Partisn calculations of 3D radiation transport benchmarks for simple geometries with void regions

Progress m Nuclear Energy, Vol. 39, No. 2, pp. 181-190, 2001 © 2001 Published by Elsevier Science Ltd. Printed in Great Britain 0149-1970/01/$ - see front matter

Pergamon www.elsevter.com/locate/pnucene

PII: S0149-1970(01)00011-7

PARTISN CALCULATIONS OF 31) RADIATION TRANSPORT BENCHMARKS FOR SIMPLE GEOMETRIES WITH VOID REGIONS

RAYMOND E. ALCOUFFE Transport Methods Group, D409 Los Alamos National Laboratory Los Alamos, NM 87545 ([email protected])

A b s t r a c t - We have solved the 3D radiation transport benchmark problems proposed by Prof Kobayashi using the transport code PARTISN. We have used different angular discretizations represented by the Sn method of various orders of n to investigate the resultant sensitivity of the solution. In addition, we have employed our first collision source method as incorporated in the code to mitigate the substantial ray effects present. We also include the computation time needed for these problems on our SGI Origin/2000 platform. © 2001 Published by Elsevier Science Ltd. Keywords: 3D Transport Calculations; Benchmark Problems; Shielding Calculations 1 INTRODUCTION Benchmark quality transport solutions of complicated three-dimensional systems are rare in the literature Takeda (1991). Kobayashi et al have defined such benchmark problems and solutions in a proposal to the OECD/NEA and are presented in Kobayashi (1997). We present here a study of those systems using the Los Alamos transport code PARTISN. The aim of this study is to compare the standard methods of solution incorporated in PARTISN to either the analytic based or Monte Carlo based solutions published in Kobayashi (2000). We decided to focus on the angular approximation since these systems contain extensive void regions and the source region is small compared to the dimensions of the entire system. This is the classical case where ray effects are very important in the discrete ordinates approximation which is the basic method used in PARTISN. In this paper we present a brief description of some of the relevant features of PARTISN, followed by a description of the discretization of the benchmark problems after which we present the calculational results of our study along with our computational times. This is followed by some conclusions. 2 PARTISN PARTISN is the follow-on code to the publicly released DANTSYS code, Alcouffe et al (1996). The main distinguishing feature is that PARTISN has parallel implementations of most of the solution algorithms and thus is operational on all ASCI platforms and has been demonstrated on up to 3200 processors with good to acceptable parallel scaling performance depending upon the specific platform. The geometries handled by the code are all structured and orthogonal meshing. These are 1D spheres, cylinders, and slabs, 2D X-Y, R-Z and R-O, and 3D X-Y-Z and R-Z-®. As pointed out above the angular discretization is Sn with provisions to treat up to S-100 in two quadrature options: either a triangular or a square arrangement of the discrete ordinates over an octant of a sphere. The angular scattering transfer function (phase function) is represented as a spherical harmonics expansion up to order n-1 of the Sn approximation. The energy variation is represented in a multigroup approximation in the usual manner. We have incorporated into PARTISN three possibilities for the spatial discretiza181

R. E Alcouffe

182

tion: (1) diamond differencing with set-to zero fixup, (2) adaptive weighted diamond (AWDD), and (3) linear discontinuous (LD). For 3D, X-Y-Z problems we also have the possibility of using exponential discontinuous spatial differencing. The solution strategy used is source (von Neumann) iteration where the source is that due to scattering and fissions. The within group scattering, which couples all the angles for a group, is solved as an inner iteration for each energy group; the fissions which couples all the groups is handled by an outer iteration procedure. The down scatter is a known source since we follow particle flow by starting from the highest energy group and proceed downwards. Upscatter is treated in the same manner as fissions in the outer iteration. Thus outer iterations are not needed if there are no fissions or upscatter. Both the inner and outer iteration convergence is accelerated by a diffusion synthetic acceleration (DSA) method. All of these features have a parallel implementation in PARTISN and all scale well to large numbers of processors as documented in Alcouffe and Baker (1999), B aker and Alcouffe (1997). Because the proposed benchmark problems are characterized by the manifestation of severe ray effects in the discrete ordinates approximation, we describe here in a little more detail the first collision source option incorporated in PARTISN to mitigate ray effects. The theory is based upon a split of the transport equation into collided and uncollided parts. That is, given the following transport equation: L

l q

• V~l/g

~

q

+~,t,g~lg(~, ~) ---- Z ]~s,l,g Z Vl, g(r)Yl (~) + Qg(t' ~) l =0 q =-1 (1)

where Y is the spherical harmonic polynomial, we split it into two transport equations for the tmcoltided and collided fluxes: ~

