See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/236368235 NESTLE: Few-group neutron diffusion equation solver utilizing the nodal expansion method for eigenvalue, adjoint, fixed-source steady-state and transient problems Article · June 1994 DOI: 10.2172/10191160 CITATIONS 65 READS 390 6 authors, including: Some of the authors of this publication are also working on these related projects: Reduced Order Modeling Based Uncertainty Quantification, Sensitivity Analysis and Model Calibration View project Paul J. Turinsky North Carolina State University 117 PUBLICATIONS 666 CITATIONS SEE PROFILE All content following this page was uploaded by Paul J. Turinsky on 27 June 2015. The user has requested enhancement of the downloaded file.
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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/236368235
NESTLE: Few-group neutron diffusion equation solver utilizing the nodal
expansion method for eigenvalue, adjoint, fixed-source steady-state and
transient problems
Article · June 1994
DOI: 10.2172/10191160
CITATIONS
65
READS
390
6 authors, including:
Some of the authors of this publication are also working on these related projects:
Reduced Order Modeling Based Uncertainty Quantification, Sensitivity Analysis and Model Calibration View project
Paul J. Turinsky
North Carolina State University
117 PUBLICATIONS 666 CITATIONS
SEE PROFILE
All content following this page was uploaded by Paul J. Turinsky on 27 June 2015.
The user has requested enhancement of the downloaded file.
II.7 Cross-Section Model......................................................................... 68II.7.a Macroscopic Model ...................................................................... 68II.7.b Microscopic Model ...................................................................... 72
II.8 Control Option Searches ................................................................... 75
II.9 Hydrodynamic Model ....................................................................... 76II.9.a Field Equations............................................................................. 76II.9.b Equation Discretization................................................................ 77II.9.c Fuel Temperature Model .............................................................. 84II.9.d Steady-State Model ...................................................................... 87II.9.e Effective Heat Transfer Coefficient Evaluation ........................... 88II.9.f Decay Heat Model ........................................................................ 89
III References........................................................................................... 91
IV User’s Guide ....................................................................................... 93
IV.1 Code Control Parameter Data File .................................................. 95
v
IV.2 Geometry Data File ....................................................................... 102IV.2.a Geometry Input ......................................................................... 107
IV.3 Cross-Section Data File................................................................. 112
IV.4 Kinetic Data File ........................................................................... 131
IV.5 Solution Method Control Data File ............................................... 133
IV.6 Initial Exposure Data File.............................................................. 137
IV.7 Initial Isotopic Number Densities Data File.................................. 139
IV.8 Pin-Power Data File ...................................................................... 140
V Programmer’s Guide ........................................................................ 142
vg= average number of neutrons created per fission
Σfg= macroscopic fission cross section [cm-1]
As with most modern nodal methods, we begin by intergrating the multi-group neu
∇– Dg∇φg Σtgφg+⋅ Σsgg′φg′χg
k----- vg′Σ fg′φg′
g′ 1=
G
∑+g′ 1=
G
∑=
r
5
s. For
, the
x-
in
diffusion equation over a material-centered spatial node which has homogenized propertie
Cartesian geometry we rewrite Eqn. (1) for the arbitrary spatial nodel,
(2)
where, , and
For simplicity, in cases where redundant equations exist in all three directions
illustrating equations will be only given in the x-direction. Using Fick’s Law, which in the
direction can be expressed as,
(3)
where,
allows Eq. (2) to be rewritten as:
(4)
Integration of Eq. (4) over the volume of nodel generates a local neutron balance equation
terms of the face-averaged net currents and the node volume average flux.
(5)
D– gl
x2
2
∂∂ φg
l r( ) Dgl
y2
2
∂∂ φg
l r( )– Dgl
z2
2
∂∂ φg
l r( ) Agl φg
l r( )+– Qgl r( )=
g 1 G,( )∈
r( ) x y z, ,( ) Vl∈≡ x y∆ z∆∆ Volume of nodel≡=
Agl Σtg
l Σsggl
–xg
l
k-----vgΣ fg
l–=
Qgl
r( ) Qgg′l φg′
lr( )
g′ g≠
G
∑ Σsgg′l φg′
lr( )
xgl
k----- vg′Σ fg′
l φg′l
r( )g′ g≠
G
∑+g′ g≠
G
∑==
jgxl
r( ) Dgl
x∂∂ φg
lr( )–=
jgxl
r( ) x-component of the net neutron current≡
x∂∂
jgxl
r( )y∂
∂jgyl
r( )z∂
∂jgzl
r( ) Agl φg
lr( )++ + Qg
lr( )=
1
xl∆
-------- Lgxl
( ) 1
yl∆
------- Lgyl
( ) 1
zl∆
------- Lgzl
( ) Agl φg
l+ + + Qg
l=
6
ted
en in
ome
It is the
lation
three-
. This
, of the
where, assuming nodel is centered around the coordinate’s origin, the volume integra
quantities are defined below:
and,
where,
Eq. (5) is known as the nodal balance equation. Now for the neutron diffusion equation writt
this form, in order to obtain the spatial neutron flux distribution, one must devise s
relationship between the node average flux and the face-averaged net (surface) currents.
equations used to compute the surface currents in Eq. (5) which distinguish one nodal formu
from another. In NEM, the widely used method of transverse-integration is used, where the
dimensional diffusion equation is integrated over the two directions transverse to each axis
generates three one-dimensional equations, one for each direction in Cartesian coordinates
following form,
φgl 1
Vl
----- φgl
r( ) x y z Node volume average flux≡ddd
z∆ l
2-------–
z∆ l
2-------
∫y∆ l
2-------–
y∆ l
2-------
∫x∆ l
2-------–
x∆ l
2-------
∫=
Qgl 1
Vl
----- Qgl
r( ) x y z Node volume average source≡ddd
z∆ l
2-------–
z∆ l
2-------
∫y∆ l
2-------–
y∆ l
2-------
∫x∆ l
2-------–
x∆ l
2-------
∫=
1
xl∆
--------Lgxl 1
xl∆
-------- Jgx+l
Jgx-l
–( ) 1
Vl
-----x∂
∂jgxl
r( ) x y zddd
z∆ l
2-------–
z∆ l
2-------
∫y∆ l
2-------–
y∆ l
2-------
∫x∆ l
2-------–
x∆ l
2-------
∫= =
Jgx ±l
Average x-directed net current on node facesx∆ l
2--------±≡
7
as a
such
(6)
where,
and,
In NEM, the one-dimensional averaged flux that appears in Eq. (6), is expanded
general polynomial,
(7)
where is the node average flux, implying for Eq. (7) to be true that must be chosen
that the basis functions satisfy
(8)
Note that for quartic NEM, the method used in NESTLE, the summation extends toN = 4. The
first four basis functions in NEM can be expressed as follows [1],
(9)
which can be shown to also satisfy the following,
xdd
jgxl
x( ) Agl φgx
l+ x( ) Qgx
lx( ) 1
yl∆
-------Lgyl
x( )–1
zl∆
-------Lgzl
x( )–=
Lgyl
x( ) 1
zl∆
-------y∂
∂jgyl
r( ) y z Average y-direction transverse leakage≡dd
yl∆2
-------–
yl∆2
-------
∫zl∆2
-------–
zl∆2
-------
∫=
Lgzl
x( ) 1
yl∆
-------z∂
∂jgzl
r( ) z y Average z-direction transverse leakage≡dd
zl∆2
-------–
zl∆2
-------
∫yl∆2
-------–
yl∆2
-------
∫=
φgxl
x( ) φgl
agxnl
f n x( )n 1=
N
∑+=
φgl
f n x( )
f n x( ) xd
x∆ l
2-------–
x∆ l
2-------
∫ 0 for n 1 ...,N,= =
x
xl∆
-------- ; 3
x
xl∆
-------- 2 1
4---;–
x
xl∆
-------- 3 1
4--- x
xl∆
-------- ;–
x
xl∆
-------- 4 3
10------ x
xl∆
-------- 2
– 180------+
f1 f2 f3 f4
8
total
e node
roup,
ides
ts from
nergy
any
h
me [3]
oment
ctions,
, and
nsion
(10)
At this point it is appropriate to consider the elementary concept of accounting for the
number of equations and that of unknowns. For a three-dimensional Cartesian geometry, th
average andN expansion coefficients in each direction appear per node per energy g
implying a total of 3N+1 equations are required. The nodal balance equation, Eq. (5), prov
one equation, where now Eqs. (3) and (7) are used to eliminate face-averaged net curren
this equation. Surface current and flux continuity provide 6 more equations per node per e
group. So forN=2, there would be an equal number of equations and unknowns without
further development. However, forN= 4, two additional unknowns are introduced for eac
direction per node per energy group. This is addressed by using a weighted residual sche
applied to Eq. (6), which in essence provides the additional equations (referred to as the m
equations) needed,
(11)
where the two weighting functions for n = 1,2 are chosen to be the same as the basis fun
namelyωn(x) = fn(x), as those used in the one-dimensional flux expansion1. Here, the first and
second (actually linear combination of zeroth and second) moments of the flux, source
leakage for each groupg are defined by,
The first term in Eq. (11) is evaluated by using Eqs. (3) and (7) and the definition of the expa
1. This constitutes amoments weightingscheme; if one usesωn(x) = fn+2(z) for n = 1,2 it is known asGaler-kin weighting. Numerical experiments favormoments weighting.
