10 -2 0.020 1.04405 x 10 -1 0.011 1.33378 x 10 -2 1.33346 x 10 -2 0.019 3.02145 x 10 -2 0.061 1.45867 x 10 -3 1.45829 x 10 -3 0.019 4.06555 x 10 -3 0.074 1.75364 x 10 -4 1.75316 x 10 -4 0.019 5.86124 x 10 -4 0.116 2.24607 x 10 -5 2.24543 x 10 - s 0.019 8.66059 x 10 - s 0.198 3.01032 x 10 -6 3.00945 x 10 -6 0.019 1.12982 x 10 -5 0.383

5, 55, 5 15,55,5 25, 55, 5 35, 55, 5 45, 55, 5 55, 55, 5 65, 55, 5 75, 55, 5 85,55,5 95, 55, 5

5.95286 x 10 -2 5.95254 x 10 -2 0.023 1.37220 x 10 -1 0.073 5.50247 x 10 -2 5.50191 x 10 -2 0.023 1.27890 x 10 -1 0.076 4.80754 x 10 -2 4.80669 x 10 -2 0.022 1.13582 x 10 -1 0.080 3.96765 x 10 -2 3.96686 x 10 -2 0.021 9.59578 x 10 -2 0.088 3.16366 x 10 -2 3.16291 x 10 -2 0.021 7.82701 x 10 -2 0.094 2.35303 x 10 -2 2.35249 x 10 -2 0.020 5.67030 x 10 -2 0.111 5.83721 x 10 -3 5.83626 x 10 -3 0.020 1.88631 x 10 -2 0.189 1.56731 x 10 -3 1.56708 x 10 -3 0.020 6.46624 x 10 -3 0.314 4.53113 x 10 -4 4.53048 x 10 -4 0.020 2.28099 x 10 -3 0.529 1.37079 x 10 -4 1.37060 x 10 -4 0.020 7.93924 x 10 -4 0.890

a) Fract ional s t andard deviat ion

3 D radiation transport benchmark problems

Table 3 Total flux for Problem 2

125

Case i (No Scat ter ing) Case ii (50% Scat ter ing)

Case Coord ina tes Analyt ica l M e t h o d Monte Carlo Me t h o d by G M V P

2A

2B

(cm) Total F lux Total Flux FSD" Total F lux FSD ( x , y , z ) ( cm-2s -1) ( em-2 s -1 ) 1~(%) (n c m - 2 s -1) l a ( % )

5 ,5 ,5 5.95659 x 10 - ° 5.94806 x 10 - ° 0.287 8.61696 x 10 - ° 0.063

5, 15,5 1.37185 x 10 - ° 1.37199 x 10 - ° 0.055 2.16123 x 10 - ° 0.015 5 ,25 ,5 5.00871 x 10 -1 5.00853 x 10 -1 0.034 8.93437 x 10 -1 0.011

5, 35,5 2.52429 x 10 -1 2.52419 x 10 - l 0.029 4.77452 x 10 -1 0.012

5 ,45 ,5 1.50260 x 10 -1 1.50256 x 10 -1 0.027 2.88719 x 10 -1 0.013

5 ,55 ,5 9.91726 x 10 -2 9.91698 x 10 -2 0.025 1.88959 x 10 -1 0.014 5 ,65 ,5 7.01791 x 10 -2 7.01774 x 10 -2 0.024 1.31026 x 10 - l 0.016

5, 75, 5 5.22062 x 10 -2 5.22050 x 10 -2 0.023 9.49890 x 10 -2 0.017

5, 85,5 4.03188 x 10 -2 4.03179 x 10 -2 0.023 7.12403 x 10 -2 0.019

5, 95, 5 3.20574 x 10 -2 3.20568 x 10 -2 0.022 5.44807 x 10 -2 0.019

5, 95, 5 3.20574 x 10 -2 3.20568 x 10 -2 0.022 5.44807 x 10 -2 0.019

15,95,5 1.70541 x 10 -3 1.70547 x 10 -3 0.040 6.58233 x 10 -3 0.244 25,95,5 1.40557 x 10 -4 1.40555 x 10 -4 0.046 1.28002 x 10 -3 0.336

35,95,5 3.27058 x 10 -5 3.27057 x 10 -5 0.044 4.13414 x 10 -4 0.363

45,95,5 1.08505 x 10 -5 1.08505 x 10 -5 0.042 1.55548 x 10 -4 0.454 55,95,5 4.14132 x 10 -6 4.14131 x 10 -6 0.039 6.02771 x 10 -5 0.599

a) Fract ional s t a n d a r d devia t ion

T a b l e 4 T o t a l f lux for P r o b l e m 3

Case i (No Scat ter ing) Case ii (50% Scat ter ing)

Case Coord ina tes Analyt ica l M e t h o d Monte Carlo Me t h o d by G M V P

3A

3B

3C

(cm) Total Flux Total F lux FSD" Total F lux FSD (x,y,z) (cm-2s -') (cm-2s -') 1~(%) (cm-2s -') 1~(%) 5 ,5 ,5 5.95659 x 10 - ° 5.93798 x 10 - ° 0.306 8.61578 x 10 - ° 0.044

