Transfer by anisotropic scattering between subsets of the unit sphere of directions in linear transport theory

Ann. nucl. Energy, Vol. 17, No. 1, pp. 1-18, 1990 Printed in Great Britain. All rights reserved

0306-4549/90 $3.00+0.00 Copyright © 1990 Pergamon Press plc

TRANSFER BY ANISOTROPIC SCATTERING BETWEEN SUBSETS OF THE UNIT SPHERE OF DIRECTIONS IN LINEAR TRANSPORT THEORY

T. TROMBETTI

Laboratorio di Ingegneria Nucleare di Montecuccolino, Universit~ di Bologna. Via dei Colli 16, 1-40136 Bologna, Italy

(Received for publication 6 October 1989)

Abstract--The exact kernel method for linear transport problems with azimuth-dependent angular fluxes is here based on the evaluation of average scattering densities (ASD's) that fully describe the neutron (or particle) transfer between subsets of the unit sphere of directions by anisotropic scattering. Reciprocity and other ASD functional properties are proved and combined with the symmetry properties of suitable SN quadrature sets. This greatly reduces the number of independent ASD's to be computed and stored. An approach to perform ASD computations with reciprocity checks is presented. The ASD expressions of the scattering source for the typical 2D geometries are explicitly given.

I. INTRODUCTION Exact

kernel

methods

{Brockmann

(1981))

have

been

developed

to

avoid

spherical

harmonics series expansions in the determination of the angular dependence of the neutron scattering source in numerical transport calculations. nonexpanded)

angular

aS(g '~ g; I).

In

(1)

fact,

dependence

when

the

of

latter

the

They account for the "exact"

group-to-group

is strongly

anisotropic

transfer

cross

or confined

(i.e.

section

to a small

y-range, early series truncation may yield unphysical oscillating and/or negative quantities (Takahashi

et al.

(1979),

Brockmann

(1981),

Ligou and Miazza (1988)).

Here 7 denotes the

cosine of the scattering angle in the laboratory system (CSL). With the direct kernel method computed at I = Q

. O m'

(Odom and Shultis (1976),

Brockmann (1981)) o 8 must be

Integration of neutrons scattered from the unit sphere ~, i.e. the m

"

sphere of the unit vectors (denoted by symbols with caret) ^ O' of R 3, are performed by means of summations

over the M discrete

(e.g. SN)

directions O m

J

, m' = 1, 2,..., M. However,

serious (storage and computational) difficulties are introduced. Nith a method based on the transformation of the scattering kernel, as the outer

integration

variable.

Again a serious difficulty

arises,

7 itself is taken since the angular

fluxes must be computed by interpolation at directions which are dependent on 7, hence not ^

coincident with the

O ma

, m' =

I, 2,

,"

., M. The I X method (Takahashi and Rusch (1979))

can also be derived in this way (Brockmann {1981)). ! ~[

17:1-A

2

T. TROMBETTI

Azimuthal (rotational) symmetry of the angular neutron flux with respect to some polar axis,

if present,

alleviates the difficulties

and favors the implementation of the above

methods (Odom and Shultis (1976), Takahashi and Rusch (1979}).

For the I* method it avoids

interpolation entirely. Hence this method could be implemented into an SN code (ONETRAN) for 1-D plane and spherical geometry (Schwenk-Ferrero (1986)) and the 3-D integrals defining the matrix elements I*(k,l,m) could be solved analytically by a general very efficient algorithm (Ligou and Miazza (1988)). "Azimuthal bridge

to the

contrary,

symmetry"

will be invoked

I* method,

azimuthal

and will always

in this paper only as a special case or as a implicitly

refer to the angular

symmetry of the scattered neutrons

distribution

flux. On the

with respect

to the

incidence direction will always be assumed. With azimuthal dependence the requirements of avoiding both the series expansion of a a (using the "exact"

transfer cross section I-dependence)

neutron fluxes might look contradictory.

and the interpolation of angular

In this paper we envisage an auxiliary problem.

We

show how by means of its solution both the above requirements can be met, i.e. an "exact" kernel method with neither azimuthal symmetry nor interpolation of the angular flux built up. Our

auxiliary

problem

is the calculation

of

the density

of neutrons

or

particles

scattered with a fixed CSL : ¥ from an "origin" ~irection ~', or a set B 0, thereof ("origin" set),

into some "destination"

set(s).

set B o of directions ~, averaged over the area(s)

of such

With these average scattering densities (ASD's) and the aa(g '* g; 7) available, the

neutron scattering source can be computed by integration over ~. This method can also be modified to allow the integration over ¥ to be performed first. The ASD is then defined for neutrons scattered with a uniform distribution of CSL for 7 ~ by c [-1,1]. This modified ASD method will be discussed in a second paper. Reciprocity

properties

storage reduction purposes. symmetry

and reciprocity

mutually

reproduced

either

for the ASD's

and used

are combined with a simultaneous

based on sorting techniques nonzero,

are proved

for numerical

checks

For one-level SN quadrature sets (Alcouffe and O'Dell

(Singleton (1969)).

integration approach

This selects, computes and stores only all

independent

ASD's.

The I* function,

as

cases

(with

special

numerical

and

(1986))

azimuthal

matrix elements symmetry)

or

as

and properties byproducts

are

(linear

combinations of ASD's).

2. THE AVERAGE SCATTERING DENSITIES AND THEIR FUNCTIONAL PROPERTIES 2 A. The Average Scattering Densities (ASD's). Let us choose on R, the sphere of the unit vectors of R 3, a polar axis with unit vector (the axis of symmetry for azimuthally symmetric problems) and consider the usual set of spherical coordinates ~ c [-1,1], @ c (-~,~], where ~ = cos e, e and ~ are colatitude and longitude.

However, we shall often refer to the cosine (here ~) of the colatitude (here e)

as the (cosine-)latitude

(leaving out the specification "cosine-"

if unnecessary) ^

as the

azimuth

or

longitude,

as usual.

We

shall

generally

take ~'

and to

^

and ~ c R

coordinates ~',@' (see Fig. 1) and ~,~ respectively, and denote this choice by

to have

Transfer by anisotropic scattering in linear transport theory

O' = ( P ' , ~ ' ) ;

3

0 = (P,@).

(1)

Let us consider on ~ a neutron undergoing scattering from a given "origin" direction ~' ~ R

with

a

CSL = 7 c [-1,1].

The

neutron

is

scattered

"uniformly"

over t h e

^

circle

^

C = C(O',7) having "center" and "radius" (say) O' and 7, i.e. the locus of all ~ ~ ~ such ^

^

that 0'. O = 7 • Hence the fraction,

F(O'~A~;

7), of that neutron scattered

into a given

^

"destination" set /tO c ~ equals the fraction of C that intersects hO.

"

F i g . 1. The i n t e r s e c t i o n s circle Bo and

of the

Fig. 2. The set B o as the

C • C(O',I) with the sets Ho.

spherical

The

sides

triangles

by t h e r e s p e c t i v e

of

the

are labelled cosines.

a n g l e s t o be i n t r o d u c e d in

i

union

of a family

H(t)

of

parallels.

The

latter

have

t e [ t ~ , t z]

(in

colatitudes

arcs

of

radians) with respect to a polar

The

^

a x i s j (which n e e d s n o t be k ) .

Eqs.

(5) and (6) a r e shown.

We now c o n s i d e r t h e bins),

i.e.

such t h a t

family •

of subsets of ~,

A(B l) ~ fB dO > 0,

if

that

have p o s i t i v e

Bi~ ~.

Likewise,

positive

length,

i

smooth c o n t i n u o u s c u r v e s on ~ h a v i n g f i n i t e

area

we c o n s i d e r L(H i) •

fH

(e.g.

angular

the family •

ds > 0,

of

i f Hi E ~.

i

Here dO and ds a r e t h e e l e m e n t a r y a r e a and a r c l e n g t h on ~. For s i m p l i c i t y we s h a l l speak o f a n g u l a r b i n s and ( s i m p l e smooth) a r c s ,

respectively,

though we a r e

not confined to

these

classes of sets. I f now hO = B° E ~ , (ASD)

we i n t r o d u c e t h e p o i n t - t o - ( a n g u l a r ) b i n

average scattering

density

4

T.

