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TITLE:
TRIANGULAR ESHMETHODS OR THE NEUTRON M F TRANSPORT EQUATION
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SUBMITTED TO:
proceedings t
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of the Axneriean Nuclear
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TRIANGULAR MESH METHODSFOR THE NEUTRONTRANSPORTEQUATION
.. .
.
by Wm. H. Reed and T. R. Hill University of California, Laboratory
Los Alarnoo Scientific
Los Alamoo, New Mexico 87544
. Tha mcthodo that cr~ta mmial
facm.
ABSTRACT axe davolopad of tha q ngular in ~his papar for cliff crane ing tha diax-y grid flux
qre
ordinates
q quationa
on a triangular
baaed on piactiiso class all
of
P.lyintar-
raprmantatiano
A mcond
flux~
The first
mathoda dioto th~ first boundaries.
mathods sro
cuasad hare asaumaa continuity claba, ?hmrical
qllows
of tht q n~ular
qcross
trfingla
clam. of m~thode, which ia shown to ba suparior flux to ba diacontinuoua tht q ccuracy ad
qcroso
tha q ngular
trlangla
of
thcs~
rosulta
illustrating
stability
is also dlacuosd.
~%!
1. tial grid. right
INTRODUCTION
A two-dimensional
#
(x, y) neutron codes,
transport all
code based on a triangular mesh
spa-
mesh is currently advantages angles~ hexagonal By an orthogonal
under development grtd,
at Los Alamos. in which all exactly easily
This code w*11 offer mesh l~nes meet at elements; mesh by coluwith triadded cost. of is
,
several
over present
of which use an orthogonal
we mean a grid seactors fourg
Many nuclear geometries
are designed six,
with hexagonal
these
can be represented can be approxhatcd of a triangular The order but involves
with a triangular Furthermore, and accurately
su-~dividing plicated angles ~
each hexagon into
or more triangles
curved geometries flexibility
The increased Description no longar trian@e iteration taining gular pl~cit tics linear Instead,
mesh is not without because
of the mesh is more :mplicated, straightforward
faces
q
the Y and y coordinates the direction
each vertex must be given.
in which the mesh unknowns are solved determining
must be made repeatedly
of flow across: times,
Such determinations
in the innermost computation
,
loops of a transport finite-dimensional
code, and $hev may increase some effective
The purpose of this grid, Difference which we will
paper is to present
~ ! new 8ch~@8 for ob- F into two broad In an im1
of
approximate ions to the trannport schemes for the transport refer to as implicit the direction the unknowns. to obtain is made to solve
is, in
equat ton on a trianmethods,
of
equat ~on fall or explicit
categories@
method no attempt of the equation, variational algebraic
that
in the direction
the charactorieare streamtng~ a @et Lo then’
An which neutrons
methods or Galerkin for all methods,
methodo ara used to determine quat~one Th~o sat of q solution. the final
equations by direct hand,
SOIVS”, usually
An explic~t for tho unStatd but ~ Of course,
~ I
~
method, on the other knowm tn tho direction an explicit thi~
.wecpa once through
charactcriatics
the m~ah~ solving phaae apace.
algabraic
qt
in which neutrons
to solvin~
are %Xearniug ~ More properly throu~h or
method follewe
is qlso equivalent
~ set of linear
q quation.,
‘
1
hero tho matrix to be inverted Perhaps the clearoot
following way.
is triangular,
least
block triangular. ic coupled
for
dtatinction
between tho two methods can, be made In the method o particular baclwrd ma-h call
‘
An
In an explicit qll,
only to ‘
tho.e
mesh c*11o visible m~thod couples
of
when looking
q ions the characteristics.
scheme is
qn
implicit examples
q djacent
qre
meeh CO1lS with no rcuard in Ref. 1.
it
the diract~on method; thoroughly
of thm eharac t er istics. implicit Although
Th~ diamond dif feranca
explicit
mathods
givm
both q xpJ,i( ~~ znd tiplic meohos in x-y Seometry, and even fewer with ptocawioe
hava
methodo hava baan studied
very
few trianguhr been
tsstad o
mesh methOd8 have
q ctually
trial
2 proposes Ohntohi
the
,
q
linear
functtons
for
opatlal
‘ I
I t
I
“ITFN
‘
dependence Several mall ical faces.
of the
flux coupled with a di~crete but he does not give numerical
ordinates results
treatment supporting
of the anthe method.
