NEACRP-L- 2''

Application of the Finite' Element Method to the Three-Dimensional

Neutron Diffusion Equation

Paper submitted to the NEACRP 21th Meeting

Tokai, November 1978 *

Takeharu ISE, Toshio YAMAZAKI* and Yasuaki N A U W R A

Japan Atomic Energy Research Institute

* Sumitomo Heavy Industries, Ltd.

Application of the F in i t e Element Method t o the Three-Dimensional

Neutron Diffusion Equation

Takeharu ISE*, Toshio YAMAZAKI** and Yasuaki NAKAtlARA

Tokai Research Establishment, Japan Atomic Energy Research I n s t i t u t e

Tokai-mura, Naga-gun, Ibaraki-ken, Japan

Abstract

The f i n i t e element method (FEM) was applied t o t he solut ion of three-

dimensional neutron diffusion equation i n order t o get a p r o f i t from the

geometrical f l e x i b i l i t y of the FEM.

To circumvent the computer l imi ta t ions a r i s ing from the three-dimen-

s ional problem, newly developed program (FEM-BABEL) has been equipped with -

the following features: Combination of prism- and box-formed f i n i t e

elements, the SOR method applied successively t o x-y planar un i t , accelera-

t ions of inner i t e r a t i ons by the coarse mesh rebalancing technique and of

outer i t e r a t i ons by the SOR extrapolation and the auxi l iary program produc-

ing mesh network.

Comparisons a r e made with the conventional f i n i t e difference CITATION

by solving an ana ly t ica l solvable problem, a coupled reactor and a pres-

surized water reactor . Results have shown t h a t the present f i n i t e element

method has advantages over t he f i n i t e difference method f o r solving r e a l i s t i c -

three-dimensional problems i n view of preciseness and computing cost .

* Present address: Japan Nuclear Ship Development Agency, Torano-mon,

Minato-ku, Tokyo

** Sumitomo Heavy Industr ies , Ltd., Ohte-machi, Chiyoda-ku, Tokyo

1. Introduction

The finite element method (FEM)') has recently been shown to offer

attractive solution procedure for neutron diffusion2%') and transport

problems2,8*11) as well as structure analysis.12) One of the most

advantages is that FEM can fit precisely to any complicated geometry

(geometrical flexibility). We therefore have investigated the applicability

of FEM to the numerical solution of the multigroup neutron diffusion equa-

tion for the three-dimensional reactor configuration.

Three-dimensional diffusion calculation is the most practical way to

a take into account all the essentitl features of realistic nuclear design

problems. Among various kinds of three-dimensional elements like tetra-

hedron, hexahedron, quadrilateral prism, triangular prism (prism shape),

rectangular prism (box shape) and so on,13) the prism- or box-shaped element

is considered to be most appropriate for describe the reactor geometry from - the viewpoint of preciseness and computer economy in balance. This is a

reason why we adopt FEM having the prism- and box-shaped elements in order

to analyze a practical problem.

~owever'one must pay two penalties for these advantages; One is that

0 a million unknowns may be needed.to solve accurately the system equations

for realistic three-dimensional problems. Moreover the coefficient matrices

have sparse and irregular structure. Another is that the program user

must specify a large amount of input data inherent in FEM. Thus in our

newly developed program (FEM- BABEL)^) we have adopted the iterative method

(successive over-relaxation method)14) for paying the former penalty and

we have produced the finite element mesh generator15) as an auxiliary program

for the latter.

The verification and applicability of the program have been demonstrated

by solving the exact problem, a coupled core reactor and a pressurized

"

water reactor i n comparison with the conventional f i n i t e dif ference cal-

culations (CITATION). 16)

2. Solution of the Neutron Diffusion Equation by the F in i t e Element

Method

I n the general multigroup formalism, the neutron d i f fus ion equation

i s represented by a coupled system of d i f f e r e n t i a l equations on t h e sca la r

f lux 4 ,

where the notation is conventional.

