THE SOLUTION OF THE XYLTIGROUP NELTRON TIUNSPORT EQUATIOX ESIX S?HERICAL HAFClONICS J K Fletcher LXAEA Risley, Warrington Cheshire, England 4 A solution of the multi-group neutron transport equation in up to ihree space dimensions is presented. ,' The flux Og(~,g) at point z in direction 2 for energy group g is expanded in a series of unnormalised spherical harmonics. Thus where 6 and 0 are the .sxial and azimuthal ogles of i,,Pz(cose) the associated Legendre poiynoxials and N an arbitrary odd xiumber constant for all g. Using the various recurrence formulae of the Pz(cos6) a linked set of first order differential equations is obtained for the moments ~~m(~),~m(~). Terms with odd'1 are eliminated resulting ia a second order system which is solved by two methods. The first is a finite difference formulation using an iterative procedure, secondly, in XX! and KY geometry a finite element solution is given. Results for a test problem using both methods are exhibited and compared.
Transcript
THE SOLUTION OF THE XYLTIGROUP NELTRON TIUNSPORT EQUATIOX ESIX S?HERICAL
HAFClONICS
J K Fletcher LXAEA
Risley, Warrington Cheshire, England
4 A solution of the multi-group neutron transport equation in up to ihree space dimensions is presented.
,' The flux Og(~,g) at point z in direction 2
for energy group g is expanded in a series of unnormalised spherical harmonics. Thus
where 6 and 0 are the .sxial and azimuthal ogles of i,,Pz(cose) the associated
Legendre poiynoxials and N an arbitrary odd xiumber constant for all g.
Using the various recurrence formulae of the Pz(cos6) a linked set of first order differential equations is obtained for the moments ~~m(~),~m(~).
Terms with odd'1 are eliminated resulting ia a second order system which is solved by two methods. The first is a finite difference formulation using an iterative procedure, secondly, in XX! and KY geometry a finite element solution is given. Results for a test problem using both methods are exhibited and compared.
THR SOLUTION OF TRE XULTIGROKJP NSLTRON TRiXS?ORT EQIJATIQN USING SPRERICAL HARJlONICS
INTRODUCTION
The work to be described began in 1970 prompted by the fact that, as will be indicated in the metho,d section, diffusion theory is readily deduced from the transport equation when og(z,fi) the neutron flux at x in direction 2 for energy group g is approximated by a linear combination of seroth and first order spherical harmonics and it seemed reasonable to increase the solution accuracy by including more of these functions in the expression. Thus 4 '.
where 9 and 0 are the axial and azimuthal angles of unit vector &P$cos0) are associated Legendre polynomials which when multiplied by cos(m#) or sin(m0) form unnormalised spherical harm0nics.l $frn(~) and 7 (i-1 are the moments of the series and N the order of the approximation deno Ed-by PN. !!
When (1) was inserted in the multigroup transport equation a system of first order differential equations resulted. Computer size at that time, even in the one dimensional case, necessitated an iterative solution for there to be .sny practical applications. Instabilities and lack of convergence with the first order system solutions attempted led to an investigation of the second order equations resulting when moments with odd 1 are eliminated. This
approach had already been proposed2J3 and the equations, being of diffusion form, could be solved by es~tablished stable and~efficient methods then
'a~ilable~'~. .o
An algorithm for one dimensional plsne geometry and any order N was developed and then P3 XY successfully programmed. It became apparent however that any higher XY or even a P3 XYZ solution involved a great deal of tedious algebra to eliminate odd moments. A small computer program written to do this initially for P3 XYZ resulted in a generalised procedure to generate second order equations for any N.
RZ and spherical geometries were then included with a transformation built in to deal with the $ terms, where r is the radial coiordinate, that occur
and make the Pl approximation take the same form as diffusion theory.
Initially all solutions were based on a mesh centred finite difference treatment5. Latterly a finite element solution has been formulatede for yXZ and XY geometries using a weighted residual technique. The motive for this is the ability of the element method to deal with non-rectangular meshes and awkward geometrical shapes.
