Determination of the expansion coefficients of a 1-D multigroup diffusion nodal model

Ann. nucl. Energy, Vol. 19, No. 8, pp. 471-472, 1992

0306-4549/92 $5.00+0.00 Copyright © 1992 Pergamon Press Ltd

Printed in Great Britain. All rights reserved

TECHNICAL NOTE D E T E R M I N A T I O N OF THE E X P A N S I O N COEFFICIENTS OF A 1-D M U L T I G R O U P D I F F U S I O N N O D A L M O D E L M . FEIZ I a n d A. F. ROHACH 2 ~Penn State University, Beaver Campus, Brodhead Road, Monaca, PA 15061, U.S.A. ZNuclear Engineering Program, Department of Mechanical Engineering, Iowa State University, Ames, IA 50010, U.S.A.

(Received for publication 22 February 1992) Abstract--This paper describes the determination of the unknown polynomial expansion coefficients in a diffusion theory nodal model. The expansion coefficients are determined from a linear system of equations using the source iteration technique. The Gaussian elimination method, consisting of the forward elimination and the backward substitution processes is used to solve the system of equations. Using the method described in this paper, application of the forward elimination process is only necessary at the first iteration, provided that the material properties and/or the geometry of the problem remain constant. As a result, the number of calculations required to determine the unknown polynomial coefficients, using the Gaussian elimination process, can be decreased. This method is especially advantageous in multidimensional nodal models and when high orders o f the polynomial expansion are used.

Dg(i)[15a~(i)+42a~(i)l-rffZ~(i)a~(i)=-q2c~(i),

INTRODUCTION

A polynomial nodal model has been described for the I-D multigroup diffusion equation (Feiz and Rohach, 1989). The development of the nodal model consisted of dividing the region into different nodes. The neutron flux for group g of node i was expanded using the Legendre polynomial functions. A linear system o f equations was obtained from the least-squares minimization of the expansion residuals over each node and the nodal interface conditions. The unknown polynomial coefficients were then evaluated iteratively from the system of equations. The Gaussian elimination method, consisting o f the forward elimination and the backward substitution processes (Duderstadt and Hamilton, 1976), was used to solve the system of equations in the aforementioned nodal model. The application o f both processes of the Gaussian elimination method after each iteration is computationally expensive, especially if high orders o f the polynomial expansion are required. This paper describes a method by which the forward elimination process is applied at the first iteration only. As a result, the number of required calculations to determine the polynomial coefficients decreases, which is computationally desirable.

Dg(i)[35a~4(i)l-q~Egt (i)a~(i) = --q~c~(i),

(3)

Dg(i)[63a~(i)]-q~YYt (i)a~(i) = --rl~c~(i),

(4)

- [~ + ~]a'~(,)- [~ + ~-, 6 D g ( i ) ] Ja,(0 g " + 15D~'(i)

l

1

~4~a~ ~i~ l

[~,1

3Dg(i)'] . . . .

L

-

+~-.

]a~(O=~'-, .

(6)

where

~ZgVEf'(i)+Y.f¢(i)

c~(i) =

a~(i)

(7)

and

THEORY NODAL MODEL

To illustrate the process by which the unknown polynomial coefficients are evaluated in any iteration, a fifthorder polynomial expansion is used in the diffusion theory nodal model. For the fifth-order expansion, the equations obtained from Table 2 o f the Feiz and Rohach (1989) paper are :

Dg(i)[3a~(i)+ lOag4(i)]--~l~Y°t (i)a~(i) = --q~c~(i),

~,..

and

9'=1

DIFFUSION

(2)

(1) 471

JS(i+ I) ± Jg+ (i - 1) 6~-

2

Equations (1)-(6) can be put into a matrix form. However, close examination of these equations reveals that the odd and even coefficients are independent of each other. Therefore, the matrix can be divided into two smaller ones containing the odd and even coefficients. These matrices are :

472

Technical Note

_ 1 4

1 4

3Da(i) 2rh

10Dg(i)]

1

,i

/ F,s(/)]

-r/~Zt°(i)

3Og(i)

10D'(i)

/

0

--q~Zf(i)

35Dg(i)

] Lag4(i)J

=

I ] -,,%g(i)

(8)