.
gular variables, explicit
methods are given in Ref. 3. 4 Very good results have been obtained with implicit problems, that is, for problem in which the total geometries that phase space is about 1000 or fewer. such complicated arc needed to describe accurately
methods for relatively number of cells many real thousand the total size several in the physspatial numeffi-
space-angle meoh cells ber of cells ciently. gular flux. function
Unfortunately,
problems involve
the system boundaries variables, tens problems involves of several the storage required
and inter‘
With a relatively
crude mesh for the angular implicit implicit methods can solve methods require locations
. ,/, 4 “
in phase space can bo on the order
of thousands. anflux : ‘I
It is not known whether Furthermore, values,
of this several required
of the complete unknown for a exceed for the angular locations~, so that disc
Since each cell
in phaae space us~ally
the number of storage of coefficients
for a single
energy group can axceed, 100tOOO. The storage has a band width of about IO* core capacities
of all
100,000 by 100,000 matrix assuming the matrix the fast storage must be used. and extended
can exceed one m~llion modern computers,
These requirw~mts
Present two d inwnsional even fot a one-group problem. 1 5 codes such as TWOTRAN which are based oh wplic It methods etorg only the scalar flux md enough moments of the angular Therefore, tained tranefer all parameters
me
flux
to generate
the
scat
t
ar ing source. be coe‘
i I
r ertinent
to a single
energy group ran usually program is obtained
in fast core, problem
We ms concerned
so that in thio problem.
a more fficient
and data for larga
that
mlnlmized. paper wtth methode that are suitable
For the q bove reasons,
~t
complicated
phyuical
appears
explicit
I
mothodc may bu uuper ior to impllcit er onLy q xp2i@it methods in this Althoush tt imgular triangular tal linao,
q
methods for q uch problem. In this paper are applicable triangular
qll
vert~ceo
Thuo we ctonsid-, to lte
q
paper. general meohea * A regular on horizonby heisting An axample of
the riiothods dwcloped
mesh, we consider so that horizontal vwtex
q
here only %esular!t by requtring band. o~ trian@es
Nota that
that
I , I
mesh Ls characterized
are formed p ad trian@cs.
that each Interior ouch equilateral There
flcation
be common to six adjacent
mesh is Riven in Fi$~ 1. and that
qgQ
we do not raquire
re@m
non-rectan@ar t~ !an?d.ar
domain 4. qllowed *
qr only
that trLan@e@ ba ‘ I
two ueaoonc why we con.id meoh.
moheo ~ Fhot,
than specif Amtion
q peoiof
q
2 g ? 3 t E
of q rosular
mesh ia mush simplar of the vertieeo
Ueneral
trian@ar
OnAYthree pieeea of data arc required t the meoh
opaatnum (Ay) ~ t %he x aoordhatm
q long mch horizontal
The orimtatim
Una o
q the orientation nd
of the first
trAan@o on @aeh bmd.
of
tlm
..
!-
..-
!
Fig, fitst points triangle
1.
A typical
regular
triangular
m~mh. whether determined the triangle uniquely L ~ is
L. .
on each band can be specified down. A regular ‘triangular
by indicating mesh
is
up or points
by the above data. The secomi reason ad to out decision sarily a defin~te order
why we consider
only regular methods.
triangular Explictt
meshes is relatmethods necesso that these
to consider
onLy expllcit
sweep the mesh in the d trection in which the triangles of neutron
of the characteristics, must be solved~ triangLe mesh. boundurtes
This order” depend q and is not of flow acrosa through for a
in
upon the d~rcction straightforward bout.iaries
flow across and decisions
for
I
I ....
as for an orthogonal
Thus the diraction
must be determined mesh than
made as to how to progreo~ and time-consuming mesh ~ Since ordinate loop. d~rectk~ of a transport triangular
tho msah c Them decfaione gcmeral tria~ular dec~aiono
If
are much more complicated
I
a regular
the order thc~~
which the mesh is awapt differs rnuot ba made repetitively such dcclsions complicated,
with each discrete in tha Innermost would ba prohibitively
.-
cods.
=pensive.
W. raotr%ct
I ,
IX
q
THEORY Th@oxm velocity nautron t?anaport squat ion can ba wrAt t q in x-y goomatry
I I
as
I
I
whort WQ hmw writtm
qo
tha ucattm iw C f i-iono
aontoxt, S would
q!
lnhomo~anaouc sources
dmply’
S,
In iB multisroup
also include
muraes duo to q $attarin$
O,.-.. .*-, s I
!.
--
~ti
and fission approximation
in other
groups.
We will
utilize
the standard
discrete
ordinates
to the above equation,
thus we write
~ I I I
a+
Um$
i)+
‘ % ‘$
flux points
+fqm(x, y)
i
SJX9Y)
9
(2)
to $(x,y,Mm,~) and a set of’
;
I
where the angular M quadrature the standard Ref. basic 5.