0

UC

UC

V~lg + ~t,g~lg (r, ~) = Qg(L ~2) L l c c n " V~g + ]~t,g~lg(}, ~'~) = Z ~s,l,g Z ~qlCl,'ff(})Yq(n) + l=0 q=-l L l uc, q Z ]~s,l,g Z Vl, g (~)Y/~(~'~) 1=0 q=-I (2) The first of these equations is thus solved for the uncollided flux and that solution is put into the second equation as the first collision scattering source. This second equation is then solved for the collided flux. The advantage of this procedure is that the uncollided flux equation can be solved with a method that is more appropriate for the situation of a localized source. Since it does not involve any iterations due to collisions, an analytic or semi-analytic method can be used. In PARTISN if the source is defined for an extended volume within the system, we use a ray tracing method for the solution. That is the transport equation is solved for a collection of rays whose origin is randomly distributed over the source region and whose extent goes to the exterior boundary of the problem. The angular direction of each of the rays is chosen from a random distribution. For many problems this is a powerful method for mitigating ray effects that result from a localized source in a system. The rays are done in batches each of which consist of a few thousand rays at a time. We typically use 20 batches. The batch method is used so that a variance of the flux can be computed on the solution mesh thus giving an assessment of the goodness of the solution. The ray tracing method in PARTISN is not yet parallelized and thus our use of it involves only a single processor on our SGI ORIGIN/2000.

PARTISN calculanons

3 DISCRETIZATION SELECTIONS FOR OUR STUDY For all of the problems, the total cross section is 0.1 cm -1 in the non-void regions. The output edit requirements from the benchmark specifications and spatial discretization accuracy considerations suggest a uniform mesh spacing of no more than 2 cm (the edit points must lie at the center of a mesh interval). Given the specified total cross section, the mesh cells are small in terms of mean free paths (0.2). With this fine uniform mesh, the spatial discretizarion we chose for the no-scattering cases is LD while AWDD is accurate for the cases with scattering. This leads us then to the choice of the angular approximation. As mentioned above for these types of problems with the localized source in void or low scattering regions, we prefer the use of the first collision method. In the case of the benchmark problems with no scattering, this is the total solution and hence if enough rays are chosen, we should get very close to the analytical solution provided. In this study we used from 600000 to 106 rays which gets an acceptable solution. Nevertheless, we also investigated what Sn order we would have to use to approach the analytic results. We show that one needs a quite high order Sn order to achieve acceptable accuracy; this is the main result of this study. In the following the pure Sn calculations were done using our parallel algorithms on 4 processors of our SGI ORIGIN/2000 computer while all first collision (FC) calculations were done on 1 processor since this method is not parallel in PARTISN yet. 4 CALCULATIONAL RESULTS In the following we present graphical results of the solution to problem 1 a which has no scattering and its companion with 50% scatter which we designate as problem lb. We show both the flux traverses and the relative error with respect to the reference analytical solution for the various Sn orders used. The first collision source calculation used 600,000 rays in both cases la and lb. As defined above these 600,000 rays are in 20 batches each of which consists of 30,000 rays. The analyric solution is provided as the reference and is designated by X on the traverse graphs.

183

R E. Alcouffe

184

f

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. . . . .

a3Ot~d •

",--x

.

__-~-Z_:

^

i

\ 12 0

20

40 Y ( o m ) 60

80

100

Fig. 1 a. Flux from 1 a traverse along Y at X=5, Z=5.