5, 15,5 1.37185 x 10 - ° 1.37272 x 10 - ° 0.052 2.16130 x 10 - ° 0.010 5 ,25 ,5 5.00871 x 10 -1 5.01097 x 10 -1 0.032 8.93784 x 10 -1 0.008

5 ,35 ,5 2.52429 x 10 -1 2.52517 x 10 -1 0.027 4.78052 x 10 -1 0.008

5 ,45 ,5 1.50260 x 10 -1 1.50305 x 10 -1 0.025 2.89424 x 10 - l 0.009 5 ,55 ,5 9.91726 x 10 -2 9.91991 x 10 -2 0.024 1.92698 x 10 -1 0.010

5, 65, 5 4.22623 x 10 -2 4.22728 x 10 -2 0.023 1.04982 x 10 -1 0.077

5 ,75 ,5 1.14703 x 10 -2 1.14730 x 10 -2 0.022 3.37544 x 10 -2 0.107

5 ,85 ,5 3.24662 x 10 -3 3.24736 x 10 -3 0.021 1.08158 x 10 -2 0.163

5 ,95 ,5 9.48324 x 10 -4 9.48534 x 10 -4 0.021 3.39632 x 10 -3 0.275

5, 55, 5 9.91726 x 10 -2 9.91991 x 10 -2 0.024 1.92698 x 10 -1 0.010

15,55,5 2.45041 x 10 -2 2.45184 x 10 -2 0.035 6.72147 x 10 -2 0.019

25,55,5 4.54477 x 10 -3 4.54737 x 10 -3 0.037 2.21799 x 10 -2 0.028

35, 55, 5 1.42960 x 10 -3 1.43035 x 10 -3 0.034 9.90646 x 10 -3 0.033

45,55,5 2.64846 x 10 -4 2.64959 x 10 -4 0.032 3.39066 x 10 -3 0.195

55,55,5 9.14210 x 10 -5 9.14525 x 10 -5 0.029 1.05629 x 10 -3 0.327

5,95,35 3.27058 x 10 -5 3.27087 x 10 -5 0.045 3.44804 x 10 -4 0.793

15,95,35 2.68415 x 10 -5 2.68518 x 10 -5 0.047 2.91825 x 10 -4 0.659

25,95,35 1.70019 x 10 -5 1.70104 x 10 -5 0.047 2.05793 x 10 -4 0.529

35,95,35 3.37981 x 10 -5 3.38219 x 10 - s 0.043 2.62086 x 10 -4 0.075 45, 95, 35 6.04893 × 10 -6 6.05329 × 10 -6 0.042 1.05367 x 10 -4 0.402

55, 95, 35 3.36460 x 10 -6 3.36587 x 10 -6 0.028 4.44962 x 10 -5 0.440

a) Fract ional s t a n d a r d devia t ion

126 K. Kobaya~hl et al.

3.2 Monte Carlo Results by GMVP

Monte Carlo calculations were also performed using GMVP (Mori and Nakagawa, 1994), where a point- detector estimator was used to tally the total flux at the given calculation points for all cases. In the pure absorber cases, 107 histories were used, and for the 50% scattering cases, 109, l0 s and 2 × l0 s histories for Problems 1, 2 and 3, respectively. Calculated total fluxes are shown in Tables 2 ~ 4. The total fluxes for the pure absorber cases are in very good agreement with the analytical results, which shows that the program GMVP is reliable for this kind of problems. Konno (2001) also calculated the total flux using MCNP4B2, and his results agree well with the present results within statistical errors.

4 BENCHMARK RESULTS

As shown in Table 5, there are eight contributions for the present benchmarks. Six contributions were obtained by using discrete ordinates method programs. They are: TORT by Azmy et el. (2001) at ORNL, TORT with FNSUNCL3 (Kosako and Konno,1998) by Konno (2001) at JAERI, PARTISN by Alcouffe (2001) at LANL, PENTRAN by Haghighat and Sjoden (2001) at PSU and USAF respectively, IDT by Zmijarevic and Sanchez (2001), and MCCG3D by Suslov (2001) at CEA. The other two contri- butions were obtained using the spherical harmonics method, EVENT by Oliveira et al. (2001) at IC, and ARDRA by Brown et al. (2001) at LLNL.

In the results by Konno, there are two cases, one in which only TORT is used, and the other in which TORT is used together with the program FNSUNCL3 and for which the first flight collision source is calculated. In the finite element-spherical harmonics solutions by EVENT, the ray-tracing method has been used in the void region, and the flux values are quoted only for points within the domain of the non- void region, since fluxes in the ray-tracing regions were not available with the current implementation.