TROMBETTI

Dpe(O',Bo;~) = --I F(O'-Bo,~).

(2)

A(B o ) If we have H o E ~, we consider the set ~

E ~ bounded by H o and an arc lying on one side of

=

Ho, at an infinitesimal distance dt. Then A(A~)

L(Ho)dt.

^

^

DpL(Q',Ho;~ ) =

tim

dr'0

We define the point-to-line ASD

^

DpB(O',~ ;~).

(3)

^

In this case F(O'~AO,

~)

is the sum of as many contributions

as are the intersections of

C(O',7) with H o. Each contribution equals dt [2~ Isin X i l / 1 - ~ ] -I, if Xi(O',Ho,~)

is the

angle with which the i-th, Oi, of the I intersections occurs. Hence

Dpt(O', HO;~)

=

in Fig. 1 we show the intersections of C - C(3',y): (there

may be

nonnegative

any nonnegative

number of them).

(4)

^ 2~ / 1 - 7 ~ L(H o) {=' {sin Xi(O',Ho,¥) }

even

number

of

Each intersection

them);

6, 3

with the boundary of Bo

0 , 0 z with

Ho

(there

is a vertex of a spherical

other two vertices being fixed at k, 0'. The sides of the triangles

may be any

triangle,

the

shown in Fig. I are

labelled by their cosines. All triangles have two sides of fixed cosine, ~' and I.

To f i n d t h e ASD's f o r t h e s e t s spherical

trigonometry)

of Ho w i t h C. E . g . ,

BO o r Ho, t h e t r i a n g l e s

f o r t h e a n g l e s ~^, ~ e '

in Fig.

a r e t o be s o l v e d ( r e s o r t i n g

" ' " a t O' o r ~1' ~ z '

"''

to

at the intersections

1 we have ^

F(O'*Bo;¥) = (~B-~^)/2~;

(5)

X1 = 61 + ~1 - ~ / 2

(6)

t o be i n t r o d u c e d i n Eqs~^(2) f o r DpB and (3) f o r DpL. The i n c l i n a t i o n

71 o f Ho w i t h r e s p e c t

t o t h e a r c o f m e r i d i a n kO 1 i s o f c o u r s e n e e d e d . To d e r i v e an " i n v e r s e " a family H(t),

t E^[tl,t2],

o f Eq. (3) we c o n s i d e r t h e a n g u l a r b i n Bo a s b e i n g t h e u n i o n o f of arcs of parallels,

t o any p o l a r a x i s j and t h e d i s t a n c e

where t E [ 0 , ~ ]

is the colatitude

between t h e end p o i n t s o f H ( t ) ,

H(t + dt)

referred

is O(dt)

for

dt * 0 except, possibly, for a finite number of values of t. In Fig. 2 we show the minimum ^

and maximum colatitudes, t I and t 2 (in radians), of points of B o with respect to j. Then t

A (Bo) = ftZdt L[H(t)]

(7)

1

^

D P B ( O ' ' B o ; ¥ ) = A(Bo)

t - 1 2

^

ft

dt L[H(t)] DpL[~',H(t);~].

(8)

1

This

can

(appearing

also at

be

interpreted

the r.h.s,

(appearing at the l.h.s°: longer as concentrated within

some " o r i g i n "

as

according s e e Eqs.

an

integral

t o Eqs. (2),

(5)).

in O', but rather set

AO' c ~ .

Let u s now t h i n k

The u n d e r l y i n g

be a p i e c e w i s e

between

such

angles

(6) and b e i n g f u n c t i o n s

(extracted

n e u t r o n f l u x w i t h i n AO' i s no more r e s t r i c t i v e when a z i m u t h - i n d e p e n d e n t ,

relation

(4),

function

71

of the scattered

n e u t r o n no

assumption

uniformly distributed of

than the assumption that

constant

~,

and ~^, ~n

from a p o p u l a t i o n ) related

as

of t)

constant

angular

the angular

of ~' E (-1,1).

flux,

An e q u i v a l e n t

Transfer by anisotropic scattering in linear transport theory

5

assumption is implicit, e.g., in the scattering source computation for azimuthally symmetric fluxes by the I" method (Ligou and Miazza (1988), Eqs.

(2),(3),(4); Takahashi and Rusch

(1979), Eq. (75)). ^

If A~' = B o' c ~, we define the bin-to-bin (B~ to B o) ASD

DBs(B~,Bo;7) =

1

fs' d~' Dpe(~',So;~) =

A(B~) =

1

o

fs~ dO' F(Q' ~ Bo;7 ).

(9)

A(Bo ) A(B~) If ~'

= Ho' ~ ~, we d e f i n e the l i n e - t o - l i n e

(H~ to Ho) ASD ^

1 f"6 ds DPL(O' ,Ho;7) L(H~)

DLL(Ho'Ho;¥) :

(10)

where ds is the elementary arc length on H o at ~'. Combining Eqs. (9), (8) and (I0) yields

Dse(B~,Bo;7) = A(Bo )-I A(B~) =I. t )

t

2

"Jr' dr' L[H'(t')] ft I

2 dt L[H(t)] DLL[ H' (t),H(t);~] )

(11)

I

where B 0' ~ • is constructed as the union of a family H'(t'), t' ~ [t~,t ' 2], in the way (and with the same hypotheses) B o was constructed from H(t). Up to now we have introduced the four kinds of ASD we shall mainly deal with. They are identified by the function symbol D and two (PB, PL, BB, LL) mnemonic subscripts (P-point B-bin L-line) referring the first to the origin, the second to the destination set C0-set, D-set). The definition of other kinds (DBL, Dis) of ASD's is straightforward. Moreover let us merely hint how other kinds of densities might be introduced. E.g. if,^ for some coordinate system it,u) on ~, H o is an arc of a coordinate line, t = to, then A~ in Eq. (3) can be the set bounded by H o and the twin arc of coordinate line, t = to+ dt. Only when the coordinate t is a one-to-one function of the latitude ~ with respect to some polar axis, is this definition of a point-to-coordinate line density, Dpc , equivalent to the definition of DpL.

2 B. Functional properties of the ASD's. Useful

functional

properties:

-i)

reciprocity relations;

-ii)

linear combinations;

-iii) normalization conditions, follow from the ASD definitions. i) Reciprocity relations will first be derived. To transform the last side of Eq. (9) let us choose an azimuth, ~, on the circle C z C(~',¥). E.g., ~s and ~I in Fig. 1 are the azimuths of ~s and 01 . See also Fig. 3 where the set B o ^¢ ^~ a n d ^

the circle C ^are projected on a plane

orthogonal to O' (shown in the case 7 > 0 and O.O' = v > 0 for all O c Bo). We consider the arc X being the intersection of C and B o and define on C a function Y(~) to equal 1 on X and 0 on its complement, X = C\X. If we now take on ~ a system of polar coordinates with ~' as the polar axis, any point ~ ~ is identified by the (cosine-)latitude v = ~'-~ and the longitude ~. Then we get

6

T. TROMBETTI

where M(~) c [ - 1 , 1 ] fraction

of C that

y ( ~ ) = I . ( v ) ) 6 ( u - y ) du,

(12)

i s t h e s e t o f v a l u e s of v s u c h t h a t

t h e p o i n t Q = (u,%0) ~ Bo. Hence t h e

intersects

Bo can be e x p r e s s e d as 2~

^

2~

1

1

F ( O ' ~ B o ; ~ ) = 2-~ f o Y(%0) d%O = ~-~ f o d~V IM(~) du 5 ( u - 7 ) . The d o u b l e i n t e g r a l (u,~).

is clearly

Coming back t o t h e o r i g i n a l ^

an i n t e g r a l coordinates

o v e r Bo e x p r e s s e d

(13a)

in the polar coordinates

and p o l a r a x i s k i n Fig.

1, t h e i n t e g r a l

is

^

to be done for Q • B o, with O.O' = ~. Finally we get

^ F(Q'--,Bo;~) = ~1

I s dO o

5(6'

¥).

