FINITE ELEMENT SOLUTION OF A THREE-DIMENSIONAL NONLINEAR REACTOR

DYNAMICS PROBLEM WITH FEEDBACK

Eulogio Conception Bermudes

"ntraTCMNOE SCHOOL

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESISFINITE ELEMENT SOLUTION OF A THREE

DIMENSIONAL NONLINEAR REACTORDYNAMICS PROBLEM WITH FEEDBACK

by

Eulogio Conception Bermudes

December 1976

Thesis AdvisorThesis Advisor

D . SalinasD. H. Nguyen

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Finite Element Solution of a Three-Dimensional Nonlinear ReactorDynamics Problem with Feedback

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This work examines the three-dimensional dynamic responseof a nonlinear fast reactor with temperature-dependent feed-back and delayed neutrons when subjected to uniform and localdisturbances. The finite element method was employed to re-duce the partial differential reactor equation to a system ofordinary differential equations which can be numerically in-

]

tegrated. A program for the numerical solution of large

nn fom"**\J I JAN 73

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SECURITY CLASSIFICATION OF THIS PAOE (When Dot* BnfarW)

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sparse systems of stiff differential equations developed byFranke and based on Gear's method solved the reduced neutrondynamics equation. Although a study of convergence by refining element mesh sizes was not carried out, the crude fi-nite element mesh utilized yielded the correct trend ofneutron dynamic behavior.

DD Form 14731 Jan 73

b/ N U102-014-6601 secuhity classification or this p»cErw«n Dif# fni.rtd)

FINITE ELEMENT SOLUTION OF A THREE-DIMENSIONAL NONLINEAR REACTOR

DYNAMICS PROBLEM WITH FEEDBACK

by

Eulogio Conception BermudesLieutenant, United Spates Navy

B.S., United States Naval Academy, 1970

Submitted in partial fulfillment of therequirements for the degrees of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

and

MECHANICAL ENGINEER

from the

NAVAL POSTGRADUATE SCHOOLDecember 1976

ABSTRACT

This work examines the three-dimensional dynamic response

of a nonlinear fast reactor with temperature-dependent feed-

back and delayed neutrons when subjected to uniform and local

disturbances. The finite element method was employed to re-

duce the partial differential reactor equation to a system of

ordinary differential equations which can be numerically in-

tegrated. A program for the numerical solution of large

sparse systems of stiff differential equations developed by

Franke and based on Gear's method solved the reduced neutron

dynamics equation. Although a study of convergence by refin-

ing element mesh sizes was not carried out, the crude fi-

nite element mesh utilized yielded the correct trend of neu-

tron dynamic behavior.

TABLE OF CONTENTS

I. INTRODUCTION ------ 10

II. THE NUCLEAR REACTOR WITH TEMPERATUREDEPENDENT FEEDBACK AND DELAYED NEUTRONS - - - - 11

A. THE PHYSICAL SYSTEM 11

B. PROMPT AND DELAYED NEUTRONS 14

C. DOPPLER EFFECT AND TEMPERATUREFEEDBACK MODEL 16

D. FIELD EQUATIONS 20

III. FINITE ELEMENT - 23

A. INTRODUCTION 23

B. THE METHOD OF GALERKIN 24

C. THE ELEMENT 26

D. DIVISION OF THE SYSTEM' INTO ELEMENTS - - - - 35

E. COORDINATE TRANSFORMATION 42

F. CONSTRUCTION OF ELEMENT MATRICES 48

G. CONSTRUCTION OF SYSTEM MATRICES ------ 54

IV. NUMERICAL INTEGRATION 60

A. LINE AND AREA INTEGRATION --------- 60

B. NUMBER OF INTEGRATION POINTS 63

V. TEST PROBLEMS AND RESULTS - - - - 68

VI. CONCLUSIONS AND RECOMMENDATIONS - - 85

APPENDIX A - MESH I CONVECTIVITY AND COORDINATES - - 87

APPENDIX B - MESH II CONVECTIVITY AND COORDINATES - - 101

COMPUTER PROGRAM - - 121

LIST OF REFERENCES - ...-159

INITIAL DISTRIBUTION LIST - - - - - - 160

I.

II.

Ill

IV.

V.

VI.

LIST OF TABLES

Physical Coordinates --------------12Coordinates of Local Nodal Points -------33Abscissae and Weight Coefficientsof the Gaussian Quadrature Formula - - - - - - - 61

Numerical Formulas for Triangles -------- 62

Selection of Integration Points for [G..1 - - - 65

Selection of Integration Points for [GG..] - - - 673j 1

J

LIST OF FIGURES

1. Schematic of cylindrical reactor - - - - - - - -13

2. Quadratic triangular prism parent element - - - - 28

3. Definition of area coordinates --------- 29

4. Isoparametric coordinates ------------305. Element classification - - - - - - - - - - - - -34

6. Layers of mesh I ---------------- 36

7. Top nodal plane of the first layerof mesh I, z = 110 cm------------- -38

8. Middle nodal plane of the first layerof mesh I, z = 95 cm ------------- - 39

9. Bottom nodal plane of the first layerof mesh I, z = 80 cm ------------- _^q

10. Local nodal numbering of curved elements - - - -41

11. n£ coordinates in a triangle --------- -45

12. Sample grid used for illustrating OCS ----- -57

13. Neutron flux transient history at threetest points with a uniform perturbationof 10 dollar of reactivity per second - - - - - -70

14. Neutron flux transient history at varioustest points for a local central perturba-tion of 100 dollar/sec of reactivity ----- -71

15. Linear and nonlinear fluxes at (0,0,0)due to a local perturbation of 100 dollar/sec of reactivity at (0,60,40) --- 72

16. Linear and nonlinear fluxes at (60,0,0)due to a local perturbation of 100 dollar/sec of reactivity at (0,60,40) -- -73

17. Linear and nonlinear fluxes at (-60,0,80)due to a local perturbation of 100 dollar/sec at (0,60,40) ----------------74

18.

19.

20.

21 .

22.

23.

24.

25.

26.

Linear and nondue to a 50 dotion at (0,60,40) 75

Linear and nondue to a 50 dotion at (0,60,40) 76

Linear and nondue to a 50 do

Linear and nondue to a 10 do

Linear and nondue to a 10 do

Linear and nondue to a 10 do

inear fluxes at (0,0,0)lar/sec local perturba-

inear fluxes at (60,0,0)lar/sec local perturba-

inear fluxes at (-60,0,80)lar/sec local perturba-

tion at (0,60,40) 77

inear fluxes at (0,0,0)lar/sec local perturba-

tion at (0,60,40) - 78

inear fluxes at (60,0,0)lar/sec local perturba-

tion at (0,60,40) -- 79

tion at (0,60,40)

inear fluxes at (-60,0,80)lar/sec local perturba-

80

Radial flux distribution for the steadystate and 100 dollar/sec local perturbation - - - 81

Axial flux distribution for the steadystate and 100 dollar/ sec local perturbation - - - 82

Early time history of the neutron fluxat core center using mesh I and mesh II 83

ACKNOWLEDGEMENT

The author wishes to thank Professors Dong H. Nguyen and

David Salinas for their moral support and professional guid-

ance throughout this work. Sincere appreciation is also ex-

pressed to Professor Dick Franke for his priceless assistance

in the implementation of the time integration package and to

Professor Giles Cant in for his invaluable counsel on finite

elements. Special thanks is given to Ed Donnellan, Kris

Butler, and Manus Anderson at the NPS Computer Center for

their patience and cooperation in running the long computer

programs required by this work. Their willingness to help

and their professionalism, I am sure, are recognized and

greatly appreciated by NPS students. Finally, I wish to thank

my wife, Carmen, who had served outstandingly as mother and

father to our children during the course of this study.

I. INTRODUCTION

The nuclear reactor with delayed neutrons and tempera-

ture-dependent feedback is a nonlinear system, whose response

to uniform and local disturbances differs greatly from that

of a linear reactor. The prompt temperature feedback model

and one average group of delayed neutrons were incorporated

in this work. Practically all neutron dynamics analysis to-

day deals with two-dimensional geometry, implying a symmetry

in neutron dynamics behavior during transient [1,2]. This

symmetry assumption, however, could be unrealistic in the

safety analysis of nuclear reactors. A unique feature of

this work is the consideration of the three-dimensional depen-

dence of the neutron flux. No symmetry assumptions were im-

posed on the problem. This work investigated neutron dynamics

under uniform and local disturbances in the core. A uniform

initial flux throughout the interior of the reactor was im-

posed. The finite element method (FEM) was employed to solve

this nonlinear initial-boundary value problem. The FEM is

quite effective in handling discontinuous forcing functions

thereby making it particularly suited for examining the ef-

fects of localized perturbations and space-dependent feedbacks

10

II. THE NUCLEAR REACTOR WITH TEMPERATURE

DEPENDENT FEEDBACK AND DELAYED NEUTRONS

A. THE PHYSICAL SYSTEM

The system under consideration is a fast reactor of cy-

lindrical geometry that is composed of two different regions.