f nx
l∆2
--------± 0 for n 3 4,==
ω< n x( )xd
djgxl
x( ) Agl φgxn
l+>, Qgxn
l 1
yl∆
-------Lgyxnl
–1
zl∆
-------Lgzxnl
–=
ωn x( ) φgxl
x( ), >< ωn x( ) Qgxl
x( ), >< ωn x( ) Lgyl
x( ), >< ωn x( ) Lgzl
x( ),< >
φ lgxn Ql
gxn Llgyxn Llgzxn
9
re the
known,
dratic
the y-
e
g the x-
odes.
coefficients, and completing the integration (i.e. inner product) analytically.
One last point which needs to be addressed before Eq. (11) can be solved a
transverse leakage terms appearing on the right hand side. Their spatial dependency is un
so their “shape” must be approximated. The most popular approximation in NEM is the qua
transverse leakage approximation. For example, the x-direction spatial dependence of
direction transverse leakage is approximated by,
(12)
where is the average y-directed leakage in nodel, and the coefficients and can b
expressed in terms of average y-directed leakages of the two nearest-neighbor nodes alon
direction (i.e. nodesl-1 andl+1) so as to preserve the node average leakages of these three n
The quadratic expansion coefficients can be shown to be given by,
(13)
(14)
where,
(15)
Lgyl
x( ) Lgyl
ρgy1l
f 1 x( ) ρgy2l
f 2 x( )+ +≅
Lgyl
ρgy1l ρgy2
l
ρgy1l
gl
xl∆( ) Lg
l 1+Lg
l–( ) x
l∆ 2 xl 1–∆+( ) x
l∆ xl 1–∆+( ) Lgy
lLgy
l 1––( ) x
l∆ 2 xl 1+∆+( ) x
l∆ xl 1+∆+( )+[ ]=
ρgy2l
gl
xl∆( )
2Lgy
l 1+Lgy
l–( ) x
l∆ xl 1–∆+( ) Lgy
l 1–Lgy
l–( ) x
l∆ xl 1+∆+( )+[ ]=
gl
xl∆ x
l 1+∆+( ) xl∆ x
l 1–∆+( ) xl 1–∆ x
l∆ xl 1+∆+ +( )[ ]
1–=
10
the
nd to
on,
mith
de
rather
(
oup
tegy,
alled
ing
oved
mated
dated
This
orces
rents
rage
forms;
usable
e base
actor
II.1.b. Non-Linear Iterative Strategy
The most common manner of solving the matrix system associated with NEM is
response-matrix formulation. To minimize computer run time and memory requirements, a
facilitate the capability to solve either the NEM or Finite Difference Method (FDM) formulati
the non-linear iterative strategy is employed in NESTLE. This technique was developed by S
[4,5,6] and successfully implemented into the Studsvik QPANDA and SIMULATE co
packages. The documentation available on this technique is scarce, but it turns out to be
simplistic and almost trivial to implement in a FDM code which utilizes the box-schemei.e.
material-centered).
The basic idea is applicable to the standard FDM solution algorithm of the multi-gr
diffusion equation. Solving the FDM based equation utilizing an outer-inner iterative stra
every outer iterations (where is somewhat arbitrary but can be optimized) the so-c
“two-node problem” calculation (a spatially-decoupled NEM calculation spanning two adjoin
nodes) is performed for every interface (for all nodes and in all directions) to provide an impr
estimate of the net surface current at that particular interface. Subsequently, the NEM esti
net surface currents are used to update (i.e. change) the original FDM diffusion coupling
coefficients. Outer iterations of the FDM based equation are then continued utilizing the up
FDM coupling coefficients for outer iterations. The entire process is then repeated.
procedure of updating the FDM couplings is a convergent technique which progressively f
the FDM equation to yield the higher-order NEM predicted values of the net surface cur
while satisfying the nodal balance Eq. (5), thus yielding the NEM results for the node-ave
flux and fundamental mode eigenvalue. The advantages of this technique come in many
the storage requirements are minimal because the two-node problem arrays are re-
(disposable) at each interface, the rate of convergence is nearly comparable to that of th
FDM algorithm being used, the number of iteratively determined unknowns is reduced by a f
N∆ 0 N∆ 0
N0∆
11
e of
n be
icity,
ions
terms
he x-
/node
with
ment
cond
of 6 (node flux vs. partial surface current), and the simplicity of the algorithm and eas
implementation, compared to any other nodal technique, is far superior.
The two-node problem produces an 8G X 8G linear system of equations which ca
constructed by applying the standard NEM relations to two adjoining nodes. For simpl
consider two arbitrary adjoining nodes in the x-direction. Denote these notes asl andl+1:
Substitution of the one-dimensional expansion, Eq. (7), into Fick’s law yields express
for the average x-direction net surface currents at the left(-) and right(+) interfaces of nodel,
(16)
Now, assume the node average flux, criticality constant, and all transverse direction
are known from a previous iteration; then, the total number of unknowns associated with t
direction two node problem is 8G, which corresponds to the 4 expansion coefficients/group
(x) G groups (x) two nodes. The 8G constraint equations are obtained as follows. We begin
the substitution of Eq. (16) into the nodal balance equation for node l, to yield the zeroth mo
constraints (G equations/node),
(17)
A similar substitution into the moment-weighted equation, Eq. (11), yields the first and se
moment constraints (2G equations/node),
(18)
(19)
Nodel
Nodel+1x- x+
jgx±l D– g
l
xl∆
---------- agx1l
3agx2l 1
2---agx3
l 15---agx4
l±+±≡
D– gl
xl∆ x
l∆---------------- 6agx2
l 25---agx4
l+
1
yl∆
-------Lgyl
–1
zl∆
-------Lgzl
Agl φg
lQgg′
l
g′ g≠
G
∑ φg′l
+––=
60
xl∆
--------Dg
l
xl∆
-------- Agl
+ agx3l
Qgg′l
ag′x3l
g′ g≠
G
∑ 10Aglagx1
l– 10 Qgg′
lag′x1
l
g′ g≠
G
∑+– 101yl∆
--------ρgy1l 1
zl∆-------ρgz1
l+
=
140
xl∆
---------Dg
l
xl∆
-------- Agl
+ agx4l
Qgg′l
ag′x4l
g′ g≠
G
∑ 35Aglagx2
l– 35 Qgg′
lag′x2
l
g′ g≠
G
∑+– 351yl∆
--------ρgy2l 1
zl∆-------ρgz2
l+
=
12
The
using
ons)
q. (7),
sics
of the
tially
t each
t the
t of a
ly [7].
