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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, farzad.rahnema@nre.gatech.eduINTRODUCTION

In a recent paper [1], 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 [3]. 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

developed.

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.

NOTATIONAL CONVENTIONS

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 eachsurface 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,

denoted OV.

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 [1], which may

be expressed as a sum of coarse-mesh functionals(1)

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 functionalsF, = V, + S,, where

(2a)

'Current Address: 'Los Alamos National Laboratory, Transport Methods Group, P.O. Box 1663 MS D409, Los Alamos, NM 87545,

swmosher@lanl.gov731

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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 September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.