The reactor core or region I is cylinderical in shape and is

fueled by U-235. Region II or the reflector region complete-

ly surrounds the core and is composed of U-238. Both regions

were assumed to be homogeneous. Table I lists the physical

properties and geometry for each region. A schematic of the

fast reactor geomenry is shown in figure 1.

Temperature and delayed neutron effects were taken into

account. Also, a one-velocity or one-group model was assumed,

thereby making the velocity independent of spatial or tem-

poral effects. The delayed neutrons were considered only in

region I since region II was assumed to be a non-multiplying

medium. The temperature effects were also assumed to be only

in the core region.

In general form, the one-velocity neutron diffusion equa-

tion is [ 3

]

|N(?,t) = vDv2Na t

- Z vN + S(r,t) (1)

where f = {x ,y ,z)

D = neutron diffusion coefficient

Z = macroscopic absorption cross-sectiona

N(r,t)dV = number of neutrons in a volume element dV at

a point r at time t

11

TABLE I

Physical Constants

SYMBOL DEFINITION

h radius of Region I

R-TT total reactor radius

^H.II

DII

aII

v

h(§)

a

X

height of Region I

total reactor height

neutron speed

core neutron diffusion coefficient

reflector neutron diffusioncoefficient

core neutron absorption crosssection

reflector neutron absorptioncross section

number of neutrons per fission

critical fission cross section

delayed neutron fraction;dollar reactivity

fission energy

modified convection heattransfer coefficient

temperature coefficient

abundance-weighted mean decayconstant

VALUE

60 cm

90 cm

160 cm

220 cm

74.8x10 cm/ sec

0.913 cm

1.200 cm'

-1

-1

0.01401 cm

0.0040 cm"1

2.54

0.005736 cm

0.00642

7.652xl0"12

cal/fission

0.06322

cal/ (cm sec C)

-0.004/uC

0.4350 sec-1

12

X

a. top view

I - coreII - reflector

> x

b. front view

Figure 1. Schematic of cylindrical reactor

13

vDV 2 NdV =

I vN dVa

number of neutrons diffusing into dV perunit time at time t

number of neutrons absorbed in dV perunit time at time t

S(r,t)dV = number of neutrons produced in dV perunit time at time t

The neutron number density is related to the neutron flux by

the expression [4]

4>(r,t) = vN(r,t) (2)

where <|>(r,t) is the flux at time t . The neutron diffu-

sion equation in terms of the flux is depicted by

1 !|<r.t)= DV 2

* - za* + S(r.t) (3)

B. PROMPT AND DELAYED NEUTRONS

The source or production term in equation (3) is composed

of the contributions of the prompt and delayed neutrons. The

majority of the fission neutrons are prompt neutrons that

appear almost instantaneously (within 10" second) on fission

Assuming a fast neutron non-leakage probability of unity, the

prompt neutron source is described by

Sp(r,t) = (l-8)K

oo(?,t) <f>(r,t) (4)

where

K (r.t) = infinite medium multiplication factor

6 = total fraction of delayed neutrons

14

The fraction of delayed neutrons is very small (note

that 6 = 0.00642 from Table I). However, they have a very

significant effect on the reactivity because their mean life-

times are long. Without delayed neutrons, reactor control

would not be possible. These delayed neutrons are born in

the decay by neutron emission of nuclei produced following

the 3-decay of certain fission fragments. For example, the

8 7 8fi6-decay of the fission fragment Br leads to Kr plus a

8 7neutron. Nuclei such as Br whose production in fission

eventually leads to the emission of a delayed neutron are

called delayed neutron precursors [4].

There are six main groups of delayed neutrons. Each

group is classified according to its decay constant. The de

layed neutron source term is portrayed by

6

Ei = l

Sd(r,t) = £ C.(r,t) A. (5)

whereJ. L.

X. = decay constant of the i group+ h

C.(r,t) = density of the i precursor

Assuming that the fission fragments do not migrate appre^

ciable distances and assuming a non-circulating fuel reactor

[3], the precursor density is delineated by

8T" Mi °° a

y li (6)

where 6- is the fraction of delayed neutrons of the i

.th

15

group. The solution to the precursor equation is in terms

of a time integral expressed by

z

C^f.t) = B^y e"X

1(t" t '

, KCD(r i tMr.t)dt (7)

Inserting equation (7) in equation (5) yields the delayed

neutron production term as

V ?' t} = L Vi s

a f e_X

i(t - t,)

Koo(r,t) (})(r,t)dt'

(8)

For convenience the six main groups of delayed neutrons

were considered as one group. This was accomplished by us-

ing the abundance-weighted mean decay constant defined by

6

X -iE ^i Wi = l

This is a reasonable approximation since many of the impor-

tant phenomena of nuclear reactor dynamics can be character-

ized satisfactorily by combining all the emitters or precur-

sors into one, two, or, at most, three effective groups [3].

Replacing X. with the abundance-weighted mean decay con-

stant showed the delayed neutron source term to be

Sd(r,t) = SAZ

a f e"X(t ' t

'

) Koo(r,t)ct.(r,t)dt (10)

C. DOPPLER EFFECT AND TEMPERATURE FEEDBACK MODEL

The Doppler effect in fast reactors is due to the tempera

ture broadening of many closely spaced high-energy resonances

in both the fission and parasitic-absorption cross-sections.

16

These resonances basically mean that as the temperature in-

creases, the number of neutrons that are absorbed increases

and, similarly, the number of fission events increases.

These nonproductive and productive processes compete in a

complicated manner, and the net effect may be either an in-

crease or decrease in reactivity [3]. For most fast reac-

tors, the effect is negative, and it will be so assumed in

this work.

The reactivity change with respect to the fuel tempera-

ture change is modeled by [2]

dK

dT*T-

3/2+ bT"

1+ cT^" 1 ) (11)

where a, b, and c are parameters determined from experimen-

tal or neutronic calculations, m is an integer, and T is

the current fuel temperature. For most fast reactor cores

(ceramic-fueled cores), TdK^/dT is yery close to being con-

stant over a wide range of temperatures. Therefore, it is

assumed that a and c are zero and the Doppler coefficient

is expressed by

dKb = T

dT(12)

The following initial conditions were used

Kj~r>0+

) = r

T(r,0+

)= T

(12a)

(12b)

17

The reactivity model is then expressed by

Kw (r,t) = K° + b £n(f-) (13)

where

= multiplication factor at steady-state

= number of neutrons produced per fission

= original fuel temperatures

= Doppler constant

The definition of K° is00

VZo _

(14)

where

If = critical fission cross-section

c2 = macroscopic absorption cross-section of the

core

The rise in fuel temperature is depicted by

(r,t) = T(r,t) - TQ(r) (15)

This is also described by the integral [5]

(r,t) /" f(t-t') *(r t t)dt' (16)

where f(t-t') is the feedback kernel and is dependent upon

the type of temperature feedback model used. ^(r,t) is the

18

rise in neutron flux above the steady state and is expressed

by

*(r,t) = 4>(?,t) -<J>

Q(r) (17)

where 4>(r) is the neutron flux at steady state.

There are three types of temperature feedback models that

can be considered here. The first is known as "Newton's

feedback" which determines the reactor temperature by Newton's

law of cooling. The second temperature feedback model is

called the adiabatic feedback model. This represents the

temperature for the loss of coolant case. The third model is

the prompt temperature feedback model in which the fuel tem-

perature follows the behavior of the neutron flux without de-

lay [5,6]. The prompt feedback model was employed in this

analysis. The feedback kernel for this temperature feedback

model is

f(t-t') = £ 6(t-t') (18)

where K is an energy production operator with units of °C

per unit flux and y , a dimensionless quantity, is related

to the mean time for heat transfer to coolant.

