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Solution of the multigroup neutron diffusion equations by the finite element method
Misfeldt, I.
Publication date:1975
Document VersionPublisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):Misfeldt, I. (1975). Solution of the multigroup neutron diffusion equations by the finite element method. RisøNational Laboratory. Risø-M No. 1809
A. E.K.Risø Risø-M-GI 09
*0
.8 CC
Titte and author(s)
Solution of the Muitigroup Neutron Diff\
Equations by the Finite Element Method
by
s; o:
Ib Misfeldt
13 pages + tables 4- illustrations
Date 2}th July ^975
Department or group
Department of fieactor Technology
Groups own registration number(s)
i
Abstract
The finite element method is applied to the
multigroup neutron diffusion equations. General
formulas are given for Lagrange type elements.
Description of some 2D- and one JO computer code is
given. Several investigations are described and the
results are presented. The capability of the 3D
code is demonstrated on the 3D IAEA Benchmark prob
lem. Suggestions for further work are given.
Copies to
Abstract to
Available on request from the Library of the Danish Atomic Energy Commission (Atomenergikommissionens Bibliotek), Riss, Roskilde, Denmark. Telephone; (OS) 35 51 01, ext, 334, telex; 5072.
ism 87 550 0351 6
COSTBHTS
Page
1. Short about the Element Hethod 2
1.1 General 2
1.2 Element Interpolation Functions .................. k
2. The Developed Programs 7
2 2.1 FBU 7
2.2 Problems to be solved before the Extension to 3D.. 7
2.J FEME 7
3. Investigations carried through 8
3*1 First Investigation 9
3.2 Second Investigation 10
3.3 Third Investigation 10
3.4 Comparative Study on the way 11
4. Future Work 11
References 13
Appendix A, 2D IAEA Benchaark problem
Appendix B, 3D IAEA Benchmark problem
2nd order FEM calculation
Appendix C 3D IAEA Benchaark problem
3rd order FEN calculation
- 2 -
SHORT ABOUT THE ELEMENT METHOD
l.t General
I have »forked on the solution of the mult igroup diffusion
theory equation by the finite element method.
The inhomogeneous equation for one energy group is
-yCDCrjvfCrj) - M,L}$(I.' ' ZiL] = ° in •-' (-' '•'
and
D*£* XT * T^£-^r_; * qCjr) = 0 (2)
on the boundary T of Ii.
In the finite element method the domain ,* is divided into a e
number of subdomains, each corresponding tc an element, e.
Within each element the flux, $(£_) is approximated by a
polynomial, ©(r_). The coefficients in the polynomial are connected
with the flux or the derivative cf the flux, at corner points and
other fixed points (called nodes or flux-points) in the element.
The flux in these fixed points (tderivative) is collected in a vector,
<£ and the element interpolation function, ^J.r) is defined by the T ce
equation 0(r> = £ (r) • £ .
A characteristic for the element interpolation function is
M.(r.) = 1 and !».(r.) = 0, j * i where r. is the flux-point with flux i—i 1 "i "-1
(or derivative of the flux) corresponding to o..
It is shown (ref. 1) that the one group diffusion equation (1)
i« the Euler equation for the functional
/ < • ^•7 = n \ (D rv»r tvM • srr - *¥^
• i/2 I <yf2 • 2$*> d r . (3)
iS3rw^iB
The term S(r_) gives the coupling between the groups.
In variational calculation it is shown that the best approxi-
sation to the solution of the equations (1) + (2) within a given
class of functions is the function which minimiaes the functional
(3)> Vhith the finite eleaent approximation this minimum requirement
leads to the equations
H A • F = 0 (4)
(B) = Z (He> r , s = 1 , 2 .
(F)r . E (Fe) r r = 1, 2, ..., M
e
the summations run over all elements in Q, K is the total number of
nodes (flux-points),
ie . De J (yNT)T (vs 1 ) da • E° J Cjr HT)da + y" J
!* - - JH Sda + ie I 2. c V
( H M T ) d r ,
-Q' tf
S = total number of groups
T is the external boundary for the element e. This is shown
is details in ref. 2.
To solve the equations CO on* must know the element interp
olation function, £(r_). When the interpolation function is fixed
the integrals can be evaluated and the system of linear equations
isA-a,-«i-,-..