Tab le 5 3D Transport Benchmark Results

Name Program Method Problems Mesh Width Computer (am)

Azmy, et al. TORT $16 LN ~ scattering 10/9 Cray Y/MP cases only 4 tasks

Konno TORT $16 all cases 2 FUJITSU TORT with S16 all cases 2 AP3000/24 FNSUNCL3 FCS b 0.25

Alcouffe PARTISN $8, FC b all cases 2 SGI Haghighat, et el. PENTRAN $2o, ADS c no scattering Variable(2 - 10) IBM SP2

$12, ADS c scattering 1.111 IBM SP2 Zmijarevic, et el. IDT S16, LC d all cases (10/9) DEC

Extrapolated Alpha 4100 Suslov MCCG3D RT ~, SC/, DD9 all cases 2.5 SP2

Oliveira, et el. EVENT Pg, RT ~ all cases 1.43 - 2 COMPAQ AXP 1000

Brown, et al. ARDRA P21 all cases 1.04-2 IBM ASCI Blue-Pacific

a) Linear Nodal b) First Collision Source c) Adaptive Differencing Strategy using the DTW and EDW schemes d) Linear Characteristic e) Ray Tracing f) Step Characteristic g) Diamond Difference

3D radiation transport benchmark problems 127

Benchmark results for Problem (i) of no scattering and (ii) of 50% scattering for three Problems 1,2 and 3 are shown in Figs. 11 ..~ 42. Shown in these Figures are the scattering cases calculated with TORT by Azmy, and the no-scattering cases calculated with TORT by Konno. In Figs. 12, 14, etc., the ratios of the fluxes to the reference values (i.e. the exact values by the analytical method for the pure absorber cases, and the Monte Carlo values for the scattering cases) are shown in order to make clear the difference from the reference values.

5 DISCUSSION AND CONCLUSION

The benchmark results shown in Figs. 11 .-~ 42 are fairly close to the reference solutions, although some preliminary results have some discrepancies with regard to the reference solutions. In the case of the pure absorber, the apparent discrepancies of the discrete ordinates program TORT(JAERI) from the exact values for problems 1Ci, 2Bi and 3Ci may be due to the ray effect. Namely, even the Sis method gives appreciable errors due to ray effects.

In the cases with 50% scattering, the total fluxes for the problems, for example, 1Cii, 2Bii and 3Cii are larger by about a factor 10 than those for pure absorber cases of 1Ci, 2Bi and 3Ci, respectively. Namely, the number of scattered neutrons is larger by about a factor 10 than the neutrons which come directly from the source, and the ray effect becomes smaller even in TORT. However, discrepancies from the exact values are fairly large for the problem of the dog leg duct, problems 3Ci and 3Cii. In the discrete ordinates programs TORT, PENTRAN and IDT, the first collision source was not used. In PENTRAN, the ray effect is remedied by using appropriate angular and spatial meshes and mesh widths, and PENTRAN's unique differencing formulations including adaptive differencing strategy with directional theta-weighted (DTW) and exponential directional weighted (EDW) schemes and Taylor projection mesh coupling (TPMC)(Haghighat and Sjoden, 2001).

In other discrete ordinates programs, TORT with FNSUNCL3, PARTISN and MCCG3D, the first collision source was used to remedy the ray effect, which is seen in figures given by Konno (2001) to be very successful. Namely, the use of the first collision source for the discrete ordinates method improved the accuracy appreciably for both problems of pure absorber and 50% scattering cases. In particular, the results of MCCG3D and TORT with FNSUNCL3 are in excellent agreement, within an error of 1% .--5% with the reference solution in most cases and can be considered as an independent confirmation of the reference solution. It should be noted that, for the pure absorber problems, the first collision source method should give in principle exact fluxes and this fact does not demonstrate the accuracy of the discrete ordinate method itself.

An advantage of the spherical harmonics method is that the equations are invariant under rotation of the coordinates and do not depend on the direction of the coordinates, that should give no ray effect. In order to overcome the difficulty in deriving the discretized equations of the spherical harmonics method for void problems, the ray-tracing method was used for the void region in the spherical harmonics program EVENT, and the flux in non-void region was coupled with the current in the void region at the void-material interface. Using this method, the accuracy of the current spherical harmonics method program was improved. In the program ARDRA, the discrete ordinates equations with a fictitious source were solved in such a way that the equations became equivalent to those of the spherical harmonics method, which were used to solve two dimensional spherical harmonics equations (Reed, 1972, Jung et al. 1974).

As seen in the figures, the accuracy of the discrete ordinates method with the first collision source is best for the present benchmark problems. It is expected that the present benchmark problem would help further improvement of 3D transport programs based on the spherical harmonics method as well as the discrete ordinates method.

