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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

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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 .

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• 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

Sfwlrj iwftMM j 9 . . 0

i

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0, L1.2 vl„

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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 0

. " 5

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

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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-