O

20

40 Y ( ¢ m ) 60

80

1o.0

Fig. lb. Flux error from l a traverse along Y at X=5, Z=5.

1o' o5 :

~

FCS

~ ' ~

°~"1°=

4

o

60

,,

i

i

80

1

060

X (om)

Fig. lc. Flux from l a traverse along X at Y=55, Z=5.

. . . . . . . . . . . . . . . 20 40 X ( o m ) 60

8o

, ,

O9

ft~ s12t-wa :20t-wa ~ot~, 1140t-Wd

-

20

40

~0

.~

FC$

,~

i ~. _ _ I ....

0

100

Fig. ld. Flux error from l a traverse along X at Y=55, Z=5.

o8

x-

, ,

80 100 120 D l e g o n a l {ore)

, 140

160

180

Fig. le. Flux from l a traverse along the diagonal.

oa

/~

os;,,, 'io''' '4'0''' '.o''' 'io''' ~0;" ~2; :,4;'' i.;" ~80 Diagonal

(ore)

Fig. lf. Flux error from l a traverse along the diagonal.

PARTISN calculations

185 O,4

,

F~ll

O~ O~ 025

0.2

l ....

I'/r '

,~

.15

[~

__.---4-~

"

~

104

1040

20

40

(;0

80

10(

20

Y(~)

Fig. 2& Flux from l b traverse along Y at X=5, Z=5.

40 Y (om) eo

so

~0o

Fig. 2b. Flux error from l b traverse along Y at X=5, Z=5.

O.3 0.25 02

•

\

,

"'"

"~ . . . .

_ -2- -

lO~

.~"

L

V

\/"!

--..-o- - 88o ~0

/t

"& x \

/t1\

\

e 40 X ( o m )

eo

10o

8o

X 1~1

Fig. 2c. Flux from l b traverse along X at Y=55, Z=5.

Fig. 2d. Flux error from l b traverse along X at Y=55, Z=5.

:g

....

ff /' / /

i- _-5 - i--J 2 !~-! l ~ - ; ! - L_~_-<_~ _

_ c = ....

/ ! [ L

.~

s.~ 20

40

: E : : - = : - 5 - _j ~

eo

100

Dkgomd(~,.}

120

~.~

~

,/.A ~

A n. v. :\-,e" •

.......... 140

160

18(

Fig. 2e. Flux from l b traverse along the diagonal.

2o'" ,o""6o Dbo.;":.~ ~2~";,~",.~ ~ g o n M (¢m) Fig. 25. F l u x error from l b traverse along the diagonal.

The results from the la problem show that the first collision calculation (which because there is no scattering is the entire PARTISN solution) gives excellent agreement with the reference analytic solution. By excellent agreement we mean that the solution lies within 10% of the reference solution. In order to achieve a high degree of accuracy, the Sn order has to be around $40 which means 1680 angles in our quadrature arrangement. In the lb case the required Sn order can be reduced to achieve an acceptable solution (within 25% of the analytical solution) but we still need either an $20 or an $30 approximation which means 440 to 960 angles. This is a much higher number of angles than is

R.E. A&ouffe

186

typically used to compute shielding problems. We should also point out that for the scattering cases (lb traverses), the FC3-S12 means that the uncollided flux is computed from 600,000 rays and the collides flux is computed using the S12 approximation. Also for some points the FC method solution strays outside the 10% boundaries. Problem 2 presents a challenge in computing the flux in a voided, narrow channel. We present the results below in the same format as for problem 1 where 2a is the problem without scattering and 2b is the same problem with 50% scattering. For the no scattering case the first collision method with 106 rays (FC5) gives an acceptable answer whereas we require an $50 (2600 angles) calculation with the pure discrete ordinate approximation. With scattering, we can use an $30 calculation while the FC3S12 method (600000 rays for the uncolfided) does well with an S12 approximation for the collided flux.

0e

;

L---.--Fel 0"4_-'-'~-~

o,!

I:! I I I

II

,"[',':iii:ii', ri~ I-,.

!

-o.~

--

-- _ . . . . . . . . . . . . . = --

10" O '

'

'

' 20

40

Y (orn)

80

80

Fig. 3& Flux from 2a traverse along Y at X=5, Z=5.