128 K. Kobayasht et al.

(n c m -2 S -1)

10 a

10 °

10 -1

10 -2

10 -3

10 -4

10 -5

' ) ' I ' I ' I '

+ GMVP (JAERI) ~ × MCNP4B (JAERI) D MCCG3D RT/SC (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) • v IDT LC (CEA) v c, FNSUNCL3 (JAERI) zx TORT (JAERI) ,x o EVENT (IC) • ARDRA (LLNL) •

, I i I L I i I

0 20 40 60 80 y (cm)

F i g . 11. P rob lem 1Ai "No scattering" (x = z = 5 cm)

i 100

m

1.5

1

0.5

0

-0.5

I ' I ' I ' I ' I

+ x 12

0

0

v

0

@

&

v • • @ ®

• zx

& GMVP (JAERI) v MCNP4B (JAERI) MCCG3D RT/SC (IPPE, UTK,CEA) PENTRAN glSAF, PSL0 PARTISN FC (LANL) IDT LC (CEA) FNSUNCL3 (}AERD TORT (JAERI) EVENT (IC) ARDRA (LLNL) t I L

20 i , I

40 60 y (cm)

@ v @

@

• 0

I I

80 100

F ig . 12. Relative flux of problem 1Ai (x = z = 5 cm)

3D radtation ozm~port benchmark problems 129

(n cm -2 S -1)

101

10 °

10 -1

10 -2

10 -3

10 -4

I t I B I ' I

31•..

"'Q, ......... ,$ .......

--q---- GMVP (JAERI) X MCNP4B (JAERI) t3 MCCG3D RT/DD 0PPE,UTK,CEA) o PENTRAN (USA.F, PSU) t' TORT LN (ORNL) © PARTISN FC (LANL) v IDT LC (CEA) c, TORT with FNSUNCL3 (JAERI) o EVENT (IC) • ARDRA (LLNL)

"-~... "4

i I i I i I , I

20 40 60 80 y (cm)

Fig. 13. Problem 1Aii "50% scattering" (x = z = 5 cm)

I

100

1.5

>

0.5

0

-0.5

X I ]

o A

0

V

®

I

0

• v A • @ @ @ ~ v O @ • []

• o 8 o 8 o

o A V •

MCNP4B (JAERI) MCCG3D RT/DD (IPPE,UTK,CEA) PENTRAN (USAF, PSU) TORT LN (ORNL) PARTISN FC (LANL) IDT LC (CEA) TORT with FNSUNCL3 (JAERI) EVENT fic) ARDRA (LLNL)

, I , I i I i I

20 40 60 80 y (cm)

Fig. 14. Relative flux of problem 1Aii (x = z = 5 cm)

i 100

130 K Kobayashi et al

(n cm -2 S -1)

101

10 °

10 -1

10 -2

10 -3

10 -4

10 -5

10 -6

10 -7

I ) I ) I ' I ' I ' I

q- GMVP (JAERI) ~ , . X MCNP4B (JAERI) ~ D MCCG3D RT/SC (IPPE,UTK,CEA) " ~ o PENTRAN (USAF, PSU) u ~ o PARTISN FC (LANL) v IDT LC (CEA) o FNSUNCL3 (JAERI) z~ TORT (JAERI) o EVENT (IC) v • ARDRA (LLNL)

i I i I i I I I i I

20 40 60 80 100 x or y or z ( c m )

Fig . 15. Problem 1Bi "No scattering" (x = y = z)

1.5

0.5

-0.5

I ' I

8

I

0

' I ' I ' I ' I

o ® 8

0 v

0 ©

0 ~7

~- GMVP (JAERI) X MCNP4B (JAERI) E] MCCG3D RT/SC (IPPE, UTK,CEA) o PENTRAN (USAF, PSU) © PARTISN FC (LANL) v IDT LC (CEA) <> FNSUNCL3 (JAERI) z~ TORT (JAERI) o EVENT fIc) • ARDRA (LLNL)

i I i I , I i I , I

20 40 60 80 100 x or y or z ( c m )

Fig . 16. Relative flux of problem 1Bi (x = y = z)

3D radtation transport benchmark problems 131

(n cm -2 S -1)

101

10 °

10 -1

10 -2

I~ 10 -3

10 -4

10 -s

10 -6

10 -7

I ' I ~ I ' I ~ I

--+-- GMVP (JAERI) X MCNP4B (JAERI) [] MCCG3D RT/DD (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) " TORT I.,N (ORNL) o PARTISN FC (LANL) v IDT LC (CEA) o TORT with FNSUNCL3 (JAERI)

EVENT (IC) • ARDRA (LLNL)

"',..

¢ I i I i l , I

20 40 60 80 x or y or z ( c m )

""-~,,

F ig . 17. P rob lem 1Bii "50% scat ter ing" (x = y = z)

I

i

100

>

1.5

0.5

0

-0.5 i

0

' I ' I ' I ' I

V z~ &

0 @

~' o 8 8 z~

× MCNP4B (JAERI) [] MCCG3D RT/DD (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) zx TORT LN (ORNL) o PARTISN FC (LANL) v IDT LC (CEA) c, TORT with FNSUNCL3 (JAERI) o EVENT (to) • ARDRA (LLNL)

i i i I , I

20 40 60 x or y or z ( c m )

I

80

F ig . 18. Relat ive flux of p rob lem 1Bii (x = y = z)

&

o O '7

r~

6 O

i 100

132

(n cm -2 S -1)

10 -I

&

~7

10 -z

N + 10 -3 ×

[ ]

