炉物理の研究 第61号 （2009年3月）＜特集１＞

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6-7-4
E-Mail Address: kobayashi.keisuke@emp.mbox.media.kyoto-u.ac.jp

炉物理の研究 第61号 （2009年3月）

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N$W&Yraak0,3+lF$k [2]#3NEWYXtrH&rO}!Oh1`N0j<sXtrH&}!N~QH7FM(U$?bNG"k#r/UK"\/ 5 n 3 |+i5iH,!9Wjs09G++lk”the International Conference
on Mathematics, Computational Methods and Reactor Physics, Honoring Enrico Sartori” N ”Thebenchmark special session” K formal invitation r9kNG"Vir7F_7$HNa$kr:$?#~TgX)X&fJ6RK)X5<rj/`17F#/GzY10/KJj"6/0N03/9nNQjGN'*}q]qDGN&f/=,GeKJk@m&HWCF$?,"=_"~VO=,K"kNGz-u1k3HK7?#o9N"wW;Wm0i`O$&FqN&fjdgXGnilF$k,"INxYN:Yr}D
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aK"?/NYsA^</dj,sF5lF-? [3]#.TO3!5"wW;Wm0i`N:YrA'C/9k?aNYsA^</dj”A proposal for
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炉物理の研究 第61号 （2009年3月）

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Oliveira et.al H Brown et.al yNO+K2o [7] G"k#chN%6B8!O[}Ng-$NOGO
M~zLKhkm9,g-$,"=lrr1k?aK first flight collision source r;Q7?W;!
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7F"%6B8!Nr,eL4BXt!NrHJk}!rHCF$k [12]#3N>[*KC(k;N`O"%6B8!Nr,eL4BXt!Nr+ig-/[JklgKg-/Jk#=lN"[},g-$NOGM~zL,2xJNOGO"3N>[*J;,g-/Jj"%6B8!Nbt?|N}+OY/Jk#f9O-BU<j(Q9!GeL4BXt!N0rr/}!rsF7?,"v~O-BU<j(Q9!Khk3!5 FFT3 KWm0i`(i<,"j"D0J,isF7?djrr/3H,Ph
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3NaN0NbFOV'*}W[1] H18G"k,"0N3P!r)W7hj,+jd97?#r/Y-?2"w}x0r!0NAKq/#3Ω · ∇fg (r, Ω) + Σ tg fg (r, Ω) =ZdΩ ′ Σ sg (Ω ← Ω′ )fg (r, Ω′ ) + sg (r, Ω),(1)4π?@7 f (r, Ω) OLV r G}~ Ω N g− 2NQY+G"GLQ Σ *hS Σ (Ω ← Ω ) O g 2
N4GLQ*hSQY Ω +i Ω XN6pGLQG"k#;N` s (r, Ω) O>2+i6p7F/kf-RHK,vG8^lkf-R*hS0t; s (r, Ω)gtg

sg′′gEg

炉物理の研究 第61号 （2009年3月）

G8^lkf-RNBG"k'sg (r, Ω) =ZdΩ ′4πχg+4πkX

Σ s (Ω, g ← Ω′ , g ′ )fg′ (r, Ω′ )g′ 6=gZdΩ ′4πXνΣ f g′ (r)fg′ (r, Ω′ ) + sEg (r, Ω).

(2)g′v*hSqXtNQY+ f (r, Ω) *hS f (r, Ω) r!0GjA9k(ef e (r, Ω) =d1
1(f (r, Ω) + f (r, −Ω)) , f d (r, Ω) = (f (r, Ω) − f (r, −Ω)) .
22(3)3lirH$J1N?aK2N-f g rJ,9kH"0 (1) O!Nh&Kq1k#Ω · ∇f d (r, Ω) + Σ t f e (r, Ω) =ZdΩ ′ Σ s (Ω ← Ω′ )f e (r, Ω′ ) + se (r, Ω),(4)ZdΩ ′ Σ s (Ω ← Ω′ )f d (r, Ω′ ) + sd (r, Ω),(5)4πΩ · ∇f e (r, Ω) + Σ t f d (r, Ω) =4πd?@7"v*hSqXtN; s (r, Ω) *hS s (r, Ω) O0 (3) Nh&KjA5lk#
v*hSqXtNQY+reL4BXtrHCF!Nh&K8+9k'e