By denoting the external boundary of a domain R by aeR, Eq. (1) should

s a t i s f y t he general form of boundary condition-

where a/an representsthe outward normal der ivat ive a t aeR with u n i t vector

n. I n applying the f i n i t e element method, the domain R is assumed the union ., ,/

of a f i n i t e number N of contiguous small subdomain R i which is ca l led the

f i n i t e element hereaf ter ,

I f the subdomain possesses the in te r face a i R , then Eq. (1) requires t he

usual in te r face conditions with respect t o neutron f l u x and current:

a 4 $g(r) and (Dg -) ( r ) a r e continuous across aiR. an (3 )

Now t o solve the generalized eigenvalue problem described by Eqs. (1)

- (3) fo r Keff and the corresponding pos i t ive eigenfunction { ~ ~ ( r ) } , we

adopt the usual outer i t e r a t i o n procedure. A t each s t ep of t h i s outer

i t e r a t i on , we thus solve G uncoupled sel f -adjoint e l l i p t i c boundary value

problems of the matrix form (n: i t e r a t ed index):

where

D = diag [Dg],

. X = col { x g ,

s = col {vXf 1 , , g

The conservation of the self-adjoint character provides t he Galerkin-type

approximation procedure f o r the numerical so lu t ion of t he e l l i p t i c boundary

value problem.p)

Here we define the Galerkin-type approximation t o the present problem.

L t ~ ~ ' ( 0 ) be Sobolev space, the weak form of the o r ig ina l problem is given by

where

Let MN be any finite-dimensional subspace w21(R), then our aim is t o solve

A the approximate problem having a unique solut ion 4 i n MN,

a(;,+) = @,+) f o r a l l +Cr) G MN . (6)

* ..

In order t o uniquely determine + cMN, we have chosen l i nea r Lagrange

polynomials which we take the reference points a t the nodes of g r id over

t he subdomain. Since w e associate one degree of freedom with each node,

the dimension of the finite-dimensional subspace MN is precisely N or the

t o t a l number of nodes i n R.

We now associate with each node (global index i ) i n R a ba s i s u i which

possesses the minimum support. That is, u i vanishes outside the union of

the f i n i t e element t o which the i- th node belongs. Futhermore u i takes the

a value 1 a t the i - th node and the value 0 a t a l l other nodes within its

support. Then t h e so lu t ion 4 o f f t h e approximate problem Eq. (6) w i l l be

of the form

- where {ui} is the bas i s of the space MN t o be constructed and the coef f ic ien t

, Iqgi} represents the value of $ a t t he node. I n other words, we have t he

iden t i ty :

Our goal i s now t o determine the basis U i n terms of the nodal parameter

q (q is the generalized coordinate i n t he f i n i t e element terminology).

Substi tuting Eq. (7) i n to Eq. (6) and taking successively each of

the basis functions of MN for +, we obtain a l i nea r system of equations:

from which the unknown nodal parameter q can be solved. This approximate

approach is commonly referred t o as the f i n i t e element method.

We now consider a pa r t i t i on 52 in to prism- and box-shaped subdomains. -

The union of these subdomains i s a rb i t ra ry except t he r e s t r i c t i o n tha t

every portion of any material in te r face l i e s on an edge of a subdomain.

It i s observed t h a t the b i l i nea r form a(ui,uj) is non-zero only i f the

subdomain of u i and uj possesses a t l e a s t one subdomain i n common, Denote

by rij the s e t of indices (y) of these subdomains (Ty) which belong t o t he

intersect ion of the subdomains of ui and u Then we obtain j.

where aY(ui,'uj) is the in tegra l of Eq. (6) over t he subdomain Ty . Using

Eq. ( lo) , we obtain the following equation on the subdomain Ty correspond-

ing t o Eq. (7): ,

3. Description of the Computer Program

Since we have chosen prism- and box-shaped elements, the basis function

may be par t i t ioned in to ax ia l and planar components:

I

u(x,y,z) = f(z).u(x,y) . (12)

Then the Galerkin approximation derived i n the previous sect ion leads t o

the system equation to be solved as follows:

[H]gI@F = {Slg , for g = 1, 2, ..., G ,

in. which

and y is the superscript which denotes the prism- or box-shaped element.