0

0

v

10 -4 o

@

10 -5 0

K. Kobayashi et al.

I I I I I

- - Analytical Solution GMVP (JAERI) MCNP4B (JAERI) MCCG3D RT/SC (IPPE,UTK,CEA) PENTRAN (USAF, PSU) PARTISN PC (LANL) IDT LC (CEA) FNSUNCL3 (JAERI) TORT (JAERI) EVENT (IC) ARDRA (LLNL)

, I , I , I , I

20 40 60 80 x ( c m )

Fig . 19. Problem 1Ci "No scattering" (y = 55 cm, z = 5 cm)

I 113(

o

1.5

0.5

-0.5

@

I I I I

@ @ ,, • A (~3 0 @ o o ~ @ @

°

© v v

+ GMVP (JAERI) × MCNP4B (JAERI) t3 MCCG3D RT/SC 0PPE, UTK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) v IDT LC (CEA) o FNSUNCL3 (JAERI)

TORT (JAERI) @ EVENT (IC) • ARDRA (LLNL)

2O I , I , I

40 60 80 x (cm)

F i g . 20. Re la t ive flux of p roblem 1Ci (y = 55 cm, z = 5 cm)

@

e

I

100

3 D radiatmn transport benchmark problems 133

(n cm -2 s -1)

10 - ]

10 -2

10 -3

10 -4

I ' I ' I ' I ' I

",% %

%,%

--+-- G M V P (JAERI) X MCNP4B (JAERI) rn MCCG3D RTIDD (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) zx TORT LN (ORNL) v PARTISN FC (LANL) o IDT LC (CEA) <> TORT with FNSUNCL3 (JAERI) o EVENT (IC) • ARDRA (LLNL)

t I I I I I I I

20 40 60 80 x (cm)

Fig. 21. Problem 1Cii "50% scattering" (y = 55 cm, z = 5 cm)

I

100

CM

1.5

0.5

--0.5 i 0

I l I ~ I i I ' I

@ @ @

o 0 /k o

X MCNP4B (JAERI) n MCCO3D RT/DD (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) A TORT LN (ORNL) v PARTISN FC (LANL) O IDT LC (CEA) <> TORT with FNSUNCL3 (JAERI) @ EVENT (IC) • ARDRA (LLNL)

i I I I I I I I

20 40 60 80 x (cm)

Fig. 22. Relative flux of problem 1Cii (y = 55 cm, z = 5 cm)

I

I

100

10 °

K. Kobayashi et al.

(n cm -2 S -l) 101 ,

10 -1

10 -2

+ GMVP (JAERI) ~ zx × MCNP4B (JAERI) v v t3 MCCG3D RT/SC (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) v IDT LC (CEA) O FNSUNCL3 (JAERI) zx TORT (JAERI) © EVENT fiC) • ARDRA (LLNL)

I ' I ' I ' I ' I

10 -3 i 0

, I i I , I i

20 40 60 y (cm)

134

I i I

80 100

Fig. 23. Problem 2Ai "No scattering" (x = z = 5 cm)

,g

1.5

1

0.5

-0.5

! ' I ' I ' I ' I ' I v

~7 A

A V

-I- GMVP (JAERI) a X MCNP4B (JAERI) v [] MCCG3D RT/SC 0PPE, UTK,CEA) o PENTRAN (USAF, PSLr) o PARTISN FC (LANL) v IDT LC (CEA) o r~StrNCL3 (JAERr) zx TORT (JAER1) © EVENT (IC) • ARDRA (LLNL)

l I i I I I , I

20 40 60 80 y ( c m )

Fig. 24. Relative flux of problem 2Ai (x = z = 5 cm)

m

I

100

3D radiation transport benchmark problems 135

m

(n cm -2 S -l)

101

10 °

10 -1

10 -2

' I

i

' I ' I ' I ' I

"~...

--~-- GMVP (JAERD MCNP4B (JAER]) ,~ ...... ..~.

[] MCCG3D RT/DD (IPPE,UTK,CEA) ~Y ...... . . ~1,,-. za

A° PENTRAN (USAF, P S U ) T o R T LN (ORNL) " ....... ~ ' " ' " - - -

o PARTISN FC (I..,ANL) v IDT LC (CEA) o o TORT with FNSUNCL3 QAER_~

EVENT (IC) • ARDRA (LLNL)

0 20 40 60 80 100 y ( c m )

Fig. 25. Problem 2Aii "50% scattering" (x = z = 5 cm)

"~ 0.5

m 0

1.5 '1

-0.5 0

' I ' I i I ~ |

v V •

z~

0 o v 6

A

x MCNP4B (JAERI) [] MCCG3D RT/DD (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) a TORT I.,N (ORNL) o PARTISN FC (LANL) v IDT LC (CEA) o TORT with FNSUNCL3 (JAERI) o EVENT (iC) • ARDRA (LLNL)

i I L I i i I I

20 40 60 80 y ( c m )