.6

(13b)

As a corollary we obtain from Eq. (9) the symmetric expression

DsB(Bo,Bo;~) and, c o n s e q u e n t l y ,

:

1 J's' d Q ' / s dO 2~I A(B O) A(B O) 0 O

the important reciprocity Dss(Bo,Bo,7)

6(Q'.Q

¥)

-

(14)

relation

(15a)

= Dss(Bo,Bo,~).

T

(u~)

/%

a,

;,"

b

-r-

z~

z~÷

Tu 1

4,

C

Fig.

3.

~..Po(Uo=O) I

\

~

!

~u

zc

/ Z~ Po(-U2) 1__ po(-u3)

Fig. 4. ^

C ( Q ' , ~ ) and

~o i n a p r o j e c t i o n Q' (shown i n t h e ^

\

a-'

%

The c i r c l e

Po(u,,

z,+

} "t~:o E

;% (u2)

the

set

orthogonal to case

~>0

and

The partitions of the ~-interval [-I,I]

into the set

and of the unit

family

sphere

1/

A ±, into

^

P = O.O' > 0 f o r all 8 e Bo).

the

set

family

Z:

(n = 1, 2, 3, 4) f o r N = 2N'= 8. Po(±~n ) i s ~/ = ± ~ . n

the

parallel

at

Transfer by anisotropic scattering in linear transport theory Using t h e r e s p e c t i v e d e f i n i t i o n s ,

reciprocity

7

i s e a s i l y e x t e n d e d to

DBL(Bo,H; ) = DLB(H;,Bo)

(15b)

DLL(Ho,H ; ) = DLL(H;,Ho).

(15c)

Here and in the sequel the argument ? is omitted whenever clarity is not compromised. ii) Linear combinations generate new ASD's. If B i ~ $, with i e Ic (set of integer nonzero indices), are disjoint sets, and B 0 their union, then

DpB(~',Bo)= A(Bo)-I ~ A(Bi) SpB(~',Bi).

(16a)

iElc

R e p l a c i n g B and • w i t h H and Z in t h e above assumption we g e t DpL(~',H O) = L(Ho )-~ ~ L(H l) D p L ( ~ ' , H i ) .

(16b)

iCI¢ ^ with respect to Q ' , Eq. (16a) o v e r Bo, ~ ~, gq. (16b) o v e r HoJ ~ H, we g e t

Integrating

Dss(B~,BO) = A(Bo )'1

~ A(Bt) Vss(B~,Hi)

(17a)

iEIc

DLL(H~,Ho) = L(Ho )-1 ~ L(Ht) DLL(H~,Ht)

(17b)

iElc

iii) Normalization conditions. A(~) = 4- implies, for all ~' c ~, B °, e $, H °J e ~, ^

4 - DpB(~',U) = 4ff Dss(B~,~) = 4 . DLB(H~,~) = 1. Then,

i f Po(P) d e n o t e s t h e

g e t from Eq. ( 8 ) ,

(full)

parallel

1

Let

us

index s e t s

p,

with L[Po(p)] = 2 - ( 1 - p 2 ) ~/z,

we

f o r a l l ~ ' ~ E, Ho ~ ^

1

2, f _ 1 D p L [ ~ ' , P o ( P ) ]

i g (m,p,s,t)

of latitude

(18)

further

introduce

d~ = 2 , /_IDLL[H~,Po(p)] d~ = 1,

a

partition

of

~

into

sets

(19) Bi = Bms tp E ~,

with

c ( ~ x ~ x ff x ~) • 3. Here x d e n o t e s t h e C a r t e s i a n product and ~ , ~, Y, f a r e

( t h e domains of m, p, s, t ) .

The m u l t i index n o t a t i o n should b e s t a g r e e with t h e

symmetry properties of certain partitions, (cf See. 3). Combining Eqs. (18) and (16a), (17a)

with Ic = 3 y i e l d s f o r a l l Bo A(Bi) D p s ( ~ ' , B t ) = ~ A(B l) DBB(B;,B,) = 1. i~

(20)

ie3

3. ASD FOR SN SETS SN q u a d r a t u r e s e t s f o r problems w i t h r o t a t i o n a l nodes Pn and w e i g h t s wn. We s h a l l

symmetry c o n s i s t of a s e t of N l a t i t u d e

c o n s i d e r a symmetric s e t w i t h N = 2N' ( e v e n ) ,

Pn = - P - n '

wn = w-n n o r m a l i z e d to 1 f o r n = 1, 2, . . . , N' (wn+ w-n n o r m a l i z e d to 2). To each node-weight pair, (p±n,w) we shall associate the subinterval A±n of [-i,I] and the subset Z~ of ~ as they are shown for N = 8 in Fig. 4, i.e. with

8

T. TROMBETT1 A±n= {P : ~JJ ~ (un_t,~n)};

~o =

The

families

of

sets

A±n

UN,: i;

(22)

Z±n = {~ = (P'@) e U: p e a±n }.

(23)

(of

~n = ~n_1+

(21)

0,

wn ~

]A±n I : w n)

lengths

and

n = 1, 2, .... , N' form N-partitions of the p-interval

Z±n (of

areas

A(Z~) = 2nWn),

for

[-1,1] and of R, respectively.

The ASD for the above sets are discussed in Appendix A. For problems with no rotational symmetry a wide variety of SN sets has been developed. The theory of Sec. 2 will be applied here to the so-called one-level equal

number

of nodes

(generally

2N : 4N',

though

we shall

use

(l-L) sets, with an

4R to add one degree

of

freedom) s y m m e t r i c a l l y a r r a n g e d o v e r e a c h one from t h e N l e v e l s ( A l c o u f f e and O ' D e l l (1986), F i g . 9 w i t h d a t a o f Table 9). These s e t s

are suited

to

Appendix B). The p o l a r a x i s , plane,

and

consistency

the

ID-cylindrical

our

and 2D-xy geometries

(see

k, is usually parallel to the cylinder axis or normal to the xy

corresponding

with

and also to 2D-rz

^

latitude

previous

denoted

notation.

In

by

~,

though

ID-cylindrical

we

and

shall 2D-rz

keep

using

geometries

p

for

the

0-n

azimuth plane is the one through the aforesaid axes. In this case we have the SN node set {Oim ~st = [spm , t(2p-1)n/4R]} mp the quadruplet

(m,p,s,t),

i ~ 3 = ~xYxYxY

(all

are

index

sets),

where i denotes

m e ~ m (I, 2,..., N'},

p ~ • • {1, 2, ..., 2R), s ~ Y, t e Y, Y the set of the two indices {+1,-1} or simply (+,-}. ^

With the same Pn' Wn as

for rotationally

symmetric

sets, the weight of ~i

is w: = wm/4R,

independent of p,s,t (weights normalized to 2 over all i c 3). To each node Oi we shall associate the set Bi : B mst = {~ E ~ : sp E Am, t~ • (@p 1,~p)} p

(24a)

~p= p~/2R, A ( B : : ) = 2~w: = ~[Aml/2R.

(24b)

with

The s e t f a m i l y Bmp 8t ' ( m , p , s , t ) case

2R = 2N'= N = 8.

ordinates, of ~

i.e.

i s shown i n C a r t e s i a n c o o r d i n a t e s ~,P in F i g .

shows t h e

and some s e t s Bmp st.

except for

poles,

This

~ 3,

values

of

p,t

in

abscissas,

of

m,s

in

i f l i m i t e d t o -~B< ¢ ~ Cs= ~, i s a o n e - t o - o n e map

t h e u p p e r and lower s ^i d e s , ^

p = ~±N.= ±1,

which map t h e

north

and s o u t h

t h e u n i t v e c t o r s ~ = k and - k .

However t h e

strip

~ ~ (~8,¢9]

is

c ( - ¢ ~ , - ~ 7 ] and c o n t a i n s t h e s e t s longitude

applicable

This f i g u r e ,

5 for the

with

representative

respect

to

of the r e l a t i v e

the

B:~ - B:8. strip

in

Fig.

5.

It

coincides

Such c o i n c i d e n t s t r i p s

~ c (0,~1].