*m(x,y) is an approximation used in two-d imensioml also contains discrete as Sm(x,y) are generated. details. of a discrete
(pm, IL) have been selected.
quadrature
For a de~ailed description of discrete ordinates codes see of how the sources
i
This reference
a good description
which we have written familiarity Our task task reduces single visiLle of omitting
with stawdard
We assume the reader has a ordinates codes and take the liberty (in x and y) approximate ion
,
I
some of these
is now the development mesh.
1 ‘
to Eq. (2) on a triangular triangle9
Since we consider only ewplicit methoda$ this’ i to the problem of generating an approximation to ~m(x,y) over a ~ assuming
must
that
$m(x,y) is known on the triangle fim determined one or two faces
boundaries
when looking
that
along the direction be considered: of the ttiangle.
by IA=and ~. may be visible
two cases
There are ! i depending
2.
i
on the orientation
These two cases are depicted
in Fig.
.
am
q
h
Fis. 2. by
q
/
/
L
* / 42m
\ ,,
.
~
,,i I
L
Orientotion I (one face vhible)
Orimtation 2 (two feces visible)
w~th ‘ ,
I
Tha two po$.iblc orient ttoxm of a trlan@e reopect to q direction A~.
qowms
that
tI
I
m
All m~thods dovaloped in thio papar tria, iglc is gtvm
low-ordor
the an$ular
~kux wer
each
I i !
2’
polynomial,
That i.,
I “
The form of the approximate
solution in this solution. all
~mover
the entire across
system is completely triangle that boundaries.
of
,
determined
by specifying
the continuity
conditions The fitst
The two methods considered imposed on the approximate lar flux be continuous
whatsoever
paper differ boundaries
only in the degree method requires
con~inuity the anguof reflux is ,/, #
across across
but does not require that is,
continuity
any derivatives quirements allowed
of the solution. across
The second method imposes no continuity boundaries, triangle the angular all boundariesside
triangle
to be d~8C0ntinu0u8 ~m. that
The flux on the the boundary of the boundary. flux~ in ~
boundary is to be the limit in the direction We reiterate The order
of the angular
flux as one approaches on the other iu this representations investigated
The jump then occurs piece-:?.se polynomial is arbitraxy, ,
the two methods considered
papet are explicit of the angdar of higher numerically
methods which utilize ordex polynomials the next section There are, wpressed. i. ical meaning. points inconvenient
of the polynomials
of
and the effectiveness
is
such as cubics th~s paper* of course,
and quartics
many ways in which a polynomial of Eq. (3) is certainly because the coefficients
in x and y can be A,, have 14ttle of the where N these
in-
I I
The representation for our purposes
We
the most common, but it physI
I I
prefer,
Instead ~ to use a Lagrange repres&tation have bben placed on the triangle because lees there of Interest,
of
polynomials
with which we work.
N+L (N+2) Let us assume that a set of K ~ ( ~
[
distinct points
(xig yi)
is the order dependent
qt
of tha polynomial below. of order
to be cepreaented. less
The placement We define
~
ia dlacusmd polynomial (Xt~yi)
We use K points of’ order
are K linearly
than or equal N.
the polyrIomlal
to the K
I .
I
Li(x, y) as the unique polynomial
che
than or equal N that is unity K - 1 points ~ We refer If the points aro l~nearly leas (XigYi) have independent
of
, ,..
I
I
point
and Lo zero at t}ie other then the Lagtanga
polynomials
LL so def Ined as Lagrange for tho q pace of
polyriatfiiala. polynomials
bean choson properly,
q form a basis nd
I
polynomials
of order
thau or equal N.
I
Thu8 wa can r~~laca RX,Y) m
Eq. (3) by tha fol.low~ng equation with no lost
z
*ml
suppt.saed
K
viL@Y)
t
,
%
content? (4)
b.
I
1
I
w~er.
bovo q quation, tho ceeffithe subscrtpt m. In the q as the valu. of ?’(xoY) at th. po~~t (xioYi)s eicints ti call be interpreted for ?’4 which laada hme the notation ?’4. Tt ir this physical interprato~ion
w. hava
!l!
us to the Lagranga r;prementation There
qre
for
u t (x. y).
I
many arrangement.
imiepmdenc~
of K points on
d the
q
tria@a
that
I
uniquenasa qnd llnear
LaUrange
polynomialo~
I
:
‘ ‘ particular boundaries distributed
placement simple. uniformly
of these
points
which makes the treatment polynomial, we place at each vertex. of the triangle.
of the triangle on each are points
For an 1!fth order v~th a point in the interior points
N+l points
face of the triangle, the placement of these
The remaining Figure polynomials.
3 illustrates
for a few low order
A
N=l
.A
e
N=3
. A .A*
N=2
q q
-.
I }-..
I
. ..
~.= 4
Fig,
3.
