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TITLE: THE

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FNERGY DEPENDENT FINITE ELEMENT !WT,”:3DFOR

TWO-EIMENSIONA1. DIFFUSION PROBLEMS I

HiclekoKomoriyaEBASCO Services, Inc.New York, New York 10006

Wallace F. WaltersLos Alamos Scientific Laboratory

TO: ANS Mathematics and CornputatlonsTopical Meeting

~[~YCh 28-30, 1977Tucson, Arizona

By nccepturwe of this arlicle for puhllcntion, thepublishrr rrwngnizes the Governmmt’s (lic~nse) rlghtoIn nny copyright und the Government md Its nuthorinwlrepresentutlveu hnve unrestricted rlgh:toreproducelnwhole or in part oald article under any copyrightoem.med by the puMIYher.

The I,os Altimos Hrlentiflc Lnbwrat(m rrqucsls thnt thepubli~her identify this nrticle FIIIwork performed undertheauupiccsoflhe USHRDA.

Ianmss6i*ntific [a bermtory

of the Unlvorsity of California

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BLANK PAGE

. . .. ... ... .., -., . .. ... . ,..,

THE ENERGY DEPENDENT FINITE ELEMENT METHOD FOR

TWO-DINENS1ONAL DIFFUSION PROBLEMS*

Hldeko KomoriyaEBASCO Services, Inc.

New York, New York 10006

and

Wallace F. WaltersTheoretical Division

University of CaliforniaLos Alamos, New Mexico 87545

Send proofs to Wallace F. WaltersP. O. BOX 1663, MS-269Los Alamos Scientific LaboratoryLos Alamos, New Mexico 87545

CONTRIBUTED PAPER——

Mathematics and ComputationsDivision Topical MeetingMarch 28-30Tucson, Arizona

17 pages3 tables5 figures

,-

.,.,

*Work performed under the auspices of thn U. S. Energy Rcsenrch andDevelopment Administration.

THE ENERGY DEPENDENT FINITE ELEMENT METHOD FOR

TWO-DIMENSIONAL DIFFUSION PROBLEMS*

Hideku KomariyaEBASCO Services, Inc.

New York, New York 10006

and

Wallace F. WaltersTheoretical Division

University of CaliforniaLos Alamos, New Mexico 87545

ABSTRAC”r

Tileeffectiveness of the cuergy dependent finite element method

(EDFEM) as applied to two-dimensional mul~igroup diff(,sion problems

is investi~:atcd. The EDFliMcouples the finite element method (FEM)

formalism with the energy dependent. element size scheme. The EDFEM

allows the elements to straddle m:ltcrial interfaces if certain condi-

tions are satisfied; this makes this methwd especially suitable for

heterogeneous reactor calculntlons. Comaprisons of the results obtained

by the KDFEM, the l?EM,and the finite difference method for a ZION-I

PWR mo4el arc?prcsentecl. A sifinlflcant reduction of the total number

of unknowns involved in the problem is accomplished using the EDFEM

which ylclds n rcductinn of the computing time by 30%.

—— —.*Work performed under the auspices of the U. S. Energy Research andDevelopment AdminsitratLon.

-2-

1. INTRODUCTION

The numerical solutian of the neutron

frequently required for reactor analysis.

diffusion cquuttoc is

Consequently, there are

continui~g efforts ttireduce the computer running tins required for

these solutions wh%le maintaining the degree of accuracy neec!edin

nuclear reactor analysis. Specifically, finite element methods

(FEM) have been investigated as a potential ❑ethod for reducing

1-5computation time. Since the FEM has been obsarved to yield a

high order of convergence for practical mesh sizes, the FEM is ideally

suited for coarse-mesh calculations.

Energy dependent spa~ial mesh scht.meshave also been investigated

6-8as a method for reducing computing time. In these schemes, fine-

mesh arrangements arc used in the lower energy neutron groups while

relatively coarse meshes arc used in the fast groups. This scaling

is used because the neutron diffllsion length, a characteristic scale

length, is generally smallest in the lower cllcrgyp,roups ilrid largest

in the fast groups. In this fashion the ratio of mesh size to group

diffusion length may be maintained roughly constant from group to

Energy deprndent spatial mesh schemes using either the FEM(3,9

group.