Hence

the

with

the

are s h i f t e d

region

k = k

mp8

strip

by n

~ c [0,99]

l o c a t i o n s o f a l l p o s s i b l e p a i r s o f s e t s g m~t. The s e t s p

r e g i o n , f o r which t = +, p = 1, 2 , . . . ,

so t h a t ±k = 1, 2 , . . . ,

included

in is

in t h i s

2R+l = 9, a r e l a b e l e d in Fig. 5 w i t h a u n i q u e index k m s [(m-1)(2R+l)+p]

(25)

N ' ( 2 R + I ) , and m = mW ! Quot ( I k l - l , 2 R + l ) + l ;

(26a)

P = PW i M°d(Ikl-l'2R+l)+l;

(26b)

Transfer by anisotropic scattering in linear transport theory

9

'°

%+

4

~4

- ~ + 3

U3

29

30

31

32

33

34

35

36

19

20

21

22

23

24

25

26

27

10

11

12

131

14

15

16

1

2

3

4

5

6

7

I,3

I

÷p 2 tt

+

28

D

~ U

1

A

17

E

F

8

9

0 11

I

Fig.

5- The s e t

each level,

18

-~

f a m i l y Bm8tp f o r t h e 1-L $8 s e t w i t h 2R = N = 8 n o d e s on

napped o n t o t h e ( ~ , p ) p l a n e .

a e s h l a b e l s ±1, ±2, . . . ,

The p - a x i s

is not to scale.

The

±36 a r e t h e v a l u e s o f k = k pa, Eq. ( 2 5 ) . E . g . ,

sone neshes are harked as follows: ÷

D: B 2 , z.

S:

.

;

E: BZ, + s÷ " ~1?

,

-

B2, 2 ,

T:

F: B ~~ e- • BZ+ s÷ " ~lS

;

÷

•

B2, s •

B_I 7 ;

;

-'4

B2, e

U:

B2, 9 • B_I e.

s = s k • sign(k), where

{Not

argunents. i'

is

the

integer

We t h e n

define

quotient

and

Mod

the

Bk = B i = Betmp w i t h

(26c) rest

of

the

i = (m,p,s,t)

division

and

between

the

two

Bk. = Bl ' = Bm,p .='t'

with

= (R',p',s',t'). The s e t

family,

Bi= BmStp

for

i • (m,p,s,t)

E 3,

c (-~2R,~2~] , o r ¢ c (-¢zR_1,~2~+1] and p ~ [ - 1 , a t most s e t s o f measure z e r o , a s f o r a l l sets possess several

partitions

invariance properties

-i)

Invariance with respect to s',s

through

the

depends

longitude

on s ' , s

shift

only

between

n e a s u r e d by an i n t e g e r p * - I = 0, 1 , . . . , p* = i t p - t ' p ' p

which

fills

up

the

is a 4NR-partition

in this

paper).

whole

rectangle

of ~ (neglecting

The ASD's i n v o l v i n g s u c h

t h a t w i l l now be d i s c u s s e d and p r o v e v e r y h e l p f u l

in reducing both the computational effort

DDB(Bi,,Bi)

1],

and t h e s t o r a g e

requirenents.

an__ddv ' , t ' , v . t . through

Bi.

their

and B I.

product

Along t h e

s's

and

shortest

on

p',t',p,t

path

the

only

shift

is

2R, w i t h

I + H(tt'),

= 4R + 2 - I t p - t ' p ' l ,

if if

Itp-t'p'l Itp-t'p'

~ 2R ;

(27a)

I • 2R+l;

(27b)

H(x) = 1 o r 0 f o r x>0 o r xS0 ( H e a v i s i d e f u n c t i o n )

(28)

10

T. TROMBETTI -ii) Symmetry with resnect to the center, O, of~/. DBB(Bi,,Bi;7)

is

invariant

under

the

replacement

of

7 and B L with

-7

~nd

the

set

symmetric to B i with respect to O:

Bst ,

-s,-t

DBB(Bi'' mp; ) = DDB(Bi''B m,ZR+l-p ;-~)'

(29)

-iii) Partial reciprocity. Accounting for symmetry,

reciprocity

is seen to apply also to a subset of one or two

indices: DBD(B::t' st st' 't e't' e t p,,Bmp) = DBB(Bmp,,B:,p) = DBB(Bm p,,Bm,p). -iv)

(30)

Composite i n v a r i a n c e .

Combining all above relations we get

,'t"

et.h~)

÷ +

DBB(Bm,p.,Bmp,

."+

= DBB(B,t,B n

q;~)

(31)

where h = 11, ~ e [0,1] and n' = min(m',m); n = max(m',m);

(32a)

s'' = s'sh; q = p* if h = I;

q = 2R+2-p* S i n c e q = 1, 2 , . . . , the

factor

2R+1 and n a n ' ,

32N'(2R)Z/(N'+1)(2R+1),

2R = 2N'= N. E . g . , Even a f t e r

this

this

factor

(32b,c)

if h = - 1 .

(32d)

E q . ( 3 1 ) r e d u c e s t h e number o f i n d e p e n d e n t ASD's by

i.e

by

32 Na/(N+I)(N+2)

for

standard

SN s e t s

with

redundant,

i.e.

i s a b o u t 182 f o r SB o r 935 f o r $32.

impressive

reduction

many o f

the

r e m a i n i n g DBB a r e

t h o s e t h a t e q u a l z e r o . They can be d i s c a r d e d as shown i n t h e n e x t s e c t i o n .

4. SELECTION, COMPUTATION AND STORAGE OF NONZERO ASD's. A method to single out, compute and store only the nonzero independent ASD's will now be proposed. We shall make reference to Fig. 6 which covers (for 2R = 2N'= N = 8) the region @ e [0~@9] , (i.e.

BS*mp = Bk'

k = kmps,

t = +. p = I, 2,...9)

Eq.

(25).

of

The i n t e g e r s

Fig.

5.

written

The

as

meshes

in the

mesh l a b e l s

are

net

are

values

the

of

k.

sets

The

i n d e p e n d e n t ASD's d e f i n e d by Eq. (31) a r e t h e v a l u e s o f

Dss(Bm, t , ÷÷ BS+;a)mp = DeB(Bk.,Bk;a)

(33)

for a e [0,i], k' = km. il = I, 10, 19, 28 (i.e. B k, = B+m'1' + m'=I,2,3,4 ~ N') and all k ~ -k', k a k'. However we shall include also all k ~ 0 for which 1 $ Ikl < k'. Then checks based on partial reciprocity, Eq. (30)

+ * BS+) -B ÷÷ , + DBB(Bm'I' mp = DBS(ml,Bm,p) will

test

the

accuracy

of

the

whole

approach.

(34) E.g.,

we

must

find

DBB(BI,B±z i) = DBB(BIg,B±3). Eq. (34) means that only the N/2 meshes labelled as k' = I, I0, 19, 28 in Fig. 5 must be taken as origin sets. All the (2R+I)N meshes in Fig. 6 will he taken as destination

sets

Transfer by anisotropic scattering in linear transport theory

"1" + 4

+

3

-t-+-+

2

+

I

I. '°

I U '"I?I

i

"1 " '1

I

-1~,, -12 /-,31

I

[pl I,l,l.l,l,l.l,l Fig.

6.

a1 >

The

/J' , (x2

half <

/J' ,

circles

l.] HC1, HC2 from

respectively,

s c a l e and t h e l o c a t i o n s o f t h e p o i n t s for ^

the

,

sake

÷

of

illustration.

÷

~± E Bm. 1 • Bk , ( o r i g i n s e t ) f o r which e i t h e r

the

circles

' = ( / J^' , ~ ) .^ and l~+

C(~',ai).

The p -^a x i s

~o' ~A' QS' " ' ' '

Referring

to

the

~

Eq.

with

is not to

are selected we

have

w i t h m' = 1, k ' = 1. The d e s t i n a t i o n

sets

HC1 o r HC2 g i v e n o n z e r o c o n t r i b u t i o n s

(33)

t o t h e ASD's a r e

l a b e l l e d by t h e i n d i c e s k = kmp 8, Eq. ( 2 5 ) . T r a n s l a t i o n a l o n g t h e ~ - a x i s of

the

amount ~ - ¢ _ ' = ¢ 1 - 2 ¢ :

direction

for

each

m',

would y i e l d

the

half

circles

with

origin

a t ~)'. ÷

to

enable

checks,

though

(2R+l)(H-2m'+2)

would

suffice

for

m' = 1, 2, 3, 4 = N/2. The DBB in Eq. (33) a r e o b t a i n e d from Eqs. (9) and (24) by n u m e r i c a l i n t e g r a t i o n : + *

,+

2 R

DBB(Bm,I,Bmp;

~)

=

g

.

