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Transport Methods: General
using non-variational inner iterations with Rayleigh
quotient outer iterations [1-3], hereafter referred to as the
NV+RQ method. Hence, it is efficient to use the
converged NV+RQ solution as an initial guess for the
DFE method, which is far more computationally
expensive per iteration. This scheme is also much more
numerically stable than starting the DFE iterations from a
simple initial guess.
The DFE method was implemented in the multigroup
discrete ordinates approximation in both one- and two-
dimensional Cartesian geometries. In 1-D, the discrete
Legendre polynomials (DLPs) were used as an orthogonal
basis for the polar dependent response function boundary
conditions, as in previous work . In 2-D, a triple
product of DLPs were used for the spatial, azimuthal, and
polar dependent boundary conditions . Both response
functions and reference eigenvalue solutions were
computed using fine-mesh discrete ordinate codes. The 2-
D calculations were performed using the DORT code 
distributed with the DOORS 3.2 package . In all cases,
response functions were computed at 10% k intervals
(e.g., 0.9, 1.0, 1.1) and updated using linear interpolation
during the coarse-mesh calculations.
A substantial increase in accuracy was obtained for
the one-dimensional benchmark problems presented in
, relative to the NV+RQ method. With a 0'h order polar
expansion in the most heterogeneous problem, the relative
errors in average and maximum region-wise fission
density were reduced from 2.13% to 0.80% and -4.19% to
-2.13%, respectively. Similar reductions in error were
obtained for the other problems. No numerical difficulties
were encountered in solving this 1-D problem set.
A modified version of the two-dimensional Henry-
Worley benchmark problem  was constructed to
provide a more realistic test of the new method. The
quarter-core configuration is illustrated in Figure 1. The
problem is composed of three coarse mesh types: type A,
a fuel node containing a cruciform-shaped control
material; type B, the same fuel node containing a
cruciform-shaped water region; and type W, a
homogeneous water (reflector) node. The two-group
material cross sections are available in reference .
A comparison of the results generated by the NV+RQ
and DFE methods is shown in Table I. A significant
reduction in both average and maximum fission density
relative errors was obtained for the expansion orders
shown. However, the method failed to converge for
several order combinations, even though an accurate
solution was used an initial guess. Unfortunately, the
source of this problem has not yet been isolated. Though
the DFE method is not yet robust, it has had much greater
success than the original FE method, and shows enough
promise to motivate further development.
w w w w
A0 B0 A0
B0 A B0 W
- A - & - a - a
Figure 1. Modified Henry-Worley Problem
Table I. Percent Relative Errors for the MHW Problem
order NV+RQ Method DFE Method
k Avg FD Max FD Avg FD Max FD
0,0,0a -0.40 1.92 -5.48 0.95 -3.82
0,0,1 -0.43 1.86 -5.31 0.72 3.15
1,0,0 0.10 0.95 -3.67 0.47 1.15
2,0,0 0.15 1.39 -4.81 0.47 -1.41
2,1,0 -0.03 0.58 -2.22 0.13 -0.33
2,2,0 0.00 0.36 -1.41 0.04 -0.12
a i j,k = i: spatial order, j: azimuthal order, k: polar order
The authors gratefully acknowledge the support of
this work by the Department of Energy's Nuclear Energy
Research Initiative under project number 2002-081.
1. D. ILAS, F. RAHNEMA, Transport Theory and
Statistical Physics, 32, 441 (2003).
2. S. W. MOSHER, F. RAHNEMA, "An Intra-Nodal
Flux Expansion for a Heterogeneous Coarse Mesh
Discrete Ordinates Method," Proceedings ofANS
Nuclear Mathematical Methods Sciences: A Century
in Review, A Century Anew, April 6-10, Gatlinburg,
3. S.W. MOSHER, Ph.D Thesis, Georgia Institute of
4. W.A. RHOADES, R.L.CHILDS, Nuclear Science
and Engineering, 99, 88 (1988).
5. Radiation Safety Information Computational Center,
Code Package CCC-650, Oak Ridge National
6. K.S. SMITH, Ph.D. Thesis, Massachusetts Institute
of Technology (1980).
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Chen, Bin; Lin, Jung-Fu; Chen, Jiuhua; Zhang, Hengzhong & Zeng, Qiaoshi. Synchrotron-based high-pressure research in materials science, article, June 1, 2016; (digital.library.unt.edu/ark:/67531/metadc929793/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.