Inserting equation (18) in equation (16) and performing

the integration produced a rise in temperature of

(r,t) = £ *(r.t) (19)

19

From equation (19) the temperature of the fuel is

T(r,t) = S <Kr,t) + T (r) (20)

The ratio of the current temperature to the steady-state tem-

perature is therefore

yT_ (21)

In this work T was considered to be constant throughout

the core. Incorporating equation (21) into equation (13)

gave the reactivity model of

KKjr.t) = K° + b £n [^- #(r,t) + 1] (22)

D. FIELD EQUATIONS

Before establishing the field equations it is desirable

to express equation (3) in terms of the rise in flux. Using

equation (17) in equation (3) and grouping terms yielded the

following diffusion equation:

t

1 1| = [Dv a *-za« +0-e)K

ooza* + exz

ayV^-^Kjiidt']o

+ [DV 2VVo +(1 - 6)K°°ZaV SXS

a 7 e^^^^KJ^t']

(23)

The second bracketed term in equation (23) is identically

equal to zero since it is the steady state portion of the

20

diffusion equation. The rise in neutron flux above its

steady state value is therefore expressed, for the core, by

7 ft= DV^ * V +

< ] -e) KooM

+ 6XZa /

-X(t-t') KJ df (24)

and, for the reflector region, by

iff = DV^ - Zfl* (25)

Inserting the reactivity model into equation (24) yielded

1||. DV 2^ - Z^ (1-3) E

aK>

t

+ (l-6)bZ Un[^- + i]} <j, + 3XK° f e"X ^ t " t, ^dt'

Yo V

+ exzb { f e"X(t " t

'

) [Jln(^- + 1)] iMt'} (26)a

./yl

o

For the reflector, the last four terms of equation (26) are

zero. The effects of the temperature on the delayed neutrons

were neglected in this work. The field equations can now be

expressed, for the core, as

|| - vDV 2^ + [vE

a- v(l-B) vZ

f]ifi

+ [-(l-e)vZ b] [iln(£f + 1)]^Y o

+ C-6XvEfv] [f e"

X(t " t, ^df] = (27)

21

and, for the reflector, as

I"!- vDV 2

^ + vE \\>= (28)

The non-linear terms of the core field equation will be

linearized accordingly. In more compact form, equations (27)

and (28) became

113t

clV 2i|; + c2ip + c4[£n(£f +l)3^+c5[/ e"

X ( t-t ' ^dt ' ] =T

(29)

and

dip

| - clV 2i|> + c2ip = (30)

where the meanings of the coefficients cl , c2, etc., are

obvious for the core and reflector.

Equations (29) and (30) were subjected to the following

conditions:

boundary condition:

* R(?

B.t) (3D

where rR

are coordinates of points on the outer surface of

the reflector and the subscript R refers to the reflector

continuity of flux:

^ R(r,t) = *

c(rj,t) (32)

where rT

are coordinates of points on the core-reflector

interface and the subscript c refers to the core.

22

III. FINITE ELEMENT

A. INTRODUCTION

The application of the finite elements of nonlinear con-

tinua has been mostly in the field of solid mechanics.

Prior work [1] using the finite elements of nonlinear con-

tinua on a nonlinear reactor dynamics problem has been suc-

cessful. The FEM in this work was utilized to reduce a non-

linear partial differential equation of the nuclear reactor

to a system of nonlinear ordinary differential equations in

time. The time integration was accomplished by using a com-

puter program for the numerical solution of stiff differen-

tial equations developed by Franke [7]. In order to minimize

computer storage requirements, an optimum compacting scheme

(OCS), described by Ref. 8, was adopted.

The finite element models of operator equations are gen-

erally classified into three categories: a) the variational

finite element models such as the Ritz method, b) the weigh-

ted residuals method such as the method of Galerkin, and c)

the direct finite element models which are not based on

functional minimization. From experience in structural

mechanics, the most effective method for generating accept-

able finite element models of nonlinear equations is the

Galerkin method [1]. This work adopted the method of Galer-

kin in a finite element approximation over the spatial

domain of the field equations.

23

B. THE METHOD OF GALERKIN

The Galerkin method is a special case of the method of

weighted residuals. It involves a rational choice of

weighting function that is consistent with the type of fi-

nite element approximation considered. Indeed, the weight-

ing functions chosen are the basis or shape functions em-

ployed in the finite element approximation. There are two

favorable characteristics of the Galerkin method which

makes it attractive. The first attribute is its amenability

to integration by parts. This supplied the freedom of using

a lower order finite element than might be otherwise possible

The second favorable characteristic of the Galerkin method

is that the symmetric operators in the field equations trans-

form into symmetric matrix operators. Both these attributes

are attractive for computational purposes.

Consider the i ni ti al -boundary-val ue problem

H-l*- '(r.t) (33)

where <£ contains the nonlinear operators. According to

the spatial finite element discretization, the solution of

equation (33) is in the form of an union of an N-term approx

i mat ion given by

R

* (r,t) s i(r,t) = U tyAt) G,(r) , J=1,2,...,N (34)J = l

where N is the number of system degrees of freedom (i.e.,

number of coordinates), and G,(r) are the system or global

basis functions which span the space of the approximate

24

solution ^(r,t) [1]. The Einstein summation is used. The

global basic functions are "pyramid functions", each of

which has a prescribed functional description over a sub-

domain of the system and is zero elsewhere [9]. The unknown

coordinate functions #j(t) are the time-dependent magni-

tudes of the approximated flux $(r,t) and/or its deriva-

tives at discrete nodal points [10].

The residual function, R(r,t) , is defined such that it

is identical to zero when ij;(r,t) is equal to the exact solu

tion. The residual function is expressed by

R(r,t) - $ -j£J - f (35)

The Galerkin orthogonality condition (using the basis func-

tions as weight functions), when applied to the residual func

tion, requires that

JGjR(?,t

Vol

)dVol = , 1 = 1 ,2,. . . ,N (36)

From the field equations, the residual function for the core

is

R(r,t)= || - clV 2^ + c2i + c4[iin(^ + 1)] *

e"^*-*') idf] (37)

and for the reflector

R(r,t) = || - clV 2^ + c2i (38)

25

Using equation (34) and applying Galerkin's orthogonality

condition produced a core equation of

/•Vol

{GJ^J(t)-V 2

GJ(cl -^) j + Gj(c2^)j

+ [£n(^- GKtK

+ 1)] GJ(c4-i|;)

J

+ [r e

-x(t-t-(c5«^)j dt']Gj} = (39)

where I = 1 ,2, . . . ,N

J = 1 ,2,. . . ,N

K = 1 ,2,... ,N

The reflector has a similar equation without the nonlinearity

and is expressed by

J ejcejijtt ) - V2GJ(cl-^)

J+ Gj(c2«^)j} dVol = (40)

Vol

C. THE ELEMENT

A three-dimensional quadratic isoparametric element was

employed in this work. The parent element is a triangular

prism or solid wedge with straight sides. The element shape

functions are expressed in terms of area coordinates in the

plane of the triangle and by an isoparametric coordinate

26

along the prism axis. Figure 2 shows the parent element.

This element was chosen because of the ease with which it

fits the cylindrical structure when it is transformed into

a curved element. This type of element has been used before

as filler elements [11].

The area coordinates are defined by area ratios. Con-

sider the triangle shown in figure 3. An arbitrary point P

within the triangle defines three subareas designated by A..,

A« , and A~. The ratio of each of the subareas to the total

area is known as an area coordinate. In equation form, the

area coordinates are

L1

= A^A

L2

= A2/A

L3

= A3/A

(41a)

(41b)

(41c)

where A is total area of the triangle. L, , Lp, and L-

are the natural coordinates for a triangle. The requirement

that the sum of the subareas be equal to the total area is

obviously satisfied by the identity

Ll

+ L2

+ L3

= ] (42)

In the plane of the triangle, only two of the area coordi-

nates are independent.

The isoparametric coordinates are best visualized by con

sidering the rectangular prism shown in figure 4. Isopara-

metric coordinates are normalized coordinates such that

their values on the faces of the rectangle are + 1. The SnC

axes are in general not orthogonal. They are orthogonal

27

Figure 2. Quadratic triangular prism parent element

28

Y

X

Figure 3. Definition of area coordinates

29

= 1

= 1

e - -l

Figure 4. Isoparametric coordinates

30

only in the special case of a rectangular prism element

[12]. The element basis functions use the ? coordinate

in the prism axis and the L, , L? , and l_

3coordinates in

the plane of the triangle.

The parent element, as shown in figure 2, has 15 local

nodal points around its periphery. As such, there are 15

basis functions which are given below [11]:

Corner nodes: (nodes

Ni

" l Li<

2LrN3

= lL2(2L

2-

h - \ L3(2L 3"

N10

= ?h< 2Lr

14lL

3(2L

3-

, 3, 5, 10, 12, 14)

5) - ^L,(l-t

H) - 2-4U-5

-c) - [l, (h-Z) - \ L

2(l-c

-e) - \ l3ii-5

Midside nodes of rectangles: (nodes 7, 8, 9)

Ny

= L-j (1-C2

)

N8

= L2(l-c 2

)

Ng

= L3

(1 -c

2)

Midside nodes of triangles (nodes 2, 4, 6, 11 , 13, 15)

N2

= 2 Ll L2

N4

= 2L2L3

N6

= 2L3L

1

N n=

2Ll L2

N13

- 2L2L3

N15

- 2L 3Ll

-C)

-C)

-C)

31

The coordinates of each local- node, in terms of !_, , l_

2, L-,

and c are listed in table II. These element basis functions

< N > define the geometry of the element. Note that they

satisfy the relationship

1 , at node i

, at other nodes (43)

On the element level, the variation of the unknown func

tion tfj is approximated by

^e

= < N '> U }e

(44)

where < N'> = row vector of element shape functions

{ $ } = column vector of time-dependent nodal

values of $e

To satisfy continuity requirements, the shape functions

< N'> have to be such that the continuity of the unknown

function ty is preserved in the parent coordinates [11].