- •» -
(*0 can be solved*
1.2 Element interpolation functions
X have decided only to consider element interpolation function
of Lagrange type. They use only the flux and not the derivatives as
parameters, whicb for me seems more reasonable than involving derivates
leading to difficulties at comerpointu between materials with dif
ferent diffusion coefficients, though :he difficulties have been
treated (e.g. in ref* 6), it is only possible for rectangular mesh
and I did not want to exclude the improvement obtainable by triangular
mesh.
Fig. 1 shows Xdgrange• triangles and rectangles of dimension
s s 1, 2, and 3.
p 1
? 1 *
O <l I
m • •*
i t • • 1
I h — H l II——<
II II II 1
Ik i o 1
flux-point, nod«
rig. 1. Lagrange triangles and rsctanglss of order
1, 2, and 3,
I?:;'-*-1 - 5 -
If we consider a triangle in the barycentric coordinate syste
defined v.zt fig. 2, we can easily show that fKV., L~. L_) for a
Lagrange triangle of order a ia defined by
t p q r
1 p.q,r 1* 2' 3 P ^ T s = 0 u 0 u, 0
s+t+u _(s) c(t) „(u) Ts .t u
S C j ) is the Stirlingnuaber of the 1st kind.
;= P. s (£-, , —) is the fixed fluxpoint corresponding to N, , i m a m x the point with the flux &,* This is shown in details in ref. 2,
For a rectangle of order • the used coordinate sy s tea5 is
natural coordinates as defined on fig. 3-
The interpolation function N_ (j• «J ) is defined by
"i • V , <)• l> - y t > • ',«>
where
with
» K S w Æ r s ^ i'te'= .
v»>
a,p
(-;)-' " p»V»-p)f
S ( > 1 > + P B+1 *
these interpolat ion
« . p «IP '
functions the
) j
r 1
equation C) i s eas i ly set up.
1 P«M'^P!WW».tWMw' 'J'" '.y»j«"wp'k W*^.«pf*ttl l Ijpiipil^lSI
- 6 -
( 1 , V L3)
area P.. P£ P
hatched area total area
analogous for I»2 and L
Fig* 2* Barycentric coordinates.
P, = (1* 1) in natural coordinates
analogous for P1, P, and P.
Fig* ?. Natural coordinates
^Bte S"S ~£*
- 7 -
2. THE DEVELOPED PROGRAMS
During ay work with the solution of the aultigroup diffusion
theory equation I have developed several programs. In the following
chapter I will give a short description of the aoet important Teraions.
8.1 FEHA
«' The prograa uses Lagrange triangles of arbitrary order, the
4- peup equations are solved by direct methods taking adTantages of
the sparsity in the matrices involved.
, This method seeaed very effective until the number of unknowns
exceeded 1000 then the execution tiaes rose drastically, in fact I
could not exceed HOC unknowns in a night's calculation.
The prograa is described in details in ref. 2.
2.2 Problems to be solved before the exter»ton to 3D.
< . The number of unknowns for a realistic 3D calculation was
expected to be 10 000 - 30 000 or even more, therefore it was clear
that new solution techniques had to be found.
An other problem was the element shape, if I continued with
the triangular shape (tetrahedron) in three dimensions the input-
SjV, .preparation would become very complicated unless the program could
automatically divide box-formed meshes into tetrahedrons.
A third problem was the very large matrices that arise fro«
the finite-element approximation in 3D calculations, as an exaaple
toe matrix (H) for a 3rd order calculation with about 10 000 un
knowns per group contains about 10 elements f 0 per group.
a<3 nam
The solution of the problems was carried through in 2 diaenaions, leading to the program FEME.
The direct solution of the group equations was replaced by a
pointvise succeesive overrelaxation. This method was found to be
superior to the direct solution in most cases, only in very small
unrealistic examples, the direct solution was faster. I can refer to
• calculation which I and O.K. Kristiansen perforasd on a Oeraan
- 8 -
Benchmark problem, representing a reactor where one control rod was
accidentally removed, leading to a very high local reactivity. On
this exaaple the second eigenvalue was very near 1 which made it
practically iapossible for the direct method to converge (more than
1000 outer iterations were needed even though extrapolation was used).
With a good overrelaxation factor the iterative solution converged
on less than 100 outer iterations with one inner per outer iteration.
If I perforaed many (5 - 20) inner iterations per outer the same
difficulty as with the direct solution arose.