=2.
Similar equations can be written for node l+1, producing a total of 6G equations.
continuity of net surface current constraints at the interface (G equations) are obtained by
Eq. (16) at the adjoining interface of the two nodes,
(20)
Last, the continuity (or discontinuity) of surface-averaged flux constraints (G equati
are obtained by equating the surface-averaged fluxes of the two adjoining nodes by using E
(21)
where and are the Discontinuity Factors (DFs) obtained from lattice phy
calculations. Do note that continuity conditions are never imposed on the outside surfaces
two-node problem, since the two-node problem is deliberately formulated to be spa
decoupled. Continuity is assured in the formulation of the FDM based equations.
Eqs.(7) through (21) constitute the 8G system of equations needed to be solved a
interface. This matrix system, after taking advantage of its reducability and by noting tha
even-moment expansion coefficients don’t change whether the node is on the left or righ
two-node problem, can be reduced to smaller systems which can be solved quite efficient
The following table illustrates this more efficient arrangement of unknowns for the case of G
D– gl
xl∆
---------- agx1l
3agx2l agx3
l
2----------
agx4l
5----------+ + +
D– gl
xl 1+∆
--------------- agx1l 1+
3agx2l 1+ agx3
l 1+
2-----------
agx4l 1+
5-----------–+–=
dgx +l φg
l agx1l
2----------
agx2l
2----------+ + dgx –
l 1+ φgl 1+ agx1
l 1+
2-----------
agx2l 1+
2-----------+–=
dgx ±l
dgx ±l 1+
13
8G
duce
cient
*Refers to order of polynomial that transverse integrated
flux expansion coefficient is associated with.
In NESTLE, the two-node problems are solved by utilizing the analytic solution to the 8G X
matrix system. This was accomplished by employing symbolic manipulator software to pro
the FORTRAN code segment used in NESTLE. This approach is computationally more effi
Table 1: Non zero entries in the 16 by 16 two-node NEM problem.
Eqn Grp Nod a b c d e f g h i j k l m n o p
0th Moment 1 l x x
0th Moment 2 l x x
2nd Moment 1 l x x x x
2nd Moment 2 l x x x x
0th Moment 1 l+1 x x
0th Moment 2 l+1 x x
2nd Moment 1 l+1 x x x x
2nd Moment 2 l+1 x x x x
1st Moment 1 l x x x x
1st Moment 2 l x x x x
1st Moment 1 l+1 x x x x
1st Moment 2 l+1 x x x x
Cur Con 1 x x x x x x x x
Cur Con 2 x x x x x x x x
Flx Dis 1 x x x x
Flx Dis 2 x x x x
UNKNOWN NODE GROUP EXP. COEF.*
a l 1 2b l 2 2c l 1 4d l 2 4e l+1 1 2f l+1 2 2g l+1 1 4h l+1 2 4i l 1 1j l 2 1k l 1 3l l 2 3m l+1 1 1n l+1 2 1o l+1 1 3p l+1 2 3
14
to
-node
-node
e
n all
wn in
nt, the
be
is
FDM
ere to
is
dard
LE,
d. The
rface
than utilizing a direct matrix solver (e.g. LU decomposition); however, it limits the values of G
those directly programmed for. Also note that on boundaries special treatments of the two
problems are required. Depending upon the specified boundary condition (BC), one
problems may originate (e.g. zero flux BC), or on interior axis geometry unfolding may b
required to create a two-node problem (e.g. cyclic BC).
Solutions of the two-node problems provide NEM evaluated values of the currents o
surfaces for specified values of the node average fluxes [recall they were assumed kno
solving the two-node problems]. To correct the FDM based expression for the surface curre
following approach is utilized. The coupling coefficient update to the FDM equation can
implemented by simply expressing the FDM net surface current at thex+ face of nodel as
follows,
(22)
The first term on the RHS is the normal FDM approximation for a box scheme, where
the actual FDM diffusion coupling coefficient between nodes l and l+1,
(23)
The second term on the RHS represents the nonlinear NEM correction applied to the
scheme. The (+) sign between the flux values in the second term of Eq. (22) is purposely th
improve the convergence behavior of the nonlinear iterative method [8]. Note that if
zero, which it initially is in NESTLE’s implementation, then Eq. (22) corresponds to the stan
FDM definition of the net surface current. This is the basis for the FDM option within NEST
where now two-node problem solves and coupling coefficients updates are never complete
value of is determined by setting Eq. (22) equal to the NEM two-node predicted su
Jgx +l FDM, Dgx +
l FDM,
xl∆ x
l 1+∆+2
-----------------------------
-----------------------------– φgl 1+
φgl
–[ ]Dgx +
l NEM,
xl∆ x
l 1+∆+2
-----------------------------
----------------------------- φgl 1+
φgl
+[ ]–=
Dgx +l FDM,
Dgx +l FDM, Dg
lDg
l 1+x
l∆ xl 1+∆+( )
Dgl
xl∆ Dg
l 1+x
l 1+∆+------------------------------------------------------=
Dgx +l NEM,
Dgx +l NEM,
15
r this
one
n for
inally,
n the
current value, using the associated node average flux values in Eq. (22) and solving fo
quantity.
Summarizing, to apply a NEM update after outer iterations of the FDM routine,
solves the two-node problem at a given interface, then (with the expansion coefficients know
that interface) one calculates the NEM estimate of the net surface current using Eqn.(16) F
one equates this result to Eq. (22), and solves for the value of which will be used i
subsequent set of FDM iterations.
N0∆
Dgx +l NEM,
16
DM
ting,
stem.
tion,
and
tions
lems
the
TLE’s
the
onal
(23)
. (8)
II.2. Outer-Inner Solution Method for FDM Equations
The only large matrix that requires solution for the non-linear iterative method is the F
representation of the multi-group diffusion equation. Much work has been done on formula
understanding and implementing the iterative solution of this large, sparse matrix sy
NESTLE takes advantage of this wealth of knowledge in its iterative solution implementa
utilizing an outer-inner iterative strategy.
The “Outer-Inner Method” refers to outer iterations to update the fission source term
inner iteration to approximately solve the resulting fixed source problem. The outer itera
correspond to a “Power Method.” This method can be applied to both Fixed Source Prob
[FSP] and the Associated Eigenvalue Problem [AEVP]. Shortly it will be shown that both
fixed source steady-state and transient problems are representable as FSP in NES
formulation. Although the AEVP involves additional calculations for the eigenvalue, basically
iteration schemes for both problems are similar. We will discuss the AEVP first.
