mathematical basis is given for a value optimization criterion bz Monte Carlo long-distance radiation transport simulation. The basis is the tendency to preserve ...
Atomic Energy. Vol 86, No~ 2. 1999
A PERFORMANCE SIMULATION TRANSPORT
CRITERION
FOR VALUE
BASED ON CONTRIBUTON THEORY
UDC 539.16.04(075.8)
N. M. Borisov and M. P. Panin
mathematical basis is given for a value optimization criterion bz Monte Carlo long-distance radiation transport simulation. The basis is the tendency to preserve the number of hTtersections between random paths and services separating the source and detector. The direct and conjugate contributon transport equations are considered, and it is shown that there is a relationship between the contributon simulation and the value one in the direct and conjugate cases. The contributon criterion is applied to the performance of direct attd conjugate value sbnulation. The computational performance of the conjugate method is higher because of the more accurate approximation for the value function, and the performance criterion confirms this.
Value optimization represents a general approach in Monte Carlo simulation for long-distance radiation transport, in which the value of direct collisions is provided by the conjugate flux density, while that of the conjugate collisions is provided by the direct one [1]. There are thus inexplicit bilinear combinations in value simulation, which correspond to the solutions of the direct and conjugate transport equations. The contributon concept is used in transport theory: particles whose kinetic equations contain such a bilinear combination. It has been derived [2, 3] by multiplying the direct integrodifferential kinetic equation by the conjugate flux density, and the conjugate one by the direct one, and then subtracting the second product from the first: (Vf~r
I2)r
= j'j'0(E',f2" ~ e,~lr)r
e, f2)-Oc(r,E, ta),
(i)
in which the contributon flux density 9 = cp~+, source function Sc = ~0+S, and sink function Dc = cpD are introduced by the functions for the real particles: the direct flux density ~0, value ~0+, source S, and detector response D from unit path length. The contributon differential cross section is introduced as 0(E', ~" ~ E, f21r) -- Es(E', f2" ~ E, ~lr)q~+(r, E, ~)/r247
E', f2'),
where the integral cross section 0 s is the integral with respect to 0 over the output arguments. There is an analogy between value optimization for direct or conjugate particle transport and that for contributon transport. We modify (1) by introducing the contributon absorption cross section 0 a = DI~p+. We interpret 0 s as the contributon scattering cross section, and thus derive the total cross section 0 t = 0 a + 0 s, which gives the equation (Vf2r r, E, f2)) + 0 t (r, E, ~ ) r
E, x"2)=
Moscow Physics Research Institute. Translated from Atomnaya t~nergiya, Vol. 86, No. 2, pp. 103-107, February, 1999. Original aaicle submitted January 27, 1999. 1063-4258/99/8602-0105522.00 9 1999 Kluwer Academic/Plenum Publishers
105
= f f O(E',O'~ E, f21r)q~(r,E i ,O 9 )d E 9 dO i +Sc(r,E,[2),
(2
which may be called the direct contributon transport equauon, since the interaction cross section is displaced by analogy with direct value simulation, i.e., in proportion to the conjugate flux density. We proceed in a similar fashion and introduce conjugately displaced scattering cross sections for contributons:
O+(E',O "--->E,~zlr)= Zs(E,-~---) E ,-O ,,] r)cp(r,E,-O)/~(r,E 9,-~'), and the absorption cross section 0a+ (r, E, f2)= S(r, E,-O)/cp(r, E , - O ) and total interaction cross section e + = 0~, + 0 F, which leads to a conjugate formulation of the contributon transport equation, since now the scattering cross section is displaced by analogy with the conjugate value simulation: -[Vf/r
E, f~)] + e + (r, E, f2) r r, E, f~) =
= I I 0 + ( E ' , O " ---* E, O I r ) ~ ( r , E',f'Z')dE'd.Q'+ Dc(r,E,-O).