The shape functions < N > which characterize the element

geometry and the shape functions < N'> which describe the

unknown function do not necessarily have to be the same.

There is no requirement that the nodal values be associated

with the same nodes which were used to define the element

geometry, though in practice it is often the case. Consider

for example, the illustrations in figure 5, If the nodes

defining the element geometry and the nodes defining the

32

TABLE II

Coordinates of Local Nodal Points

Local Node h L2

L3

5

1 1

2 H h

3 1

4 h h

5 1

6 h H

7 1

8 1

9 1

10 1 -1

11 h h -1

12 1 -1

13 H h -1

14 1 -1

15 h h -1

33

(a) Isoparametric

(b) Super-parametric

o

(c) Sub-parametric

geometry nodes

variable function nodes

Figure 5. Element classification

34

unknown function are identical, the element is known as an

isoparametric element. This means that the shape functions

describing the geometry and the shape functions describing

the unknown function ty are equal, or

< N'> = < N > (45)

If there are more nodes defining the geometry than nodes de-

fining the variable function, the element is called a super-

parametric element. Using more nodal points to define the

unknown function than to describe the geometry of the element

results in a subparametri c element [11]. This work utilized

the isoparametric element classification.

D. DIVISION OF THE SYSTEM INTO ELEMENTS

In three-dimensional space, the division of the sys-

tem into discrete elements is difficult to visualize. It is

virtually impossible to show every nodal point of the system

in one schematic. To present a clear view of the discretized

domain, a "layer" approach was adopted. The first finite

element grid or mesh employed here consisted of 128 elements.

Under this grid (mesh I), the reactor was divided into four

layers as shown in figure 6. Each layer was composed of 32

elements. The first and fourth layers were each 30 cm in

height and each contained entirely reflector elements. The

second and third layers were each 80 cm in height and together

they encompassed the entire core plus the remaining reflector

elements. Each layer, in turn, was partitioned into three

horizontal (xy) planes. The top plane included all the global

35

z

Layer 1

Layer 2

II

Layer 3 II

Z=110

Z=80

z=o

_ Z=-80

Layer 4 II

I - core II - reflector

Figure 6. Layers of mesh I

—Z=-110

36

or system nodes corresponding to local or element nodes 1

through 6. The middle plane contained all the system nodes

corresponding to the local nodes 7, 8, and 9. System nodes

corresponding to element nodal points 10 through 15 comprised

the bottom plane. To fix ideas, the three nodal planes of

the first layer of mesh I are shown in figures 7, 8, and 9.

In this work there was only one curved side which was in

the plane of the triangle, as shown in figure 10. The first,

seventh, and tenth element nodes were each arbitrarily assigned

to have as an opposite side the curved side of the triangle.

The remaining local nodes in each of the respective planes

of the element were then numbered consecutively in the counter-

clockwise direction.

At this point it is appropriate to define connectivity.

The connectivity of an element is a row vector array that

'relates the local nodes to system nodal points. The connec-

tivity lists the system nodal points that are "connected" to

form the element domain in the sequence of local nodal number-

ing. Thus, the connectivity matrix for mesh I is a matrix

of size 128 x 15. To illustrate, the connectivity of element

number 106 is

<235, 210, 193, 194, 195, 211, 266, 176, 177, 283, 152, 10, 11, 12, 53>

A second finite element mesh (mesh II) consisting of 192

elements was developed. It contained the same number of ele-

ments per layer as mesh I. However, mesh II has six layers.

The second and third layers of mesh I were each divided in

37

264 233 234

259 239

258

257 u

256

255

240

241

242

243

250 249248

Number - element number

Figure 7. Top nodal plane of the first layer of mesh I,

z = 110 cm

38

265

278

277 '

276

268

269

270

274

Number - element number

273

Figure 8. Middle nodal plane of the first layer ofmesh I, z = 95 cm

39

312 281 282

307

306

305

304

303

287

233

<>289

290

291

298 297 296

Number - element number

Figure 9. Bottom nodal plane of the first layer ofmesh I , z = 80 cm

40

Figure 10. Local nodal numbering of curved

el ements

41

half, thus forming the two additional layers of mesh II.

The connectivity matrix and the coordinates of each global

node for mesh I are given in Appendix A. Appendix B lists

the connectivity matrix and nodal coordinates of mesh II.

The reader can construct the different layers and elements

without much difficulty by using the nodal coordinates and

the connectivity matrix. A mesh generator was not utilized

in this work.

E. COORDINATE TRANSFORMATION

The use of a quadratic or higher order element permits

the transformation or mapping of the straight-sided parent

element into an element with curved sides. Distorted or

curved elements provide a better fit to curved domains than

linear elements, and thus a smaller number of elements is

required to represent the structure adequately.

The transformation from cartesian coordinates to curvi-

linear coordinates can be accomplished by employing a one-to

one correspondence defined by [11]

x

y

z

= f or f

1

L(46)

The element shape functions are utilized to achieve this

transformation via the relation

y

/ Z N.x.l l

i

z M,y1

I ?

N^, i=l,2 (47)

42

where n is the number of element nodes, and the element

shape functions N. are in terms of local coordinates. Each

set of local coordinates corresponds to only one set of car-

tesian coordinates.

In performing the transformation, the compatibility re-

quirement must be met. The transformation into the new,

curved elements should leave no gaps between adjacent elements

If two adjacent elements are generated from parents in which

the element shape functions satisfy continuity requirements,

then the curved elements will be contiguous [11]. For the

isoparametric element, uniqueness of coordinates ensures com-

patibility. Continuity is assured when adjacent elements

are given the same sets of coordinates at common nodes.

Since the element shape functions are in terms of local

coordinates, an element of volume, dxdydz, must be transformed

into an element of volume expressed in local coordinates.

This is achieved through the use of the Jacobian matrix de-

fined below. Using the chain rule, the relationship between

£ » n » C and a corresponding set of cartesian coordinates

x,y,z is

/3N.^

3£

3N.

S 3n•

9Ni

19C

i

^x 3y 3z

3C ' 3? ' 9£

3x 3y 3z

3n 3n ' 3n

3x 3y 3_z

35 • 3C * 3C

/3N. N

3x

3

'->3y

3N.

(48)

43

The Jacob i an matrix [J] is defined as

[J] =

3£

3x3n

3x_

3^

ax3?

3n

3C

11

9_z

3n

9z_

3C

From equation (47), the Jacobian matrix becomes

[0] =

3N. 3N

j a£~x

i 5 al"

3N

l

3N.

3n~X

i

3N

3N

13n

3N

3?

3n~

z .

l

zi

i

3N.l

V 3C j 3Cy

i ' ? 3C

(49)

(50)

The determinant of the Jacobian is used to transform the vol

ume of element in cartesian coordinates to local coordinates

For a volume of element [11],

dxdydz = det [J] d^dndc (51)

Note that the determinant of the Jacobian is a variable for

elements of curved geometry. Only in the case of straight-

sided elements is the determinant of the Jacobian a constant.

In the plane of the triangle, the area coordinates (L,,

L?,L

3) number one more than the cartesian coordinates (x,y).

Thus, L^ is defined as a dependent variable. This establishes

the origin of the £n coordinate system at corner point 3, as

44

illustrated in figure 11. Recall that the £n axes need

not be orthogonal . As such

5 = L

n = L,

(52a)

(52b)

Using equation (42)

L3

= l - 5 - n (52c)

Applying the chain rule yields

3N. 8Ni

3L1

3N. 3l_2

2H. 3L3

i? 3L]

3£ 3L2

3? 3L3

3?(53a)

3N. 3N. 3L, 3N. 3L 3N . 3L-l _ + ]_ _2 . 3

3n 3L, 3ri 3Lp 3n 3L~ 3n(53b)

Using equations (52a), (52b), and (52c) in equations (53a)

and (53b) gives

3N.l

3N. 3N

3£ 3L, 3L.(54a)

3N. 3N. 3N.

3rj 3 L 3L o(54b)

45

Figure 11. n? coordinates in a triangle

46

Now the Jacobian matrix can be evaluated using the shape func

tions Ni

= Ni(L

1,L

2,L

3,c) . Inserting equations (54a) and

(54b) in equation (50) produces

3N. 3N. 3N. 3N. 3N. 3N.

f(3q--3q)xi • f aq-

- iLj)y i ^ - 3Ej)z

i

ti(L}

L2,Lv z)l =

3N. 3N. 3N. 3N. 3N. 3N.