X decided to use boxformed elements in the 3D program instead
of tetrahedrons, this leads to a loss in flexibility, but the program
became much easier to use.
The rectangular elements were examined in two dimensions and
I found about the same accuracy as if the rectangles were divided
into two triangles.
The third problem, the large matrices, was handled in the
following manner. I did naturally take full advantage of the sparsity
of the matrices, furthermore their elements are stored on backing
storage and accessed absolutely sequentially during the iterations,
but during the set up of the matrices I access the element unordered.
This did run very well even for very large 2D problems and on the
basis of this 2D program I decided to develop the 3D version.
During the 3D calculations I soon ran into problems because
of the large matrices, during the set up of the matrices. I did
solve these problems in a rather untraditional way, but on computers
with more backing storage these problems could easily be solveu in
a aore efficient way.
3. INVESTIGATIONS CARRIED THROUGH
The goal for my investigations with finite element methods
was, if possible, to develop a fast and accurate 3D-fluxcalculation
program.
There had already been done a lot of work in this field with
2D-codee, for example the investigations described in ref. 3, ref. 5
and ref. 6. What I missed in these investigations was realistic
- 9 -
examples on light water reactors, which mainly is the field in which
we are working at Rise. An other lack was flux comparisons, I do
not believe that a high precision on the eigenvalue necessarily
, swans a high precision on the calculated flux.
^»1 First investigation
The first investigation, carried through with FBMA, is de
scribed in details in ref. 2. The goal was to complete the inves
tigations found in the literature with respect to our interests*
tin basis on this we should decide whether or not we would work on
a 3D prograa.
I only examined the accuracy of the solution as function of
•eshsize for a rectangular division because of the expected diffi
culties with tetrahedrons in a 3D calculation.
The investigation included 2 realistic examples, a 2 group
calculation on the aidplane of the Connecticut Yankee reactor and a
2D version of the 3D IAEA Benchmark. Further I included 3 smaller
examples with varying discontinuities caused by different diffusion
coefficients In core and reflector.
The result of this investigation was.
1) On "nice" problems (small difference in diffusion coefficients)
the high order finite element approximations provide a more accurate
solution within the same execution time« but with increasing dio-
eonuities the accuracy of the high order methods decrease faster
than for the low order methods (FDT as well as FEM 1st order).
2) The realistic problems seemed to be "nice" enough for the high
order methods to converge faster (as function of meshsize) than the
low order methods when considering calculation times, but the im
provements were less than expected at the beginning of the investi
gation.
All in all I did from my first investigation conclude that
the finite element method does not constitute an alternative to the
fast nodal methods, but the method is superior to the low order FDT pro
grams with respect to calculation time when high precision is wanted.
- 10 -
3.2 Second investigation
The next phase in my work was the development of the program
FEMB as described in chapter 2.
With this program I performed some calculations on planes in
the 3D IAEA Benchmark problem in order to estimate the precision of
a calculation on the 3D IAEA Benchmark problem.
One of the considered planes was the core midplane, where I
naturally used the 2D IAEA specification as published in ref. 7.
As reference solution I used an extrapolated FDT solution performed
by G.K. Kristiansen. The results from this calculation are described
in appendix A*
This investigation showed that my first investigation was too
optimistic in its conclusions considering FEM, but still the FEM
program seemed to be a good deal faster than the FDT programs, so I
continued the work with the 3D version based on the prograa FEMB.
3.3 Third investigation
After the development and testing of the 3D program 1 had
planned some calculations on the 3D IAEA Benchmark problem. The
goal of these calculations was to provide a reliable reference sol
ution to the problem. In order to get a good estimate of the errors
in the flux-distribution I had performed the described 2D calculations*
The largest calculation on the 3D IAEA Benchmark problem is
described in appendix B, it is a calculation with 16 x 16 x 13 >eeh
and a second order approximation. The mesh used in the directions
is chosen so that the max. error in the planes considered (in 2D cal
culations), was about 3«5 % (relative to ^ in the group), accord
ing to the 2D calculations I assume the max. error to be less than
5% on the 3D calculation described in appendix B.
this calculation is until now the most accurate calculation
performed on this problem. Naturally there will be some doubt about
this result, because the program is new, including risk for errors,
and the method is not too well known considering accuracy for re
alistic examples, especialy not in three dimensions. I hope that a
large FDF calculation with coaparable accuracy will soon be available
and support my results and estimations.