(3)
It is evident that the points of contributon generation and absorption are interchanged by comparison with (2). Equations (2) and (3) resemble the direct and conjugate Boltzmann equations in form, but with displaced cross sections, source, detector function, and self-conjugate flux density r --- ~+. To handle these by Monte Carlo methods, we introduce the contributon collision density: that for the direct entering ones ~P(r, E, f2) = Or(r, E, ~) r E, f2) and that for the emerging + ones X(r, E, s and that for the conjugate entering ones X+(r, E, f2) = Or(r, E, fl) ~+(r, E, -f2) and that for the emerging ones hU+(r, E, ~). We derive the following relationship for the direct collision densities:
9 (r,E,O)=ITc( r " ~ r IE,.Q.)X(r ",E, f2)dr "; (4) X(r, E , ~ ) = I I Cc( E"~z" "~ E,121 r) W(r,E',O')dE" dO" + Sc(r, E,O),
and for the conjugate ones
[ X+(r'E'f2)=ITc+(r''-erlE'fl)~+(r''E'fl)ar';
,
9 (5)
W+(r, E,D.) = I I C~(E', O" ~ E,x'-21r) X+(r, E',f2")dE dI'2 + Dc(r, E,.Q),
in which r-i"
s
0r(r, E,O) exp[-Zc(r" ~ r I E)]5(O - _i r- - - ~ ) Tc(r'--~ r I E,O) =
(6)
Ir-r'l 2
is the direct contributon transport kernel and "rc is the optical distance, with
Cc(E'.O' ~ E, O I r) = O(E',~" --~ E.f21 r ) / e r ( r , E',O') 106
(7)
the direct kernel for contributon collisions, and
+
0r (r, E, f2) exp[-l:c (r +
p...~
r - r
r I E)}~5(~2- I r _ r, I)
s
T. (r -4 r l E , f2)=
(8)
I r - r' 12
the conjugate contributon transport kernel, Zc+ the optical distance, and
C~(E',f2' -4 E,s r)= 0+ (E',f2 ' -4 E, f21 r)/0~'(r, E',f2")
(9)
the conjugate contributon collision kernel. When one simulates direct contributon transport, the random paths begin in the source, and then the hypothetical particles reach the detector without absorption and vanish in it. In conjugate contributon transport, the particles are generated in the detector and vanish in the source. The contributon collision densities W, X, X", W+ derived from (4) and (5) have the physical meanings of the contributions to the functional from all the paths (direct and conjugate respectively) that have points of collision in unit phase volume around the given phase point. The scalar contributon flux density through a certain surface is J = ~ln, ~1, in which n is the normal, and this has the meaning of the contribution to the functional from all paths that intersect unit area of that surface. Although there are essential differences [3] between contributon simulation and value simulation, there is a close relationship between them. We show that our scheme of (4)-(9) for contributon transport has the same density distribution as in value simulation. We use the displaced value transport models defined in [1 ]: direct one containing source function, transport kernel, collision kernel, and absorption function respectively:
f0(r, E, f2) = S(r,E,~)?~'(r,E,~);
~ ( r ' --~ rl E, f2) = T(r" -4 r l E , ~ ) w*(r, E,f/).
z'(r',E,~)'
q ( e ' , n ' - 4 E, n l r) = c(E'.n'-4 e, n l r) x'(r,e;n) ; ~1/ (r,E .rE)
D(r, E, fl)
fa (r'E'fl)= Z l ( r , E ) g * ( r , E,f/) and the conjugate one f0+ (r, E , ~ ) = D ( r , E , - ~ ) ( g t + ) * ( r , E , ~ ) ;
(~+)'(r,E,
f2) .
T/+(r ' -4 r l E, f2) = T(r' -4 r l E, f2) ( ~ + ) ' ( r ' , E, xQ) '
C+(E',E2" -~ E,~2I r) = C + ( E ' , ~ ' -4 E, f 2 t r ) ( ~ + ) ' ( r , E ,
f2) .
( Z + ) ' ( r , E ' , ~ ')
+
.f,, (r,E,f/) = *
+
~
+
S(r, E,-f~) I:, (r, E)(Z +)" (r, E,f~)"
*
Here ~g . X'. ( t g ) ,(X ) are the values of the incoming and outgoing direct and conlugate collisions respectively, while
T, C,
and C + are the transport kernel and the direct and conjugate collision kernels respecuvely. 107
TABLE 1. Calculations on Dose Buildup Factors for Various Forms of Monte Carlo Method and Low Thresholds Wl for the Weighting Windows Method
IV#
Ba
o, %
FOM, tel. units
Direct
I. 10-5
53.6
5.5
0.37
Conjugate [4] data
1.10-3
55.2
5.2
1.0
I. 10-2 1.10-.6
66.3 62.5
11.7 4.5
0.34 5.8
1.10-3
61.2
7.8
2.3
-
60.6
It has been shown [1] that we have as follows for the direct and conjugate values:
;r (r, E, f~) = r
(r, E, g2); ( ~ + ) ' ( r , E , ~ ) - ~(r, E,-E~);
V* (r, E,f~) = f f C(E,I) ---) E',12"lr)z*(r,E',f2")dE'df2" + D(r,E, E2)IZt(r,E);
(X+)'@,E,f~)=ffc+(E,O~ E , ~ e
9
+
*
#
9
9
#
lr)(~/ ) ( r , E , f ~ ) d E dr2 +S(r,E,-f~)/gt(r,E ).