Returning to Eq. (5), the FDM representation of this equation in three-dimensi
Cartesian geometry within homogenous nodel can be expressed as follows:
(24)
where the non-zero values of the coupling coefficients are obtained via Eqs.(22) and
and L denotes the total number of nodes. Substituting in the definitions for and into Eq
and rearranging terms we obtain
(25)
This equation can be written in terms of matrix notation spanning the spatial domain as
(26)
Cgl l ′, φg
l ′
l ′ 1=
L
∑ Agl φg
l+ Qg
l=
Cgl l ′,
Agl
Qgl
Cgl l ′,
l ′ l≠
L
∑ φgl ′
Σtg
l Σsgg
lCg
l l,+–( )+ φg
lΣsgg′
l
g′ g≠
G
∑ φg′l
–xg
l
k----- υg′
g′ 1=
G
∑ Σ fg′l φg′
l=
Agφg Σsgg′φg′g′ g≠
G
∑–1k---xg υg′Σ fg′
l φg′g′ 1=
G
∑=
17
has a
X L)
GL)
ps.
y Eq.
es for
ive
e rate
ups,
plies
where the “bar” over the node average flux value now denotes a column vector. Matrix
seven-banded matrix structure for three-dimensional Cartesian geometry. In turn, the G (L
matrix systems expressed by Eq. (26) can be collected to write the following single (GL X
matrix system.
(27)
The matrix is block lower triangular in structure for that portion applicable to the fast grou
The outer-inner iteration process is summarized as follows: For the AEVP specified b
(27), given an arbitrary initial vector , the outer iterations generate successive estimat
the flux vector by the process
(28)
where how the criticality constant (i.e.eigenvalue) is updated will be discussed later. The iterat
matrix associated with the outer iterations is
(29)
The properties of the iterative matrix has a significant role in determining the convergenc
of the power iterations [9,10].
In solving Eq. (28), advantage is taken of the structure of the matrix. For the fast gro
solving from low to high energy group number results in energy group decoupling. This im
that we may solve a system of linear equations of the form
(30)
where,
(31)
Ag
Aφ 1k---Fφ=
A
φ0( )
φ
φq( ) 1
kq 1–( )---------------A
1–Fφ
q 1–( )=
Q A1–F=
Q
A
Agφgq( )
Sgq( )
=
Sgq( ) Σsgg′φg′
q( ) 1
kq 1–( )---------------+
g′ g≠
G
∑ χg νg′Σ f g′φg′q 1–( )
g′ 1=
G
∑=
18
oups
g Eqn.
ups’
attering
lor
ions.
ree-
direct
of
llows.
For the thermal groups, NESTLE assumes the group fluxes for all other thermal gr
except the one being updated are known. This produces energy group decoupling, allowin
(30) to be utilized. So called “scattering” iterations are then completed after all thermal gro
fluxes are updated. Stationary acceleration is employed to accelerate convergence of the sc
iterations.
II.2.a. Inner Iteration Acceleration
To solve Eq. (30) we introduce the inner iterations. In this work we employ a Multi-Co
Point or Line SOR Method, depending upon problem geometry, for the inner iterat
Specifically, a Red-Black Point or Line SOR method is used in NESTLE for two or th
dimensional Cartesian geometry, respectively. For one-dimensional Cartesian geometry, a
matrix solve is utilized since the group-wise A matrix is triangular allowing employment
Gaussian elimination.
Mathematically, this approach is a multi-splitting method and can be expressed as fo
(32)
where,
(33)
and
(34)
(35)
φ φp where vectorφp spans nodes of color ''p''⊕=
φpm 1+( )
Bp1–
S Cpp′φp′m 1+( )
Cpp′φp′m( )
p′ p 1+=
P
∑+p′ 1=
p 1–
∑+= for p 1,2,...,P=
A Ap and non-square matrixAp equals rows ofA that span nodes of color ''p''⊗=
Ap Bp Cpp′
p′ p≠
P
∑–= for p = 1,2,...,P
φpm 1+( )
φpm( )
ω φpm 1+( ) φp
m( )–( )+=
19
in the
cheme
that
per
error
that the
since
ese
ptable.
ss-
he
arizes
lue of
nergy
Note that the group g and outer iteration count (q) indices have been suppressed for clarity
above equations. The matrix is square and has either a diagonal structure for the point s
or block diagonal structure composed of tridiagonal blocks for the line scheme. This implies
the action of indicated in Eqs. (32) is simple to evaluate. A total of inner iterations
outer iterations are completed, this value determined such that the specified relative
reduction from the 0th iterative error for the inner iterations is achieved.
To a priori determine the value of the optimum relaxation parameter,ω and [which
are energy group dependent but dependence notation has been surpressed], it is assumed
iterative matrix associated with this inner iterative method is symmetrizable. This is not true
the NEM corrections to the FDM coupling coefficients invalidate symmetry; however, th
corrections have been found to be relatively small so the symmetrizable assumption is acce
Making this assumption, we can expressω in terms of the spectral radius of the associated Gua
Seidel iteration matrix, , as follows,
(36)
Clearly . Therefore, calculation of the spectral radius of t
associated Gauss-Seidel iterative matrix is the heart of this procedure. The following summ
the details of the computational procedure used in NESTLE to obtain an estimate of the va
ω, which is based upon the DIF3D methodology [10]. These steps are completed for each e
group.
Bp
Bp1–
NI∆
NI∆
ρ LG-S
( )
ω 2
1 1 ρ LG-S( )–[ ]
1/2+
------------------------------------------------=
LG-S
LSOR
ω( )with ω 1= =
20
st
d
0.1 Step 1.Starting with an arbitrary non-negative initial guess vector , complete at lea
ten Gauss-Seidel iterations in solving the following equation.
0.2 Step 2.Following each iteration withm >10, estimate the upper and lower bounds of the
spectral radii using the following equations.
Compute the corresponding relaxation factors given by
0.3 Step 3.Terminate iteration when either
or mequals a specified upper limit [10,11]. The optimum factorω is then set toω(m). This
test forces tighter convergence ofω when is close to unity to ensure the require
numerical accuracy is achieved.
x0( )
Ax 0=
λ m( ) xm( )
xm( )
,⟨ ⟩
xm( )
xm 1–( )
,⟨ ⟩-----------------------------------≡
λm( )
MAXixi
m( )
xim 1–( )----------------≡
λ m( )MINi
xim( )
xim 1–( )----------------≡
ω m( ) 2
1 1 λ m( )–[ ]
1/2+
---------------------------------------≡
ω m( ) 2
1 1 λm( )
–[ ]1/2
+---------------------------------------≡
ω m( ) 2
1 1 λ m( )–[ ]
1/2+
---------------------------------------≡
ω m( ) ω m( )–
2 ω m( )–
5--------------------<
ρ LG-S( )
21
uch
d of
may
ation of
ner of
antial
ns has
roup.
tio of
s are
0.4 Step 4.Determine the number of inner iterations required for each outer iteration , s
that the value of satisfies the following equation:
where
and denotes the desired relative error reduction from the initial iteration to the en
-th iteration. It is suggested that a very small number for not be used since it
force excessive inner iterations [10].
The advantages of these accelerations strategies are clear. The automated determin
the optimum overrelaxation factors relieves users of the burden of the trial and error man
specifying optimum parameters for a large class of reactor models. In addition, subst
computational time can be saved since the need to check the convergence of inner iteratio
been removed by using a fixed number of predetermined inner iterations for each energy g
The outer iterations defined by Eq. (28) are slow to converge, since the dominance ra
the iterative matrix, Eq. (29), is close to one. Two complementary acceleration technique
utilized in NESTLE to accelerate the outer iterations of the AEVP.