I performed two further calculations on the problem, a 2nd
order and a 3rd order coarse aesh calculation, the result from the
3rd order calculation is shown in appendix C.
3.1* Comparative study on the way
A coaparative study of different aethod, FDT-cornerpoint,
FDT-midtpoint and FSH is at the aoacnt carried out bj Mr. G.K. Kri
stiansen, Jtis*. The results of this work will soon be available.
1. FUTURE WORK
In the future work with FBI progress we will concentrate on
prograas suitable for 3&-overall calculations.
Until now we hare in no way taken advantage of iaproveaents,
which can be achieved by the flexible division of the reactor with
a triangular aesh. The gain obtained by this seeas to be quite large
(see f.ex. ref. 3 and ref. 5 ) .
Ve have planned to develop a 3D-code using prisa-foraed el-
eaents (triangles in the xy-plane, rectangles in the xz~ and the ya-
alane). This should at the expense of soae annual work be able to
reduce the calculation tiae.
The last part of the work with the 3D-codes is planned to be
ah optiaisation of the code.
There are several ways which can lead to a faster code, I
will aention soae.
Most of the calculation tiae is spent during the iterations,
bare the solving of the equations and the calculation of the right
•tie (?) takes almost all the tiae. The calculation of £ can be
accelerated sacrificing soae generality (scattering froa all groups
to all groups, fission neutrons born in all groups). If the order of
approximation is fixed, a lot of calculation can be saved, I have
tried this for 1st order approxiaation, leading to a aaving of about
80ft cpa-tiae. The gain will be less in higher ordsr calculations, I
expect about 50« for 2nd order calculation etc.
An other way to accelerate the calculation of £, la at the ex
pense of »or« storage, here the gain should be about 50*, but at the
aoaent this would not be advaisable at our computer.
- 12 -
The total nuaber of iterations sight be reduced by introduction
of acre efficient iteration techniques. But as long as the siaple
pointvise successive overrelaxation is able to finish within about
100 iterations, it will probably be difficult to find a auch faster
aethod.
- 13 -
REFERENCES
1. Garrett Birkhoff, The Numerical Solution of Eliptic
Equations, SIAM, Philadelphia.
2. I. Kisfeldt, Kaster thesis.
J. L.A. Semanza, E.E. Lewis and E.C. Rossow, The Application
of the Finite Element Method to the Multigroup Neutron
Diffusion Equation.
Nuclear Science and Engineering: *f7, 502-310 (1972)
k. O.C. Zienkiewics, The Finite Element Method in Engineering
Science.
5. H.G. Kaper, G.K. Leaf and A.J. Lindeman, A Timing Comparison
Study for some High Order Finite Element Approximation Pro
cedures and a Low Order Finite Difference Approximation
Procedure for the Numerical Solution of the Kultigroup Neutron
Diffusion Equation.
Nuclear Science and Engineering: V), 27-^8 (1972)*
6» C M . Kan g and K.F. Hansen, Finite Eleaent Methods for
Reactor Analysis.
Nuclear Science and Engineering: 51. ^56-495 (1973).
7* M.R. Wagner, Current Trends in Multidimensional Static
fieactor Calculations.
Proceedings of the Conference on Computational Methods in
Nuclear Engineering, April 15-1?, 1975. Session I.
9^^§l^^tl»?" h
APPENDIX A
RP-3-75 Danish Atonic Energy Commission k j u n e 1975
Research Establishment Bise IM/rj
Department of Reactor Technology NEACRP-L-I3S
Reactor Physics Section
2D IAEA Benchmark problem
by
lb Hisfeldt
This is an internal r.port. It may contain results or conclusions that
are only preliminary and should therefore be treated accordingly. It
is not to be reproduced nor quoted in publications or forwarded to
persona unauthorized to receive it.
1 -
In ccTine-ctton with the calculation on the JD Benchmark problea we mvc performed soaic i D-calculat ion:;- The -'U proble« i s the one specified m »-ef. 1 t'ron which fig* T i s copied.
To get n reasonably accurate independent reference solution Mr. O.K. Kristiansen has performed a series of FBT-ealculations and from them obtained a solution by Richardson-extrapolation in each power assembly.