L-q:c~(i)/ and 1 4

Da(i) 2r/i

--q:Z~(i) o

1 4

6Dg(i) 2rh

1

15D g (i)l

15D#(i)

42Dg (i)

-,1~y4(i)

63Dg(i)

] La~(i)l

ward elimination process. In fact, matrix [L] contains all the elementary operations that are necessary to convert matrix [M] into an upper-triangular matrix. The backward substitution process is now applied to equation (12) to solve for the [X]*) values using the [S](p- ~)values. Once the unknown coefficients, [X](p), are found in a given iteration the nodal interface conditions and the eigenvalue are calculated changing the source vector. As a result, an iterative process is applied and continued until a given convergence criterion is satisfied. Since the elements in matrix [M] do not change unless the material properties and/or the geometry of the problem change, matrices [M'] and [L] will remain constant. Therefore, the forward elimination process is necessary before the first iteration only. The backward substitution process, however, is required at every iteration since the source vector, which is dependent on the eigenvalue and interface conditions, is updated following each iteration. For computational storage efficiency, the matrices [M'] and [L] can be combined in the same array as [Ms] = [ M ' ] + [ L ] - [ I I .

(9)

L - q7cg (i)J Obviously, the systems in equations (8) and (9) are small and can be solved by simple elimination techniques. As a result, applying both the forward elimination and backward substitution processes at every iteration is not computationally expensive. However, for multidimensional diffusion and transport theory models, or in cases where high orders of the polynomial expansion are required, the systems of equations are much larger and warrant the use of the technique that is described in the following section.

The process just described for calculating the unknown polynomial coefficients o f the diffusion theory nodal model can be used in the transport theory nodal model developed by Feiz and Rohach (1988). Unlike the diffusion theory nodal model, the coefficients of the polynomial expansion in the transport model cannot be separated into odd and even systems. In addition, for a given polynomial expansion order, the number of equations involved in the transport model is much greater than the number of equations in the diffusion model. As a result, the coefficient matrix, [M], is much larger in the transport model, and the application of the process becomes even more computationally efficient.

GAUSSIAN ELIMINATION METHOD

SUMMARY

The system shown in equation (8) or (9) is written as follows :

The determination of polynomial expansion coefficients of a diffusion theory nodal model was discussed. The coefficients are required after each iteration in the computational solution process. Due to the nature of the simultaneous equations, the even and odd coefficients are separated into different systems. The Gaussian elimination process was used to solve the equations. The forward elimination process is only necessary at the beginning of the iterative process as long as the geometry and/or material properties in each node do not change. However, the backward substitution process is necessary in every iteration. Using the process described in this paper, the number of calculations required to determine the unknown polynomial coefficients can be reduced. The advantage of using this process becomes apparent in multidimensional diffusion and transport models, and when high-order polynomial expansions are used.

[ M I [ X F ) = [s] (~- ')

(lO)

where the superscript (p) refers to a given iteration while (p:- 1) refers to the previous iteration. The elements of the matrix [M] and vectors [X](e) and [S](p-1) correspond to the values shown in equation (8) or (9). The node and group notations have been dropped from the matrix and the vectors in equation (10). It is, however, understood that the solution obtained from equation (10) is for group 9 of node i. The matrix [M] depends only on the material properties and the geometry of the problem. The vector [S], however, depends on the eigenvalue o f the problem through equation (7). The aforementioned characteristics of matrix [M] and vector [S] are important for the solution of the system of equations as discussed below. Equation (10) is rewritten in the following form: [MI[X] (p' = [ZlIS] (p ",

(11)

where [I] is the identity matrix. If the forward elimination process is applied to both sides of equation (11), the following would result : [M,][X]~O) = [L][SlW-1)

(12)

where the matrix [M'] is an upper-triangular matrix with diagonal elements equal to unity. The matrix [L] is a lowertriangular matrix obtained as the result of applying the for-

Acknowledgement--The first author wishes to acknowledge Ms Mary Catania from the University of Central Florida (BCC) for editing not only the present paper but previous papers and works of the authors. REFERENCES

Duderstadt J. and Hamilton L. (1976) Nuclear Reactor Analysis. Wiley, New York. Feiz M. and Rohach A. (1988) Ann. nucl. Energy 15, 389. Feiz M. and Rohach A. (1989) Ann. nucl. Energy 16, 521.