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Transport Methods: General
A Decoupled Finite Element Heterogeneous Coarse Mesh Transport Method
Scott W. Mosher' and Farzad Rahnema
Georgia Institute of Technology, Nuclear and Radiological Engineering / Medical Physics Programs
of the G. W. Woodruff School, Atlanta, GA 30332, email@example.com
In a recent paper , an original finite element (FE)
method was presented for solving eigenvalue transport
problems on a coarse spatial mesh. The method employed
a surface Green's function expansion of the angular flux
trial functions, so that heterogeneous coarse-meshes could
be treated with relative ease. Numerical problems were
solved using the multigroup discrete ordinates
approximation in one-dimensional (1-D) slab geometry.
Unfortunately, difficulties were encountered in finding
solutions to the algebraic finite element equations, which
led to sizeable angular flux discontinuities at coarse-mesh
interfaces and significant errors. For this reason, a non-
variational iterative technique was ultimately favored for
converging the angular flux distribution, and was used in
conjunction with a Rayleigh quotient for converging the
eigenvalue [1, 2].
In this paper, a new derivation of finite element
equations is presented, which seems to offer a remedy for
at least some of the numerical ills that plagued the
previous work. First, the equations are derived in terms of
a generalized response function expansion . This
allows a more efficient response basis to be employed and
vastly reduces the overall computational effort without a
substantial loss of accuracy. Second, the tight coupling
between coarse-meshes in the original equations is
effectively broken by assuming that an accurate estimate
of the flux distribution entering a given coarse-mesh is
known. With an additional assumption that an accurate
eigenvalue estimate is known, an iterative approach to
solving these decoupled finite element (DFE) equations is
The DFE method has been applied to both 1- and 2-D
heterogeneous coarse-mesh problems with a far greater
degree of success than the original FE method. However,
some numerical difficulties remain to be overcome before
the new approach can be considered robust.
Consider a system, V, composed of a collection of
heterogeneous coarse-meshes V, (where i = 1, 2, ... I). Let
the boundary of V,, denoted &V,, be divided into flat
surfaces OV, (where s = 1, 2, ...) with outward unit
normal vectors denoted nis . It is assumed that each
surface is either: 1) an interface, separating V, from one
adjacent coarse-mesh, or 2) a boundary segment,
coinciding with a section of the external boundary of V,
For convenience, a concise phase-space notation is
used. The collection of independent variables (c,Q,E)
with the spatial variable restricted to coarse-mesh V, is
denoted by w,. The subscripts in the symbol w-s denote a
restriction of the spatial variable to the surface 8 V,s. The
negative superscript denotes a restriction of the angular
variable to the directions entering V, (i.e., so that
n, Q 2 < 0 ). The combination of angular superscripts and
surface subscripts is not arbitrary (on any quantity).
Reversing the order of the subscripts, reverses the
negative and positive angular half-spaces (e.g.,
ns = -nf ). The seemingly redundant combinations that
result from this convention (e.g., w and wI ) are used
uniquely in identifying limiting values of the trial
functions, which are allowed to be discontinuous on
interfaces. Specifically, p(w-) denotes the limiting
angular flux distribution as w-s is approached from inside
V,, while p(w ) denotes the limit as approached from
outside. Both limiting values are considered to lie
precisely on w-s in the surface integrals developed later.
DECOUPLED FINITE ELEMENT EQUATIONS
The starting point of the derivation is the variational
principle developed by Ilas and Rahnema , which may
be expressed as a sum of coarse-mesh functionals
F9, rP*; A] F, (w,), *(w,);l
where 9 and p* denote the forward and adjoint angular
flux trial functions and A denotes the trial estimate of 1/k.
The coarse-mesh functionals are themselves conveniently
expressed as a sum of volume and surface functionals
F, = V, + S,, where
'Current Address: 'Los Alamos National Laboratory, Transport Methods Group, P.O. Box 1663 MS D409, Los Alamos, NM 87545,
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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/m1/2/: accessed December 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.