6,7or the finite difference method (l?DM) have been developed for one-

dimensional diffusion problcins. Results obtained usinR rne ener~y

dependent finite element method (EDFi?M)as applied to one-dimensional

prob.lcms in Rcfs. 8 and 9 demonstrated that reduced computing times

could be obtained without sacrificj.ng accur;lcy using this new method.

Tl]ispaper reports the results obtained by applying the method to a

I

-3-

2pair of two-dimensional problems. The two-group

using Lagrange pieccwise biquadratic functions as

equati.vns are solved

a basis.

1.1. MATHEMATICAL FORMJLATION

The FEM is an extension of Rayleigh-Ritz-Galcrkin methods for ap-

proximation solutions to problems involving differential equations.

Instead of solving the differential equatien directly this method casts

the equation in a variational or weak form. The functional for a given

equation is not unique. The determination of the associated functional

for a given problem is the first step is gcne:ating the soiution. The

basic formalism of the EDFEM is identical to that of the FEM.

Considered the generalized m;ltrix representation of the multigroup

diffusion equations,

ksT@-V*WO +x@ =~;

and of the multigroup adjoint equation

for G groups where,

0=12 G

Col(l$”,$ , ....@ )

J = Col (l$il,0*2, ...SJG)

D= diag (D1,D2, .... DG)

z = diag (X1,X2, .... xc)

12x

G=Col (x ,x , ....)()

(1)

(2)

direct neutron flux

adjoint neutron flux

the diffusion coefficients

group removal and scatteringcross sections

the prompt neutron ~pectrum

-4-

S m c~l (s1, S2, .... SG)

k

The Dlrichlet condition is

@+(r),O(r) =

where a$leis the

the product of fl.ssioncrosssection aridnumber of neutronper fission

neutron multiplication constant.

The associated

O for rca~e

external boundary of domain !2.

functional is

tT@T/k)@ +Vo “Dw]dn,

or

a (~+T@1 - b($+T@)/k = J.

The functional,Eq. (4),is not a quadratic functional but a billne.ar

functional. The bilinear forms arc

(3)

(4)

(5)

(6)

(7)

The weak form of the problem is now an

-tand QT have the same properties. ~ and @8,

x H~(fl)G times. H~is the Sobolev space of

eigenvalue problem. Both @

where H = H:(Q) x H~(!’2)x ...

functloos which vanish on

2!2 and whose first derivatives are square integrable ic.0.

in terms of trial functions ~,~1 Eq. (5) is rewritten

a(i+T,;) - b(~:Ti)/k = :,

-5-

wheru

N

Q=z

$iui, (

i+l

N;$ =

I

;t ~i“

I

i=1

The Ui are the basis functions often called the element functions and

the expansion functions ii are the noljalvalues of the fluxes.

The variation of ~ with respect to @-tT

is required to vanish. Then

a(Ui,;) – b(uii)/k = O, i=l,2,0 ... N. (11)

Therefore the function uhicllminimizes the functional, Eq. (4) is also

a solution of the weak form, Eq. (11). After the discretization

proceduru is perfolrned, Eq. (1) yic’lds the mat:ix equation

fig~g= Q8 (12)

for each energy group g where Ag is given by N x N matrix Q in an N x 1

matri~ and Ox is a column vector for N space point unknowns.

Each element Agij

of Ag is obtained from

j = i, ifi, i<l;

(9)

(lo)

(13)

and Q1 from

(14)

h=.1

-6-

The rationale bchinti the EDFEM is to keep the size of A minimal

as an alternative methnd of reducing the computing time. In the

steady state probLem, the ~oefficicnt matrix A for each energy group

needs be assembled and inverted only once. However, for time

dependent cases, the computing time rcqutred to assemble and Invert

A(t) for each time step becomes a major concern.

The characteristic features of matrix A are that it is symmetric,

positive definite, diagonally dominant, and highly banded. Any one

or more of these characteristics can be used advant:ngeously to speed

up the numerical computation. The Choleski direct inversion method

is used here.