The [QIY and [BJY are the integrals over finite elements:

Accordingly the three-dimensional [91Y and [61Y may be expressed by using

the two-dimensional ones as follows,

@ Let I and I1 be the planar indices over elements as shown in Fig. 1.

Then the matrices contained in Eq. (13) are written by the submatrices

having planar indices as follows:

9

9

and

where

In addition, QI, PI, BI and DI are expressed by two-dimensional submatrices:

Y where QY (2) and B~(~) are two-dimensional Q and BY, respectively. In

other words, QY(~) and By(') correspond to triangular and rectangular

elements which have already shown in the appendix of Reference 7 for detail

or in Reference 3.

Our ultimate object is to solve Eq. (13). Since the global matrix

[H I possesses the tridiagonal structure which is composed of planar sub-

matrices, Eq. (13) is solved by the successive over-relaxation method

(SOR)14) in which the inner iterations are successively performed within

t he core memory f o r each planar layer. This planar-like SOR w i l l give a

l a rge help from the viewpoint of l e s s storage l imita t ions t o computers.

The inner i t e r a t i ons a r e accelerated by means of the coarse mesh

rebalancing technique17) and the outer i t e ra t ions by means of the extrapola-

t i on based on SOR. The program is a l l wri t ten i n the FORTRAN-IV language

implementing on the FACOM 230175 operating system, equipped with the f r e e

f i e l d FIDO format, the complete r e s t a r t i ng procedure, the automatic mesh

generator b u i l t i n fo r a regular mesh network, and the auxi l iary mesh

generator with half of the graphic display f o r a pressurized water reactor.

4. Numerical Tests and Results

In t h i s section, we present the r e s u l t s of a detai led comparison study

of the three-dimensional f i n i t e element calculations f o r t he numerical

solut ion of the neutron diffusion equation t o show the r e l i a b i l i t y of the -

computer program FEN-BABEL. The comparisons a r e made for preciseness and

computing cost of the eigenvalues and the eigenfuctions between the f i n i t e

element program FEN-BABEL and the f i n i t e difference program CITATION. For

t h i s purpose-the numerical t e s t s r e f e r t o the three d i f fe ren t types of

e reactor configurations: a homogeneous cubic reactor problem which is

exactly solvable (an exact probl~m), a simplified two-zone reactor with two ./ *

energy groups (a coupled reactor problem), and a l i t t l e modified IAEA

three-dimensional benchmark problem (a pressurized water reactor problem).

Material constants f o r the t e s t problems a re referred t o Reference 4 f o r

both the exact problem and the coupled reactcr problem and t o Reference 6

f o r the pressurized water reactor problem.

To ver i fy the computer program, i t w i l l be best t o deal with t he

analyt ical ly solvable problem f o r a homogeneous cubic reactor i n a two-

energy-group model. The reactor configuration is i l l u s t r a t e d i n Fig. 2.

For program check of FEM-BABEL, we have used both t h i s exact solution and

the f i n i t e difference solution. Tables 1 and 2 show tha t the present

f i n i t e element program gives precise solutions f o r both eigenvalues (Keff)

and eigenfunctions (fluxes). It is noted tha t the exact eigenvalue l i e s

between the f i n i t e element and the f i n i t e dif ference solutions. It is

shown i n Table 3 t ha t t he f i n i t e element calculations have the advantage

of computing cost over t he f i n i t e difference calculations.