DERIVATIOK OF THE 110?1ENT EQVATIOXS
The neutron transport equation is expressed in the form2
z JJ us(CJ',$,E',E,~)0(~,~';E')d$'dE' + S&&E) (2)
E' 2'
where ut(z,E) is the total cross section for energy E, CT~(~~,~~E~,E,;) the scattering from SJ',E' to $E including fission and S(L,C?,E) any external source.
The multigroup approximation is made by dividing the range of E into a series of intervals Egml, Eg where g goes from 1 to NG and Eo is the maximum energy.
Assuming that quantities in Lath interval are constant or that the form of 0(~,fi,E) is known to sufficient accuracy 2 is integrated from E to E and putting g-1 !3
J E g-1
E 'Jt(Z,E)O(z,$E)dE = utg($O;(@)
i3 .
=E J G~g,g(~',~,~)Og,(~,~')~' g'=l fl!
g=l, NG
The method presented in this paper solves the above equation group by group beginning at g=l using the latest values of 0 ,(~,fi)
E. to take
advantage of the predominant dowr.scatter and itera ug if there is any fission source or up scatter. Convergence for such a scheme is assumed and hence all source terms may be added to Sg(z,SJ) and the g tubscript omitted.so that the equatitin for solution is:
‘0110000zi
Accepting tha: scatter is a function 6nly of tha angle between 2' and fi the following expansion is made
to transform the products caused by expression 5 back to spherical harmonics. Since these functions are independent each coefficient must be zero and the moment equations below are found
The dependence on 2 or x,y,z is understood and o1 = ot(~) - U:(L) which results
because of the orthogonality of spherical harmonics. St m and S'Im come from
the expansion of S(x,g) in the same manner as the flux. *
There is a proviso that in the n=l equations coefficients of qIo are
multiplied by two. This is caused by not using the third recurrence
where the dash denotes differentiation with respect to cos@ gives
BOUNn4RY CONDITIONS
Three types of boundary are considered
i At a reflective boundary tha moment is zero if the harmonic is odd across it and the normal derivative is zero in the even case.
ii At material interfaces moments are continuous because the flux is continuous and the harmonics are independent,
iii The correct condition is zero flux for 2.n < 0 where 2 is the outward normal to the surface. In this method the diffusion theory condition
~,v$OO = - x 3uc) $00
where X i& arbitrary and elm = o for all other 1m is employed. The reason
is programming simplicitj and if surface fluxes are required some mean free paths of pure absorber can be put round the problem.
ONE DIMENSIONAL PLANE GEOMETRY
The operator fi.V takes the form cos0 se so that if the L r terms in equation
10 are ignored and the variable changed to a the plane geometry moment equation is
When all quantities apart from *I are known then 11 is recognisable as a diffusion equation with an external source. by a finite difference approximation5.
This is solved straightforwardly The problem is divided into a series
of intervals with length hi and mesh points located at their centres. Equation 11 is integrated over each interval assuming dqI is approximated
az- by 5
%i*l - @ii)
hf+l + hf as appropriate dealing similarly with the I+2 terms. At
material in&faces, if o and o are the cross-sections each side of the boundary between i and i+ f: then ~~%z the continuitv of the odd moments and expanding qIi and $li+l in a Taylor second orders and above, an average
series about that boundary ignoring value
.6 is found to multiply the approximation to % . Gradients at the bounds of the problem are deduced from i) or iii) ofczhe boundary conditions.
The result is an equation with a tri-diagonal coefficient matrix for qIi which can be solved by forward substitution backward difference5 or any suitable method.
Beginning with aguess of qii, say $loi = 1 and qLi = 0 otherwise any fission
source is calculated then equation 11 is scLved in the aanne: described for 1 = 0, 2, . . N-l.
The fission source is now recalculated. In the case of no external source (as in diffusion theory) there is an eigenvalue which is taken to divide the fission source. This is adjusted by the ratio of the scalar product of the new fission source with itself to that with the previous value. The evaluati~on of the moments is repeated and the whole procedure iterated until flux changes are less than required accuracy criteria.
The multigroup solution follows exactly the same path except that the scatter source is added before equation 11 is solved.
It stay be thought that for the one group case it is unnecessary to treat the fission source separately when there is no eigenvalue search. This is so but the algorithm is programmed generally to accommodate the multigroup problem where the fission spectrum is spread over several groups and cannot be included in the ol.