Fig. 26. Relative flux of problem 2Aii (x = z = 5 cm)

v I

)(

o

o

o

I

100

136

(n cm -2 S -1)

lO-1 I

K. Kobayashi et al.

10 .2

10 -3

~ 1 0 - 4

10 -5

10 -6

10 -7 I

0

V

A

- - Analytical Solution + GMVP (JAERI) ~ • X MCNP4B (JAERI) A [] MCCG3D RT/SC (IPPE, UTK,CEA) o PENTRAN (USAF, PSU) O PARTISN FC (LANL) ~ v v IDT LC (CEA) o FNSUNCL3 (JAERI)

TORT (JAERI) @ EVENT (IC) • ARDRA (LLNL)

I I I i I

20 40 60 I

x (cm)

Fig . 27. Problem 2Bi "No scat tering" (y = 95 cm, z = 5 cm)

I I i I i I

- 1

- 2

@ @

@

@ O

o ~

A V

A V

z,

+ GMVP (JAERD ¢ × MCNP4B (JAERr) [] MCCG3D RT/SC (IPPE,UTK,CEA) o PENTRAN (USAF, PSL0 o PARTISN FC (LANL) v IDT LC (CEA) o FNSUNCL3 (JAERD A TORT (JAERI)

EVENT (IC) • ARDRA (LLNL)

0 2O i I

4O x (cm)

Fig . 28. Relative flux of problem 2Bi (y = 95 cm, z = 5 cm)

I

6O

3D radtatton transport benchmark problems 137

(n cm -2 S - l )

10 -1

10 -2

10 -3

10 -4

10 -5

I ' I ' I ' I

" % .

- -+-- GMVP (JAERI) . . . . . . . . X MCNP4B (JAERI) ..... . ~ [] MCCG3D RT/DD (IPPE,UTK,CEA) ~ " ........ o PENTRAN (USAF, PSU) --. . . A TORT LN (ORNL) .... ~ . . . o PARTISN FC (LANL) . . . . . . . . . . v IDT LC (CEA) c, TORT with FNSUNCL3 (JAERI) .... "~ @ EVENT (IC) • ARDRA (LLNL)

I I I i I i

0 20 40 x (cm)

Fig. 29. Problem 2Bii "50% scattering" (y = 95 cm, z = 5 cm)

I 60

>

1.5

0.5

0

-0.5

I ~ ' I ; I ' I @

@ ©

A a~ @

@

0 e

©

× MCNP4B (JAERI) [] MCCG3D RT/DD (IPPE,LrrK,CEA) o PENTRAN (USAF, PSU) z~ TORT 1_2,I (ORNL) © PARTISN FC (LANL) ,7 IDT LC (CEA)

TORT with FNSUNCL3 (JAERI) @ EVENT (IC) • ARDRA (LLNL)

i I i i 20 40

x (cm)

Fig. 30. Relative flux of problem 2Bii (y = 95 cm, z = 5 cm)

I 6O

138

m

(n cm -2 S -1)

101

10 °

10 -I

10 -2

10 -3

10 -4

K. Kobayashi et al

I ~ I r I ' l ' I ' I

- - Analytical Solution + GMVP ( JAE RI ) ~ z~ X MCNP4B (JAERI) D MCCG3D RT/SC (IPPE,UTK,CEA) J t ~ o PENTRAN (USAF, PSU) ~ o PARTISN FC (LANL) v IDT LC (CEA) • <> FNSUNCL3 (JAERI) ~ J z~ TORT (JAERI) q @ EVEm (lc) ~ ] • ARDRA r

I i I , I i I i I I I

0 20 40 60 80 100 y (cm)

Fig. 31. Problem 3Ai "No scattering" (x = z = 5 cm)

2 I ' I I ' I ' I

o

m

1.5

0.5

0

-0.5 0

~7

ZX

+ GMVP (JAERI) X MCNP4B (JAERI) [] MCCG3D RT/SC (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) v IDT LC (CEA)

FNSUNCL3 (JAERI) z~ TORT (JAERI) ® EVENT (IC) • ARDRA (LLNL)

i t i I

20

• ~ @ ~3 @

0

0 ©

v •

v

i i i I

40 60 80 y (cm)

Fig. 32. Relative flux of problem 3Ai (x = z = 5 cm)

i

100

3 D radiatton transport benchmark problems 139

(n cm -2 S -l)

101

10 °

10 -1

10 -2

10 -3

I ' I ' I ' I ' I

o°-~-. .

X [] o /x

O

~7 <>

@

"m.