Table * ."howo the results fros these calculations. In table 1 are further hown sone calculations with KERB, a twodinen.sion.il f in i te eienent ftux calculation progra* using a rectangular mer>h. From ref. T are taken two calculations with FEhVD, a f in i t e eU-nent flux calculation progran using a triangular Bech. The error« shown in .able 1 are defined:
j * Hnx ( lPi ' P i . r g r l ) „ ,00 r i i tr«*r
l p - p A
c*«*•« H—LiI£lL >• 1 0 ° i Max,re f
(P. in the a:-.:;f*mbly power, and i donotcf* the asscnbly)
Tile error in th.- flux, defined a,-:
"Htf-^.r.r l £ , B,,x (_LJL L > « too
r A* T a.-uc,r«-r
(i dpnotoi; th<: f.fKici-point, r, in ;: ftroun-index) if. t.ypicnl ty found 1.0 be twice .in bitf an £ . in the power.
In Fir.. •' the error m; function or the eesh i s r.hovn for the diffrrrnl. en I <:u I ,'it i on:;.
's^>e*»'W!2***[T-
SÉ>iuj$i,Virfik™'i»- -
Solatioa Ho
1 teedii
2 "
3 k
5 6 ? ran 8
9 10 FEH2D
11 "
12 FEW
13
description
17 x 17
Th x J".
68 x 63
1J6 * 136
170 X 170
exetrapolatcd value
i . order 9 x 9 (19 x
2. order 18 x 18 (37
?. order 36 x 36 (73
19)*
x 3?)* x 73)"
2. order 182 flu poicts 1/8 core
2. order €06 fluxpoints 1/8 core
3. order 9 x 9 (28 x
3- order 13 x 18 (55
28)*
.95)*
B 67C0 cpu-tiae
H ai
7.1 " 30 »
110
11.5
*.4 -
23-3 »
98.P »
18.9 "
90.6 "
TVSDIM is a FDT floxcalculation prograa using corner aesh points.
FTUB ie a finite-eleænt flux calculation prograa using Lagrange inter
polation in a rectangular meet.
FBM 2D ie a finite eleaent flux calculation prograa using Lagrange
interpolation in a triangular aesh. See ref. 1.
* 9 x 9 (19 x 19) aeans 9 aesh in each direction and 19 fluxpoints
because of the second order aproxiaation etc.
- J -
REFERENCES
Proceedings o f t h e Conference on C0HPØTATIOIAL METHODS IS
NØCLEAR ENGINEERING
Apri l 15 - 1 7 , 1975
SESSION I
M.B. WAGNER: Current Trends in H t t l t i d i . e n a i o n a l S t a t i c Heactor
C a l c u l a t i o n s .
f'atvci'r- a! vanioil/ init'Utl tod htcffftbt/ III
• W K M W M M M e « mat
*mr Octwt: H$m iu leuwtt Law Octet: fotl isnsUr HtBttffcitiei
tteadinr {Mittens
Crttrat Swieirit« j j t . 0
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i
1 1 M*
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0, L1.2 vl„
1 1.5 2 1.5 3 1.5 « 2.0 5 2.0
o.» 0.* a.« 0.3. 0.3
0.02 0.02 0.02 0.0t 0.0*
0.01 0.01 0.01 0 0
0.18 0.085 0.13 0.01 0.055
0.135 0.135 0.135 0 0
F M I I Fwl2 Fwl 2 • M feflistar toll. . Bet
Xi " '•" ' X) • 0.0 > v t „ > 0 i l l regie«
b U i 29 IAEA ItKki^k PreUw rsprawrt« »i<!j!ir,: 1 . ISO '.= vith coutnt 2 •*
wi l l butillnj C" • 0.0 a 10 for i l l rrjlmi »«* Mar« growl
FiO.1 33 IAEA Bintli-iri Prjbl« SpcciflcitiM
method program po in ts s o l . n o .
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5-1
064
C35
» 1
' 3 4
- 6 5
342
' 7 8
o ? i
976
7 0 '
' 9 1
»o«
908
853
476
685
60S
5 9 *
' . 7
0 . 7
FEHB
j i x 2 3
' 2
- 5 -• 2 0 7
• - 3 -
• • 9 1
c ' O
9 3 '
9»0
777
' 4 ' 4
• - 6 0
1299
' 0 6 0
' 036
»77
"55
1*5!
' 331
' • 7 3
•072
984
7 2 *
•182
959
907
366
- 7 6
684
621
60?