L−1

f e (r, Ω) =lX
1 X(2l + 1)flm (r)Ylm (Ω),
4π l=0(6)m=−levenLlX
1 X
f d (r, Ω) =(2l + 1)flm (r)Ylm (Ω),
4π l=1(7)m=−lodd?@7"8+N!t L OqtG"kH>j7F$k#
P aw0rJiK3/?aK".lNY/Hk J (r, Ω) r!0L

J (r, Ω) ≡ Ωf (r, Ω),(8)

GjA7"eL4BXtrQ$F!0NAK8+9k#
J (r, Ω) ≡ Ωf (r, Ω) =Lll=0

m=−lX
1 X
(2l + 1)
J lm (r)Ylm (Ω).
4π(9)

0 (6)"(7) *hS (9) rH&H"0 (4) *hS0 (5) +i!0,@ilk#e∇J dlm (r) + Σ l flm(r) = selm (r),(10)d∇J elm (r) + Σ l flm(r) = sdlm (r).

(11)

0 (10) *hS (11) rGLQN[Jk-&r4sGQ,9k3HKhjQYf-R+Nb<asH
KX9k*A-&roH7F!0,@ilk#
n · J (r) *hS n · J (r) O*A-&G"3(12)33G"n O-&Kb>J1LY/HkG"k#0 (12) N-&roOeL4BXtN2=0rH&
H!NAKq1k'
n · J (r) *hS f (r) O l = 0, 2, · · · , L − 1, −l <(13)=m<= l KP7F"3.
dlmdlmelmelm

炉物理の研究 第61号 （2009年3月）-&ro (13) O"z 4}~N1!5NNONlgO"f (r) O l ,qtN~b"3G"k,"2*hS3!5NNOGOlLKqt!N f (r) O"3G"k3HO31J$"9JoA"lLKqt!
Nb<asHO"3GOJ$3Hr(7F$k#
eL4BXtN#G&r Y (Ω) r]1FQ,7F".lNY/HkNb<asHO!0Nh&K
q1k#
l0dlm

∗lm

Z

J elm (r) =∗dΩYlm(Ω)J e (r, Ω) =4π∗dΩYlm(Ω)Ωf e (r, Ω)

4π′

LlX
X

=Z′′′′

C llmm fle′ m′ (r),(14)l′ =0 m′ =−l′even′LlX
X

J dlm (r) =C llmm fld′ m′ (r).

(15)

l′ =1 m′ =−l′odd

33G"8+8t Cl′ m ′lm

OY/HkG!0G?(ilk#′′2lC llmm =Y/Hk Cl′ m ′lm′+14πZ∗dΩYlm(Ω)ΩYl′ m′ (Ω).

(16)

4π

NWGOc(P!0G?(ilk#Z
2l ′ + 1∗dΩYlm(Ω)Ωx Yl′ m′ (Ω),
4π4πZ2l ′ + 1∗
l′ m ′
Clm2=dΩYlm(Ω)Ωy Yl′ m′ (Ω).
4π
4π′′lmClm1=(17)3liN0rH$"qt!Nb<asH f (r) rCn7"vt!N_,Dkh&K9kH"vt!Nb<asH f (r) KX9k!0N2,Ny,}x0,@ilk'dlmelmge(−∇Dlm ∇ + Σl )flm(r) = ŝlm (r) + s̃lm (r).

33G" D OPQTsG"=NPQWGO D(ilk#lml′ m′ p′Dlmp=′′LX

lX′′l mClmp′′l′′ =1 m′′ =−l′′odd^?

l′ m′ p′lmp ,p = p′ = 1, 2, 3,Ll
X
X′′C llmm ∇l′ =1 m′ =−l′odd′s̃lm (r) =LlXX
X X

l′ =0 m′ =−l′ p6=p′ p′ 6=pevenG"j"=liO!0G?