The triangular point low order-polynomials.
arrangement
for a few
Because a polys.omlal uniquely
in x and y of order points flux on the boutiary, these
N is determined without regard
on a straight fcr the other
ltne points~ poly-
by N+l distinct the boutiary
on the line,
the boundary flux
can be deter-
1 ..
I ,
mined by the N+l points
Furthermore,
is given by the unique one-dimensional N+l points. description
I
! ,
4
nomial which passes
through
We have now given a complete solution this for the two methods, What remains is Senerated arrangement
We simply
of the form of the approximate that we discuss In --~
eonthuous
cam.
and discontinuous, We ccnaider
i I
paper.
to be describad in uch indicated
q ssumg
that
is ~h~ manner in which this f~ra : the
approxi-
mate solution method. The point littl~
q ffort.
qre
cort~fnuoue
i in Fig.
~’
3 and the representation to be Impoad incomi~ calls boundnrieo
of
of
Eq. tri-
..
2
J. .
(4) for 7(x, y) over a triangle
allow continuity on qll
upon V(X, y) with
q
q ngla
known Zrom prior the call. upon
Of
calculation
q qn
i: q djacemt triansle
1
t
or from oyotem boundwhich the neutron Ls
qn
my data.
An incoming boutiar y is courm,
bol~ndaxy across
flow is into
incoming baundary •~d tha def inttion consideration.
for onu cell
out-
~oirag boundary for tha q djacsnc ary depmda
tho
cell, (la ud~r
of qn incoming
bound1s. t
direction
Since there can bc one
~b
os two incoming boundaries, (h) are determined lC-(N+l)or the orientation an@e. there Table I that desirable K-
eithet
N+l Ot 2N+1 of the coefficients at the boundaries. This leaves ~’ per triangle,
~i of Eq. a total of upon
from continuity 2.
(2N+1) unknown coefficients of Figs is tabulated polynomials
depending
Let NN equal the number of unknowns in a given triin Table I for a few cases. and a triangle We believe this situation of order W@~ from see to be ungreater than with two incoming boundaries
This parameter for linear
are no unknowns to be determined. and thus restrict our attention method.
to polynomials
~r equal two for the continuous
TABLEI
THENUM3ER UNKNOWNS IN A TRIANGLE OF NN
AS A FUNCTIONOF ORIENTATION? ANDORDER POLYNOMIAL OF ORIENTATION ORDEROF. POLYNOMIAL 1 2 3 1 2 2
3
~
1
3 6
0
1 3
q
Wa must now derive triangle. the solution diraction succaasivcly triangle. WU y). (x, takea is haerted
a set of NN equqtions in the discrete
for
the NNunknowns on the given manner. equation equation The assumed form Of for the particular 48 then multiplied over the the ‘ . and are denoted functions,
This is accomplished Qmutier consideration.
in the following ordinate The resulting functions
by each of a tiet of NN weight For the moment the weight With
q
functions Mepandent
and Integrated weight
are arbitrary
proper gives
choice
of
linearly
abov. procdm$ the form
the desired
sot of NN equatione.
This set of equatiom
f.-
(5)
j-l,
2*oo@NN
*
*
where the inner product of interest. Note that presents fluxes.
(a,b)
represents
the integral
of ab over the triangle on the left side
some of the coefficients algebraic
vi appearing
of Eq. (5) are known from boundary data, an NN by NN linear With a proper choice of weight
so that in reality this equation re. system of equations for the unknown functions this system is nonsingular for small linear systems,
and can be solved routinely such as Gaussian A good choice method. less tation of these functions sults these cocrse, We believe thanor elimination of weight that
by any method appropriate functions is crucial functions verify
to the success the triangle that there
of the above of ordes NN are reof in
,, ./.
4
the best
weight
are the polynomials
equal N-1 or N-29 depending in either that case,
upon whether we obtain choice
is of orien-
1 or 2, respectively. polynomials as we have unknowns.
One can easily so that
are precisely functions
the same number of weight of weight Numerical It is,
Another possible are unity that
the Lagrange polynomials in the next section Lagrange w~ights possible results with either across lying errors
at the unknown points. the method dots polynomial the resulting weights.
indicate
not perform as well with method ia unstable, the mesh.
as with the low order so that problem, are amplified
to choose weights bearing on this
the sense that theoretical stability continuous ever, points
as one sweeps through of weights.
We have no an in-
but we have never abserved is allowed 3. In this
of the above two choices boundaries.
We consider
next the second method in which the flux triangle We again arrangement of Fig.
to be discase, how-
use the Lagxnnge representat~on of as actually lying In this arrangethan
of Eq. (4) for the flux and the point &n the interior ciated
mmt
on the tr ~.angle boundar~es b. t arb~trarily i~tto two or We attempt It is clear splits
axe thought
more
of the tr.tan~le trjangles. method.