NI∆
NI∆
LSOR
ω( )( )NI 1–∆
LG S–
⋅ t2 NI 1–∆2
t2 NI∆2
+[ ]1/2
εin≤=
t NI∆ ω 1–[ ]NI 1–∆
2------------------
ρ LG S–
( )[ ]1/2
1 NI 1–∆( )+ 1 ρ LG-S
( )–( )1/2
[ ]=
εin
NI∆ εin
22
ased
. No
s the
yshev
ctors
od [9,
ressed
hod.
ction
yshev
in the
,
lied to
h to
II.2.b. Outer Iteration Acceleration
The outer iterations for the AEVP are accelerated by using either a polynomial b
acceleration method or an eigenvalue shift acceleration method with flux extrapolation
knowledge of higher eigenvalues are required to utilize either method. We first discus
polynomial based acceleration method, which utilizes Chebyshev polynomials. Cheb
polynomials [12] are used to obtain the best linear combinations of the previous iterative ve
so as to minimize the error. The method implemented is the Chebyshev Semi-Iterative meth
10, 11, 13]. In this method, the error vector associated with the acceleration method is exp
in terms of a linear combination of the error vectors of the underlying interactive met
Acceleration of the iteration is achieved by minimizing the error vector by appropriate sele
of the expansion coefficients, which is determined to be those associated with Cheb
polynomials. Further details of the mathematical background of this method can be found
related references [9, 10].
Since the rate of convergence in the AEVP is dependent on the dominance ratio
the Chebyshev acceleration method detailed in Refs. [9, 10, 11, 13] can therefore be app
iterations,
(37)
provided that a suitable estimate of is obtained. NESTLE follows the DIF3D approac
solve the AEVP in which we accelerate the fission sourceΨ [13], whereΨ is defined as
(38)
The accelerated iterative procedure can then be expressed as follows:
(39)
σ Q
φq( ) 1
kq 1–( )---------------Qφ
q 1–( )=
σ Q
Ψ νg′Σ fg′φg′g′ 1=
G
∑ νΣ f φ= =
Ψn* p+( ) 1
kn* p 1–+( )
------------------------QΨn* p 1–+( )
=
23
yshev
to be
ented
trix,
are
ower
tone
are
where
(40)
(41)
and
and p denotes the successive fission source iterations employed within a Cheb
cycle (i.e.since last updating the estimate of ). Note the dominance ratio needs
estimated in order for the scheme to work. This is accomplished using the procedure implem
in DIF3D [10] as now outlined. Do note that versus is the relevant outer iterative ma
since fission source versus flux extrapolation is employed.
Since an accurate estimate of is not known when the outer iterations
commenced, a “boot-strap” process is required. By performing a limited number of p
iterations, a reasonable initial estimate of is obtained. Only when all but the first over
mode are essentially damped out, high-order cycles based on accurate estimates of
Write steady-state restart file (“Y”/”N”) at burnup step IBU
IF (ISAVE(IBU).EQ. “Y”) THEN READ
OUT(IBU) (A40)
Steady-state restart file name [to write] at burnup step IBU
ENDIF ISAVE(IBU)
ENDDO IBU ...Burnup step loop
IF (ITRAN.EQ. “Y”) THEN READ
ISAVETR (A5)
Write transient restart file (“Y”/”N”)
ENDIF ITRAN
IF (ISAVETR.EQ. “Y”) THEN READ
OUTTR (A40)
Transient restart file name [to write]. This file is written at every time indicated
the kinetic file input variable TIMEPR(IT), that indicates times when outp
should be produced. Note transient restart file is overwritten each time to
space.
ENDIF ISAVETR
IEXP (A5)
Initial node-wise exposure map available (“Y”/ “N”)
IF (IEXP.EQ. “Y”) THEN READ
FINITEXP (A40)
99
-at the
nal
)
rm
Name of the file containing the initial node-wise exposures (Unit=79)
ENDIF IEXP
OEOC (A5)
End of depletion ASCII restart file option (“Y” / “N”)
If OEOC = “Y”, NESTLE will write new code control parameter file (CNTRL file), nodewise exposure map, and node-wise number density map (for microscopic depletion)end of the depletion. This option is valid for a steady-state problem only.
Note: the following pin power reconstruction option is currently available for hexago
geometry only.
PPR (A5)
Pin power reconstruction option (“Y”/ “N”)
IF (PPR.EQ. “Y”) THEN READ
FPIN (A5)
Corner point discontinuity factors and pin-wise form factors available (“Y”/ “N”
IF (FPIN.EQ. “Y”) THEN READ
INPPIN (A40)
File name containing the corner point discontinuity factors and pin-wise fo
factors (Unit=81)
ELSEIF (FPIN.EQ. “N”) THEN READ
NHRPIN
Number of radial rings of pins surrounding center pin.
Number of pins within the assembly = NPIN
When PPR = “Y” and FPIN = “N”, NESTLE will set all corner point
NPIN 6 ii 1=
NHRPIN 1–
∑ 1+=
100
discontinuity factors and form factor values to 1.0.
ENDIF FPIN
OUTPPR (A5)
Output pin-wise power values (“Y” / “N”)
IF (OUTPPR.EQ. “Y”) THEN READ
OUTPIN (A40)
Pin power output file name (Unit=82)
ENDIF OUTPPR
ENDIF PPR
101
ions.
IV.2. Geometry Data File
(Unit 2 => “GEOM”)
NSHAP (A5)
Node shape in radial plane
Hexagonal (“HEXA”)
Cartesian (“CART”)
IDRUN (A5)
Core symmetry
IF (NSHAP .EQ. “HEXA”)
Full core (“FCORE”)
One-third core (“TCORE”) 5 to 9 o’clock
One-sixth core (“SCORE”) 5 to 7 o’clock
One-dimensional axial core (“AXIAL”)
IF (NSHAP. EQ. “CART”)
Full core (“FCORE”)
Half core (“HCORE”) 3 to 9 o’clock
Quarter core (“QCORE”) 3 to 6 o’clock
One-dimensional axial core (“AXIAL”)
ENDIF NSHAP
IF (NSHAP.EQ."CART") THEN READ
NX,NY
The x and y total mesh numbers applicable to initial homogenous material reg
NMULXY
Number of mesh to create from each input x and y material mesh.
ELSEIF (NSHAP.EQ."HEXA") THEN READ
102
all
n.
al
e
NHR
Number of radial rings of bundles (assemblies) surrounding center bundle.
ENDIF NSHAP
NZ
The z total mesh number applicable to initial material regions.
NMULZ
Number of mesh to create from each input z material mesh.
NFIGURE
Number of different radial configurations (basic figures) of core materials over
Figure 8 (cont): Radial material geometry figures for different core geometries andsymmetries
111
IV.3. Cross-Section Data File
(Unit 3 => “XSECT” & Unit 33 => “AXSCIN”)
NG
Total number of energy groups.
NGT
Number of thermal energy groups.
ICOLXY
Total number of material colors (i.e. cross section sets).
NBUMAX
Maximum of the number of burnup mask values input for the cross-sections.
IF(NXSEC.EQ. “Y”) THEN READ
NPREC
Number of delayed neutron precursor groups.
NDECAY
Number of decay heat precursor groups.
NTERMMACRO,NTERMMACRI
Number of cross-section coefficients input for macroscopic cross-sections
w/o and w/ rods in.