Wm

~

(35a)

(m,p,s);

dij

i,j dlj(m,p,s)

^

84

= wpiw~jF(O~j~ B p ; ~ ) ;

(35b)

~wp,= ~w,j= 1. P J Here wpiw@j is the weight of the integration node ~'ij = (Pi,@j) 6

(36) B m'i" + ÷

Since m' and ~ are

fixed for the moment, they are not explicitly indicated as arguments of dij. ~e make the node set to be symmetric with respect to @ = @I/2 (the symmetry meridian of the origin set B m,~) + +

and associate symmetric nodes in pairs. Dropping indices i,j we denote any chosen pair

by D'+ = (P',@~) and D'_ = (P',@'),_ where @~+@: = @i" The method is in t.o steps, each one repeated for each pair of integration nodes. 8te) ~

works on the circles C± = C(D~,a) onto which the neutrons are uniformly scat-

12

T. TROMBETTI ^

t e r e d from O~ w i t h a CSL = u. The c u r v e s HC1 and HC2 i n F i g . 6 map h a l f o f C_ f o r ~ = ul> ~ ' and u = u2< ~ ' longitude ~+

respectively. ~ for all

Both c u r v e s ( w i t h e n d s ~ 0 , 0 x ) c o n t a i n a t l e a s t

~ > 0 sufficiently

one p o i n t w i t h

s m a l l . HC1 d o e s n o t s e p a r a t e t h e two p o l e s ;

HC2

d o e s , hence i s S - s h a p e d , s i n c e a 2 < Y ' . During s t e p - i ) for all parallels and i s

using spherical

intersections

Q^, ~ s ' ~ c '

trigonometry

""

and m e r i d i a n s i n F i g . 6. Then F i n Eq.

associated

k = 28, 2 9 , . . .

to

the

o r k = 36, 3 5 , . . .

o f C± w i t h a h a l f - m e r i d i a n we a s s o c i a t e

together

the

=

t o t h e same i n t e g r a t i o n

~2R÷l-p'

aB

all

Cp(~)

intersected

point,

either

sharing the ^

~ '+ o r Q ' ,

= cos2(~p - ~ ) . svae c ( ~ ' ) .

To

For u

p

save

> ~'

1

and t h e same m e r i d i a n ,

and some m e r i d i a n , e . g .

e.g.

~ = ~p, t h e o t h e r one t o

C2R+l_p(~).

=

o u t p u t s two v e c t o r s d 2 and k 2 o f e q u a l l e n g t h .

the nonzero quantities

(5)

mesh B:~_ = B~ ( e . g . ,

b u t remark t h a t f o r t h e i n t e r -

( l o n g i t u d e ~p) we n e e d C p ( ~ ) two i n t e r s e c t i o n s

= ~p. For u 2 < ~ ' t h e y p e r t a i n one t o ~

Hence s t e p - i )

""

i s o b t a i n e d a s s u g g e s t e d by Eq. the

i n F i g . 6 ) . We omit d e t a i l s

section

0'+ and ~

(35b)

i n d e x k = k=p n i d e n t i f y i n g

efforts

they pertain

we d e t e r m i n e t h e a n g l e s ~^, ~ s ' ~ c '

( c f . F i g . 1) o f an HCi ( i = 1, 2 ) , w i t h t h e n e t o f a l l

d i j ( m , p , s ) , Eq. ( 3 5 b ) ,

The e l e m e n t s o f d 2 a r e

f o r t h e p a i r o f n o d e s ~ ' = ~ ' and ~ ' = ij

+

lj

-'

Those o f k 2 a r e ( i n t h e same o r d e r ) t h e i n d i c e s k = k p 8. ~tep -ii)

adds up t h e a f o r e s a i d q u a n t i t i e s

r e n t v a l u e o f t h e sumDation i n Eq.

(35a).

dij(m,p,s)

to the p r e v i o u s l y cumulated cur-

Also t h e n o n z e r o e l e m e n t s o f t h e l a t t e r

r e a d y s t o r e d i n a t w o - v e c t o r form d l , k 1 ( s a y ) . Now t h e two v e c t o r s ( o f i n d i c e s ) , are joined

i n t o a unique two-block v e c t o r .

order of the algebraic latter.

values of its

sorted

(Singleton

a r e s e a r c h e d and a l l

Finally,

but one e l e m e n t s from each group d e l e t e d .

in t h e b l o c k v e c t o r

and - i i ) +

+

The l o c a t i o n s o f t h i s

group

(dl,d2)

after

permutations.

The e l e m e n t s o f t h i s

sum.

are repeated for all

w i t h two r e s u l t i n g

pernutation of the

d z and i s s u b j e c t e d t o

o f v a r i o u s n o d e s t o one and t h e s a l e Dee occupy t h e c o r -

group a r e t o be r e p l a c e d by t h e i r Steps -i)

in a s c e n d i n g

g r o u p s o f c o n t i g u o u s e q u a l v a l u e s in t h e s o r t e d i n d e x v e c t o r

denote that nonzero contributions responding locations

k 1 and k z ,

(1969))

e l e m e n t s , which i m p l i e s a c e r t a i n

Then a t w o - b l o c k v e c t o r i s formed a l s o from t h e v e c t o r s d l ,

t h e same p e r m u t a t i o n .

left

This i s

are al-

vectors,

one o f

nodes, ~'

. For g i v e n m' and ~ we

ij

i n d i c e s kmp 8 and ( a f t e r

multiplication

are thus by 2R/nwm)

8+

one o f ASD's, Dss(Bm.l,Bmp;~). Any two e l e m e n t s o c c u p y i n g c o r r e s p o n d i n g l o c a t i o n s i n t h e two vectors refer different Only

t o t h e same t r i p l e t

(m,p,s),

l o c a t i o n s r e f e r t o two d i f f e r e n t the

nonzero

ASD's

and

ascending order of the algebraic retrieved using the auxiliary

their

hence t o t h e same d e s t i n a t i o n such t r i p l e t s

indices

k

mps

all

"channels"

(m,p,s)

v a l u e s o f such i n d i c e s .

thus

that

vector of indices k

computed

Any i n t e r e s t i n g

and

stored,

in

ASD can l a t e r

be

t h e ASD's has two main f e a t u r e s .

is the multichannel numerical integration give nonzero c o n t r i b u t i o n s

T h i s g r e a t l y compacts t h e r e q u i r e d o p e r a t i o n s . auxiliary

are

s e t Bm8+. Any two p

sets.

index v e c t o r .

The above a p p r o a c h t o compute, s t o r e and r e t r i e v e computational feature

and d e s t i n a t i o n

mp8

The

a c t i n g simultaneously over

and d i s c a r d i n g a l l

The s t o r i n g - r e t r i e v i n g

other

feature,

channels.

b a s e d on t h e

p r o v e s v e r y s i m p l e and g e n e r a l . However a more c o n v e n t i o n a l

a p p r o a c h can be u s e d f o r t h e o n e - l e v e l SN s e t s c o n s i d e r e d . To i l l u s t r a t e

the

latter

i d e a by an example,

we remark t h a t

if

on t h e

same l i n e

of

Trans~rby anisotropic ~attefinginlincartransporttheory Fig.

6 two d e s t i n a t i o n ~ s e t s ~

between them, i . e . lines

there

mediate line, terization

have n o n z e r o ASD's,

there

is at

least

o f a l l and o n l y t h e d e s t i n a t i o n a

chosen subinterval

further

of

B22

s e t s w i t h n o n z e r o ASD's, e . g .

am = - 2 , - 1 , ÷ 1 ,

We remark t h a t features,

B13 and B17,

B14, B I s , B l s , have n o n z e r o ASD's ( f o r f i x e d m ' , a ) .

destination

are

e.g.