C1OSS to the boundary. points to illustrate that this point
manner each boundary potnt with different in Fige 40
for
which are each asso-
For thie method the total equal
to K times
number of unknowns is larger number
of
that
the conttnuoua is in fact
the number of unknowns per tr tangles in the mesh.
direction
the total
.@
w! +“1 ?’, .-v .-
t’ , .
“. * ,., .
Fig, 4.
A
typical point q rrangement for the d iocont inuoue mathod, Boutiary point. ar- actually arbitrarily close to the boundary,
*9
We now proceed form of the solution which gives boundaries equation sng~e.
qre
tn precisely
the same manner as in method one. is Inserted at these func~ions algebraic of weight in the transport plus a Dirac delta function
The assumed equations
in a given triangle
a smooth function
at the incoming The resulting over the trithese for equations t:hia method inis
due to the jump dlacontinuity is multiplied by NN = K weight an NN by NN linear choice
boundaries. and integrated functions Note that
Again we obtain
system for the NN unknowns
in each triangle, nonsingular
and with a proper and can be solved functions of order less
for the unknowns. than or equal N.
tho number of weight d~pemle~t polynomials
therefore
required
is equal to the number of linearly Our choice of basis spanning
immaterial, less
and any set of functions
the space of polynominon-polynomial that are used.
al
of order
than or equal N will for either
of
give the same answer when used as weight ~he use of more complicated Again, we find weight experimentally functions
functions.
weight
We have not investigattu method is stable
functions discontinuous
our methods.
this
when polynomial
111. NUMERICALESULTS R
A
one-group, with this test
isotropic
scattering,
discrete section,
ordinates
code was written numerical
to
implement the method a of Sec. II. obtained code for several was used in all The first tahmd with these calculations~
In this
we present
results
simple problems. exhibit
An S2 angular the accuracy that
quadrature can be obcontain200
problem was designed methods. It consists The source 5 and a similar in this papw
to
of a one mean free and constant
path square
ing 8 pure absorber.
is isotropic 800 triangle the epatial
over the square t and tri-
boundary cond it ions are vacuum.
Cdcuiatione were performed using the
order N varying differenclng
q n@c mesh of
qd
Fig.
mesh for both the continuous from one to four.
of the transport
discontinuous
m~thod~ with the polynomial
Because w~ emphasize equation,
tho
we choose to compare our computed remlte from conaiderat Ateelf. The exact solution absorption the percent
with the exact ion any
qrrors
eolution introduced
of
Sn oquat~ona t thus eliminating easily for thio total
of
in
the
tho Sn q pproximation tained
of the S2 equations computed error
from
can be ob.. #
‘/
simple homogeneous probl,em. absorption this table
In Table II we present S2 solution. docrease~
porcanta~e dif f orenc o between the total
oolutiwia and the
our numerical We rapidly places a. than
computed from the exact that and that an.were
of
nota from the results
tho
‘d
polynomial
order
is increased
for
hlsh order accurate
polynomial
method& appear
to ba more q fficient Wo qleo aothods
qre
obtaining
to many 4ecimal the high order
low urdw polynomial method..
note
from the result.
Table 11 that accurate.
polynomtil
no more than ser.ond order
Thlo is saan in th~ followi~
Fig.
5.
Pure aboorblng
square
200 trian@e
mesh.
TABLE IX
,
P~CENTAQ~ ENRORSIN TOTAL ABSORPTION FOR A PIKCEWISE POLYNOMIAL AWROXINATION TO TM SOLUTIONOFTI{B TRANSPORT M?UATION , , ?OR A PURE ABSORBER 1A SQUAM MITll N1?ORM IN U . SOURCE, WEICNTFUNCTIONSARE LOWORDERrOLMON1AL8.
200 200 200 800 800 800 200 m aoo 200 000 800 800
Cont$nuoua Conttnuou@ Continuous CtmtMuouo
2 39 4 9
0,52
qa
0005202 ,Oooslg ,000062 ,Oolm ,000006 .900012
,006!)65 ,000294
q,
m.
1*5O m q
4,A2 1,98 toe
qao
,
5*4S sac
130s7 ma
lb’
DAnaon&Snuou8
1
q 67
4.79 1s.44 S.61 6.90 No30ma
qoa
Diaaonthuouo D&oaontinuouw DSacontinuous Ohaontinuouc OAnamtinumm Dloootttinuou@ ,
a 3 4 1 a s
1084 Daa aao
qaa
caa ma
0000175 ,000044 ,001330 ,00006s ,00002?
b
manner halving that
q
An tncreaue
of
in the numbnr of triangles yields
from 200 to 800 represents of polynomial
four
a in
mesh spacings.