01100007
A further point is that, exc+t in the oze group case the iteration is a dual one. That is moments are not calcul.a:ed exactly for a giver. group but evaluated once in the order 1=0,2,. . .:i- 1 before proceeding to cbe next gro”p . This is the procedure followed in all geometries and Rio conyergence problems have occurred.
The equivalence of Pl and diffusion rheory mentioned in the introduction can be quite clearly shown in equation 11 with 1=C and X=1
It can be seen from 11 that N may be odd or e;en. However the same number of equations is solved whether N has th& ‘form 2n or 2n+l so the higher approximation that is N odd is usually taken.
27 GEOMETRY
The moment equations are found Ky ignoring all derivatives with respect to z and noticing that the flux is even about the LY plane so that odd f+m does not occur as these harmonics are odd. If ~1, for clarity, are assumed constant with IJ =ml,f>l and.S(z,s) isotropic then the P3 approximation on f eliminating moments with odd ! gives the following four equations,
$ Woo - E B$+o + 9 py22 - 5uL2 Y22 = 0
These are treated as in one dimension except that the intervals are now rectangles and the integral is over dxdy. A problem is the representation of the mixed derivatives. a21axay is considered for a general even moment Z(x,y). .so. that~ ..
JJ it? dxdy =k $$ dy -h , g dy
cell axay
with S’,S the surfaces parallel to the y axis is required
Denote x meshes by h: and y meshes by hy. At cell i,j, S is the boundary of
length h{ betveen i,jand i+l,j. j
To approximate in finite differences to this integral equate odd moments and repeating the procedure of the one dimensional case-it is -found that a term
0.11clQ0QEi
,’ L’,
occtlrs. Tne differential is approximated as the mean of that at i,j and i+l,j thus giving
where the subscripts cn f denote location, second orders have been neglected
an' ul,i,ja u.t,i+l,j are the relevant cross sections.
When the difference equations for each moment are written down; assuming other quantities are known, the coefficients form a block tri-diagonal matrix in which the matrices on the diagonal are themselves tri-diagonal. This system is conveniently solved by the over-relaxation iterative technique*'s. The rest of the calculation follows the one dimensional method.
So far second order equations have not been too difficult to derive. However for the ~5 case there are nine moments each involving a great deal of algebra to eliminate odd moments. Similar difffculties occur with P3 XYZ. where there are six even moments.
A computer program developed in FORTRAN primarily to find the equations for P3 XYZ resulted in an algorithm that was quite genera17.
The manual calculation is simulated almost exactly. The moment equation for I=m=O is represented in the computer. This can be visualised by regarding the equation as three rows or vectors. The top row contains the moments in the equation. The second row the appropriste coefficient and the third row
the differential signified by an integer ie, 1 for $-, 2 for'%, -jY
3 for ?&.
For each odd moment in the first row the preceding prescription is followed and the appropriate coefficient from the original equation multiplies the middle row of the second system and similarly with the derivatives giving 1,
2, 3, 4, 6 or 9 where for instance 3 represents & a2 and 9 3. The
resulting coefficient is stored in an array at a location depending on the moment in the top line of 'the second equation and the derived differential. Th%s is perfcrmed for all the odd monents in the first equation. ,,It could happea thatthe'.same' location fnthe ~effic~entarray ?~s.s-pec+ied more-.... ~. than once and of course the cumulative sum is taken.
The whole array is now printed out giving the second order equation of & and this is repeated for each even 1 moment up to N-l.
Concurrently with the printing the numbers are stored in a suitable serial order for use in the flux calculation which can now proceed in exactly the same manner as the ~3 solution.
XY and XYZ can also be used with equilateral triangular cells and triangular prisms respectively. The only difference is that some cell boundaries are not parallel to the axes and combinations of derivatives with respect to x and y occur here. The a direction is not affected as prism faces are always parallel or perpendicular to this, axis.