1 l′ m ′C ′′ ′′ ′ , for p, p′ = 1, 2, 3,
Σl′′ l m p′ŝlm (r) = selm (r) −(18)′′ ′lmp
Dlmp

(19)


1 ds ′ ′ (r) ,Σ l′ l m(20)

(21)

∂∂xp


1 ∂ efl′ m′ (r) ,Σl′ ∂xp′

G"k#?@7"x = x, x = y yG"k#0 (18) OH6}x0H18Ar7F*j"H6}x0H1MN"Q,N}!GuVB8KX7F%6=7"r/3H,Phk#0 (18) O"(l, m) = (0, 0) +i (l, m) = (2, 0), (l, m) = (2, 1) yHg!b!N0rr$Fb!NbasHraa"@il?b!Nb<asHr&UN`Xe~7FT/#
12炉物理の研究 第61号 （2009年3月）

3N?|Obt?| (inner iteration) H0t?| (outer iteration) NVGT&NG"fV?|HFs
G$k#
0 (18) N (l, m) = (0, 0) N~N8UOLoNH6}x0H18G"j"&UN;N`O?N;N`Kb!N`+iJkUC*J`,CoCF$k,"=lOH6}x0+iN:lr=7"H6}x0+iN:l,g-$[IUC*J`Og-/Jk#%6B8!GO1l2bN6p`,&UK"kNG6p`,g-$xbt?|N}+OY/Jk,"eL4BXt!GO3N&UN6p`O8_7J$NG"bt?|O6pNg-5K5X8G"j"6p`,%6B8!Nh&K}+rY/9k3HOJ$#0 (18) N2,y,}x0rxQ7F"2!5 x − y B8N0rr/ PLXY Wm0i` [13]"3!
5 x − y − z B8Ndjrr/ PLXYZ Wm0i` [14, 15] ,nil?#-BU<j(Q9!
GLQONh V bGOljG"kH7"0 (10) K e kr r]1FNh V GQ,9kH"!0,@ilk#Z
4−ieedSe−ikr n · J dlm (r) + Σl Flm(k) = Slm(k).

ikJ dlm (k) +(22)S

33G"S ONh V N=LG"Y/Hk n O=N=LKb>J1LY/HkG"k#Nh V GN-BU<j(Q9OQ9Qia<?< k rHCF!0GjA5lk'
Z

dre−ikr flm (r),(23)

F (slm (r)) ≡ Slm (k) =dre−ikr slm (r),VZ
F (J lm (r)) ≡ J (k) =dre−ikr J lm (r).

(24)

F (flm (r)) ≡ Flm (k) =

VZ

(25)

V

0 (22) H1MK0 (11) hj"^?"0 (14) *hS (15) hj"!0,@ilk'
ikJ elm (k) +J elm (k) =Z

dddSe−ikr n · J elm (r) + Σl Flm(k) = Slm(k).

(26)S′LlX
X′′′′

C llmm Fle′ m′ (k),(27)

l′ =0 m′ =−l′even′J dlm (k) =LlX
XC llmm Fld′ m′ (k).

(28)

l′ =1 m′ =−l′odd

0 (22) *hS (26) +iqt!Nb<asH F (k) rCn9kH"0 (18) Kjv9k!0,@ilk#dlm

3
X

lmp 2
Dlmp
kp + Σ lp=1+ZS!

eFlm(k) −3X
p=1lmp
Dlmp
ikp

ZedSnp e−ikr flm(r)

SdSe−ikr n · J dlm (r) = Ŝlm (k) + S̃lm (k),(29)炉物理の研究 第61号 （2009年3月）?@70 (20) *hS (21) Kjv9k`O′eŜlm (k) = Slm(k) −LlX
X′′l′ =1 m′ =−l′odd′S̃lm (k) =LlX
XX X′′lmDlmp
ikp

l′ =0 m′ =−l′ p6=p′ p′ 6=peven

G"j"k = k , k = k yG"k#Q9Qia<? k r!01x

1 dS ′ ′ (k).
Σ l′ l m(30)1ikp′ Fle′ m′ (k),Σ l′(31)

C llmm iky2

3X

lmp 2
Dlmp
kp + Σ l = 0,(32)p=1

r~?9h&KhakH"0 (29) O!0KJk#−3X
p=1

lmp
Dlmp
ikp

ZedSnp e−ikr flm(r) +SZdSe−ikr n · J dlm (r) = Ŝlm (k) + S̃lm (k).