Conventionally, the FEM maintains the element sizes the same in

all energy groups so that the spatial arrangerneut of nodal points is

identical for all groups. Elements arc refined uniformly through all

energy groups. The philosophy of the EDFEM is to refine rm.ly those

energy groups whose characteristic

with the diffusion length of other

requires inner products of element

diffusion length arc small comparerl

energy groups. Since the EDFEM

functions whose supports Q. and1

fkjarc not equal, the larger element edge size is alwzys an integer

multiple of the smaller one for simplicity.

The solution is obtained by first solving the fast-group equatlan

on its coarse mesh. Tllc thermal gro[lp term in tilefast source is handled

in a straight forward manner. The fast-group flux is represented by

biquadratic polynomials in each clement, and therefore quadratic

interpolation can he used to obtain fast group fluxes at spatial. points

corres~o[ding to tileflncr thermal mesh nodes. These values are then

-7-

used to generate new fast elements having the same support as the thermal

elements. This representation is used for fast group fluxes which appear

in the thermal group source.

The treatment of element ~izcs is clearly an important difference

between FEM and the EDFEM. The EDFE also allows clement boundaries to

cross more material interfaces than does the FEM. However, element

boundaries must lie on the mat~rial interfaces when adjacent diffusion

coefficients differ drastically. In the fast groups. jumps in the

diffusion coefficient are generally smaller from reglo.~ to region while

in ~llethermal groups this is not Lhe case. Since an energy dependent

mesh is used, the coarse fast elements may encompass several materi,ll

interfaces while the finer thermal elements do not. The solution then

does not possess a continuous first derivative inside the coarse fast

element.

Assume that a finite element subspace S!!:11of degree k - 1 is used

with the forms a(u,v), b(u,v) given in Eq. (11). If the coefficients

in the multigroup Eq. (1) are smooth, then10

11$-011 ~~hk-ss 1

and

(k -2k-2

~) < C2h .

Since the smoothness conJition is violated in heterogeneous elements,

tlmse estimates for order of convergence do not hold.

-8-

Wll~n the fast elcmt?nt is lnllt)mo~rmeuug, the dlf fusl.t]ncoc[ficicnts

and otllezcross sections differ wiLhin the elornent, errors are intro-

duced ~nto the computed fast flux. Since the multigroup equations

are coupled this error propngatcs into all other group fluxes. It

is expected however, that the accuracy of the sc)lution will not be

severely degraded when the di.scontinuities arc small. Therefore, the

placement of fast group elements requires a careful study of problem

configuration.

III. RESULTS AND ll”fSCl!SSION

The two thermal reactor proble:ns investigated arc (i) a simple two

L-eg”io.l model consisting of a core aud reflector,end (ii) an inllomogencous

model of the ZION-I PNR core. Quadratic T,agrange polynomials art?selected

for basis functions. The EDFX cumputer program was US(:U for numerical

experimentation on the Burroughs 6700 computer.

The basic data for the first problem is given in Table I. This

reactor configuration was examined in the x-y domain shown in Fig. 1.

This problem is a two-dimensional extension of the problem discussed

in Ref. 8. Notice the existence of a singular poirit,a so called “corner

singularity”.

The L/N 1-1 designation refers to the ccis~ in which the fast and

thermal group clement cd~es are of equal length. The L/N 1-2 case is the

refinement applied only to the thermal group who~e element edges are

one hali the size of those in the fast group.

-9-

An impor~nnt observation can be nadr from FiEs. 2a acd 2h. TileL/4

1-2 thermal flux rli::tributionagree~ well with the L/8 1-1 thermal flux

distribution. In addition, the L/4 1-2 flux profile exhibits better

afircementwith the L/8 1-1 flux profile than dues the L/6 1-1 profile.

It is important to rccognizc the total number of unknowns involved in

these runs. There are 320 unknowns involved in the 1,/41-2 computation

whereas 518 unk[lownsarc involved in t!le 1./8l-l computation; accordingly,

this reduction of nearly 38% in the Lotal number of unknowns leads to

a reduction of approximately 33% computing time required to set up the

problem by the EDFE proRram. Eigenvallie r~sllltsar~~jnc!ic;trcdillTahlP lIT.