3N. 3N.

5^i3N.

?( 3T)Z

i

(55)

To summarize, suppose it is required to transform the

integral

= J F'(L1

,L2,L

3,c) dxdy dz (56)

Vol

to an integral entirely in terms of local coordinates. The

determinant of equation (55) is then utilized to give

11 1-L

I "ffj F,

(L1

,L25 L3,ddet[J(L 15 L

2,L

3^)]dL

1

dL2dc

" ] o o (57)

Equation (56) is now in a form suitable for numerical inte

gration .

The development of coordinate transformation to this

point has been under the general assumption that all the

sides of the element are curved. In this work, the only

47

curved side is in the plane of the triangle. The sides of

the element along the prism axis are straight. As such, a

modified Jacobian matrix can be employed, thereby reducing

the number of calculations to be performed. This modified

Jacobian is the 2x2 matrix defined by

[0*] -

3N. 3N. 3N. 3N.

3N. 3N. 3N. 3N.

?( 3I7- 3L7

)xi - 5

( aU7- 3T>ii

Ul"l

(58)

Along the prism axis or 5 direction, it can be shown that

dz = £ d^ (59)

where h = height of the element. The volume relationship

is then given by

dxdydz = | det[J ] dl^dL^ (60)

The integral of equation (56) then assumes the form of

1 1 1-L,

I s \fff F,

(L1

,L25 L3S Odet[J*(L1

,L2,L

3,d]dL

1

,dL2,dc (61)

-loo

Equation (61) is the basis of numerical integration applied

in this work.

F. CONSTRUCTION OF ELEMENT MATRICES

The system matrix operators can be constructed through

the use of the global basis functions Gj or through the

48

application of the element shape functions. Although the

global basis functions were used to demonstrate the method

of Galerkin, the construction of the system matrices in this

work was achieved through element considerations. The solu-

tion of the unknown variable 4> within the element domain

was approximated by

iQ

= £ N. n>*(t) , i = l,2,...,n'i = l

(62)

where n is the number of element nodal points, are

the element shape functions, and i^.(t) are the time-depen

dent nodal magnitudes of ie

. The element contribution to

the system matrix operators is defined by Galerkin's ortho-

gonality condition expressed by

Vol

N . Re

dVol = , j = l ,2,. . . ,ne

(63)

p " "" ewhere R is defined by replacing ty with \p in equations

(37) and (38), and the integration is over the element volume

Using equation (62), the element contribution is portrayed

for the core by

[ f N.N.dVol]{^.e(t)}-[ f N,V 2 N,dVol]{(cl •/).}+[ f N.N.dVol]{(c2v|;

e) .}

Vol Vol Vol

+ [ / N.N.JlnO + -f- Z N. >|,.

e(t))dVol] {(c4^e

).}

VolJ ^

yTo k

k k 1

+ [ f N.N.dVol] { / e"1 ^' 1 ^ (c5^(t') ).dt'} = (64)

* j i * i

Vol °

49

where i,j,k = 1,2,. ..,15 , and cl , c2, etc., are constants

at node i. The bracketed expressions represent square

matrices of size 15x15, and the braced expressions represent

column vectors of size 15x1. For the reflector, the last

two terms of equation (64) are zero. Before any operation

can be performed on equation (64), the nonlinear terms (last

two terms) must be "linearized" and the term with the V 2

operator must be integrated by parts.

K15

The nonlinear feedback term, £n[l + -— £ N.ij;. (t)] ,

k=lwas linearized by using predicted values

of the unknown function at time t. These predicted values

come from the time integration scheme by Franke [7].

Basically, the integration scheme utilizes a predictor-cor-

rector method which predicts values of the unknown function

at the next time by using the derivatives of the function.

PAdopting the predicted values \\). enabled integration over

space. The term in equation (64) involving the nonlinear

feedback can be written as

[/NjN.HnO + -^<H1^

P+ N^

2

P+ . . .+ N^-,/) )dVol ] C(c4-ip

e)1}

Vol°

The last tern of equation (64) describes the delayed

neutron contribution. In general,

t.n t. *n-l r+i

e-Xtn/eXt '

^.e(t')dt' = e^VW [e^n-l J e

Xt '

*.e(f)dt']

o

(65)-Xt f+ e n J e

Xt '

*.e(f )df

'n-1

where n is number of time steps, t is the current time,

50

and t , is the previous time. To approximate the integrals

pin equation (65), the predicted values ty . were employed

in a simple trapezoidal rule. The trapezoidal rule was be-

lieved to be sufficient since very small time steps were uti-

lized in this work. For example, at time t2

,

(SUM^ = (previous SUM^e"*" + j(elH

^.e(t

]

)+ *.P

)

where

H = current time step = t« - t.

-U r 2 ^Xt ' ,e

(SUM.) = e"At

2 / LeAt

<j,,

e(t')dt'

(previous SUM.) = e

C 1 -

*1 J eAt '

ip.e(t')dt

The previous sum is formed by accumulating the trapezoidal

integration of each time step. In general then, for time tn

(SUM.) = (previous SUM.)e"XH

+ j(eXH

^i

e( t

fl. lJ+*

1

P) (66)

The delayed neutron term in equation (64) can be expressed

as

[ f N.N.dVol] {(C5-SUM) .}

Vol

51

In order to bring equation (64) to final form, the 7 2

operator was integrated by parts. Since the flux at the sur

face of the reactor is zero, integration by parts yields

r - 3N. 3N, 3N, 3N. 3N. 3N.

/ y^dxdydz - - / (^ _i + _i _i + _i _i)dxdyd2 (67)

Vol Vol

In vector notation, equation (67) is expressed as

Vol

f N.V 2 N.dxdydz = -/Vol

3x

3N.l

"3x

3N.

' 3y

3N.

' 32^ 1

3N.J

3y

3N.J

> dxdydz (68)

3z

Using the chain rule produces

/ 3Ni

\

"3x~

3y /

3N.

3z

8L,

3x

3L2

' 3x, o

3L1

3y

3L2

' 3y, o

, o' 3z

(W 3NiA

3N. 3N.

V

3L?

" "3T7;

/"2

3N.3N.

(69)

Letting the 3x3 matrix above be [B1

], the V2 term becomes

/Vol

.V2 N. dxdydz =

J i

3N. 3N. 3N.

"/ <(~3L7

" lUT^ICTVol '

3 l

em. 3N

7 3C/

,-|T,

U\3L

3N. 3N3N. 3N.

3L3

' '3C

3N.J

'dxdydz

(70)

52

where [B 1

] is the transpose of [B'] . By applying the

chain rule in equation (47) and using equation (59), it can

be shown that for this work

[B'] =

1 1

3N 3N, C 3N, 3N, C

3T^X

1 '•• lT^ x15

j l 8L2

xl ••• 3l_

2

Xic)

1 1

3N

%^ +'" +3L

1

y15

; hl2

y^

"•3L

2

,

, o

yi C )

2' h

(71)

,

[B1

] can be derived from equation (71).

Note from equation (64) that there are three basic ele-

ment matrices which are defined as follows after applying

equation (60):

[ G ii ]= til! N

jNl det[J

*]dL

l

dL2dC (72)

-loo

-loo1 1 1-L

1

3N.

3N .

(-

3N 3N 3N 3N

U7)'(H7-^'^ )>[B,]iiT

3L

3N .

I)3U ;

3

3N. 3N.

£ B 'i i ^3u " 3rt } ldet[J ]dL

1

dL2d?

3N .

J(73)

h J- 1-

1" 1

!

[GGG...] = £/// NjN.Jln(l+^-(N

1i|;

1

K+...+N

15i|;

1J))det[J IdL^d?

(74)

-loo

53

Element matrices [S..] and [GG..] are independent of

time. However, [GGG..] is time dependent due to the utili-

zation of the predicted values i> . which changes with time.

In terms of these three basic element matrices, equation (64)

is

EGjjHif} + [GG^.Kcl.^).} + [G lUcZ.^)^

+ [GGG^.] {(c4-ii;e

)i

} + [Gj.] {(cS-SUM)^ = (75)

The last two terms of equation (75) are zero for the reflector

G. CONSTRUCTION OF THE SYSTEM MATRICES

The 15x15 coefficient element matrices were calculated

according to equations (72), (73), and (74); and the results

were collected element by element into the corresponding sys-

tem coefficient matrices. The system coefficient matrix

[BIGG] is developed from [£..] , [BIGGG] from [GG..] ,

and [BIGH] from [GGG^.] . [BIGG] and [BIGGG] are inde-

pendent of time and can be constructed once and for all from

geometry considerations. [BIGH] is dependent on both geom-

etry and time due to the time dependence of the predicted

flux utilized in the feedback term. Thus, [BIGH] is recal-

culated at each time increment.