RZ GEO?KTRY 1 Mae new feature in equation 9 is the ; term, this is followed by each location
in the coefficient array mentioned in the 1YZ derivatcon now having four values. The general operators are
The first product is rearranged to
L r
la This implies the usual ; s r $ operator in the l-=0 equation so that Pl
and diffusion theory have the same form and the four coefficients ac, (b-a)c2 ad and (b-a)d are stored. The second product is treated in the same manner but the third left unmodified. With these changes the procedures outlined previously are sufficient to set up difference equations with the exception
of the integral of $ over 2 cell containing r=O. The moment is assumed ~,
.constant in each cell so the logarithm of infinity results on integration. However the flux is finite at r=O so that when this term occurs that moment concerned must be zero on the z axis or other moments will be infinite and hence its contribution can be neglected.
SPHERICAL AND PIANE GEOXETRY
The techniques already described are sufficient to deal with these geometries. Although it is not strictly necessary to use a program to eliminate odd moments a neater flux solution from a programming point of view results if the coefficients are precalculated.
EXAMPLES
The problem is described in FIG 1. 'l%ere is no scatter so that an accurate answer can be found by integrating over the source volume, eg l
where
r = (Cx - x'12 + cy - y'F+ (2 - zv+
and S = 111.44
The scalar flux along BC for several PN approximations and the exact calculation are shown in TABLE 1. For the P7 case FIGURES Z-6 display the even flux moments. It is seen that the P7 results are close to the exact result aven though the flux is contained in the forward solid angle subtended ~ by the source to the point and hence is highly anisotropic. No convergence problems occurredand the P7 case took about 3 minutes on an IBM 370/168 using a 30 x 30 .2 cm mesh. The problem was extended to a six centimetre square because of the approximate surface boundary conditions.
Although this paper is concerned mainly with a description of the method some indication will now be given of the range of applicability.
Tn plane geometry up to a I'23 calculation with sixty mesh points has been solved.
For RZ geometry when studying neutron streaming, react<on rates and control rod worths ?5 approximations in 37 energy groups using about a 1000 mesh points have been performed.
A ?3 triangular prism problem with ZS,OO&mesh points and 16 energy groups has converged.
Running times increase by a factor of about 2% in two dimensions between successive PN approximations. In three dimensions the ratio is 3-4 thus a P5 calculation takes about ten times as long as diffusion theory.
These factors are less than the increase in the number of moments over diffusion theory and appear.to be caused by faster solutions of the higher moments occurring in the inner iterations.
FINITE ELEMENT SOLUTION
Since the inception of this PN method considerable effort has been expended by others in solving the diffusion equation by the finite element method. Instead of approximating to differentials using Taylor s&es expansions neglecting higher orders 55 in finite differences a polynomial approximation is made to the flux in a series of discrete volumes into &ich the problem has been divided. The coefficients of the polynomials are expressed in terms
,~ of fluxes at set nodes on the cells and matrix equations are derived for these nodal values. This formulation has the advantage that no restrictions are put on the shape of the cells and hence awkward geometries can be treated.
XY7, and XY geometries have been solved in the PN approximation using this approach. The basic assumption of the finite difference procedure is kept that is convergence will result when each moment equation is solved in turn taking other terms as known and iterating.
a= with VI2 = aim ox a2 a2 7 + 'fm p + 'fm 3
a and @ take the values one to three and x1 = *fm, *;* = Yfm. 'Irn includes two and qirn
If equation
J V “fm
11 is true then
dy a2qitrnt
f'm'cq3 ax ax dV = 0 f 'm'Cr@y cz @~
(13)
= x, x2 = y, x3 = z. y is one or all sources
01100011
where Wim is any f*unction and V the volume of the problem, The idea is to make apprcximations to \J1m and qlm derive a matrix equation ror qlm.
then use the arbitrariness of Xim to
The only condition put on polynomial expressions for W1m and qlm is that when the problem is discretised there is continuity across internal boundaries. Second order derivatives occur in equation 13. If the approximating functions are continuous, first order derivatives may n,ot be and second orders would then be singular on the boundary. To preclude any integrals of a delta function type 13 is usually put in firstorder form using Gauss theorem or integration by parts. Thus on dividing the problem into cells.
and transforming
with 1'm' terms taken in the direction x cx and where S
j are external surfaces
since the continuity of moments eliminates internal terms if W1m is continuous.
Some of these surface integrals may be expressed differently. For example since 2.V *oo = - 3 uo 1 $oo (see boundary conditions) a term of the form
is found. However there are those of the type.
which remain.