13

[0,1]

can

one such s e t .

o f Dee in Eq.

be e a s i l y

all

sets

I f on two d i f f e r e n t

and B13, a l s o on e a c h i n t e r T h i s a l l o w s an e a s y c h a r a c -

s e t s w i t h n o n z e r o ASD's,

integration

then also

(35a)

performed using

f o r f i x e d m' and ~.

with respect

to a over a

any a p p r o a c h w i t h

the

above

which a v o i d s w a s t e o f c o m p u t a t i o n s w i t h z e r o e l e m e n t s .

5. NUMERICAL EXAMPLES We r e p o r t

below t h e r e s u l t s

o f sample c a l c u l a t i o n s

f o r a 1-L $8 s e t

number, 4R = 2N = 16, o f q u a d r a t u r e p o i n t s on each l e v e l . and w e i g h t s

wpi ,

w@j

Legendre i n t e g r a t i o n s Pi9 ' ~j

ly spaced

used with Eq.(35b)

over the intervals with constant

are

the

(am_i,~)

wpi , w~j

The i n t e g r a t i o n

with a constant points

same a s needed f o r

and ( 0 , 9 1 ) .

~'i ' ~j'

standard

Gauss-

Other c h o i c e s , as e . g .

equal-

have a l s o p r o v e d s a t i s f a c t o r y .

The r e s u l t s

r e p o r t e d h e r e a r e from a 20 x 20 g r i d (~l,@l).' ' Table

1 shows a l l

n o n z e r o e l e m e n t s Dee,

p = 1, p = 2R+l = 9, f o r a = 0 . 9 .

Eq.

(331,

I t a l s o shows a l l

triplets

if

2 s p ~ 2R = 8,

or

Dee/2,

if

( m , p , s ) and ( c o m p o s i t e ) i n d i c e s

k ffi knp a f o r which Dee# O. These e l e m e n t s and i n d i c e s form t h e v e c t o r s o u t p u t by t h e twostep

a p p r o a c h o f Sec.

satisfy

4.

The e x t e n t

t o which t h e

lines

for

which m < m',

i.e.

]k[ < k '

t h e c h e c k , Eq. ( 3 4 ) , g i v e s an i n d i c a t i o n o f t h e a c c u r a c y o b t a i n e d .

Let us now come back t o p r o b l e l s the set

f a m i l y Z±n ' Eq.

(23).

w i t h a z i m u t h a l symmetry and t h e p a r t i t i o n

From Eqs.

(17a),

(AT) and (27)

we a r g u e t h a t

of ~ into

Dee(Z +m,,z~;~)

e q u a l s t h e a v e r a g e A 2 e ( m ' , s m ; a ) o f 2R = 8 e l e m e n t s r e p o r t e d ( o r m i s s i n g i f z e r o ) on t h e l i n e (sl,m')

o f Table 2. E x i s t i n g e l e m e n t s o f t h e column p = 2R+1 = 9 a r e t o be added t o column

p = 1 before averaging. If we use

Eq.

(A9)

with an

interval

of

integration

r

small

enough

to assume

the

n

integrand to be nearly constant,

for 7 = ~ ~ r , we get n

I*(sm,m',n) ~ 2~ A2R(s',sm,a).

(37)

Table 2 compares the values given by Eq. (37) and Table I, for ~ = ~ = 0.9, with the ones of I (k,l,m) found by Ligou and Niazza (1988), where m stands for our n and specifically refers to the 7-integration

interval r

= (0.89, 0.91).

Symmetry and reciprocity combined

that each line can be attributed to four pairs (s'm',sm),

two of which are reported in the

table. The first (column 3) is taken from Table 1, the second

corresponds

to

the

pair

k) = 1 , 2 , . . . , N ' , N ' + I , N ' + 2 , . . . , N respectively

(1,k) are

one

(column 4) is the one that

Miazza

(1988),

where

(or s m ) = - N ' , - N ' + l , . . . , - 1 , 1 , 2

represents

a

by-product

two columns o f Table 2 l o o k s s a t i s f a c t o r y . of

the

roughly speaking,

one o n l y

Instead,

17:1-C

and

used f o r our s ' m '

azimuthal dependence (and, node,

integration

Ligou

1 (or .... ,N',

( h e r e N = 2N'= 8 ) .

The a g r e e m e n t between t h e f i r s t first

of

ensure

~ = 0.9).

the

numerical

integrations

reports

an i n t e g r a t i o n

s e c o n d one

in 3 dimensions (thus accounting a l s o f o r

is

the the

result full

required over r of

n

In f a c t , to

with

performed with

an e x a c t

interval

deal

the

analytical

r n = (0.89,0.91))

14

T. TROMBETTI +

Table

1.

The n o n z e r o

2 ~ p ~ 8)

or

values

Dss/2

(with

combinations (m,p,s). k = k

, Eq.

mps

of l i n e s

m'

k'

that

sm

(25),

of

p = 1,

also

$+

p = 9)

The e l e m e n t s o f are

+

DsB(B , t , B m p ; a = 0 . 9 ) ,

the

for

Eq.

all

index vectors

shown. L a b e l s

(33)

(with

possible

index

k ' = km, l l

(A) t h r o u g h (F)

and

mark p a i r s

s h o u l d be e q u a l by r e c i p r o c i t y .

p =

1

k=28

4

2

3

29

30

5

31

32

6

7

8

9

33

34

35

36

4 28 3 i0 -s 8 I0 -s O. 0144 0.9182 1.5696 1.0003 0.7025 0.5755 0.2697 3 19 +4 0.3854 0.9824 1. 1007 0 . 6 1 0 5 0 . 0 3 6 8 ( F ) . . . . . . . 2 10 0.1246 0.0960 0. 0002 ( S ) . k=19

21

20

22

23

4 28 0 , 3 9 2 4 0 . 9 8 3 2 1. 1026 0.6090 0.0369 (F) 3 19 +3 0 . 3 7 3 7 I . 2808 0.0908 2 i0 0.4197 1.0182 0. 2234 (D) 1 1 0.0949 0.0407 (c) -

k=10

11

-

12

4 28 0.1175 0.0963 3 19 +2 0 . 4 1 8 8 1 . 0 1 7 8 2 10 [0.0033 0 . 8 6 0 1 1 1 0.4405 0.8113 k=l

-

0 . 0 0 0 2 (E) 0 . 2 2 3 5 (D) 0.3265 0 . 0 3 5 9 (B)

2

3

3 19 0.0941 0.0403 (C) 2 10 ÷1 0 . 4 4 4 4 0 . 8 1 1 7 0 . 0 3 5 8 (B) 1 1 0.0152 0.8937 0.1110 k=-I

-2

-3

2 10 -1 0 . 0 7 9 9 0 . 0 2 8 9 (A) 1 1 0.4496 0.7330 0.0143 k=-10 1

-Ii

1 -2 0.0823 0.0292

(A) -

with no azimuthal dependence.

6. CONCLUSIONS The p r o b a b i l i t y over a suitable

CSL = 7 ( u n i f o r m l y terms of

average

one-dimensional properties

f o r one n e u t r o n e i t h e r

origin

subset

in azimuth) scattering

or

c o n c e n t r a t e d on a p o i n t o r u n i f o r m l y d i s t r i b u t e d

( O - s e t ) o f t h e u n i t s p h e r e ~ c ~3 t o be s c a t t e r e d into

some d e s t i n a t i o n

densities,

two-dimensional

sets

ASD's. such

o f t h e ASD's, s u c h a s r e c i p r o c i t y t

set

(D-set)

O- a n d D - s e t s as

smooth a r c s

normalization

(O-D-sets) or

with a given

of ~ has been

in

may be m e a s u r a b l e

angular

and r e l a t i o n s

found

bins.