The ZION-I PWR model represents a more stringent numerlcaL test if

the EDFEN bcczusd of i~s hetcrozeneity khich 1s rxpected to dictate the

slow/fast mesh ratio rather than a simple, ratio of djffusion lengths.

The model contains sc:veral core r~’flc”ctorcGrnrrs wilirh intrmlucc “strong”

singularity.cs in the flux. In all cases fzst croup elements include

several fuel rcglons. The model cnnfl.guration is indicated in Fig. 3,

and the cross sections :ippear in Table 111.

Tilethermal neutron flux alon~ the core center line is shown in

Fig. 4. The L/4 1-1 case consists of a 4 x 4 element arrfingement in

both groups. Here L is the half width of the reactor. The <ntervals

on the abscissa correspond to fuel enrichment zones with the cxccption

of the last interval which is the water reflector. The near perfect

agreement between t!lL? thermal flux shapes of the L/4 1-2 run and

L/8 1-1 run is evident in Fig. 4. The value of keff corresponding to

the L/8 1-1 run is 1.275 02; the L/4 1-2 valve is 1.274 53, and the

T./41-1 value is 1.275 29. These can be compared to a benchmark value

-1o-

of 1.275 08 resultlng from a fine mesh Citation run with 5 624 unknowns/

group.

The fas~ flux shapes obtained with the L/4 1-1, L/4 1-2, and 1./8

1-1 runs are observed to be “smooth” , as compared to the thermal flux,

and in good agreement with one another. This indicates that the L/4

refinement is sufficient for the fast group. The L/4 1-1 therranlgroup

flux plot of Fig. 4 indicates th.~tthe refinement is not sufficiently

fine for the thermal group. Notice that even though the flux eigen-

function of the L/4 1-1 run is not highly accurate the value of’keff

is. This is a result of the variational principal ember.iicdin the FEM.

For the L/8 1-1 run there are 256 unknowns in each group while

foz the L/4 1-2 run there are 256 unknowns in the thermal group and

64 in the fast group. It is observed that a reductLon in total ex-

ecution time of 35X is achieved in the EDF!IL/4 1-2 run as compared

to the conventional L/8 1-1 run. These comments refer to the use of

direct inversion. When inner iterctiuns are used, as they are for

problems with a.large number of unknowns, the time savings will be

7even greater.

REFERENCES

1. L. O. Deppe and K. F. Hanson, “Application of th~ Ffnitc ElementMethod to Two-Dimensior.al Diffusion Problurn,” Nucl. Sic. Eng. 54,456 (1974).

—

2. Chang hfuKang and K. F. Hanson, “Finite Element Ilethods EorReactor Analysis, “ Nus!,. Sci. Eng. 51, 456 (1973).—

3. H. G. Kaper, G. K. Lezf, i~ndA. J. Lindeman, “The Use of Intcr-pol,atc)ryPolynomials for n Finite Element Solution of the Multi-group Neutron Diffur,ion Equation, “ Trans, fun.Nucl. Sot. 1.5,298 (1972).--

-11-

4. L. A. Semanza, E. E. Lewis, end E. C. Rossow, “The Applicat:.onof the Finite Element Method to the Multigroup Neutron DiffusionEquations,” Nut]. Sic, Eng. 47, 302 (1972).

5. W. F. Walters and Glenn D. Miller, “Quadratic Finite Element fors-y and r-z Geometries,” I?roc.Cornp.Methods inNucl. Eng. ~, 39 (1975).

6. M. NateLson, J. B. Yasinsky, and D. Rampolla, “An Energy DependentSpatial Mesh Approximation for Static Few-Croup Diffusion Equations,”Trans. Am. Nucl. Sot. I-6, 143 (1973).

7. J. B. Yasinsky and M. Natclson, “An Energy Dependent Spatial MeshApproximation for Neutron Group Diffusion l?roblema,” WAPD-TM-1062,Bettis Atomic Power Laboratory (1973).

8. W. F. Walters and Hideko Komoriya, “An Energy-Dependent FiniteElement Methods for Few-Group Diffusion Equations,” Trans. Am.Nucl. SOC. 21_, 224 (1975).