XY GEOMETFCY
The problem is divided into quadrilaterals and moments assumed to take the form in the ith region.
*I* = ai + bix + ciy + dixy
These have the required continuitya across boundaries. Let qlrnk be the values
cf the moment at the four vertices or nodes of the cell where k is either a local nunbering 1 to 4 or some global value for the whole problem, then for each value of k
As .d ~~ is required to be continuous put
wink= a; + b;xk + c;yk + d; \yk
The constants in the above equations can be found in terms of the qirnk and "1rnk' Reordering the terms there results
*hrl fk(x%Y) qlrnk 9 '@ =xfk(x,y) WIrnk
8 k k
These are inserted in equation 14 and the integrals evaluated' for terms up to x2y2 over Vi and Sj where applicable. The result can be expressed in
terms of coefficients-of Wirnk thus
L*ere the gkC91ms ~~,m,n,ssia) are linear :o!nbinations of the $'lmn and
source terms @I,*,*, Film from 14. Since tb2 Yfmb are arbitrary each % coefficient must be zero to satisfy equation 15 and the result may be written
are matrices and equation 16 is solved using Gaussian elimination'. As stated in the opening paragraph of this section each moment is evaluated in the same manner iterating until convergence criteria are satisfied.
.The source problem of FIGDRB 1 has been solved by this metbod using the same nesh size as the finite difference solution, that is 900 square cells of side .2 centimetres. Table 2 shows values of scalar flux for the P7 approximation along AC for both solution algorithms and a graph of these results is given in FIGDPxE 7. The running time is about a factor of five longer than finite differences but no attempt at code efficiency has yet been made.
A solution was also obtained using triangular cells or elements but this gave odd results along AB and AD of FIGIJRE 1 for the scalar flux outside the source region. A negative flux gradient perpendicular to the boundary occurred and it should be sero. The cause of this has not been investigated.
~01100013
XC2 GEOHETRY
The most recent work is an XYZ solution using the approximation
4flrni =ai i bix + ciy + diz + eixy + fixz + giyz + hixyz
A sxxh more complex programming task results but following the pattern for XY geometry Pl and P3 solutions have been obtained for a small test problem.
This consisted of a unit cube reflected on three faces with 64 .Z cm cubic cells, no absorption and transport and production cross sections of unity. The eigenvalues given are
Pl .7793314
P3 .?098760
with J, = .5 on the other three faces for $ oo and all other moments zero there.
A solution of the multigroup transport equation has been found by expressing the flux as a series of spherical harmonics. Good agreement with an exact qalculation has been obta&d for an XY test problem in either a finite difference or finite element formulation. The method works in many geoznetry options and is not dependent on the number of energy groups. Finally to deal with awkward or irregular boun.daries finite element solutions have been demonstrated in XT and XYZ geometries.
References
I. H Margenau and G ?I Murphy. The Mtithematics of ?hysics and Chemistry, D Van Nostrand, Princeton, New Jersey (1956) Chapter 3.
2 M Clark Jr and K F Hansen. Numerical Xethods of Reactor ha~ySiS.
Academic Press, New York (1964).
3 B Davison. Neutron Transport Theory. Oxford University Press- Oxford (1957) Chapter 10.
R S Varg&Matrix Iterative Analysis. ?rentice Hall (1962) Englewood Cliffs New Jersey.Chapter 9.
3 J K Fletcher. CTD. A Computer Program to Solve the Three dimensional Multigroup Diffusion Equation. TRG Report 2344(R) 1973 United Kingdom Atomic Energy Authority.
qe J K Fletcher. A Solution of the Multigroup Transport Equation Using Spherical Harmonics. TRG Report 2547(R) 1974. United Kingdom Atomic Energy Authority.
7 J K Fletcher. Further Work on the Solution of the Multigroup Transport Equation Using Spherical Harmonics TRG Repnrt 2849 (RI 1976 United Kingdom Atomic Energy Authority.
3 J K Fletcher. A Finite Element Option for the MARC Transport/ Diffusion Theory Code. ND:R-56O(R) lYS0. United Kingdom Atomic Energy Authority.
9 A M Weinberg and E P Wigner. The Physical Theory of Neutron Chain Reactors.. Chicago University Press (1958) Chicago. Chapter 9