We see from Table II fos any order about a factor examination
of
such a halving of mesh apacinga
of
teduction in their
the percent predictions for this otder
error.
of total
Thus all absorption
these methods are second order rates. that
of
accurate
A closer these point values
of the flux flux
shapes
problem yields in their
the tesult prediction
methods are
An fac~ only first but are total
,4. .!-
accurate
the scalar
second ord~x accurate absorption w~ighting large
of
when pred Icting
II
any integral using
parameter low order absorption
such as the
. .
or an oigenvalue. of Table
were obtained
The results the continuous low order
polynomials at least method,
ae in twice as
functions.
Use of the Lagrange errore in Table II.
polynomials
as weishting
functions the choice
method yields
In the total
a8 those xeported The results
of
For the discontinuous a clear
or Lagrange weights Table
la immaterial. eupcriority order that of either tho methods. substantially use. Although the diocontinuoua than are the the d iecont inuous we find with
11 do not indicate
continuous continuous much lass that
that
methods or the discontinuous methods~ the latter computation time~ advuntagee utiltze
methocle ax. somewhat more accurate
for a given polynomial ws do believo
fewer unknowns and requirs In particular,
Never khelese,
methods possess
which racommend their
the discontinuous tho q cceleration problems
Transport
methods aro more qtablo method known qs coarse These claims will
than the cent inuouo mathode qnd works better be eubstantlated rapreocntationa by th~ nsxt few
of th~
mesh rabalanco
thg diucontinuoua teat
o
methods.
thgory mathods bad treatin$ raprcaantatZon includa fluxca The diamond d$fforence
of
on continuous thick
flux,
flux
hava great linear,
to
difficulty
optically
reg~ono without and tha tandancy LD wall known.
fixup
uainB
q
fins
meoh spactng~
scheme can be darived
th
by using Ttaaasport
a piec.wioe codes
those
conttnuou.
of th~o mtthod ellminata
produce flux o.clllatlons method q lway.
qnd th~ negative
of
An such re8iono
qoms
basad on thi. oarillatfons tho bahavior
iinear,
typo of
schemo to
they
pxoduco, whenever pos.ibla~ in tha total spatial cros.
mction.
To examine probTha
,J8
10
the discontinuous Wth a hundrad+old mathnd o wing total
method. under such conditions increas~
of
$ tho first
lem was repeated
discontinuous
q n qror
tha 200 trian@a 0.0027%. uming
achame),
q
nmh of pi~. 5,
P,
gave
tha
qrror
in
the
q baorpt~on
f srence of
The TW@TRAN coda (based on
eontinuo!ia,
diamond dif is plotted
in th~ total
abaarpt~on in FiR.
O*29%. Th~ scalar
60
J*OO rquaro maoh, gave qn flux q long one half of th
is oboerved
the
t’
mntu osaillato
plant
Th@TW#TRAN solution
to Th~
f
qbout the infinita
mm}] q olution
in
th
medium .olution
(O.O1)O whreaa
if
diacontinuouo, .duo luxos
triangular
rapidly
damps to tho hfinitc
medium q olutiont cell
oacillat ion
TW6TRAN aolut ion would ba mor~ qpparent
.
I
I
1.
I
0.01c
7“0
—
0
0
W05
~ 0
. ‘t = @O/cm
Dkcontinuous,Linear, 200A TWOTRAN ,100 ~’S
# o—
0
qi~=o
s=.
10
I I I
03
q
0.4
0.5
x (cm)
Fig.
6.
center plane ScalAr flux for
Problem 2*
**
WQZO plotted 3, dia~ammed
q
The atabiltty
of the d~sconthuoua
method is demonstrated
again
by Problem
7. The triangular mesh calculat ione ware performed with q 200 trtinggle mesh identical to FLg* 5. The TW@RAN mesh conaiated of 225 equally spaced squareo~ Scalar f Iuxea along one half of the cent et plane are plotted in Fig, 80 The continuous triangular mesh ocheme exhibits large, slowly damped oscillatlou. Although the linear discont inuoua method results in ncgativ~
fluxes,
in Fig.
they are relatively fixup An this
caae.
small
in magnitude
and rapidly
damped.
Ths fluxes
nesative
flux
Zn TW#TRAN eliminates
the difficulties
of negative
and oscillattone
S“o
q
k
The
‘cm
--
lfi~. 7. Geometry for Problan 3,
q bility
q
of
q
triangular
nmh to treat
to the mmad
curved
boundaries
q ccurately
re-
is
illustrated
FiB.
by Probl~
4, d iagrammed F*u, in ma.hec q hewn Ln FLu..
9. Tho orthogonal
TW@TRAN mech of akin
10 ~ives
poor q pproxkation
boundary of the Sntertor
th
Cion.