As shown i n Fig. 3, t he coupled reactor configuration w i l l provide

saddle shaped neutron f lux dis t r ibut ions . Such f l u x behaviors a r e sub-

s t a n t i a l l y d i f fe ren t from those i n another reactors. Since t h i s configura-

t i on i s symmetric with respect t o the two diagonals ( the 45-deg symmetry),

the present f i n i t e element program can take advantage of t h i s symmetry by

use of both prism- and box-shaped elements. Table 4 shows tha t the f i n i t e

element solutions a r e i n an excellent agreement with the reference value -

(from experience f o r the exact problem we take a s the reference value the

average of the two numerical solutions f o r the mesh s i z e of 2.5 cm). It

is shown i n Fig. 4 t ha t both the numerical solut ions a r e i n a good agree-

ment on the +lux d is t r ibu t ion . Table 5 shows t h a t the f i n i t e element cal-

e culations have an advantage of computing cost over the f i n i t e difference

a l so f o r a coupled reactor. ,,

Final ly we discuss the efficiency of the program f o r a r e a l sca le

pressurized water reactor i l l u s t r a t e d i n Fig. 5. This problem comes from

a s l i g h t modification of the IAEA water reactor problem for reason of con-

sistency t o the boundary condition of the f i n i t e difference calculation.

That is , discussion is performed on the octant reactor configuration i n the

f i n i t e element calculat ions but on the quarter reactor configuration i n

the f i n i t e difference calculations. The mesh s i z e a r e taken 5 cm on x-y

plane and 10cm i n Z direction.

Figures 6 and 7 i l l u s t r a t e comparisons of rad ia l power d i s t r ibu t ions

and ax ia l power d i s t r ibu t ions , respectively. It is l i k e l y tha t the f i n i t e

element solutions show a be t t e r performance than the f i n i t e difference

solutions especially near the core-reflector interface. Table 6 l ists

comparison of outer i t e r a t i v e performance and Table 7 comparison of the

solut ion techniques. The r e su l t s show tha t the f i n i t e element solutions

take advantage over the f i n i t e difference solutions about the i t e r a t i v e

performance and the storage requirements, but disadvantage about the

computing time because of difference between the solution techniques on -

data processing.

5. Conclusions

A d Through both the ana ly t ica l and numerical-, i t may be con-

cluded that the three-dimensional f i n i t e element calculations a r e acceptable

from viewpoints of preciseness and computing cost. It i s seen tha t the

i t e r a t i v e method adopted i s effect ive, too. It is noted tha t i n the

present t e s t s we do not use the coarse mesh rebalancing accelerations b u i l t

i n t he progiam ( i t is reported tha t the acceleration is very e f fec t ive f o r

a p rac t ica l By using t h i s technique the computing time w i l l

be reduced t o some extent compared t o t he present calculation. Now an 2'

improvement t o the i t e r a t i v e method is t r i ed by one i n our laboratory.

The auxi l iary program on f i n i t e element mesh generation already developed

by us w i l l give the users a help t o prepare a ia rge amount of data.

References

1 ) Strang G. and Fix G . J . : F i n i t e Element Method," Prentice-

Hall (1971)

2) Ohnishi T.: "Application of the F in i t e Element Solution Techniques t o

Neutron Diffusion and Transport Equations," CONF-710302, Vol.11, 273

(1971)

3) Semenza L.A., LewTs E.E. and Rossow E.C.: "The appl icat ion of the

f i n i t e element method t o t he multigroup neutron d i f fus ion equation,"

Nucl. Sci , Eng., 47, 302 C19.72)

41 Kaper H.G., Leaf G.K, and Lindeman A.J.: "A timing comparison study

fo r some high order f i n i t e element approximation procedures and a

low order f i n i t e di f ference approximation procedures f o r the numerical

so lu t ion of t he multigroup neutron dif fusion equation," Nucl. Sci.

