Acceleration of step and linear discontinuous schemes for the method of characteristics in DRAGON5

Đăng ngày 7/3/2019 10:07:22 PM | Thể loại: | Lần tải: 0 | Lần xem: 10 | Page: 8 | FileSize: 0.77 M | File type: PDF
Acceleration of step and linear discontinuous schemes for the method of characteristics in DRAGON5. The applicability of the algebraic collapsing acceleration (ACA) technique to the method of characteristics (MOC) in cases with scattering anisotropy and/or linear sources was investigated. Previously, the ACA was proven successful in cases with isotropic scattering and uniform (step) sources. A presentation is first made of the MOC implementation, available in the DRAGON5 code.


  
Contents lists available at ScienceDirect
Nuclear Engineering and Technology
journal homepage: www.elsevier.com/locate/net
Technical Note
Acceleration of step and linear discontinuous schemes for the method
of characteristics in DRAGON5
Alain Hebert
Ecole Polytechnique de Montreal, P.O. Box 6079 Station Centre-Ville, Montreal, Quebec H3C 3A7, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
The applicability of the algebraic collapsing acceleration (ACA) technique to the method of characteristics
Received 21 June 2017
Accepted 7 July 2017
Available online 25 July 2017
(MOC) in cases with scattering anisotropy and/or linear sources was investigated. Previously, the ACA was
proven successful in cases with isotropic scattering and uniform (step) sources. A presentation is rst made
of the MOC implementation, available in the DRAGON5 code. Two categories of schemes are available for
Keywords:
Method of characteristics
Linear discontinuous source
Algebraic collapsing acceleration
Generalized minimal residual acceleration
method
integrating the propagation equations: (1) the rst category is based on exact integration and leads to the
classical step characteristics (SC) and linear discontinuous characteristics (LDC) schemes and (2) the second
category leads to diamond differencing schemes of various orders in space. The acceleration of these MOC
schemes using a combination of the generalized minimal residual [GMRES(m)] method preconditioned
with the ACA technique was focused on. Numerical results are provided for a two-dimensional (2D) eight-
symmetry pressurized water reactor (PWR) assembly mockup in the context of the DRAGON5 code.
DRAGON5 code
© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the
1. Introduction
iterations in a production code such as DRAGON5 [17]. In this paper,
the focus is on the acceleration of the scattering iterations which
This paper is related to the application of the method of char-
are required with the application of the MOC. Two acceleration
acteristics (MOC) for solving the neutron transport equation [1].
techniques are investigated: (1) the generalized minimal residual
This method iteratively solves the transport equation in terms of
[GMRES(m)] method [18] and (2) the algebraic collapsing acceler-
the angular moments of regional uxes by summation upon a
ation (ACA) method [4]. The application of these techniques was
tracking. Solution of the characteristics form of the transport
demonstrated on a two-dimensional (2D) eight-symmetry pres-
equation is performed over each track as a function of a polynomial
surized water reactor (PWR) assembly mockup in the context of the
approximation for the neutron source along this track. Most pro-
DRAGON5 code. It is shown that the acceleration remains effective
duction implementations of the MOC are based either on a
in spite of the introduction of scattering anisotropy and linear
discontinuous at-source spatial approximation [2e7] or on a
sources.
discontinuous linear-source spatial approximation [8e14] along
the tracks. We investigated a new class of linear characteristics
schemes along cyclic tracks for solving the transport equation for
2. Theory
neutral particles with scattering anisotropy. These algorithms rely
on step and linear discontinuous exact integration, as described in
[15,16]. These schemes are based on linear discontinuous co-
efcients that are derived through the application of approxima-
tions describing the mesh-averaged spatial ux moments in terms
of spatial source moments and of the beginning- and end-of-
segment ux values. The linear discontinuous characteristics
(LDC) scheme is inherently conservative, a property that facilitates
its practical implementation and the acceleration of its scattering
A brief introduction of the MOC formalism is given. The
boundary conditions treatment along with the details on the iter-
ative strategy are not reported in the present paper but can be
found in [1]. The classical step characteristics (SC) and at-source
diamond differencing (DD0) schemes are also presented in this
book. Emphasis is put on the characteristic form of the transport
equation which arises from the trajectory-based formulation of the
ux integration. The conservation principle is formulated within
this framework.
The backward characteristic form of the transport was obtained
in Section 3.2.1 of [1]. The one-speed and steady-state form of this
E-mail address: alain.hebert@polymtl.ca.
equation is written as:
1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