Tha triangular
11, 12, ad 13 q pproximate Tha total
qrrors
uular boundary in q much more qccurate faohtin. variouo modol. i. tabulated h Table 111. Tha
tho
tho
q baorpt
qro
ion
for the
@ven
the qrrors
q baorpt ion from
the moat q caurate
modal, namely th~ 648 trian@e
mesh with
the
diaaont inuouo. cubic d if f qcence q hum W. cm that tho aont huoua quadratia aeh-e civ~o rnisntfieantly lQCO q eouratc ab.otption rate. than
.
I
I I --Disccmthtcm Lineur, ZOO k ——w thtintmus, Qmkufic ,2001 . * ‘Twotran4 225 C
--
---~
#----
7-mm.
.
.-* . *
---4
8
-
Cq =
I .CxM3n
=
ex~= 0.95Am = I*O s
o.o/ixn
0.0
0
* 01
s = I . 03 02 x km) “
q
1 I
04
q
. 5
Fig.$. Centerplaneacdar flmx fox Robla
3.
41
tha diacontiwous$ ia tho result far quatc tively
of
squara Equivalent in Table III Iinar achama~ Tho TW@TRAN tho problm in which the circular &rea is converted to a Both Tl@TRAN raa~lts
grid to treat
@quul arm.
in.dicata iteration
tha inability
accurately.
of
q
relacode can
coaroa orthogonal
Convergmco
curved
boundaries
of tha inner thick
times. tha use of
or within-group
qn
in a tranaport
near unity
is
ba *1OWif optically
In such situation eonabl~ comae zon~ by obtainad. computation maah rddanca.
q
reglona
with @cattaring
acceleration
ratio
aro praeent. for ream~thods is mesh zonoa is plus can yield ‘
with
tochiqua
essential
(ha of tha most effective
acceleration balanca thi~
5
that
This method multiplies zone choam
q ourco. q0
the fluxas lwutron
in each coarse ov~r all
leakage
factor for
that
By neutron mu~t
balanca,
the
wa mean that
6
for every zono the
q bsorption
a divugont
q qual
It is known that to tho stability
accduation
q lgorithm
qppoaro
in some casas.
Tha convmgenca
of the accdaratad m~thod, algonicely
itaration
to ba related
of tha difference
th. mora qtabla •ch~oa yialding tho mora rapidly conwrgont accalaratd 6 rithmao For this reason W* expect our diacontinuoue mathoda to coupla with tha rdalance q ccclaration techniqua
to yield
q ro
a
rapidly
to
convmgmt
teat this
algohypoth-
Tho next probla.ns frc~
ua~t qourca
daaigned
path q quaro with 8 scattering
moah of Fig.
ratio
of 0.999,
q
throughout
th~
region,
and vacuum boundary conditions. 5 wtra usd
The linear
for the tri-
diacontinuoua mgular tion.
mathod qwl the 200 triangl,
mash calculation.
A 121 square umh waa used for and CDC-7600 computation
-0
q ra
tha TW@XAN calculatim
r,qu~red
The numbar of Itorationo
for
q
point-wi.9
oehmoe ~
flux conv~rgenc~
to 10
given
in Tabla IV i’or savaral
rcbalanca
Tha.* q chemaa diffu
only
in thair
definition
of
q
coaraa flna
mgah zona.
Each In
triangla ati
is
q
q oparata
coarse
mash xona in what wa call system comprlaca
m~sh rabalavcao
q
moah robalanc~. coaraa m~ah sons,
system of
wholo oysta
rdalanca
q
the q tirs
bad
single
linear
each band i.
sons
of
In band rabalanca. or fins
for
q
For the cao~ quations usd
*toration.. for tho
q lsabraic
must b~ q olvad
tha rdalance
fmtora.
An it~rativo
mathod ia
to 00Ivo thasa
earlier
1. tht cornmrgonca praci.ion of th~sa q quations, and c r~bal Sine@ a tight convtrgancc on tha rabalanca factora is unnacaaaary inner itarationa,
q
variabla \
rabalanca
proaioion
was axarmimd in
‘r
q bal
-
Ocol *
ma%
ll-fil
,
i
with
The fi
q re
the fine mesh rebalance
procedure flna
factora
from the previous
factore is was alao taken
qs
inner
iteration.