Important

b e t w e e n ASD's f o r

1- and 2 - d i m e n s i o n a l O - D - s e t s h a v e b e e n p r o v e d i n a g e n e r a l way ( S e c t i o n 2) i n d e p e n d e n t l y o f

Transfer by anisotropic scattering in linear transport theory T a b l e 2. Comparison o f I ~ f o l l o w i n g column l a b e l s . ( 2 ) , ( 5 ) : Ligou and Miazza column ( 1 ) . Col. ( 3 ) : T a b l e

l*(m',m,n)

the

knowledge

portions

of

the

of coordinate

(1)

(2)

3.96681 2.44738 1.37097 0.17343 1.30489 0.93462 0.10658 1.01145 0.80115 0.94006 0.08764

3.97042 2.44508 1.37336 0.17322 1.30382 0.93638 0.10494 1.01178 0.80261 0.93925 0.08771

SN q u a d r a t u r e reciprocity

factor

(e.g..

number o f

2, 3 2, 2 1, 3

-3,-3 -4,-2 -3,-2 -2,-2 -3,-1

2, 1, 2, 3, 2,

1, 2

-2,-1

3, 4

1, 1

-1,-1 -1, 1 -21 1

4, 4 4, 5 3~ 5

Further

2 3 3 3 4

ASD c o n c e p t s

referring

e.g.

to

sets,

importance.

For

on e a c h l e v e l

the

are of practical

D

s't*

points

st

BB(Bm.p. JBmp;~), have been shown t o obey f u r t h e r -

which r e s u l t

i n Eqs.

suffice

to

completely

(20),

fulfilled

The n e u t r o n

which

is

characterize

an SN s e t

source

and 1 D - c y l i n d r i c a l

reduction obtained •

simultaneously

essential for

is

÷

o÷

for

for all

numerical

m,p,s.

angular

geometries

are

fluxes

obeying

expressed

Table

2)

proved properties

as

rotationally

a

numerical

(reciprocity

The n o r m a l i z a t i o n

by-product,

Eq.

symmetries

in terms of

the

and h a s s a t i s f a c t o r i l y

(37),

the

I~

matrix

t h e c h o i c e o f t h e s e t o f SN d i s c r e t e In

our

numerical

F n : {y : Y ~ ( 0 . 8 9 , 0 . 9 1 ) } , good

value

y-integration In t h i s It

of

a

directions.

example,

with

o n l y one i n t e g r a t i o n

y-averaged

performed first

ASD.

A modified

sufficiently y-point,

elements

using the (see

obtained

for

s h o u l d be i n d e p e n d e n t o f fine

~ = 0.9,

presentation

of

t o a p p l y t h e ASD method t o o t h e r

can be c h o s e n a t ~

segmentation,

is sufficient

to give a

this

with

( a s i s t h e c a s e w i t h t h e I* method) w i l l

c a s e t h e ASD i s d e f i n e d f o r a u n i f o r m d i s t r i b u t i o n is also possible

2D-xy,

reproduced

In o u r method t h e 7 - p o i n t s a

of

above ASD's i n

method o f Ligou and Miazza (1988).

With e x a c t k e r n e l methods t h e s e g m e n t a t i o n of t h e y v a r i a b l e will.

is

(36) and ( 5 ) .

the

etc.)

s y m m e t r i c a n g u l a r f l u x e s by t h e a n a l y t i c a l

and

particle-conservation,

Appendix B. The above n u m e r i c a l a p p r o a c h f o r t h e ASD c o m p u t a t i o n h a s been t e s t e d theoretically

of the

from t h e

o n l y t h e n o n - z e r o v a l u e s o f Dss(Bm.1,B p ; a ) ,

by o u r p r o c e d u r e b a s e d on Eqs. s u c h a s ( 3 5 ) ,

scattering

symmetry and

(31) and (32) and r e d u c e by an i m p r e s s i v e

935 f o r $32) t h e number o f i n d e p e n d e n t ASD's. The u l t i m a t e

gq.

2D-rz

1, 1 1, 2

4 4 3 4

1,-1 11-2

performs the required numerical integrations

2D-r~,

(5)

-4,-4 -4,-3

4, 3, 3, 2,

t o SN q u a d r a t u r e

method o f Sec. 4 which computes and s t o r e s

automatically

(4)

w i t h an e q u a l number o f q u a d r a t u r e

relations

ASD's t h a t

condition,

1, k

(3)

ASD e x p r e s s i o n s .

related

sets

ASD's f o r t h e O - D - s e t f a m i l y , partial

s'n',sm

l i n e s have b e e n i n t r o d u c e d and a r e n o w ' b e i n g d e v e l o p e d .

Families of O-D-sets, one-level

m a t r i x e l e m e n t s found a s s p e c i f i e d by t h e Col. ( 1 ) : Eq. (37) o f t h i s p a p e r . C o l s . ( 1 9 8 8 ) , T a b l e I I , where m s t a n d s f o r n o f 1 o f t h i s p a p e r . Col. ( 4 ) : See t e x t .

I~(k,l,m)

explicit

15

of scattered

method,

be p r e s e n t e d particles

k i n d s of SN s e t s ,

the

later.

w i t h i n Fn . as o n e - l e v e l

16

T. TROMBETTI

The f u n c t i o n

in Eq.

I = 2 is best stated Independence of

(A5) was f i r s t

i n t r o d u c e d by Takahashi

et

al.

{1979).

The c o n d i t i o n

(Ligou and Hiazza (1988)) as g > 0. DpL in

Eq,

(A4)

o f @'

(tracing

P o ( p ) ) combined w i t h Eq. (10) and w i t h r e c i p r o c i t y ,

back

to

the

rotational

symmetry o f

s u g g e s t s t h a t we d e f i n e

D~p(p',p;~) E DLL[P(p',A'@),po(p);~ ] = DLL[Po(p'),po(p);¥] = = D L L [ P o ( P ' ) , P ( P , / ~ ) ; 7 ] = [ 2 n 2 g l / 2 ( P , P ' , ~ ) ] -1 all

independent of the azimuthal i n t e r v a l s

(A6)

~ ' ~ , ~0.

Moreover, a c c o r d i n g t o Eq. ( 1 1 ) , we d e f i n e D z z ( A ' p , b g ; ~ ) s DBB[Z(A.~,A,~);Zo(Ap);7 ] = D e B [ Z o ( ~ ' p ) , Z o ( ~ p ) ; ~ ] =

{ap{-'[a'~{"f dP'fdp DRR{p',p;~)

= DBB[Zo(a'~),Z(Ap,A@);~] =

A'p

(AT)

Ap

a l s o i n d e p e n d e n t o f h ' ~ , 5~. The f u n c t i o n I ( p , p ,~) and m a t r i x I * ( k , l , m ) and f u r t h e r d i s c u s s e d by Ligou and H i a z z a (1988),

i n t r o d u c e d by Takahashi and Rusch (1979) as w e l l as t h e i r

properties,

can now be

s e e n a l s o as s p e c i a l c a s e s o f ASD's: 27( D R R ( p ' , p , ¥ ) = I (P,P ,~)

2~{r}-'f F Dzz(a~,~k,~) d l

(A8)

= I*(k,l,n)

(A9)

n

and t h e g e n e r a l r e c i p r o c i t y and t h e i n t e r v a l s

F

n

and n o r m a l i z a t i o n p r o p e r t i e s

form two p a r t i t i o n s ,

Because o f t h e f a c t o r 2n a t t h e 1 . h . s , tered (take,

e.g.,

2~ n e u t r o n s ,

i.e.

Eqs.

one n e u t r o n p e r a z i m u t h a l r a d i a n . meaning a g r e e s

The a s s o r t m e n t of p h y s i c a l

(A8) and (A9) i s v e r y r i c h : We remark t h a t ,

and d i f f e r e n t i a t i o n

from t h e o r i g i n

in v i r t u e operations

(all

requires

a

spherical

of

s e t s P(~, Ll~)

s e t Po(p ' ) o r Zo(A l)

I ~ by Takahashi

equivalent)

of

and Rusch

the quantities

s i x p e r each Eq. (A6) and (A7), u s i n g r e c i p r o c i t y .

used by Wakahashi and Rusch (1979)

triangle

to

be

solved

to

get

and Brockmann (1981)

we can r e s o r t Eqs.

(A3)

and

i s e x p l a i n e d as a r e a l i z a t i o n

o f Sec. 2 B, b e f o r e t h e e x p r e s s i o n s ,

Eqs.