: i [ I i t I

i ilj i/!i I/1 I i

i

i

-°'~o

,

,

I

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bl

114

;

; i

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;

x (0.1,)

iI

r

I

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]

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I

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,\

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r.)~,.

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%;

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,\! +

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r

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! I i ~,

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i

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'.'~4

! I~-, .... It t-'~.£ ~ - - f---.4t-.+-~-"t- oi

,Jli I

i

~ i ! I

t

I

,

Fig. 3c. Flux from 2a traverse along X at Y=95, Z=5.

, 80

;

!~-:l=,-4-kli~..- : --1~,,-b,~

40

i

Hg. 3b. Flux error from 2a traverse along Y at X=5, Z=5.

1

X(*m)

I

i?,?q

il',i I

.

!i

$,!

: 'I

I,'.! I 1 I I 20

,

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~ .

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'i!,

~

' ; i

I,-,

il

i ~\,

I 1\[

1

/

i

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i L L i

--"--~. .

X (,~r~)

Fig. 3d. Flux elror from 2a traverse along X at Y--95, Z=5.

PARTISN calculations

!

~

__: _~_ m

x

187

":I

iC FC3-Si2 Illl

' :

oJ

r

-:-:

:

:

~-

:_ . = _ - . _

t

. . . . . . . .

,

20

40 Y

i

,

,

(~)

60

,

,

,

,

8o

,

,

,

,

lO0

Fig. 4a. Flux from 2b traverse along Y at X=5, Z=5. . _=

7:

;=.

" --,

....

"

.

.

.

.

-

.

i

,i,

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,,

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X

,

o

'....

/

~ 2o

'I

~D4"~

1'

i

4o V (tin) 6o

\

:

eo

1oo

Fig. 4b. Flux error from 2b traverse along Y at X=5, 7.=5.

"I

02E,

_

--

i_J

FC3-Sli

015

01

,\ i

I

!

~02S -03 I

'

a 0 0

'i

-035 o

20

4O

X (~)

6O

Fig. 4c. Flux from 2b traverse along X at Y=95, Z=5.

Fc3-s12

$12 s~ sio

0

20

4o

X (om)

6o

Fig. 4d. Flux error from 2b traverse along X at Y=95, Z=5.

The third problem is one with a three dimensional duct nmning through a block of either pure absorber for 3a or 50% scatterer for 3b. In the 3a problem the FC method does well using 1.6x106 rays (FCS) while the $30 is marginally acceptable. We see that the answers are converging even at the difficult point of Y=95 while the $12 and even $20 results are abominable. Adding 50% scattering mitigates greatly the ray effects so that the $20 and $30 results are acceptable and the FC5-$12 results are acceptable given our 10% criterion. -{L+

~:s:-

I?:

:-

- - t - + .--T.

L:ul-- ~ :

•

ill

....

~.;~

,

-,

SiO r

t

j .......

+

FCS

. . . o. .

,

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- - 0 - -

°

~ ,

,7;

,

,

.

0

.

.

.

.

.

20

.

.

.

.

.

.

40 Y (ocn) 60

,

,

,

,

,

80

,

,

,

%/5

:,.

r

,

100

Fig. 5a. Flux from 3a traverse along the channel.

20

40

Y ((.xl) 60

80

~

100

Fig. 5b. Flux error from 3a traverse along the channel.

188

R. E

Alcouffe 05

r

O,4

r

!

FOI S.IO

0:3 -

~o,

r

--o-

;

L g:

-

$1= SSO

I 'e

r ,

-o I

~0

40

X (om)

60

X (*m)

Fig. 5c. Flux from 3a traverse along X at Y=55.

Fig. 5d. Flux error from 3a traverse along X at Y=55.

Fell

o ---O-

W

S12 sso

-

I

.

. . . . .

i

,-

i

~

}

i'

,

~

-,~

'

i

(

......

r

~'~.

i

i

\\ : l

i

i

20

.

•

i

I

40

,F

i

.