0 Eng., 2, 27 U972)

5) Kang C.M. and Hansen K.F.: "Fini te element methods f o r space-time

reactor analysis," Nucl. Sci., Eng., 2, 456 (1973)

6) Misfeldt I,: "Solution of the Multigroup Neutron Diffusion Equations

by the F in i t e Element Method," RISO-M-1809 (1975) -

7) I s e T., Yamazaki T. and Nakahara Y.: "FEM-BABEL, a Computer Program

f o r Solving Three-Dimensional Neutron Diffusion Equation by the F i n i t e

Element Method." JAERI-1256 (1978)

8) Reed W.Y., H i l l T.R., Brinkley F.W. and Lathrop K.D.: "TRIPLET: a

Two-Dimensional, Multigroup, Triangular Mesh, Planar Geometry, Expl ic i t

Transport Code," LA-5428-MS (1973)

9) Lewis E.E., Miller Jr. W.F. and Henry T.P.: "A two-dimensional f i n i t e

element method f o r i n t eg ra l neutron transport calculations," Nucl. Sci.

Eng., 58, 203 (1975)

10) Martin W.R. and Duderstadt 3.3.: "Fini te element solut ions of t he

neutron transport equation with applications t o strong heterogeneit ies,"

Nucl. Sci. Eng., 62, 371 (1977)

11) Fujimura T., Tsutsui T., Horikami K., Nakahara Y. and Ohnishi T.:

I I Application of f i n i t e element method to two-dimensional multi-group

neutron transport equation i n cy l indr ica l geometry," J. Nucl. Sci. -::

Technol., 14, 541 (1977)

12) Zienkiewicz O.C.: "The F i n i t e Element Method i n Engineering and

Science," McGraw-Hill (1971)

13) Desai C.S. and Abel J.F.: "Introduction t o the F i n i t e Element Method,"

Van Nostrand (_1•÷72)

14) Varga R.A. : "Matrix I t e r a t i v e Analysis ,"Prent ice~Hall (1962)

15) I s e T. and Yamazaki T.: "LOOM-P, a F i n i t e Element Mesh Generator

Program with On-Line Graphic Display," JAERI-M 7119 (1977)

0 16) Fowler T.B., Vondy D.R. and Cunningham G.W.: "Nuclear Reactor Core

Analysis Code: CITATION," Om-TM-2496, Rev. 2 (1971)

17) Freolich R.: "A Theoretical Foundation f o r Coarse Mesh Variational

Techniques," Intern. Conf. Research Reactor Ut i l i za t ion and Reactor

Mathematics, Mexico, 219 (1967)

18) Freolich R.: "Flux Synthesis ~ethod; versus Difference Approximation

Methods f o r t he Eff ic ient Determination of Neutron Flux Dist r ibut ions

i n Fast and Thermal Reactors," IAEA-SM-154114 i n "Numerical Reactor

Calculat~ons," IAEA C1972)

Table 1 Comparison of the multiplication factors betweenlthe numerical and

analytical solutions for the exact problem

I I I Mesh size (cm) I I l Numerical

2 5 10 I solution

-~ ~

1.335506

solution

~ ~

FEM-

BABEL

Kef f

Quarter '..

Octant

Relative error

CITATION

Kef f

-

1.33537

Kef f Relative error

1,33562

Relative error

-

0.010 %

0.009%

1.33478

1.33473

1.33623

0.054 %

0.058 %

0.054%

-

1.33211

-

0.25 %

1.33842 0.22%

Table 2 Comparison of t h e second energy group f l u x 42(x,0,0)

at t h e mesh s i z e of 5 cm f o r the exact problem

Analytical, 1 1.5643 1 1.4497 1 1.1932

FEM- BABEL

I Quar te r 1 1.5644 1 1.4497 1 1.1932

Table 3 Comparison of s to rage requirements and computation times

between two numerical ca lcu la t ions f o r the exact problem

Octant

CITATION

Numerical

method

FEM- Quarter

BABEL Octant

1.5662

1.5641

I CITATION

Mesh s i z e (cm)

1.4493

1.4496

Storage CPU time Storage (words) 1 (see) 1 (words)

1.1929

1.1933

CPU time Storage CPU time (set) 1 (words). 1 (see) I

Table 4 Comparison of multiplication factor between two

numerical solutions for the coupled reactor

problem

Numerical method

FEM-BABEL

* Percents in parentheses show the relative error to the reference value which is the average of the multiplication

factors by both the programs for the mesh size of 2.5 cm.