GMVP (JAERI) .. MCNP4B (JAERI) MCCG3D RT/SC (IPPE,UTK,CEA) " " - PENTRAN (USAF, PSU) TORT LN (ORNL) PARTISN FC (LANL) IDT LC (CEA) TORT with FNSUNCL3 (JAERI) EVENT (IC) ARDRA (LLNL)

I

0 i I , I i I i I

20 40 60 80 y (cm)

Fig. 33. Problem 3Aii "50% scattering" (x = z = 5 cm)

O

i

100

1.5 D ' I ' I ' I ' I

>

"~ 0.5

0

-0.5

)< [] o z~

0

V <> @

• x7 v $ @ zx @ @

,,, ~ @ O v •

i z~ v

MCNP4B (JAERI) MCCG3D RT/SC 0PPE,UTK,CEA) PENTRAN (USAF, PSU) TORT LN (ORNL) PARTISN FIE (LANL) IDT LC (CEA) TORT with FNSUNCL3 (JAERI) EVENT (IC) ARDRA (LLNL)

i I , I i I , I

20 40 60 80 y (cm)

Fig. 34. Relative flux of problem 3Aii (x = z = 5 cm)

I

100

140

(n cm -2 S -1) 10 ° ,

K. Kobayashi et al

' I ' I ' I

10 -1

10 -2

10 -3

10 -4

10 -5

va y t i ~ Anal + GMVP (JAERI) X MCNP4B (JAERI) ~

MCCG3D RT/SC 0PPE,UTK,CEA) - ~ -- o PENTRAN (USAF, PSL0 o PARTISN FC (LANL) x , ~ c~ v IDT LC (CEA) z ~ <> FNSUNCL3 (JAERI) a TORT (JAERI) @ EVENT (IC) • ARDRA (LLNL)

i I i I , I

20 40 60 x (cm)

F i g . 35. P rob lem 3Bi "No sca t te r ing" (y = 55 cm, z = 5 cm)

C9

m

3

2

1

0

- I

-2

-3 I

0

' I ' I ' I - @

@

-[- GMVP (JAERI) × MCNP4B (JAERI) o MCCG3D RT/SC (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) v IDT LC (CEA) c. FNSUNCL3 (JAERI) a TORT (JAERI) @ EVENT (IC) • ARDRA (LLNL)

i J i I I

40 60 2O x (cm)

F i g . 36. Rela t ive flux of p rob lem 3Bi (y = 55 cm, z = 5 cm)

zk

• zx

~7

3 D radiatto'n transport benchmark problems 141

(n cm -2 S -1)

m

10 -1

10 -2

10 -3

--+-- GMVP (JAERI) X MCNP4B (JAERI) [] MCCG3D RT/SC 0PPE,UTK,CEA) o PENTRAN (USAF, PSU) a TORT LN (ORNL) o PARTISN FC (LANL) ';, IDT LC (CEA) o TORT with FNSUNCL3 (JAERI)

EVENT 0C) • ARDRA (LLNL)

i i I

20 40 x (cm)

Fig. 37. Problem 3Bii "50% scattering" (y = 55 cm, z = 5 cm)

i 60

2 I ' I ' I ' I

1.5

0.5

-0.5 i 0

X MCNP4B (JAERI) [] MCCO3D RT/SC 0PPE,UTK,CEA) o PENTRAN (USAF, PSU) " TORT LN (ORNL) o PARTISN FC (LANL) v IDT LC (CEA) o TORT with FNSUNCL3 (JAERI) o EVENT (Ic) • ARDRA (LLNL)

i I i I

20 40 x (cm)

Fig. 38. Relative flux of problem 3Bii (y = 55 cm, z = 5 cm)

I

60

142

(n cm -2 S -1) 10-3 I

K Kobayashi et al.

' I ' I

10 -4

10 -5

10 -6

10 -7

10 -s

10 -9

~7

v

0

A

V Analytical Solution

-~- GMVP (JAERI) × MCNP4B (JAERI) [] MCCG3D RT/SC (IPPE,UTK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) v IDT LC (CEA) o FNSUNCL3 (JAERI) zx TORT (JAERI) @ EVENT (IC) • ARDRA (LLNL)

20 40 x (cm)

F i g . 39. P rob l em 3Ci "No scat ter ing" (y = 95 cm, z = 35 cm)

@

=

I

60

-2

-4

I ' I ' I ' I

v

@

x7 q- GMVP (JAERI) × MCNP4B (JAERI) o MCCG3D RT/SC 0PPE, LrrK,CEA) o PENTRAN (USAF, PSU) o PARTISN FC (LANL) v IDT LC (CEA) o FNSUNCL3 (JAERI) zx TORT (JAERI) © EVENT 0(2) • ARDRA (LLNL)

@ @

I I I I J

20 40 x (cm)

F i g . 40. Re la t ive flux of p rob lem 3Ci (y = 95 cm, z = 35 cm)

i 60

(n cm -2 S -1) 10 -3 ,

3 D radtation transport benchmark problems

' ! ' I

143

10 -4

10 -5

~7

- @ . . . . . .~,

" ' - .% ~

- -+-- GMVP (JAERI) × MCNP4B (JAERD [] MCCG3D RT/SC (IPPE, UTK,CEA) o PENq~AN (USAF, PSU) zx TORT 124 (ORNL) o PARTISN FC (LANL) v IDT LC (CEA) <> TORT with FNSUNCL3 (JAERI) @ EVENT taC) • ARDRA (LLNL)

" ' . %

, I , I

20 40 x (cm)

Fig. 41. Problem 3Cii "50% scattering" (y = 95 cm, z = 35 cm)

