JAEA-Data/Code 2007-004 - Welcome to Research Group for ...

Note that among four terms appearing in the expression of Δ ij (ρ,ϕ), the first two terms have been
calculated as the last two terms of the previous k’. The calculation of Δ ij (ρ,ϕ) in the geometrical order
reduces the number of transcendental functions to be evaluated into half.
Care is taken when λ i is so small compared with unity so that differential approximations are
used, for example,
K
λ ) − K ( λ + λ ) ≅ λ × K ( λ )
(7.1-76)
i3 ( ij i3
ij i i i2
ij
On integrating Δ ij , the symmetric relation, Δ ij =Δ ji validates to eliminate the loop of the source
region in the reverse direction or to reduce the range of angular integration into half. The simple
process to replace the off-diagonal element Δ ij by (Δ ij +Δ ji )/2 covers the above saving.
A normalization so that the sum of P ij over j be unity is effective to reduce the error caused by
coarse integration mesh, and also by truncated optical distance which is terminated by the fixed
number of cells to be traced by neutron line for the perfect reflective boundary condition.
The numerical calculation of K in functions has yet to be explained. Although some rational
approximations are developed for the Bickley-Naylor functions 50) , they would be very time consuming
because they have to be used so frequently as 10 6 ∼10 7 times. In the SRAC, a quadratic interpolation is
performed numerically by using tables of a, b, c; the coefficients of three terms for the quadratic
expression of the Bickley-Naylor function. These tables list a, b and c as a function of x and n, where
ym−1
− 2ym−1/
2 + ym
am
=
2
2Δx
ym
− ym−
1
b m = − am
( xm−1
+ x
Δx
m
)
(7.1-77a)
(7.1-77b)
c
m
2
= ym−1 − bm
xm−1
− am
xm−1
(7.1-77c)
y = K
m
in
( x m )
Δx
= ( x m − x m −1 ) = 0.01
(7.1-78)
(7.1-79)
The range of the tabulation of K in (x) is 0 ≤ x11.0, K in (x) is set
to be zero, while the practical usage (by Fortran statements) may assume K in (x) vanishes if x>6.0.
Thus the Bickley-Naylor function is computed by performing twice the multiplication and twice
the summation after table-look-up:
K ( x)
= ( a x + b ) x + c
in
m
m
m
(7.1-80)
where
x
m−1
≤ x ≤ x
m
The table-look-up and the interpolation are performed in the routine itself to avoid the additional
process of calling any external subroutine.
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