NTERMCSCATRO,NTERMCSCATRI
Number of cross-section coefficients input for macroscopic scattering kernels
w/o and w/ rods in.
NTERMFPRO,NTERMFPRI
Number of cross-section coefficients input for transient fission products
microscopic absorption cross-sections w/o and w/ rods in.
[In following, IXSxxx values imply the following:
112
Base = 1,
Linear-Coolant Density (g/cm3) = 2,
Quadratic-Coolant Density (g/cm3) = 3,
Linear-Coolant Temperature (oF) = 4,
Linear in Square Root-Effect. Fuel Temperature (oF) = 5,
Linear-Soluble Poison Number Density (1/b) = 6,
Quadratic-Soluble Poison Number Density (1/b) = 7),
Cubic-Soluble Poison Number Density (1/b) = 8]
IF(NTERMMACRO.GT.0) READ
(IXSMACRO(ITERM),ITERM=1,NTERMMACRO)
Cross-section coefficients to be input for macroscopic cross-sections
w/o rods in.
IF(NTERMMACRI.GT.0) READ
(IXSMACRI(ITERM),ITERM=1,NTERMMACRI)
Cross-section coefficients to be input for macroscopic cross-sections
w/ rods in.
IF(NTERMCSCATRO.GT.0) READ
(IXSCSCATRO(ITERM),ITERM=1,NTERMCSCATRO)
Cross-section coefficients to be input for macroscopic scattering kernels
w/o rods in.
IF(NTERMCSCATRI.GT.0) READ
(IXSCSCATRI(ITERM),ITERM=1,NTERMCSCATRI)
Cross-section coefficients to be input for macroscopic scattering kernels
w/ rods in.
113
.
IF(NTERMFPRO.GT.0) READ
(IXSFPRO(ITERM),ITERM=1,NTERMFPRO)
Cross-section coefficients to be input for transient fission products
microscopic absorption cross-sections w/o rods in.
IF(NTERMFPRI.GT.0) READ
(IXSFPRI(ITERM),ITERM=1,NTERMFPRI)
Cross-section coefficients to be input for transient fission products
microscopic absorption cross-sections w/ rods in.
IMICRO (A5)
Microscopic cross-section option (“Y”/”N”).
IF(IMICRO.EQ. “Y”) THEN READ
INUMDEN (A5)
Node-wise isotopic number densities available (“Y” / “N”).
IF (INUMDEN.EQ. “Y”) THEN READ
FNUMDEN (A40)
Name of the file containing the node-wise initial isotopic number densities
NTERMMICRO
Number of cross-section coefficients input for microscopic cross-sections
w/o rods in.
NTERMMICRI
Number of cross-section coefficients input for microscopic cross-sections
w/ rods in.
IF(NTERMMICRO.GT.0) READ
(IXSMICRO(ITERM),ITERM=1,NTERMMICRO)
114
functionutronnctioned
cursor
utron
Cross-section coefficients to be input for microscopic cross-sections
w/o rods in.
IF(NTERMMICRI.GT.0) READ
(IXSMICRI(ITERM),ITERM=1,NTERMMICRI)
Cross-section coefficients to be input for microscopic cross-sections
w/ rods in.
ENDIF IMICRO
ENDIF NXSEC
AXSEC (A5)
Cross-section colors are read from different input files (“Y”/ “N”).
IF(NXSEC.EQ. “Y”) THEN READ
RLI,RLX,RLPM
Decay constants (I-135, Xe-135, Pm-149) (1/sec)
The next data, the delayed neutron precursor decay constants, can be entered as aof both isotope and delayed neutron precursor group or as a function of delayed neprecursor group only. To enter the delayed neutron precursor decay constants as a fuof both isotope and precursor group, write “ALAMDAMI” in a line preceding the delayneutron precursor decay constants input (see example below).
Delayed neutron precursor decay constants as a function of delayed neutron pregroup only:
Figure 9 (cont): Dependence diagram of the NESTLE code
.
155
XSECBU|
PINTER
XSECBUS|
PINTER
Figure 9 (cont): Dependence diagram of the NESTLE code
ellet
rrently
ations.
linear
hecks
and
essage
SUBROUTINEADJOINTThis subroutine controls the calculation of the adjoint flux.
SUBROUTINEAMFThis subroutine determines (A-F)*flux for the fixed-source scale factor method.
SUBROUTINEANMERGEThis subroutine merges alphanumeric string arrays together to form single string.
SUBROUTINEAVGEDITThis subroutine edits out radial and axial core averaged properties.
SUBROUTINEAVGEDIT-PINThis subroutine edits out maximum value of axially-averaged pin power and maximum ppower.
SUBROUTINEBURNNODEThis subroutine solves the isotopic depletion equations.
SUBROUTINEBURNNODESThis subroutine solves the isotopic depletion equations on the six surfaces of a hexagon (cuworks for NSHAP= “HEXA” only).
SUBROUTINECALACVThis subroutine determines the interaction rates required to solve the isotopic depletion equ
SUBROUTNECHAINThis subroutine uses the integrating factor technique to analytically solve a coupleddepletion chain.
SUBROUTINECHEBY1This subroutine applies the semi-implicit Chebyshev polynomial acceleration method and cfor convergence for the steady state problem.
SUBROUTINECHEBYTRThis subroutine applies the semi-implicit CHEBYCHEV polynomial acceleration methodchecks for convergence for the transient problem.
SUBROUTINECHECKThis subroutines recognizes the different non-permissible runs and flags back an error malong with suggestion for the alternative run case.
Table 5: Listing of procedures and their functions.
156
uction
odel
odelf each
ector
d. All
ll
d. All
oyed..
SUBROUTINECNTRODThis subroutine determines the fraction of control group inserted.
SUBROUTINECONVERThis subroutine converts the input data units to NESTLE’s internal working units.
SUBROUTINECORNERFLXThis subroutines determines the corner point flux values needed for the pin power reconstr(currently works for NSHAP = “HEXA” only).
SUBROUTINEDECAYHNThis subroutine solves the decay heat precursor equations.
SUBROUTINEDEPLETEThis subroutine controls the depletion, determining number densities for the microscopic mand returning the macroscopic cross section expansion coefficients.
SUBROUTINEDEPLETESThis subroutine controls the depletion, determining number densities for the microscopic mand returning the macroscopic cross section expansion coefficients at the six surfaces ohexagonal assembly (currently works for NSHAP = “HEXA” only)
SUBROUTINEDIR12FULLThis subroutine solves a 12 x 12 matrix system needed for the pin power reconstruction. Varguments are employed. All unknowns are evaluated. Full matrix structure is assumed.
SUBROUTINEDIR2FULLThis subroutine analytically solves a 2 x 2 matrix system. Vector arguments are employeunknowns are evaluated. Full matrix structure is assumed in analytic solution.
SUBROUTINEDIR4FULLThis subroutine analytically solves a 4 x4matrix system. Vector arguments are employed. Aunknowns are evaluated. Full matrix structure is assumed in analytic solution.
SUBROUTINEDIRECT2X2This subroutine analytically solves a 2 x 2 matrix system. No vector arguments are employeunknowns are evaluated. Full matrix structure is assumed in analytic solution.
SUBROUTINEDIRECT16This subroutine analytically solves a 16 x 16 matrix system. Vector arguments are emplOnly half of unknowns are evaluated. Matrix sparsity is taken advantage in analytic solution
Table 5 (cont): Listing of procedures and their functions.
157
ll
d. All
d. All
ed for
rovides
ternal
gather
SUBROUTINEDIRECT4This subroutine analytically solves a 4 x4matrix system. Vector arguments are employed. Aunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.