, 60

X (om)

Fig. 5e. Flux from 3a traverse along X at Y=95.

-

Fig. 5t". Flux error from 3a traverse along X at Y=95.

FC6-S12 S30 S~

.......

-

.\ o

20

~

Y(*m)

~

~

1~

Fig. 6a. Flux from 3b traverse along the channel.

20

40 Y ( o m )

60

SO

lO(

Fig. 6b. Flux error from 3b traverse along the channel.

PARTISN calculations

189

MC FC$-SI2 SlO S20 s12

x . . . . . . . . .

1~ 1

\~¢//

"....

~1o ~

FCS-S12

v

! :,7

I

1040

4O

20

. . . .

60

20

L

*

i

i

41o

'

'

'

i

60

X (=m)

X (cm)

Fig. 6d. Flux error from 3b traverse along X at Y=55.

Fig. 6c. Flux from 3b traverse along X at Y=55.

04

X

MC

0

Fcr*-s12 0.3

-S20 $12

FC$'S12

0

$30

A 0

S12

02

\ , , , , 0

, , , ,

+.:i I

J

I

40

20

60

i

,

J

i

i

i

J

2o

4o

J

,

i

i

X (om)

x(~)

Fig. 6e. Flux from 3b traverse along X at Y=95.

Fig. 6£ Flux error from 3b traverse along X at Y=95.

The computer time to solve each of these problems is given in the following table. The Sn problems without the first collision source option were done on 4 processors of an SGI ORIGIN/2000. The first collision option is not parallelized thus it was done on 1 processor of the same machine. The times shown are in wall clock seconds.

T A B L E 1. W A L L C L O C K T I M E F O R E A C H O F T H E A P P R O X I M A T I O N S

Wall Clock Time (seconds) i

Problem

FC a

$12

$20

I

$30

$40

$50

i

nea3d-la

123.9

7.4

19.7

43.3

nea3d-lb

271.7

48.9

100.0

209.3

nea3d-2a

122.5

4.2

8.6

17.2

nea3d-2b

153.6

14.8

74.1

34.8

nea3d-3a

122.3

2.7

7.1

15.4

nea3d-3b

236.9

15.9

31.6

74.1

a. Single processor of SGI ORIGIN/2000

83.6 44.3 42.4

190

R E AlcouJfe

5 CONCLUSIONS The benchmark problems proposed by Kobayashi et al can be successfully solved by the discrete ordinates method in a reasonable amount of computing time. However because of the localized source and the voids in the problems, we require significantly more angles to obtain an accurate solution than the $8 or S16 normally used to solve shielding problems. The accuracy can be greatly improved by using the first collision source method to account for the source near singularities in regions far from the source while handling the collided flux with a much lower Sn order ($8 - S12). Our original goal of obtaining a benchmark quality solution that is within 10% of the analytic or MC results has been met with the first collision source method at all points, and with straight Sn in all but 4 points of the prescribed edits.

REFERENCES

R.E. Alcouffe, et al (1995), DANTSYS: A Diffusion Accelerated Neutral Particle Transport Code System", Los Alamos Technical Report, LA-12969-M R. E. Alcouffe and R. S. Baker (1999), Impact of Very Large Parallel Computing Platforms on LWR Core Calculation, Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, M&C '99 - Madrid R. S. Baker and R. E Alcouffe (1997), Parallel 3-D Sn Performance for DANTSYS/MPI on the Cray T3D, Joint International Conference on Mathematical Methods and Supercomputing for Nuclear Applications. K Kobayashi (1997), A Proposal for 3D Radiation Transport Benchmarks for Simple Geometries with Void Regions, 3-D Deterministic Radiation Transport Computer Programs, OECD Proceedings, pp 403-410 K Kobayashi et al (2000), 3D RADIATION TRANSPORT BENCHMARKS FOR SIMPLE GEOMETRIES WITH VOID REGIONS, to be published Progress in Nuclear Energy. T.Takeda and H. Ikeda (1991), 3-D Neutron Transport Benchmarks, OECD/NEA Committee on Reactor Physics Report, NEACRP-L-330.