A Hybrid Monte Carlo-Deterministic Method for Global Binary Stochastic Medium Transport Problems Page: 4 of 6
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stochastic medium transport problem using Algorithm B
proposed by Zimmerman and Adams [6]. The Monte Carlo
code can use analog Monte Carlo techniques and also the
implicit capture and weight windows variance reduction
techniques. Because the flux of particles can be signifi-
cantly different for the two materials at the same spatial
location, we use material-dependent weight windows. The
weight windows are applied at collisions and at both zone
boundary and material interface crossings.
We use the approximate scalar flux distribution com-
puted using the discrete ordinates code, either AM or L-P,
with a low-order angular quadrature to automatically esti-
mate the weight windows. The center, ceiling, and floor of
the weight window in material i are computed using [2]center /> Qi (x)
max 01 (x)
ww 1fl (x) p x ww,"1"e (x),
2ucenter 1>)
2 floor __ zE " () ,
p(2a)
(2b)
(2c)where 0, (x) is the material i scalar flux distribution and
p > 1 is a scale factor. Using weight windows defined
as in Eqs. (2) results in Monte Carlo particles distributed
approximately uniformly across the problem domain [2].
When using the AM solution, the same flux is used to
generate the weight windows in both materials, so the
resulting weight windows are the same in both materials.
Because the deterministic solution is computed on a finer
spatial mesh than that used for the Monte Carlo weight
windows, the deterministic solution is volume-averaged
onto the coarser weight window spatial mesh.
NUMERICAL RESULTS
We consider a binary stochastic medium defined by
of=10/99, A0 99/100, uj 100/11, and A1 =
11/100 with a total slab width L = 25 cm. The scat-
tering ratio, c, =u/J, is the same in both materials,
co =ci 0.7. Therefore, the stochastic medium consists
of two materials with mean material slab widths of 0.1 and
1.0 mean free paths, respectively. The ensemble-averaged
macroscopic total cross section is unity. This test prob-
lem specification is a binary stochastic medium variant of
a test problem used by Becker, Wollaber, and Larsen [8] to
investigate hybrid Monte Carlo-deterministic algorithms.
For the deterministic L-P solution, we used the S2 discrete
ordinates approximation and uniform spatial zones with
Ax 0.1 cm (zones of approximately unity mean free
path in material one). We generated the deterministic AM
solution using the S4 discrete ordinates approximation and
uniform spatial zones with Ax = 0.05 cm. Determinis-
tic AM results computed using the S2 approximation were
so inaccurate that the Monte Carlo solution was too ineffi-
cient to be practical. Larger zone sizes would most likelybe sufficient for generating the approximate AM or L-P for-
ward solution but would violate the positivity requirements
of the diamond difference spatial discretization [1]. The
Monte Carlo solutions were obtained using 107 histories,
a weight window scale factor of p 4, and uniform spa-
tial zones with Ax 1.0 cm. The Monte Carlo zone size
is consistent with that used in similar studies by Becker et
al. [8], also demonstrating the ability of the hybrid method
to use disparate zone sizes in the deterministic and Monte
Carlo algorithms.
We compare the results of the hybrid Monte Carlo-
deterministic method using implicit capture and weight
windows (WW) to the results obtained using implicit cap-
ture and a Russian roulette weight cutoff (IC). We evaluate
the efficiency of the Monte Carlo methods using a material
scalar flux distribution figure of merit (FOM) defined asFOM, (x) =
R2 (x) Tep,(3)
where R2 (x) is the relative standard deviation in the
ensemble-averaged material i scalar flux distribution at a
spatial location x, and Tp is the total CPU time required
for the generation of the approximate deterministic solution
(if required) and the Monte Carlo transport process. A high
FOM value is desirable, as this indicates a small relative
variance in the flux and/or a small CPU time.
The ensemble-averaged material scalar flux distribu-
tions computed for the test problem using the AM and the
L-P models with the S4 and S2 angular approximations,
respectively, and using Monte Carlo Algorithm B are plot-
ted in Fig. 1. The L-P model demonstrates only reason-
able agreement with the more accurate Monte Carlo results
for this problem. The material scalar flux distributions and
the leakage rate at x L computed using the L-P S2 ap-
proximation exhibit relative differences with respect to the
Monte Carlo of up to approximately 40%. The scalar flux
distribution computed using the AM approximation is in
error by a few orders of magnitude over much of the spatial
domain. The inaccuracy of the AM approximation persists
for higher discrete ordinates quadrature orders, i.e. the in-
accuracy is a result of the AM approximation itself and not
the angular quadrature order. The extreme inaccuracy of
the AM approximation for this problem results in weight
window values that produce an inefficient Monte Carlo cal-
culation due to excessive particle splitting.
The material scalar flux distribution figure of merit
values are plotted in Fig. 2 for the Monte Carlo simulations
using implicit capture alone and for the simulations with
weight windows automatically generated from the AM
and L-P solutions. The FOM obtained using the hybrid
Monte Carlo-deterministic method with weight windows
computed from the approximate AM solution is larger than
the IC FOM over a limited range of the spatial domain
but is a few orders of magnitude lower over much of the
domain. The FOM obtained using the hybrid method
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Keady, K P & Brantley, P. A Hybrid Monte Carlo-Deterministic Method for Global Binary Stochastic Medium Transport Problems, article, March 4, 2010; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc1013598/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.