Non-zero contributions to a global nodal point I come

only from adjacent elements sharing that same nodal point I.

Thus, the system matrices are sparse and banded. The pro-

cess of assembling contributions from element matrices

54

requires the identification of a local nodal point (1*1,2,

...,15) with a global nodal point ( 1 = 1 ,2 , . .

.

NUMNP , where

NUMNP is the total number of system nodes). This correspond-

ence between element and global nodes is accomplished via

the connectivity matrix.

The formal treatment of the field equations in terms of

the system coefficient matrices is described by the equation

[BIGG] {iT> + [BIGGG] {(cW)j} + [BIGG] { ( c2 -^ )

x>

+ [BIGH] {(c4-iMj} + [BIGG] {(c5«SUM)j} = (76)

where the system matrices are NUMNP x NUMNP and the column

vectors are of length NUMNP x 1. However, the direct appli-

cation of equation (76) requires a large amount of computer

storage. To take advantage of the sparsity of the system

matrices, an optimum compacting scheme described by Ref. 8

was employed.

The concept behind OCS is simply to store only the non-

zero terms of a coefficient matrix. OCS requires two integer

arrays, say JB and NAME, and a vector of non-zero coeffi-

cients of the square system matrix. For purposes of illus-

tration, the square system matrix is called B, and the vector

of non-zero coefficients of B is called BB. The i integer

entry in the NUMNP x 1 JB vector is the number q.. This

number is defined by

i = l

q = 1 + £ p. , i = l ,2,... , NUMNPJ

(77)

J-lth

where p. is the number of terms in the i equation. InJ

55

other words, p. is the number of nodes that the i

thnode

"sees". JB is therefore a pointer vector of length NUMNP+1

whose i term locates the initial position in the BB vector

of the contributing coefficients to the ith

equation. TheNUMNP

NAME vector of length Mxl , where M = £ p. , consists of1-1

1

NUMNP successive vector blocks of variable length p. ,

i-1 ,2,. . . , NUMNP. The p. integer numbers in the ith

block

f hof NAME list the p. contributors to the i equation. The

Mxl BB vector contains the real non-zero coefficients of the

NUMNPxNUMNP B matrix, arranged in the same contiguous block1" h t" h

arrangement as the NAME vector. The j term in the 1

block or BB( JB( I ) + J-l) is B(I,K), where K = NAME( JB( I )+J-l

)

To illustrate, consider the grid shown in figure 12. The two

array vectors and the coefficient vector matrix of non-zero

terms are:

JB = <1, 5, 10, 13, 18, 25, 30, 33, 38, 42>

NAME = <1,2,4,5|2,3,1,6,5| — |9,8,6,5>

BB = <Bir B12

,B14

,B15

|B22

,B23

,B21

,B26

,B25 |

— |Bgg

,Bg8

,B96

,Bg5

>

In this illustration, NUMNP = 9 and M = 41 [8].

In this work, a judicious method of numbering the system

nodal points was adopted to further reduce computer storage

requirements. Since the surface boundary nodes of the reac-

tor represent zero neutron fluxes, the contributions of these

nodes to interior or non-zero nodes can be discarded. Thus,

only the interior nodal points need to be considered. These

non-zero nodes were numbered first in the finite element mesh

56

- /

/ 3_

8 /^

// 2

- /

/ 1

5 y

/ 5_

Number - element number

Figure 12. Sample grid used for illustrating OCS

57

used so that in the OCS, the number of non-zero nodes (NNZ)

replaces NUMNP.

The vectors of non-zero coefficients will be designated

BIGG, BIGGG and BIGH since the square coefficient matrices

described in equation (76) were not utilized. To illustrate

the application of OCS in the system, the following sample

program is given:

DO 450 1=1 , NNZ

JBB = JB(I)

JE = JB(I+1 )-l

DY ( I ) = 0.0

DO 500 J=JBB, JE

LL = NAME(J)

DY(I) = DY(I) + BIGG(J)*i(LL) + B IGGG ( J) *cl ( LL )

*

iMLL) + BIGG(J)*c2(LL)*<MLL) + BIGH ( J ) *c4 ( LL )*

ip(LL) + BIGG(J)*c5(LL)*SUM(LL) (78)

500 CONTINUE

450 CONTINUE

where

LL = nodal point "seen" by node I

JE-JBB = total number of nodal points that node I "sees"

DY ( I) = summation of all contributions to node I and

should sum to zero as stated by the field equa-

tion

The assumption of a homogeneous reflector was relaxed in the

application of equation (78). Interface nodes were assigned

properties of the core. Therefore, if LL is a reflector

58

node not on the core reflector interface, the last two terms

of equation (78) are nonexistent.

59

IV. NUMERICAL INTEGRATION

A. LINE AND AREA INTEGRATION

The solution of the element matrix equations was achieved

through numerical integration since an exact closed form solu-

tion cannot be established. The volume integration was accom-

pi

i

shed by using a line integration in the c-di recti on and an

area integration in the plane of the triangle. The line inte

gral is described by the Gaussian quadrature formula [12]

1

j,

(79)/-1

fU)dc a L H.f(a. )

k=lK K

where n is the number of Gauss integration points

H, = weighting coefficients

f(a.) = the function f(c) evaluated at Gauss point aR

Table III lists ak

, Hk

and n [11].

The area integration was achieved by the equation

//o o

1 ,.1-Lm , m . m1 m

f(L1

,L2,L

3)dL

1

dL2

- £ winf(L

1

m,L

2

,,,

,L3

m)

m = l

where m is the number of area integration points and wm

are the weights. The numerical integration points for the

area integration are given in Table IV which was extracted

60

TABLE III

Abscissae and Weight Coefficientsof the Gaussian Quadrature Formula

/1

f(c)dc-1

n

= £k = l

Hkf < a k>

057735 02691 89626

077459 66692 41483000000 00000 00000

86113 63115 94053033998 10435 84856

90617 98459 3866453846 93 101 0568300000 00000 00000

0-93246 95142 0315266120 93864 66265

023861 91860 83197

94910 79123 4275974153 11855 9939440584 51513 7739700000 00000 00000

096028 98564 975360-79666 64774 136270-52553 24099 1632901 8343 46424 95650

= 2

= 3

n = 4

= 5

= 6

= 7

- 8

H

I 00000 00000 00000

0-55555 55555 555560-88888 88888 88889

034785 48451 3745465214 51548 62546

0-23692 68850 561890-47862 86704 9936656888 88888 88889

017132 44923 791700-36076 15730 481390-46791 39345 72691

1 2948 49661 688700-27970 53914 892770-38183 00505 051190-41795 91836 73469

10122 85362 9037622238 10344 53374

0-3)370 66458 778870-36268 37833 78362

= 996816 02395 07626 08127 43883 61574

083603 11073 26636 01 8064 81606 c<»857

61337 14327 00590 26061 06964 v>2935

032425 34234 03809 0-31234 70770 40003000000 00000 00000 33023 93550 01260

0-97390 65285n =

1717210

006667 13443 08688086506 33666 88985 0- 14945 13491 50581

067940 95682 99024 0-21908 63625 15982

043339 53941 29247 0-26926 67193 0999614887 43389 81631 0-29552 42247 14753

61

TABLE IV

Numerical Formulas for Triangles

Order Fig- Error Points Triangular Weights

Co-ordinates 2Wk

Linear R = Oih1) i.i.i

R = 0(A 3)

a U.o 1

b o.|.i i

c i.o.l i

- A/\ 1 \ a i.U -«

Cubic /i_^4r~\ /? = (X/i*) A H.A.A H«^ c

d

A a i.i.i &/7\\ b i i. o

)

/6 \ \ c o.i.1\ A

Cubic /X-l^y. /? = (XA4

) d i.o.l J

^-"-""P^X e 1.0.0 >

^^^^-*1l\\ f 0. 1.0\ A^-£^ 9 0, 0. 1 J

A a 1.1.1 0225

/ tV b J,.0,.0iA l/\ c fl.«l.#l 13239415

Quintic />J ^ /? = (X/ih

) d 0.. 0.-31

Z?&~-^t\*\ e *1- 02, 02^^ f9

02- ^2,02

0:-02-»2

01 259391

8

with

*i - 05971587

0, = 47014206

*2 79742699

01 = 101 28651

62

from Ref. 11. The volume integration of the function

f ( Ll

,L 2' L3'^ using numerical integration is therefore

1 1 1-L

///f(L1.L

2,L

3, ? )dL

1dL

2d

• I ooc ~

m

E wk£ w

mf(L

1

m,L

2

m) L

3

m,c

k)

k=lK

m=lm

'

L 6(81)

where wk

= Hk

. In the manner of equation (81), the element

matrices become

. n m ,

CGdi

] " | E % E, wm

N (L1

m) L

2

m> L

3

m;C

k) Ni det[J ] (82)

k=l m=l

hn m

i, *[GG .] = \ £ w

k 2 wm

F (L1

m9 L

2

m,L

3

m,C

k) det[J ] (83)

k=l m=l

mm . m , m k[GGG..]=fE w

k Z wm

T (L1

m,L

2

m,L

3

m,c

K) det[J] (84)

k=l m=l

where F and T are easily derived from equations (73) and

(74), respectively. Note that N. and det[J ] are also

evaluated at each integration point. They are not shown as

such merely for the sake of convenience in writing the equa-

tions.