4 . 2
2 . 0
.5*55
' 4 3
•305
•«> ' 2 0 7
» 1 0
435
»37
760
' • 5 0
1-75
' 3 "
' 068
'03?
»53
7 4 1
•465
• • 4 2
' ' 7 3
1071
978
6»8
1 1 9 1
>66
907
850
- 7 1
686
601
,38
0 . 9
o.»
2D IAEA BENCHMARK PROBLEM
Coflparison of finite element- and
finite difference-methods.
Average assembly powers*
APPENDIX B
Danish Atomic Energy Commission BP-ff-75
Research Establishment Bise k j u n e 1075
Department of Reactor Technology IM/rj
Reactor Physics Section NEACRP-L-l'ti
3D IAEA Benchmarkproblem
2nd order FEM calculation
by
lb Misfeldt
This is an internal report. It may contain results or conclusions that
ara onlyereliminary and should therefore be treated accordingly. It
it not to be reproduced nor quoted in publications or forward** to
"arsons unauthorized to receive it.
Ct.uj -am- FEM JD
Oi'i;:mi r.otjon Risø. DK.
3D Benchnark Problem
Result Scheme 1. Pare 1
(to be filled in by type-writer)
1. Name of participant: lb Hisfeldt
Organisation; Danish Atomic Energy Commission
Address: Research Establishment Rise
SK-'tOOO Roskilde
Telephone: (33) 355101 Telex: ">31'6
2. Computer code name: FEM 3D
Description (max 20 lines):
FEK 3D is a three-dimensional finite-element flux calculation
programme. The programme uses box-formed elements with Lagrange
interpolation. The order of the interpolation might be 1., 2. or
3. order. The programme uses an ordinary power iteration technique
with one inner iteration per outer. The group equations are solved
by SOS and the iterations are accelerated by extrapolation.
References:
None so far.
Cede nnnc FEM 3D
Organisation Rise. DK.
3D Benchmark Problem
Resul t Scheme 1, Pape 2
(to be filled in by type-writer)
3* Calculation description (simplifications, noshes, iterations«
problems, etc.) (max. 15 lines):
This solution is performed with 2. order interpolation. The
aesh size is 16 x 16 x 13 (33 x 33 x 27 flux points) and 2 energy
groups are used.
The result scheme is filled in after 7. iterations and the
local error is assumed to be leas than 0,1% of $ in each group.
%,• Computer type: Burroughs 6700
Calculation timet 23 hours cpu
Computer speed relative to one or two well-known computer types:
1 hour on CDC 6600 *v 10 hours on B 6?00.
An extra page kk with information may be added if necessary.
The precision of this calculation
It is possible to estimate the precision of this 3D-calculation
from some 2D-calculntiono.
These 2D-calculate one are described in HP-3-75 (ref. 1). It is
seea that the accuracy of a calculation with the mesh used in the
3D-calculation is about 1.7* on the assembly power in the xy-plane, this
corresponds to about 3-5* in the flux in each group.
X have also performed 2D-calculatioae in the plane y = ^0 and
found that the mesh used in the 3D-calculation in the ^-direction
gives about the same accuracy as described above.
Prom these considerations I will estimate the absolute error
to be less than 3% is the whole reactor.
A PDT-calculation with the same accuracy in the flux is in
ref. 1 seen to need about 100 x 100 mesh in the xy-plane. I will
assume that the same accuracy can be obtained with about 100 mesh
in the ^-direction with a nonuniform mesh« where the meshsise around
the core-reflector boundaries is not greater than in the xy-plane.
A PDT-calculation with comparable accuracy will therefore
need about 10 mesh.
Also from ref. 1 it is seen that considerable improvements can
be obtained with the use of triangular elements in the xy-plane,
the same accuracy is obtained with almost doubled meshsize.
Ref. 1t BP-3-75.