9. Hiieko Kcmoriya, Ph.D. thesis, University of T)elaware (1976).

10. G. J. Fix, “Eigenvalue Approximation by the Finite Element Method,”Adv. Math. 10, 300 (1973).

TABJ4E I

Two-Group F:acroscopic Cross Sections

Fast G~oUP

The.mal Group

Fuel

1.5I)

0.4

r CI.L1623@

0.2

0.0

0.218

0.06

ileflector

1.2

0.15

~,~ol

0.02

O*G

0.0

5.1

‘1=1.0 X 108 cm,isec

v2=2.2 X 105 cmlsec

TABLE II

Eigenvalues l/Keff: Two-Dir.ensional, ~o-Region, Tuo-Group Problem.

Qt!adraticLasrange(l-1)

1.09230613

1.108521721.12225374

1.11341751

(Benchmrk I/Kef==1.1140943 for m=2, A=L/6).*

Quadrakie ~~adra~ic Linear* Cubic*LaSr2nge(l-2) Lti9ran5e(1-4) Eermite(m=l) He.~Ate(r2)

1.lZ752~16 1.12232815 1.0802150 1.1082321

1.11360767 1.0962251 1.1134916

1.11359092 1.1040456 1.1140943. “

..-- .-

Finite*Difference

i.0703013

1.0797120

L.ok195577

1.1105021

●See 2eS. 2.

TABLE III

. . .._

Macroscopic Czcw.s Sections

Grcup 1

Group 2

CmpOsition

3

11

12

13

15

D (cm)

1.02130o.3354a

1.4i7600.37335

1-419700.37379

1.425500.374i4

1.455400.28994

q(s) Vrf!cn‘~) &##) (k+k+l)

0.03322 @.o@.14595 0.2

0.0

0.92597 g.~~53fi0.06659 ‘ 0.1C433

0.01742

0.32576 C.0J5CIo.076C15 c.~2472 0.0i694

0.02560 0.3C5530.05359 9.14120

0.01658

0.02950 0.00.00949 0.0

0.02903”

Comme3ts

Zion I (CoreBaffle)

Zicn I(2.25%)

Zion T.(2.a3)

z~o~ ~

(3.3%)

Zion I(Waser]

$=0L

o

Reflector

.—— .. ——. —

1“1.c!1

— —. -,. L/2

--.--

‘;.=o

,=40. Ocm

#

2.0

1.8

1.6

1.4

1.21

1 1 I

o 10’.0 20 -~ 3C.O 43.3

L1.0-,w0.8 , n

0.6

0.4 1

I.

0.2

\

1+=LZ4 1-1

0.0 n=Lz4 1-29=L~6 l-i

~“0.0 10.0 20.0 20.0 4D.O

~ (~~] x (CM)

(a) (b)

Fig. 2. Relati~..c?’Wrml Fluxes (a) at y=O.Ocm and (b) at y=20. Ocm:

~fi,o-DinmcnS~onal, ~wo-Regi@n, ‘lqfi-o-Grou~Problcm.

.

1.2

I--- ..—.p-4. -—------

1II! ;I , 1!

——- .. —.-. .

I“1? II 12

I- .— — —.

I*II 12 II

I-——....—,.1,,

–“;i- .’_t!’11 12 II

t-‘1..—.—112II12-i———-----

-it’1113III —. —’1313I

[y.-=.&:—_-u------

d+ /(1).. .. --—

1-—-.“T!1 II.- -.- —..- ..

-._. ,--_-

-j ~

-.— -—

12’ 12-—

II l?!-—— -.

13 13—. —--....

.!:..ii-—.4

i:-

rfJ

—..-. —- —- .— -

IT

II—.. — -.. — .,. —.

12 : II I3 :!------ —, ..- -_. .

II I ii i IS~~11

---—.-—i,

12 III

1~ ii

1!.— .——

I—. ;

II Idii 17II,4,—-- -..-.:1

+’l? i---””13 I

●-13 13

1-----—., -----II ‘3

113 [,——---

15

+=0

Vj:n!l,clfy-— -

..

.

]t

o 19/4 l-ln 1,,/4 1.-2. . L,~c 1-J

\q iJ .

I ~J—— J ——-L 10—— 1‘—-‘---- ~;J

●