B. NUMBER OF INTEGRATION POINTS

It is difficult to estimate the number of integration

points required for good accuracy due to the complexity of

the functions involved. The basic rule that the best number

of integration points is found by trial and experience was

adopted. For the [G..] element matrix, the numbers involvedJ

63

can be approximated. This was done by letting the determi-

nant of the Jacobian equal to twice the area of the curved

triangle (checking det[J*] at each integration point showed

that this assumption was not too unreasonable). With det[J*]

outside the integration process, [G..] can be solved in

closed form by integrating out 5 from -1 to +1 and then apply

ing the closed form equation [12]

m, m„/...-, ...„ m_ m. !m9 !m«

!

L,'

L 9c

L,6 d(Area) = 2A /

' .; .; , 9 v

,

1 2 3 (m, +m«+m^+2 )

!

Area '

c 6

(85)

where m, , m9 , m~ are positive integer exponents and A is

the area of the triangle.

Five test points within the 15x15 element matrix were

selected, as listed in Table V. The values obtained from

the application of equation (85) are also given in Table V.

Three sets of area integration points, each with a different

number of c Gauss points, were used. These three sets of

area integration points, given in Table IV, are:

1. cubic order (4 points)

2. cubic order (7 points)

3. quintic order (7 points)

Using each of the three integration points above with differ-

ent £ Gauss points in equation (82) produced the results

obtained in Table V. From these results, the quintic order

area integration points were selected for the element matrix

[G..] with three c Gauss points in the prism axes. This

set of integration points was also used for [GGG..].

64

TABLE V.

Selection of Integration Points for [G..]

C pointsAreapoints 6(1,1) G(2,2) G(l, 9) 6(6,9) G(9,9)

13 cubic

(4 pts)

1415 5278 -2756 5072 8536

5ii

1817 5278 -3170 5072 10240

7it

1817 5278 -3170 5072 10240

3 cubic(7 pts)

3248 8247 -3114 4960 10243

5ii 3249 8247 -3114 4960 10240

7H

3249 8247 -3114 4960 10240

3 qui ntic

(7 pts)

2464 6600 -3144 5020 10240

5H 2464 6600 -3144 5020 10240

7ii 2464 6600 -3144 5020 10240

Approximated values 2500 6700 -3142 5026 10053

65

The numbers for the [66..] element matrix could not

be approximated. The best that could be done was to obtain

an idea of the order of magnitude of this element matrix.

Towards this end, a linear triangular element was assumed.

Using linear approximation, the order of magnitude was found

3to be about 10 . Using the three different area integration

points mentioned above with varying 5 Gauss points in equa

tion (83) yielded the results given in Table VI. Further

checks on the [66..] element matrix showed that the cubic

order with four area integration points yielded the desired

order of magnitude. Thus, the fourth order cubic with five

C Gauss points was employed for the [G6..] element matrix

A note should be mentioned here in regards to the vast dif-

ference in results obtained for [GG..1 using differentJ "i

area integration points. Most likely, it was due to the

[ B '

] and [B1

] matrices which required the inversion of

3 x 3 x

TT ' TT * etc. Or perhaps it was caused by the nature of

the hybrid element used in this work. In any case, further

investigation is warranted in this area.

66

TABLE VI

Selection of Integration Points for [GG..]

Area

5 points points GG(1 ,9) GG(3,3) GG(14,9) GG(8,8) GG(1 5,1 5)

3 cubic(4 pts)

18.4 170.1 -165 165.9 106.1

5ii 22.0 177.6 -186 195.9 106.1

7ii 22.0 177.6 -186 195.9 106.1

3 . cubic

(7 pts)

-1.03xl04

2.11xl07

1.05xl07

8.42xl07

2056

5M -1.03xl0

42.11xl0

71.05xl0

78.42xl0

72056

7ii -1.03xl0

42.11xl0

71.05xl0

78.41xl0

72056

3 quintic -52.6 1.27xl04

-6.64xl04

6.84xl04

1.51xl05

(7 pts)

5 " -52.6 1.27xl04

-6.64xl04

6.84xl04

1.51xl05

7 " -52.6 1.27xl04

-6.64xl04

6.84xl04

1.51xl05

67

V. TEST PROBLEMS AND RESULTS

The reactor was subjected to uniform and local perturba-

tions in the form of a ramp input described by

E.p = E.c + at (86)

wh ere a is the change in Ef

per unit time and Ef

is

the critical fission cross-section. £- must first be ob-

tained before the perturbations can be applied. This was ac-

complished by trial and error until a stationary solution

was reached. For mesh I, Zf

was found to be 0.0057360

per cm. No attempt was made to find Ef

for mesh II due

to time limitations. As such, the test problems outlined be-

low were applied to mesh I. Future work is planned to apply

the test problems on mesh II.

The following perturbations were applied:

a) Uniform perturbation of 10 dollar of reactivity per

second :

2f(r,t) = Z

f+ at , in the core

where a = . 005893/cm-sec

b) Local perturbation at the core center of 100 dollar

of reactivity per second:

Ef(r,t) = S

f+at5(r

Q) , in the core

68

where rQ

is (0,0,0) and a = 0. 01 5407/cm-sec

c) Local, off-center perturbation

2f(r,t) = E

f+ a

it5(r

1) , i = 1,2,3, in the core

where

a.

a.

a.

(0,60,40)

0.015407/cm-sec

0.008123/cm-sec

0.005894/cm-sec

100 dollar per second

50 dol 1 ar per second

1 dol 1 ar per second

Three test points, (0,0,0), (60,0,0), and (-60,0,80),

were selected to trace the neutron time history. For cases

a) and b), the neutron flux was plotted at each test point

during transience. This is shown in figures 13 and 14.

Case c) involved three ramp inputs and were conducted for

both the linear and nonlinear reactor equations. The linear

and nonlinear responses were compared at each test point for

each ramp input and are illustrated in figures 15 through 23

The radial and axial flux distributions at time t = 0.0123

second were plotted for the steady state and the 100 dollar

perturbation case. These are embodied in figures 24 and 25.

Finally, a neutron flux early time history between mesh I

and mesh II was plotted to check the effect of using a finer

element mesh. The result is portrayed in figure 26.

Figures 13 and 14 revealed a clear space dependence of

the neutron flux during transience as expected. Figures 15

through 23 demonstrated the effects of temperature feedback

69

COi-

+->

3

io12

.

A

/ B

1011 -

/ lie

io10

-•<IZ—-//

io9

•

1 1 1 1

10-5

io"4

io'3

time (sec)

10-2

A - neutron flux at test point (0,0,0)B - neutron flux at test point (60,0,0)C - neutron flux at test point (-60,0,80)

Figure 13. Neutron flux transient history at threetest points with a uniform perturbationof 10 dollar of reactivity per second.

70

1011

x3

Oi-

O)

1010

io9

+

10-5

10-4

10-3

time (sec)

A - neutron flux at test point (0,0,0)B - neutron flux at test point (60,0,0)C - neutron flux at test point (-60,0,80)

Figure 14. Neutron flux transient history at varioustest points for a local central perturbationof 100 dollar/sec of reactivity.

71

io12

+

Oi-

=3

c

lo1^

io10

+

109..

10-5

10-4

10-3

10-2

time (sec)

A - nonlinear neutron responseB - linear neutron response

Figure 15. Linear and nonlinear fluxes at (0,0,0) dueto a local perturbation of 100 dollar/sec ofreactivity at (0 ,60,40) .

72

io12

+

iou

t

COS-

-t->

io10

+

10 "

10-5

10-4

10-3

10-2

time (sec)

A - nonlinear neutron responseB - linear neutron response

Figure 16. Linear and nonlinear fluxes at (60,0,0) dueto a local perturbation of 100 dollar/sec at(0,60,40).

73

1012

C 1011

os-

M3

e

1010

10 -.

-+-

10-5 -4

10

time (sec)

-310

-2

A - nonlinear neutron responseB - linear neutron response

Figure 17. Linear and nonlinear fluxes at (-60,0,80)due to a local perturbation of 100 dollar/secat (0,60,40).