Orc.-itii. . t i o*\ Risø. DK
^0 Benchmark Problem
eff '
Writerf*. undo? r e l a t i v e to ff average,core
core
•idplane
10 cm'6 up
the partial ly Inserted
SMttrol absorber
Core bottom
Power
peak
C
X
80
8o 100
120
0
1.0
80
140
150
0
0
1J0
32
o-ordinat cm
y
40
80
80
40
0
0
40
0
0
0
40
56
32
CO
z
190
190
190
190
290
290
290
'90
290
20
20
178
174
Relativt
* 1
7.743
2.846
4.499
6.326
2.656
4.151
3.815 2.504
1.271
O.613
0.925
3.291
9.805
fluyoE
<Pt
1.816
0.641
1.108
1.568
0.444
0.973
0.895 0.649
0.941
0.370
0.659
2.426 . ,
2.298
Relative fluxes and pos i t ion , where <f has i t c maxiaun in the t o r e '
APPENDIX C
Danish Atomic Energy Commission
Research Establishment Rise
Department of Reactor Technology
Reactor Physics Section
V75
HP-5-75
1 J ane
IK/r.j
:IEA"R:-.-1 J9
3D IAEA Benchmarkproblem
3rd order FEM calculation
by
lb Misfeldt
This is an internal report. It may contain results or conclucions
that are only preliminary and should therefore be treated accordingly.
It is not to be reproduced nor quoted in publications or forwarded
to persons unauthorized to receive it.
Cui<: ..c.-.- FEH 3D Oi (-,..!• i ;o t :ort Ris*. DK.
5B U-..-J.-3-..-*: !•-!,•-len
Be.".^_ •*•-'•—x 1. P r» 1
( to he f i l l e d in by type-writer)
1 . tese of particip.-.nt: Ib Kisfeldt
. føganiKation: Danish Ator.ic Energy Cceoaiseion
ådåresr: Research Establishsent Rise
, Mt-'tOOO Bos::ilae
; »tlephone: (03) 355101 Telex: *5H6
2* CoBpuU-.- code nar.s: FER 3D Baacription (at* £0 line.-.):
FEH 3D i s a threc-dircnsiontl finite-element flux calculation
programme. The progracre uses hox-formcd elements with Lagrange
interpolation. The order of the intorpolatioi: s ight be 1 . , 2 . or
V ordor. The prograr.sie ures an ordinary power i teration technique
tfith one inner i terat ion rer outer* The group equations are solved
fcy 30H and the i terat ions are accelerated by extrapolation.
Seferoiicon;
Stone FO fe
Code ti«««* FBI 3D
Orc^nisulioii Bis* DK
jSi Benchmark ProPlen-
Result Scheme 1, Pare 2
(to l*e filled in by t/ji-j-vriter)
3. Calculation description (siaplificationn, meshes, iterot*oD&,
piableas, etc«) {mix, 15 lines):
This calculation is perforated with 3rd order interpolation.
The M s h used was 9 1 9 x 4 (28 x 28 x 13 flax point*). The eolation
is not yet sufficiently converged so the local error is large, but
lev« than 5*.
The aeah in the n direction is too coarse so the precision
will not be as high as predicted in RP-3-75 (6£) for the xy-plane.
These results do mostly serve the purpose t3 prove that 3rd
order rat calculation is possible on a realistic 3D-probiem.
k, Computer type: Burroughs 6700
Calculation time! 9-5 hours
Computer speed relative to one or two well-known computer types:
1 hour en CSC £600 rv 10 hours on B 6700*
An extra page AH witb information nay We added if necessary.
Code name FEM 3D
O r c i i e n t i o n R i s e ØK
3D Benchmark P r o b ' e i r
R e s u l t s S c h e - e g .k ,_ ond f l ' i x p o i n t e
k , , = 1.028? e f f
Write (0 and<£ relative to tf. *?, average, core
c o r t
• i&plane
10 C B ' S up
tbe p a r t i a l l y
I n s e r t e d
e e p i r c ' i a b s o r b e r
Core bo t tom
Ponor
peak
C o - o r d i n a t
»
80
80
100
120
0
40
80
140
150
0
0
130
34
cm
'
t o
80
80
40
0
0
40
0
0
0
40
56
34
en
* 193
Vfr
* » •
W -
¥#-
2 9 3
-&96-
-Me-29)
-S*>-
20
10
175
175
R e l a t i v t
<?!
7 .626
3 .899
4 . 3 9 5
6 .289
2 . 7 6 0
4 . 2 4 6
3-924
2 .624
1.339
692
1.092
3 .229
9 .481
' I v x e s
f?
1.796
0 . 6 0 2
1.095
1.541
0 .426
0 . 9 8 8
0 . 9 1 4
0 .646
0 .984
0 . 1 5 0
0 .346
2 . 3 8 9 ,
2 . 2 5 0 - >
Relativ« fluxes and position, where<$^ han its maximum in tbe core''
fsfw^fr^^^f^sp^^^m-
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