SUBROUTINEDIRECT8This subroutine analytically solves a 8 x 8 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.
SUBROUTINEDIRECT8BThis subroutine analytically solves a 8 x 8 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.
SUBROUTINEECHOINPThis subroutine echoes out the radial input figures.
SUBROUTINEFILE_CNTThis subroutine reads the general control input parameters.
SUBROUTINEFILE_GEOThis subroutine reads the geometrical parameters.
SUBROUTINEFILE_KINThis subroutine reads the kinetic parameters required for the transient runs.
SUBROUTINEFILE_PPRThis subroutine reads the corner point discontinuity factors and pin-wise form factors needthe pin power reconstruction (currently works for NSHAP = “HEXA” only).
SUBROUTINEFILE_PRFThis subroutine reads the parameters used to control the solution methods employed and pthe convergence criteria.
SUBROUTINEFILE_XSCThis subroutine reads the cross sections and T-H input parameters.
SUBROUTINEFLUIDCONThis subroutine calculates coolant temperature, density, and void fraction based upon inenergy.
SUBROUTINEGASCATCThis subroutine is used for the cart geometry only. It calculates the needed parameters tothe nodes with the same color together (i.e. blacks with blacks...etc). for Cartesian geometry, two
Table 5 (cont): Listing of procedures and their functions.
158
.
gatheree.
ndary
AIX
ulateve the
start
rol
ntrol,
rything
inlet
different colors are used. this will be utilized in the solution of the finite difference equations
SUBROUTINE GASCATHThis subroutine is used for the hex geometry only. It calculates the needed parameters tothe nodes with same color together (i.e. blacks with blacks...etc). For Hexagonal geometry, thrdifferent colors are used. this will be utilized in the solution of the finite difference equations
SUBROUTINEGEOMETRYThis subroutine sets up the geometry including B.C. mesh expansion and bundle bouidentification for output control.
FUNCTION GTIMEThis subroutine returns back the elapsed time (in seconds) applicable to ULTRIX oroperating system.
SUBROUTINEHYPINTThis subroutine contains three functions: COSH0, SINH1, COSH2. The three functions calcthe zeroth, first, and second moment integration of the hyperbolic functions needed to solconformal mapping based hexagonal nodal equations. (For NSHAP = “HEXA” only).
SUBROUTINEINITALThis subroutine initializes the flux, fission source, and T-H conditions guesses, or for reoption on reads in the restart file.
SUBROUTINEINIVALThis subroutine edits out the initial input data.i.e. cross section data, geometry data, and contoptions, if users choose long edits for the inputs (AL3 = “Y”).
SUBROUTINEINPDATAThis subroutine controls the overall input reading.
SUBROUTINEINPEDITThis subroutine edits out input associated with the transient case, method of solution cocontrol options, I/O file names, and if elected cross sections and T-H parameters.
SUBROUTINEINPUTCKThis subroutine checks whether alphanumeric input was correctly entered and converts eveto upper case letters.
SUBROUTINEKSEARCHThis subroutine performs criticality search on four different parameters: soluble poison,coolant temperature, power level, and control bank insertion.
Table 5 (cont): Listing of procedures and their functions.
159
laroutine
for the
letable
letable
every
SUBROUTINELAMDASUBThis subroutine determines the scale factor for the fixed-source scale factor method.
SUBROUTINELINEARThis subroutine completes the linear interpolation or extrapolation.
SUBROUTINELSORBThis subroutine calls the following two subroutines:SORCE - which calculates the RHS of the finite difference equations for a specific color.TRIDIA - which solves for the flux by solving a tridiagonal system of equations for a particucolor. The flux is also accelerated using the omegas which were precalculated in the subrLSORB0.
SUBROUTINELSORB0This subroutine calculates the number of inner per outer and the acceleration parameterscolor line SOR equations.
PROGRAMMAINThis is the main routine for NESTLE.
SUBROUTINEMFSTThis subroutine controls the multi fixed source scale factor method.
SUBROUTINEMICROXNTThis subroutine determines the microscopic cross section expansion coefficients for the depisotopes.
SUBROUTINEMICROXNTSThis subroutine determines the microscopic cross section expansion coefficients for the depisotopes on the six surface of the hexagonal assembly (NSHAP = “HEXA” only).
FUNCTION NEWPAGEStarts a new page of output.
SUBROUTINENONNEMCThis subroutine solves the NEM equations.********* For Cartesian Geometry **************The nodal method used here is nonlinear NEM, We solve a one or two node problem forinterface in the core to update the coupling coefficients.
SUBROUTINENONNEMHThis subroutine solves the NEM equations.
Table 5 (cont): Listing of procedures and their functions.
160
ometry.pling
urrent
urrent
nsion
nsion
.
********* For Hexagonal Geometry **************The nodal method used here is a conformal mapping based nodal method for hexagonal geWe solve a one or two node problem for every interface in the core to update the coucoefficients.
SUBROUTINENONNETCThis subroutine determines the currents for NEM.********* For Cartesian Geometry **************
SUBROUTINENONNETHThis subroutine determines the currents for NEM.********* For Hexagonal Geometry **************
SUBROUTINENONONECThis subroutine solves a one-node problem. It is used only for edge nodes without zero cboundary condition when updating the net currents by NONNEM.********* For Cartesian Geometry **************
SUBROUTINENONONEHThis subroutine solves a one-node problem. It is used only for edge nodes without zero cboundary condition when updating the net currents by NONNEM.********* For Hexagonal Geometry **************
SUBROUTINENONPLMCThis subroutine calculates the leakages for every node in every direction and the expacoefficients for the quadratic leakage approximation.********* For Cartesian Geometry **************
SUBROUTINENONPLMHThis subroutine calculates the leakages for every node in every direction and the expacoefficients for the quadratic leakage approximation.********* For Hexagonal Geometry **************
SUBROUTINENONTWOCThis subroutine solves the two node problem and returns the NEM current JNEM.********* For Cartesian Geometry **************
SUBROUTINENONTWOHThis subroutine solves the two node problem and returns the NEM current JNEM.********* For Hexagonal Geometry **************
SUBROUTINENORMThis subroutine normalizes the flux and fission source to a core relative average power = 1
Table 5 (cont): Listing of procedures and their functions.
161
d. All
d. All
tate
int
ient
st.
SUBROUTINENORMFSPThis subroutine normalizes the flux and fission source to an input specified scaling.
SUBROUTINEONENODE4This subroutine analytically solves a 4 x 4 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.
SUBROUTINEONENODE8This subroutine analytically solves a 8 x 8 matrix system. Vector arguments are employeunknowns are evaluated. Matrix sparsity is taken advantage in analytic solution.
SUBROUTINEOUTCYCThis subroutine writes the node-wise isotopic number densities.
SUBROUTINEOUTINThis subroutine performs outer-inner iterations utilizing FDM or NEM option for the steady ssolution.
SUBROUTINEOUTINADJThis subroutine performs outer-inner iterations utilizing FDM or NEM option for the adjosolution.
SUBROUTINEOUTINTRThis subroutine performs outer-inner iterations utilizing FDM or NEM option for the transsolution.
SUBROUTINEOUTPCRTThis subroutine outputs values to the screen (i.e.CRT).
SUBROUTINEOUTPOINTThis subroutine outputs point-wise values of parameter passed through calling argument li
SUBROUTINEOUTPUTADThis subroutine outputs the adjoint problem’s solution.
SUBROUTINEOUTPUTSSThis subroutine outputs the steady state problem’s solution.