(A4).

s i o n o f symmetry t o t h e t h i r d v a r i a b l e s this

of the general reciprocity

(A6) t h r o u g h (A9),

are actually

to

t o Eq. (4) which

symmetric dependence o f t h e I ~ f u n c t i o n and m a t r i x on p ' , p and k,1 (Takahashi e t a l . Ligou and Hiazza (1988))

in

o f t h e g e n e r a l t h e o r y i n t r o d u c e d in Sec. 2, t h e i n t e g r a t i o n

d e f i r , e t h e f u n c t i o n I ( g , p ,~) a r e n o t n e e d e d . Here, in f a c t , only

of neutrons scat-

Accounting also for the f u r t h e r

with the d e f i n i t i o n s

interpretations

~i

[-1,1].

t h e f o u r t h s i d e o f Eqs. (A6) and (A7)) i n t o t h e d e s t i n a t i o n

a v e r a g e over F n , t h i s (1979).

of the interval

the I * ' s are average d e n s i t i e s

( w i t h " t h i c k n e s s " d t ~ 0) or Z(Ak,A~) when s c a t t e r i n g affects

o f Sec. 2 B. Here t h e i n t e r v a l s

usually different,

Also

the

(1979); property

computed. E x t e n -

(~ and n) c o u l d a l s o be o b t a i n e d s i m i l a r l y .

However

i s o m i t t e d h e r e , as t h e t h i r d v a r i a b l e s o f t h e I ~ f u n c t i o n and m a t r i x must g e n e r a l l y be

assigned different

s e t s of values than the f i r s t

and second o n e s .

Transfer by anisotropic scattering in linear transport theory sets

with a different

( A l c o u f f e and O ' D e l l properties

still

properties

before

number o f (1986)).

hold.

quadrature

For s u c h s e t s

These a r e

to

points all

on e a c h

general

level

or

reciprocity

be s u p p l e m e n t e d by s p e c i f i c

l7 fully

and n o r m a l i z a t i o n

sets ASD

symmetry and r e c i p r o c i t y

t h e n o n z e r o ASD'a c a n be found by t h e s i m u l t a n e o u s

and u s e d t o e x p r e s s t h e s c a t t e r i n g

symmetric

integration

approach

source.

Acknowledgements~This work has been supported by the research funds of the Italian Ministero per la Pubblica Istruzione.

REFERENCES Alcouffe R.E. and O'Dell R.D. {1986) Transport Calculations for Nuclear Reactors. In: Y.Ronen (ed.) CRC Handbook of Nuclear Reactor Calculations, Vol. I, CRC Press, Boca Raton. Brockmann H. (1981) Nucl.Sci. Eng. 77, 377. Ligou J. and Hiazza P. (1988) Nucl. Sci. Eng. 99, 109. Odom J.P. and Shultis J.K. (1976) Nucl. Sci. Eng., 59, 278. Schwenk-Ferrero A. (1986} KfK 4163. Singleton R.C. {1969) Comm. ACM, 12, 185. Takahashi A. et. a1.(1979) J. Nucl. Sci. Techn. 16, 1. Takahashi A. and Rusch D. (1979) KfK 2832 Parts I and II.

APPENDIX A. ASD'S FOR ROTATIONALLY S M E T R I C

ANGULAR FLUX PROBLEMS.

Let us consider the families of those subsets A~,A~p of the p and ~ intervals, [-1,1] and (a,a+2x], having positive measure IAPl and [A@ I (say). For simplicity we shall speak of subintervals &p,A~ or 6'p,A'@ of [-1,1] and (a,a+2x]. By

Zo(A~)

Zo((~.~,~))

we

mean

the

union

of

the

parallels

Po(P),

for

p E Ap.

E.g.,

= z~ , n = 1,2,3,4, in Fig. 4. Then let

Pip,A@) = {O = (P,@) e po(p): @e A@},

(A1)

Z(Ap,&~p) = {O = (p,@) e Zo(Ap): ~ • A~}

(A2)

denote the subset (arc or angular bin) of Pc(W) and Zo(A~) formed by the points O E ~ having If rotational

symmetry is obeyed by the angular neutron fluxes

(according to the

transport problem setting), origin and/or destination sets enjoying this same property, as Po(P) and Zo(&P), are worth selecting. To express DpL[Q',Po(p);¥] we come back to Fig. 1 where we must set H o = Po(P). Both Po(p) and C • C(O',7) are circles. We consider the case that they have I = 2 real distinct intersections. Then in Fig. 1

W1 = Wz = x/2~ and from Eq.

(6) X I = X z = ~I = ~2" Hence,

solving for cos ~I the spherical triangle kO'O I in Fig. 1 and resorting to Eq. (4) we get s i n Xj = g l / 2 ( ~ , p , , ~ )

(1 - p 2 ) - 1 / 2 ( 1

-

12} "1/2,

j = 1,2;

(A3)

DpL(~ ' ,Po(P),7) = [2xZgl/Z(p,p',7)]-1 ,

(A4)

g(p,p',?) = 1 - p2 _ p,2 _ 72 + 2P~'7.

(A5)

with

18

T. TROMBETT1 APPENDIX B. THE SCATTERING SOURCE. We sketch the computation of the scattering source for the SN sets of Sec. 4. Let us st

denote by el(g) = @mp(g)

{g dropped when not needed} the g energy-group angular flux (per

steradian) at the discrete direction ~st , and assume in the first approximation that this mp is the constant flux value within the set B st , for all m E A,

p E ~, S E if, t E ff (see

mp

Sec. 3). We consider the typical 1- and 2-dimensional geometries. •

~

sJt '

8t

We denote by q~g ,Bm,p,~ g,Bmp) the average source component (per steradian) in group g s*t'

and in set Bstmp ' due to neutrons scattering from group g' and set Bm,p,. Then 8't"

st

~ Wm'

s't'

BmP) = ~ 2 R

q(g''Bm'p'~g'

2 ~ f l~

~m'P'(g')hffi± ~

s

s't'

,

st

d~ O (g ~g; ha) DBB(Bm,p, ,Bmp;ha).

(B1)

The superscript s in o s means scattering while the other superscripts s and s' are variables taking the values ±I. The symmetry properties of typical geometries may reduce the above expression. E.g., in 2D-xy

geometry

¢÷,t: m

p

@-,t= m

p

¢,t

(xy

mp

is a symmetry

plane

for

fluxes).

The

star

denotes

independence ( of some function, here ¢) of the replaced index. Then we need •

q ( g ' , B .S. ,, .t , g , .B,t, ,

-

The d o t d e n o t e s t h e summation o v e r t h e r e p l a c e d i n d e x ( s ' ) .

(B2)

We g e t

~w q(g''Bm'p'~g~

mp

2 R

hffi±l

m'p'

*

'P'

P

where 'p'

'

mp '

s'

DBB(Bm'p'

'

(B4)

ha).

mp

Each side of Eqs. (B2), (B3) and (B4) is i n v a r i a n t irrespective of the value (+ or -) assigned to the *, r.h.s.

provided t h i s

The main p o i n t

with

the

is the same in the two terms of each summation at the SN s e t s

adopted

is

p r o p e r t y o f Eq. (31) i n t h e c o m p u t a t i o n o f t h e r . h . s , Eqs. rules.

(B2),

(B3),

in t h e a n g u l a r

one

respectively.

(t,t')

of

the

composite

invariance

o f Eq. (B4). geometries with the

following

f l u x @ and in t h e d e s t i n a t i o n

set are

( * ) . In t h e o r i g i n s e t t h e y a r e r e p l a c e d by d o t , which a t t h e r . h . s

denotes

summation o v e r t h e i n d e x domain, { + , - } . right

use

(B4) can be a d a p t e d t o o t h e r t y p i c a l

One ( o r two) u p p e r i n d e x ( - e s )

r e p l a c e d by s t a r

the

or

both

(s,s',t,t')

Such u p p e r i n d e x ( - e s ) a r e : in

2D-xy,

2D-rz,

the left

one ( s , s ' ) ,

1D-cylindrical

the

geometries,

2D-r@ f o l l o w s t h e same r u l e a s 2D-xy. Summation f o r h = ± i s always p r e s e n t .