74

X3

O

3a;

io12

i

10U

+

io10

i

io9t

10-5

10-4

10-3

10-2

time (sec)

A - nonlinear neutron responseB - linear neutron response

Figure 18. Linear and nonlinear fluxes at (0,0,0) dueto a 50 dollar/sec local perturbation at(0,60,40).

75

1012

x3

COs-+J

3

iou

+

1010 ..

10 ••

10-5

10-4

10-3

10-2

time (sec)

nonlinear neutron responselinear neutron response

Figure 19. Linear and nonlinear fluxes at (60,0,0)due to a 50 dollar/sec local perturbationat (0,60,40)

.

76

1012

10U

+

Cos-

3V

1010 ..

io9+

10-5

10-4

10-3

10-2

time (sec)

A - nonlinear neutron responseB - linear neutron response

Figure 20. Linear and nonlinear fluxes at (-60,0,80)due to a 50 dollar/sec local perturbationat (0,60,40).

77

io12

+

<4- 10

Os-

3

io10

t

109..

10-5

10-4

10-3

10-2

time (sec)

nonlinear neutron responselinear neutron response

Figure 21

.

Linear and nonlinear fluxes at (0,0,0) dueto a 10 dollar/sec local perturbation at

(0,60,40)

.

78

1012

1011

cos_

=3

io10

+

10 ..

10-5

io"4

lo"3

time (sec)

10-2

A - nonlinear neutron responseB - linear neutron response

Figure 22. Linear and nonlinear fluxes at (60,0,0) dueto a 10 dollar/sec local perturbation at

(0,60,40).

79

1012 -.

x

r ioli

eoi-

O)

1010

10 •

10-5

io"4

time (sec)

10-3

10-2

A - nonlinear neutron responseB - linear neutron response

Figure 23. Linear and nonlinear fluxes at (-60,0,80)due to a 10 dollar/sec local perturbationat (0,60,40).

80

io12

+

X3

COS-

3a;

c

iou +

io10 +

io9 +

60 30 -30 -60

Figure 24.

x=0, z=0, y=y

9 - point of local perturbation

A - neutron flux at steady-state

B - neutron flux under local perturbation

Radial flux distribution for the steady stateand 100 dollar/sec local perturbation.

81

iou

+

X=3

<4-

O

0)

io10 +

lo9 +

+

95 80 40

x=0, y=60, z=z

-40 -80 -95

Figure 2 5

^ - point of local perturbation

A - axial flux distribution during steady state

B - axial flux distribution during perturbation

Axial flux distribution for the steady state and

100 dollar/sec local perturbation.

82

X3

O+J3CD

C

1010

io9+

-10

A

B

10

mesh I

mesh II

10

time (sec)

-910

Figure 26. Early time history of the neutron flux at corecenter using mesh I and mesh II.

83

and delayed neutrons on the flux. To obtain an idea of the

difference in magnitude between the linear and nonlinear

flux for the 100 dollar local perturbation, at time t = 10-3

second, the linear case predicted a neutron flux at the core

1

2

2center of 2.024 x 10 neutrons per cm per sec. The non-

10 2linear reactor predicted 1.85 x 10 neutrons per cm per

sec. Although the numbers are not sufficiently accurate due

to the crudeness of the finite element mesh, the significant

difference in the order of magnitude between the linear and

nonlinear neutron flux leads to the belief that the three-

dimensional finite element model utilized is predicting the

correct trend of neutron flux behavior.

The radial flux distribution for the steady state shown

in figure 24 portrays the expected flux distribution. The

radial flux distributions for the local perturbation case

show a skew distribution at the point of perturbation. The

axial flux distribution at steady state is very much symme-

tric about the center. Again, the expected skew at the

point of perturbation is there, as shown in figure 25.

Figure 26 gives an indication that the E- for finer

meshes is higher. However, curve B of figure 26 should be

extended to longer times to verify this hypothesis. Curve B,

as plotted in figure 26, used four hours of computer time

employing the H-compiler of the IBM 360/67.

84

VI. CONCLUSIONS AND RECOMMENDATIONS

The finite element mesh employed here is crude, and,

thus, results that were obtained should be considered as p o s i •

tive indicators rather than numerically conclusive facts. The

trend of neutron flux behavior is the major thrust of this

work. From the results obtained, it is concluded that the

expected patterns of neutron behavior as predicted by the

three-dimensional finite element model used do occur. These

patterns were best demonstrated by the differences between

the linear and nonlinear flux responses and by the spatial

flux distribution at steady state and during local perturba-

tion. The three-dimensional quadratic finite element model

utilized should produce better results by resorting to a

finer mesh. The draw-back to a finer mesh, of course, is

the significant increase in computer time and storage re-

quirements. Once accurate results are obtained through

finer element meshes, comparisons between three- and two-

dimensional models can be attempted.

A mesh generator was not developed for this problem. It

is recommended that this be done to ease the transition from

one mesh to another and to minimize human error. In addi-

tion, a similar calculation using a three-dimensional linear

element should be performed to corroborate the results ob-

tained here. It should be particularly noted that this type

of problem is highly sensitive to the fission cross-section.

85

In the search for E,. , it was necessary to adjust Ef

to

the sixth decimal place. There is no exact method of deriv-

ing Zf

due to the highly nonlinear aspect of the problem;

therefore, trial and error must be used. Finally, the

Gaussian quadrature used to determine the element matrices

should be investigated. The cause of the differences in

values obtained by using different number of integration

points should be established. Was it due to the integration

points, the coordinate transformation, or the element shape

functions? This is an important question upon which future

numerical results could be based.

86

to

3o

CO

<o

on to

LU •r-

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i

<uo Eq:o +->

o eC_> cu

EQ o>

< z. ^< CD

X I

t—

*

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H

r—LU >Q_ 1—

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z oo M-oX

: 1

&_

X +->

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LIST OF REFERENCES

1. Nguyen, D. H. and D. Salinas, "Finite Element Solutionof Space-Time Nonlinear Reactor Dynamics," Nucl earScience and Engineering , v. 60, p. 120-130, June 1976.

2. Hanford Engineering Development Laboratories ReportHEDL-TME74-47, Melt 1 1 1 -A Neutronics Thermal -Hydraul i c

Computer Program for Fast Reactor Safety , v. 1, byA. E. Walter and others, 1974.

3. Hetrick, D. L., Dynamics of Nuclear Reactors , p. 1-15,University of Chicago Press, 1971.

4. Lamarch, J. R. , Introduction to Nuclear Reactor Theory ,

Addis on -Wesley, 1972.

5. Nguyen, D. H., "The Time-Dependent Nuclear Reactor withFeedback," Nuclear Science and Engineering , v. 55,p. 307-319, June 1974.

6. Nguyen, D. H., "The New Equilibrium State of a PerturbedNuclear Reactor with Negative Feedback," Nuclear Scienceand Engineering , v. 52, p. 292-298, June 1973.

7. Naval Postgraduate School Report NPS-53Fe76051 , A Pro-gram for the Numerical Solution of Large Sparse Systemsof Algebraic and Implicitly Defined Stiff DifferentialEquations , by R. Franke, May 1976.

8. Naval Postgraduate School Report NPS-59Zc761 1 1 , An Opti-mum Compact Storage Scheme for Nonlinear Reactor Problemsby FEM , by D. Salinas, D. H. Nguyen, and R. Franke, 1976.

9. Norrie, D. H. and G. de Vries, The Finite ElementMethod; Fundamentals and Applications , Chapter 5,

Academic Press, 1973.

10. Salinas, D., D. H. Nguyen and T. W. Southworth, "FiniteElement Solution of a Nonlinear Nuclear Reactor DynamicsProblem," International Conference on ComputationalMethods in Nonlinear Mechanics , Austin, Texas, p. 541-

550, 1974.

11. Zienkiewicz, 0. C, The Finite Element Method in Engineering Science , McGraw-Hill, 1971.

12. Cook, R. D. , Concepts and Applications of Finite ElementAnalysis , Wiley & Sons, 1974.

159

INITIAL DISTRIBUTION LIST

1 .

2.

3.

5.

6.

7.

Defense Documentation CenterCameron StationAlexandria, Virginia 22314

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2

Library, Code 0142Naval Postgraduate SchoolMonterey, California 93940

Department Chairman, Code 69Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

Dr. D. H. Nguyen (thesis advisor)4625 Larchmont NEAlbuquerque, New Mexico 87115

Assoc. Professor D. Salinas (thesis advisor)Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

Assoc. Professor R. Franke (second reader)Department of MathematicsNaval Postgraduate SchoolMonterey, California 93940

LT E. C. Bermudes, USN3721 -D McCornack Rd.

Schofield Barracks, Hawaii

(student)

96557

Professor Gil 1 e s Cant in

Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

160

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