Mesh size: Ax(cm) xAy(cm) xAz(cm)

CITATION

Table 5 Comparison of computing cost between two numerical ... . , . ....

solutions for the coupled reactor problem

2.5x2.5x2.5

Full

45" symmetry

Numerical

method

0.983884 (O.ll%>

symmetry

2.5x2.5x5.0

-

0.981742 (0.11%) *

I CITATION

5.0x5.0x5.0

0.984248 (0.15%)

-

0.971605 (0.12%)

0.990954 (0.83%)

I

Storage CPU time Storage CPU time Storage CPU time (words) (min.) (words) (min.) (words) (min.)

0.978115 (0.48%)

0.977697 (0.52%)

Mesh size: bx(cm) x Ay(cm) x Az(cm)

2.5~2..5~2.5 - . ~.

2.5X2.5x5.0 - .

5.0X5,0X5.0

Table 6 Comparison of the iterative performance for the IAEA problem

outer iteration C 20th

outer iteration

outer iteration k 1 outer iteration

Kef f

Outer error

Inner error

Kef f

Outer error

Inner error

Keff

Outer error ,,

Inner error

Kef f

Outer error

Inner error

FEM-BABEL

1.02261(0.58%)*)

CITATION

1.02075(0.76%)*)

7.4~10-4

3.4~10~1

1.02416 (0.43%)

2.2~10'4

1.6~10-1

*) Reference value is the CITATION result of Keff = 1.028615,

with outer error of 1. x and inner error of 6. x

at 63th outer iteration.

Table 7 Comparison of the solution techniques and

the computing costs f o r the IAEA problem

I Program s i z e [ 41 kilowords ( 61 kilowords I FEM-BABEL CITATION

Solution method

Acceleration f o r :

outer i t e ra t ions

use only SOR but not coarse mesh rebalanc- ing

Storage requirements and data processing

SOR with adaptive extrapolation

fixed extrapolation by SOR with 6, = 1.7 ,

CPU time a t the same outer e r ror of 0.24%

126 kilowords; with only planar data i n memory

Chebysheu extrapola- . . . ... ,~

t i on

455 kilowords; with a l l data i n core memory

120 min. f o r 28 outer i t e r a t i ons

~

32 min. for 34 outer i t e r a t i ons

Fig. 1

1 l o Prism-shape ehmt ( b) Box-shape element I

-%

i

Local node indices on x-y planar layers f o r both t he f i n i t e

elements

Fig. 2 Reactor configuration f o r t h e exact problem

Core

- Core -

Core I .

. .

. .

. .

Blanket .

~ i g . 3 Coupled r e a c t o r conf igu ra t ion 4 Comparison of fast f l ux d i s t r i b u t i o n s for the coupled r eac to r . . . ~

~.. ~ . ~...~ . ~. ~- problem

Fig. 5 Reactor conf igura t ion of t h e IAEA problem i

4 - e P(x;y =o,z=295.0cm) e % - FEM-BABEL

15 C

Pe-rn CITATION - 0 - F .4 - - E .. e 1.0 - - .. i m > .- - 0 = 0.5 - a 3 : F u e l 2 t k b e r

4 :-Reflector .

Fig. 6 Comparison of r a d i a l power d i s t r i b u t i o n s f o r t h e IAEA problem

Fig. 7

- f [ - CITATION z 0

Distance from the bortom (a) G-ic . ... .~ .~ ...-- -~ ~. ~

. ~ ~

Comparison of axial power distributions for the IAEA I

problem I