I

60

I 2 I ' I ~ I

;>

1.5

0.5

0

-0.5

@

@

0 o

X MCNP4B (JAER1) [] MCCG3D RT/SC 0PPE,UTK,CEA) o PENTRAN (USAF, PSU)

TORT LN (ORNL) o PARTISN FC (LANL) v IDT LC (CEA) o T O R T with FNSUNCL3 (JAERD

EVENT (IC) • ARDRA (LLNL)

I I L I I

20 40 x (cm)

Fig. 42. Relative flux of problem 3Cii (y = 95 cm, z = 35 cm)

@

I

60

144 K. Kobayasht et al

A C K N O W L E D G E M E N T S

The authors wish to express their sincere thanks to Richard Sanchez for providing exact values of the pure absorber cases, by which the exact values could be confirmed by the independent method. They would like also to thank the participants for their contribution to this benchmark, E. Sartori of OECD/NEA for his support to this benchmark, and E. Kiefhaber of Forschungszentrum Karlsruhe and Y. Azmy of ORNL for their useful comments to improve the manuscript.

R E F E R E N C E S

Ackroyd, R.T. and Riyait, N.S. (1989), Iteration and Extrapolation Method for the Approximate Solution of the Even-Parity Transport Equation for Systems with Voids, Ann. nucl. Energy, 16, 1.

Alcouffe, R. (2001), PARTISN Calculations of 3D Radiation Transport Benchmarks for Simple Geometries with Void Regions, this volume of Progress in Nuclear Energy.

Azmy, Y. Y., Gallmeier, F. X. and Lillie, R. A., (2001), TORT Solutions for the 3D Radiation Transport Benchmarks for Simple Geometry with Void Region, this volume of Progress in Nuclear Energy.

Brown, P.N., Chang, B. and Hanebutte, U.R., (2001), Spherical Harmonic Solutions of the Boltzmann Transport Equation via Discrete Ordinates, this volume of Progress in Nuclear Energy.

Haghighat, A. and Sjoden, G.E. (2001), Effectiveness of PENTRANTM's Unique Numerics for Simulation of the Kobayashi Benchmarks, this volume of Progress in Nuclear Energy.

Jung, J. et al. (1974), Numerical Solutions of Discrete-Ordinate Neutron Transport Equations Equivalent to PL Approximation in X - Y Geometry, J. Nucl. Sci. Technol. 11,231.

Kobayashi, K. (1995), On the Advantage of the Finite Fourier Transformation Method for the Solution of a Multi-group Transport Equation by the Spherical Harmonics Method, Transp. Theory Statis. Phys. 24, 113.

Kobayashi, K. (1997), A Proposal for 3D Radiation Transport Benchmarks for Simple Geometries with Void Region, 3-D Deterministic Radiation Transport Computer Programs, OECD Proceedings, p.403.

Kobayashi, K., Sugimura, N. and Nagaya, Y. (1999), 3D Radiation Transport Benchmarks for Simple Geome- tries with Void Region, M ~ C ' 99 - Madmd, Mathematics and Computation, Reactor Physics and Environ- mental Analys,s in Nuclear Applications, p.657, Madrid, 27-30 September.

Konno, C. (2001), TORT Solutions with FNSUNCL3 for KOBAYASHI's 3D Benchmarks, this volume of Progress in Nuclear Energy.

Kosako, K. and Konno, C. (1998), FNSUNCL3 - GRTUNCL code for TORT, JAERI/Review 98-022, p. 140. See also Kosako, K. and Konno, C. (1999), FNSUNCL3: First Collision Source Code for TORT, Proc. of the Ninth International Conference on Radiation Shielding ICRS-9, Tsukuba, 17-22 October, Journal of Nuclear Science and Technology, Supplement 1, March 2000, p. 475.

Mori, T. and Nakagawa, M. (1994), MVP/GMVP: General Purpose Monte Carlo Codes, JAERI-Data/Code 94-007 [in Japanese].

Oliveira, C.R.E., Eaton, M.D. and Umpleby, A.P. (2001), Finite Element-Spherical Harmonics Solutions of the 3D Kobayashi Benchmarks with Ray-Tracing Void Treatment, this volume of Progress in Nuclear Energy.

Reed, WM. H. (1972), Spherical Harmonics Solutions of the Neutron Transport Equation from Discrete Ordi- nate Codes, Nucl. Sci. Eng. 49, 10.

Sartori, E. (1997), Editor, 31) Deterministic Radiation Transport Computer Program. Features, Appli- cations and Perspectives, OECD Proceedings, Nuclear Energy Agency.

Suslov, I., (2001), Improvements in the Long Characteristics Method and Their Efficiency for Deep Penetration Calculations, this volume of Progress m Nuclear Energy.

Takeda, T. and Ikeda, H. (1991), 3-D Neutron Transport Benchmarks, NEACRP-L-330.

Zmijarevic, I. and Sanchez, R. (2001), Detarministic Solutions for 3D Kobayashi Benchmarks, this volume of Progress in Nuclear Energy.