(33)

3N0N8UOf-R+*hS.lNNhN-&MN_r^sG*j"+D=liO0 (13) N-&
roKhj"3GJ1lPJiJ$LG"k#^?"0 (33) O-&MN_r^`NG2!5NdjO
1!5N"3!5NdjO2!5NdjH7Fr/3H,Phk#0 (33) rxQ7F"2!5 x − y
B8ONdjrr/ FFT2[16] *hS3!5 x − y − z B8ONdjrr/ FFT3 Wm0i` [17, 18],nil?
tMW;c!2!5\$Idj
3NaGO2!5\$IdjNtMW;kLr(=&#Ackroyd H Riyait[5] O2!5N\$IN"kJ1JNONf-R+,[r\YK!$7F$k#^ 1 K$N,x@/Hdjr(9#^N@/HNhO\$IKJCF*j"Σ = Σ = 0.005cm H7FW;7?#^ 2 K PLXY"^ 3 K FFT2 Wm0i`rHCF@il?W;kL [19] r(9#FFT KhkW;
O#sFW;7?#^ 1 N y = 0 G x− 4eN OX KhC?4f-R+N P aw+i P awNM,(5lF$k#^?"^ 4 KO"PLXY *hS FFT2 Khk P awN4f-R+,(5lF$k#3liN^K(5lF$ku@O Ackroyd y, 5NJMH7F(7? RANKERN G@?MG"k#3liN^+i"PLXY O P awN!tre2Fb Ackroyd yN?(? RANKERN Khk?
NMKaE+J$,"FFT2 NMOawN!t,e,kK>CF`iN?NMK\a7F$k3H,,+k#PLXY H FFT2 OlNhNdjGO18kL,@ilk3HO{KN+ailF$k#>
TNc$O"\$INhN"kNONh&Jg-JGLQ.cCWN"kdjG2xKJk#
PLXY H FFT2 NW;!Ng-Jc$O"0 (13) N-&ror5NKH&+]+@1G"k#
PLXY GO"0 (18) OLoNH6}x0NlgH1MN"Q,!G3+l?9,0rHCF$k,"
0 (18) ,soK#(J?aK0 (13) N-&ro,<?5lF$k]ZOJ$#
3!5YsA^</djGb-BU<j(Q9!N0 (33) rQ$? FFT 3Wm0i`NkLO
5NJrKa$Mr?(F$k,"=lO0 (13) N-&ror5NKHCF$k+i@HWolk#5ta−1

39L

9炉物理の研究 第61号 （2009年3月）

Neutron scalar flux (n cm-2s-1)

1010-1-2

10

10-3-4
0

2.5

5

7.5

10

12.5

15

17.5

Distance along outer boundary OX (cm)Fig. 1 Problem 3: Geometry for thedog-leg duct source problem

10-1

10

Neutron scalar flux (n cm-2s-1)

Neutron scalar flux (n cm-2s-1)

10-2

1010·Fig. 2 Total flux by RANKERN given byAckroyd et al. and the PLXY alongouter boundary OX-310102.557.51012.515

17.5

Distance along outer boundary OX (cm)·Fig. 3 Total flux by RANKERN andthe FFT2 along outer boundary
OX-2-3-4
0-1

10-4

Distance along outer boundary OX (cm)·Fig. 4 Total flux by RANKERN , thePLXY and FFT2 with P9 alongouter boundary OX

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NG"k,"lDRp7F*-?$#sin θ d cos θ yN3QXtNv?X*JU#O"fX;db;NH-K,&NG-/NilF$
k#7+7"sinh θ d cosh θ JINPJ~XtNv?X*JU#OaLX;G5oiJ$7"?/
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炉物理の研究 第61号 （2009年3月）

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37Fvi;FulkNOa<+<N[jbJN@HNCFSC/j7?#
77/X~7? S RN&$sI&: Vista Q=3sN`nO&#sI&: Xp KfYkHgX#(G7$K//"FORTRAN d Latex r$s9H<k9kNKbjVK,++C?#_&fG"lP
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