An q xtrapolation
on the rabalanca mash rebalance
investigated
whereby a corrected
factor
‘i
Corr &(fi =
- 1) + 1
,
Choice of a = 1 corresponds
rabalancet r~balanco
to fine
ma~h rebalance
and Q = O corrospondo
to no
An appropriately
factors
from
choson CY tmda ituation
to dampen the oscillation to the noxt~ that the scattering
of
of tha ratio ia
ona innu
Problan unity
technqiues Tabloa iteration
6 ia identical
for this N qd
to Problem 5 q xcept
shown in Tabla V. that
q
and the equare
la 100 mean fraa paths wido~
problem is larga reduction of fins
A comparison
tha rebalanca
V Indicata
in tha numbar of innu
may result
from th~ application to ba much larger
mash r~balancc.
In particuschmaa
lar,
q.
the gains appaar
rebalance procadurs
for th~ discontinuous no savings
diffarancc
thsa
opposed to the continuous
diffucnce
off us
q
q chama of TWf!)TMN.For in computation, reduction
problem.
wharaas th~
the variable extrapolation
precieion q ffects
.
significant ,
in the numbu of innar
iterations.
,
I
,
a—
.
I cm
.
I
FA~.
9. (koamtry
for Problam 4.
TWOTRAN
Mesh 100 cells
4.
‘ Fig,
10,
TWOTRANooh for Problan m
.
50 tdongle mesh
Ftg,
11,
Problem 4
50 triangle
mesh ,
150 Wlonglonmh
Fig, 12,
!& O!)hsa
4
150 trien~lo
mesh
.
.
a
648
Triangle
Mesh
FQ.
13.
Ikeblem 4
648 txiangle
mesh
.
TABLE 111 TOTALABSORPTION FORPROBLM 4
e
MODEL . TWdTRM, 100 square mesh TW6TRAN Square
AB&RPTION
% mROR
0.1064
1600 square tneah
Quadratic
3436
4559
-0*282:
-0,233%
equivalent,
O*1O59
50 triangle
mesh, Continuous. mesh, Continuous,
MO triangle
648 triangle
Quadratic
001001 7172 O*1O3O S362 0.1033 8129
0,1029 03a4
+0.344%
+Q,056%
+0,023% +Q.071% +Q.016% -0,010%
30 triangle
MO trtangle
mesh, D$scontAnuous* Linear
mesh, Discontinuous, Linear
0.1034 5378
O*1O37 1294
G48 triasglernesh,
Discontinuousd Discontinuous,
Linear
648 trian~le
web,
Cub3c
0.1036
12S3
TABL8 IV , INNER IT~TIONS ltEQUIRl!O FOR CONVERGENCE , CCMPIJTAkION 17NRATIONS — TIME (SEC), ——
ACCELERATION METHOD
Triangular No r~balanoo
Mwth
306
mbalanm
‘
23b02 6479 6,24 3@?Y 3060
UhOla
q yotam
rebalance
w 60
hd
42 41 crobal %abal’ a =0.70
54
21
-—
7.2s 2.70
—
q
TABLEV
t-
lTFJMTIONS REQUIRED FOR CONVERGENCE OF PROBLDl 6
COMPUTATION ACCELERATION METHOD
ITERATIONS > 1200
Tnt .
(SEC)
TU@TRAN, Fin. mesh dtqrnatlng Triangular No rebalance
Whole q ystm Bead rebalanm
Pina=wh
with whole systatt rebalance
Mesh
892 66,94 34,32 9 455
q
rebalance Crehl variable = 10
-3
&
rebalance,
71 75 75 a -0.70
7.99 6,90 12,39
Pins aesh
q lternating
rebalance,
with whole system rebalance Crehl,
?L~mesh
*
It@rationodivar8e
1.
WilliaIE H. Raed, Nucl.
Sci.
Eng, 4S, 309 (1971).
‘
Element Selut ion Technique to Neutron 2* T. Ohnishi , “Appltiat ion of Flnita IMffusion ad Tranepor t Equations, ” Proceedings, Conf. on New Developments in Ructor Math~tica and Applications, CONF-71O3O2, Idaho Falls (1971).
Schemes for the Transport 3. Wm. H* Reed, “Triangular Mesh Difference tion, ‘t Los Alamo. Scientific Laboratory report LA-4769(1971).
Equa-
of p~aQ4, W. ~. Miller, Jr., E. E. Lewis, and E. C. Reesow, ~lThe Appli~ation Spaco Finite Ehmmta to the Two-Dimensional Transport Equation in X-Y Geomatry, ‘t oubmitted to Nucl. Sci. Eng. for publication.
5, K, D. Lathrop qnd F. W. Brinkley, ‘fTheory and Use TlKM3tAN Program, ” Los Alamo@ Scientific Laboratory
60
of the
report
C.@neral-Geometry LA-4432(1970).
E W. H. Ihed,‘tTha f fectivcneoa
Method ~ in Transport
of
Theory, ” Nucl.
Acceleration Techniques for Iterativ@ Sci. Eng. 4S, 24S (4971).
& -.
‘# ..