The total interaction cross sections for contributons and real particles are related in such a way that
Ot(r,E,f'z)=Et(r,E)VS(r'E'D);
0~+(r, E,X"2)= Z, (r, E)
(Z+)*(r,E, f2) (V+)*(r,E, f2)'
(r,E,~)
which gives us that value simulation coincides with the contributon simulation in the direct and conjugate cases as regards the initial phase state densities of the collision kernels and absorption probabilities. Also, there is the following relation between the direct value transport kernel and the contributon one:
To(r" --+ r I E, f2) = Ti(r" --~ r I E, f2) z ' ( r ' , E, f2) x g*(r,E, f2)
We integrate the integrodifferential transport equation for the collision values -EiVz*(r, E,Q) + Et(r,E)x*(r,E,f~) = Et(r,E)~*(r,E, E2) with respect to the spatial coordinate to get that "/',.- T i. Similarly, we can show that there is identity for the conjugate transport kernels: Tc+ = T~i.
108
12
8 6 4
2! ==
0,13 0.30 0,46 0.62 0.79 X a
3.O 2.5
2.0 \ 1.5
1.0 0.5 0, 0.13 0.3O O,46 0.~ 0.79 ,
,
i
i
+
1
i
+
r
"t--X
b Fig. 1. Numbers of intersections between paths of direct particles (a) and conjugate ones (b) with planes perpendicular to the source-detector axis for W t of 10-5 (1), 10-3 (2), 10-2 (3), 10--6 (4), 10-3 (5).
This means that ideal value simulation (direct or conjugate) produces phase states that arise with the same probabilities as in the corresponding contributon simulation. This allows one to estimate the performance in value optimization from the results obtained in the contributon transport theory. In particular, the latter contains the result of constancy for the scalar contributon current J through the surfaces separating the source and the detector [2]. Then from the viewpoint of the performance criterion, the optimization is the more accurate, the better the conservation of the number of intersections for wandering particles between the random paths and the surfaces separating the source and the detector. We apply this criterion to the direct and conjugate value simulation in a test case of deep photon penetration into the atmosphere [1], where the value simulation was performed by means of roulettes and splittings, while the value function was estimated from the buildup factor. Table 1 gives calculations on the dose buildup factor for a point isotropic monoenergetic (0.5 MeV) source in an unbounded atmosphere in the case of a penetration depth of 10 times the mean free path. The error o is understood in the sense of one standard deviation, while the factor of merit FOM is taken as the reciprocal of the product of the dispersion and calculation time. For comparison, we give the calculated buildup factor obtained by the method of moments [4]. The number of published data on the direct functions is greater than that on the conjugate ones because of the better approximation to the value function for conjugate simulation, which the calculations confirm. Figure 1 shows the distributions of the numbers of intersections with planes perpendicular to the source-detector axis in value optimization with various weighting windows. In direct simulation, the distance x is reckoned from the source, while in conjugate simulation, the distance "r - x is reckoned from the detector. This distribution indicates that the weighting window in this scheme is an important regulator for the factor of merit in value optimization, which redistributes the phase states from the source to the detector. The performance criterion can form a convenient method of displaying how closely the chosen parameters approximate to the optimal ones. The best result as regards the factor of merit is obtained with maximally uniform distributions of the numbers of intersections, which tends to a plateau over a range of values for the mean free path and remains approximately constant as the detector is approached (or the source in the conjugate case). 109
REFERENCES
2.
3. 4.
II0
N. M. Borisov and M. P. Panin, "Adjoint importance Monte Carlo simulation for gamma ray deep penetration problem," Monte Carlo Meth. Applic., 3, No. 3, 241-250 (1997). M. Williams, "Generalized contributon response theory," Nucl. Sci. Eng., 108, 355-382 (1991). C. A. Aboughantous, "A contribution to Monte Carlo methods," ibid., 118, 160-177 (1994). A. Chilton, C. Eisenhauer, and G. Simmons, "Photon point source buildup factors for air, water, and iron," ibid., 73, No. 1, 97-107 (1980).