d
ffiffiffi
k k
2
Z
1
k
ffiffiffi
Z
k
2
3
[
ðTÞ
k
k
Z
1
k
ffiffiffi
Z
k
2
3
[
ðTÞ
d


ffiffiffi
k k
2
U
Q ð
Þ
S
U
 
ð Þ
3 Q
k
S
1136
A. Hebert / Nuclear Engineering and Technology 49 (2017) 1135e1142
dsfðr þ s U;UÞ þ Sðr þ s UÞ fðr þ s U;UÞ ¼ Qðr þ s U;UÞ
(1)
segment lengths [k are always dened in three-dimensions, even
for prismatic 2D geometries. The intersection points of a charac-
teristic with the region boundaries, and the corresponding angular
where r is the starting point of the particle, s is the distance trav-
ux on these boundaries, are written:
elled by the particle on its characteristic, U is the direction of the
characteristic, S(r) is the value of the macroscopic total cross sec-
rkþ1 ¼ rk þ [k U;
k ¼ 1;K:
(3)
tion at r, 4(r,U) is the particle angular ux at r, and Q(r,U) is the xed
source at r.
The spatial integration domain is partitioned into regions of
volume fVi;i ¼ 1;Ig, each of them characterized by uniform nuclear
properties and surrounded by boundary surfaces fSa;a ¼ 1;Lg. The
MOC is based on the discretization of Eq. (1) along each path of the
particle and on the integration of the ux contributions using
spatial integrals of the form:
2.1. The linear discontinuous characteristic assumption in space
A linear representation of the sources along characteristic T
based on an expansion in normalized Legendre polynomials is
introduced. This expansion is applied over segment [kðTÞ, as
pictured in Fig. 2. Its mathematical expression is:
Z Z
Vifi ¼ d3r d2U fðr;UÞ
Z Vi Z 4p
(2)
Qðrk þ s U;UÞ ¼ Qð0ÞðUÞ þ 2p3s  [kðTÞQð1ÞðUÞ;
k ¼ 1;K
(4)
¼ d4T ds cVi ðT;sÞ fðp þ sU;UÞ
Y 
where
where 4i is the average ux in region i and Y ¼ (T) is the tracking
domain.
A single characteristic T is determined by its orientation U and
[kT
Qð0ÞðUÞ ¼ [kðTÞ ds Qðrk þ s U;UÞ
0
(5)
its starting point p dened on a reference plane PU perpendicular
to T, as depicted in Fig.1. The characteristics are selected in domain
and
Y ¼ 4p PU which is characterized by an order-four differential
d4T ¼ d2U d2p. The local coordinate s denes the distance of point r
with respect to plane PU. Finally, the characteristic function
cVi ðT;sÞ ¼ 1 if point p þ sU of characteristic T is located inside Vi,
Qð1ÞðUÞ ¼ 2p3 [kT dss  [kðTÞQðrk þ s U;UÞ:
k 0
(6)
and 0 otherwise.
The MOC requires knowledge of region indices Nk and segment
lengths [k describing the overlapping of characteristic T with the
domain. This information is written ðNk;[k;k ¼ 1;KÞ where K is the
total number of regions crossed by T. It is important to note that
Knowledge of the moments of the ux over segment [kðTÞ are
required in order to compute components of the source, Qð0ÞðUÞ
and Qð1ÞðUÞ. The moments of the segment-averaged uxes are
dened as:
[kT
fð0ÞðTÞ ¼ [kðTÞ ds fðrk þ s U;UÞ
0
(7)
and
fð1ÞðTÞ ¼ 2p3 [kT dss  [kðTÞfðrk þ s U;UÞ:
k 0
(8)
Substitution of the linear source approximation Eq. (4) into the
characteristics form of the transport equation leads to:
dsfðrk þ s U;UÞ þ SNk fðrk þ s U;UÞ
¼ Qð0ÞðUÞ þ 2p3 s  [kðTÞ Qð1ÞðUÞ:
(9)
Analytical solution of Eq. (9) is written:
ð0Þ  
fðrk þ s U;UÞ ¼ fðrk;UÞ es SNk þ k 1  es SNk
pffiffiffi ð1Þ h Nk
þ 2 2es SNk þ 2 s SNk  1
Nk  i
 SNk [kðTÞ 1  es SNk :
(10)
Fig. 1. Spatial integration domain.
where fkðTÞ ¼ fðrk;UÞ.