SUBROUTINEOUTPUTTRThis subroutine outputs the transient problem’s solution.
Table 5 (cont): Listing of procedures and their functions.
162
.
group>2
ower
e two-
gian
, inlet
les the
lerkin
SUBROUTINEPEAKThis subroutine determines the total peaking factor and location.
SUBROUTINEPERTURBThis subroutine interpolates the time dependent input parameters for the transient problem
SUBROUTINEPINCOLLAPThis subroutine collapses kappa-sigma-fission cross sections and fluxes for NG>2 into two-cross sections and fluxes for the pin power reconstruction (NSHAP = “HEXA” only). NGintroduces complexities in determining pin-wise fluxes.
SUBROUTINEPINLOCThis subroutine determines the fuel pin locations within the hexagonal assembly for the pin preconstruction (NSHAP = “HEXA” only).
SUBROUTINEPINPOWERThis subroutine performs a pin power reconstruction in hexagonal geometry based upon thgroup cross sections and fluxes (NSHAP = “HEXA” only).
SUBROUTINEPINTERThis subroutine completes quadratic interpolation (or extrapolation) using Lagranpolynomial.
SUBROUTINEPOINTERThis subroutine determines the A array pointers (i.e.starting locations of arrays).
SUBROUTINEPRECRThis subroutine solves the delayed neutron precursor equations.
FUNCTION PROPPOLYThis subroutine calculates specified property as function of a stated dependence.
SUBROUTINEPSEARCHThis subroutine performs power level search on three different parameters: soluble poisoncoolant temperature, and control bank insertion.
SUBROUTINERELPOWERThis subroutine determines the total core power level accounting for decay heat and scapower density to a relative core power level = 1.
SUBROUTINERESIDDetermines the relative residual of the diffusion equation and the eigenvalue using Ga
Table 5 (cont): Listing of procedures and their functions.
163
ed by
cale
ource
ining
steady
factor
IDIA.e the, and
IDIA.Theed inouter
weighting of the diffusion operators.
SUBROUTINERSTRINGThis subroutine loads a numeric value into an alphanumeric variable.
SUBROUTINESCALAPRXThis subroutine calculates the ratio of the effective external source to the fission source usthe fixed-source scale factor method.
SUBROUTINESCALEXCTThis subroutine determines the ratio of the effective external source to (A-F)*FLUX to get sfactor update.
SUBROUTINESCALINGThis subroutine scales the flux and fission source using the scale factor from the fixed-sscale factor method.
SUBROUTINESETUP0This subroutine solves the homogeneous problem using color line G-S in support of determoptimum relaxation parameters and number of iterations per outer iteration.
SUBROUTINESFSTThis subroutine performs the single fixed-source scaling technique procedure for the FSPstate case.
SUBROUTINESHAPECORThis subroutine adjusts the flux for coolant spectral effects within the fixed-source scalemethod.
SUBROUTINESLOWTRANThis subroutine provides overall control of the transient fission product problem.
SUBROUTINESORCEThis subroutine calculates the RHS for the tridiagonal system to be solved by subroutine TRThe tridiagonal system results from using the color line SOR method which is used to solvfinite difference form of the diffusion equations. The RHS here includes fission, scatteringdiffusion terms.
SUBROUTINESORCE0This subroutine calculates the RHS for the tridiagonal system to be solved by subroutine TRThe equation to be solved is A*PHI=0 where A is the coefficient matrix and PHI is the flux.RHS here does not include any fission or scattering. It only includes diffusion terms. It is usobtaining the color line SOR extrapolation parameters and number of inner iterations per
Table 5 (cont): Listing of procedures and their functions.
164
ion in
ts.
for the
for the
r line
ys for
ctorge in
ctorge in
iteration.
SUBROUTINESPECSHFTThis subroutine determines the B**2 values used in making the coolant spectral shift correctthe fixed-source scale factor method.
SUBROUTINESTARTERThis subroutine sets initial values to initiate the transient from.
SUBROUTINESTEADYNThis subroutine provides overall control of the steady state solution.
SUBROUTINESXENONThis subroutine solves for the steady state number densities of the transient fission produc
SUBROUTINETHFDBKSThis subroutine calculates coolant internal energy, coolant density, and fuel temperaturessteady state problem.
SUBROUTINETRANSITThis subroutine provides overall control of the transient problem.
SUBROUTINETRIDIAThis subroutine solves a tridiagonal system of equation which results from using the coloSOR method. It solves for the flux of a particular color each time it is called.
SUBROUTINETRIDIA0This subroutine factors the tridiagonal matrices associated with color line and assigns to arraeach color.
SUBROUTINETWONODE8This subroutine analytically solves a 8 x 8 matrix system used in NONTWOH routine. Vearguments are employed. All unknowns are evaluated. Matrix sparsity is taken advantaanalytic solution.
SUBROUTINETWONODE16This subroutine analytically solves a 16 x 16 matrix system used in NONTWOH routine. Vearguments are employed. All unknowns are evaluated. Matrix sparsity is taken advantaanalytic solution.
Table 5 (cont): Listing of procedures and their functions.
165
e step
for
for
roup
eption.
sion
ity, fuel
ity, fuelnsion
ity, fuel
SUBROUTINETXENONThis subroutine solves for the transient number densities of the transient fission products.
SUBROUTINEUPDATEThis subroutine saves time-step values of various parameters for usage in the next timsolution.
SUBROUTINEWEILANDT1This subroutine is applicable to Wielandt shift with stationary acceleration and checksconvergence for the steady state problem.
SUBROUTINEWEILANDTRThis subroutine is applicable to Wielandt shift with stationary acceleration and checksconvergence for the transient problem.
SUBROUTINEWSHIFTThis subroutine determines Wielandt shift utilizing the Sutton method to retain energy gdecoupling at inner iterative level.
SUBROUTINEXSECBUThis subroutine determines the nuclear properties (e.g.cross section expansion coefficients) at thnode color and burnup except for the depletable isotopes modeled using the microscopic o
SUBROUTINEXSECBUCThis subroutine determines the corner point discontinuity factor and form factor expancoefficients (NSHAP = “HEXA” only).
FUNCTION XSECPOLYThis subroutine calculates cross sections accounting for coolant temperature, coolant denstemperature and soluble poison feedback corrections.
FUNCTION XSECPOLY2This subroutine calculates cross sections accounting for coolant temperature, coolant denstemperature and soluble poison feedback corrections. Difference with XSECPOLY is dimeof calling arguments arrays.
FUNCTION XSECPOLYSThis subroutine calculates cross sections accounting for coolant temperature, coolant dens
Table 5 (cont): Listing of procedures and their functions.
166
agonal
djoint
utronic
utronic
rnup
temperature and soluble poison feedback corrections on the six surfaces of the hexassembly (NSHAP = “HEXA” only).
SUBROUTINEXSFDADJThis subroutine determines the matrix transpose of the coefficient matrix required for the aflux solution.
SUBROUTINEXSFDBKThis subroutine determines the macroscopic and microscopic cross sections and other nenode values and uses them in determining the coefficient matrix.
SUBROUTINEXSFDBKSThis subroutine determines the macroscopic and microscopic cross sections and other nevalues on the six surfaces of a hexagonal assembly (NSHAP = “HEXA” only).
SUBROUTINEXSMODThis subroutine calculates the flux-volume weighted cross sections for the within-node bugradient treatment in the hexagonal nodal method (NSHAP = “HEXA” only).
Table 5 (cont): Listing of procedures and their functions.