Assembly homogenization techniques for core calculations

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Progress in Nuclear Energy 51 (2009) 14e31www.elsevier.com/locate/pnucene

Review

Assembly homogenization techniques for core calculations

Richard Sanchez*

Commissariat a l’Energie Atomique, DEN/DANS/DM2S, Service d’Etudes de Reacteurs et de Mathematiques Appliquees,CEA de Saclay, 91191 Gif-sur-Yvette Cedex, France

Abstract

We have applied the black-box paradigm to assembly homogenization and introduced current discontinuity factors (CDFs) for an arbitrarylow-order operator in the presence of boundary leakage. The CDFs preserve average reaction rates and the assembly partial currents in a givenreference situation as well for full assembly as for pin-by-pin homogenization. In the presence of surface leakage, the CDFs depend on the dis-cretization of the low-order operator but can be determined from a few calculations with the low-order operator without scattering. For diffusion-like, low-order operators, the CDFs can be advantageously replaced by flux discontinuity factors (FDFs), which also preserve partial currents.However, the effect of the FDFs is not equivalent to that of the CDFs in the final core calculation. Unlike the CDFs, that are double-valued forhomogenization with surface leakage, the FDFs are always single valued. The cases when the low-order operator is diffusion, SPN or quasidif-fusion are discussed in detail. We also show that, for full-assembly homogenization without boundary leakage, the FDFs are identical to Smith’sdiscontinuity coefficients (DCs) only if the reference calculation has also been done with diffusion. In the case of diffusion, preliminary testcalculations for small PWR motifs show that the FDFs and Smith’s DCs give close results, with a better precision for the FDFs when transporteffects are predominant.� 2009 Elsevier Ltd. All rights reserved.

Keywords: Assembly; Homogenization; Current discontinuity factors; Flux discontinuity factors; Diffusion; SPN; Quasidiffusion

1. Introduction

Homogenization techniques are frequently applied in dif-ferent domains of physics and engineering for the calculationof large complicated systems, comprising numerous heteroge-neous components, for which a direct calculation is not feasi-ble, or is too demanding in computing time and resources to beused in routine calculations. The basic idea of homogenizationis to replace the heterogeneous components with homoge-neous ones so that the calculation of the homogenized systemyields accurate average values. Because the function to becalculated is much smoother than the original one, a directconsequence of the homogenization is to reduce computingtime and memory requirements. A further gain can be obtainedby replacing the original exact operator with a low-order onethat eliminates some of the variables of the original problem.

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0149-1970/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.pnucene.2008.01.009

Such an operator has to be constructed from the analysis of thebehavior of the solutions of the exact operator over largehomogeneous components. In other words, the low-order oper-ator has to be derived as an asymptotic limit of the exact one.

Because the homogenization replaces heterogeneous com-ponents with homogeneous ones and the number of homoge-neous parameters per component is necessarily small, it isnot possible for the homogeneous calculation to preserve allthe details of the heterogeneous one. Clearly, there is a lossof information in the homogenization process and only global,average values can be preserved. Thus, the homogeneousparameters associated to each heterogeneous component oughtto be determined to preserve the reference averaged values thatwould be obtained from the hypothetical solution of the orig-inal heterogeneous system. However, since the latter solutionis not known, this poses the problem of how to obtain thereference averaged values to be preserved by homogenization.The way out of this dilemma is to determine the referenceaverage values by calculating separately each heterogeneous

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15R. Sanchez / Progress in Nuclear Energy 51 (2009) 14e31

component with the exact, reference operator. These referencecalculations have to be done for a reference problem whosesolution is close to the solution that would be obtained if theentire heterogeneous system were calculated.

However, this is not enough to ensure that the averagevalues yield by the homogenized equation are close to the ref-erence values. The reason is that the homogenization is donein a reference problem separately for each heterogeneous com-ponent and that the homogenized components do not necessar-ily reproduce the boundary values that would be predicted bythe calculation of the entire heterogeneous system. To palliatethis difficulty one resorts to the technique of black-box homog-enization, known in reactor physics as Selengut homogeniza-tion (Selengut, 1960). The basic idea is to construct eachhomogeneous component in a way that the replacement ofa heterogeneous component by the homogeneous one willnot change the global solution of the original heterogeneoussystem. Again, for the reasons earlier discussed, this aim ispartially attained by introducing discontinuity coefficients onthe surface of the homogenized component so that it repro-duces average field boundary values of the reference problem.

In this work we shall apply these ideas to the analysis of thehomogenization techniques that are currently in use for reactorcore calculations. In particular, we use the black-box homog-enization approach to establish a homogenization paradigmbased on the conservation of the partial currents in the refer-ence problem via the introduction of discontinuity factorsfor these currents. This contrasts with the currently used dis-continuity factors that aim to preserve flux boundary values(Smith, 1980, 1986).

There is a large literature concerned with assembly homog-enization techniques. Most of the earlier work in homogeniza-tion was done with analytical techniques and was based on, bytoday’s standards, rather drastic approximations: one-grouptheory, isotropic scattering and simple infinite lattices. How-ever, there is a wealth of physical insight to be found in theearly literature which has defined the ground ideas for modernhomogenization techniques based on detailed numerical calcu-lations. As early as in 1946, Feynmann and Welton (1946)used the conservation of the kN to homogenize an infinitelattice in transport theory, Plass (1952) computed diffusionlengths in a finite lattice and, inspired by the work of Benoist(1959), Leslie (1962) analyzed the effects of neutron migra-tion on the anisotropy of diffusion coefficients. In a relativelyrecent work, Williams and Wood (1971) studied anisotropy ofdiffusion in a homogeneous periodic lattice with periodic sour-ces and derived expressions for reaction rate preservinganisotropic diffusion coefficients. For more references in earlyliterature consult this later work and Williams (1971).

The ground ideas for modern homogenization methods arediscussed in the books by Henry (1975) and Stamm’ler andAbbate (1983). Also, the interested reader could profitablyconsult current research as presented in the biannual topicalmeetings of the Mathematics and Computation and the Reac-tor Physics Divisions of the American Nuclear Society. In thepresent work we shall concentrate on the two prevailing ho-mogenization techniques. These techniques are known under

the names of equivalence or super equivalence homogeniza-tion. To avoid confusion, we will use the label of equivalencetheory to denote the technique based on the sole conservationof pin-by-pin average reaction rates (Kavenoky, 1978;Kavenoky and Hebert, 1981; Sanchez et al., 1988; Hebert,1993; Hebert and Mathonniere, 1993), and reserve the nameof discontinuity factors to the techniques that preserve globalassembly reaction rates and, somewhat in the spirit of black-box homogenization, flux boundary values (Koebke, 1978,1981; Smith, 1980, 1986).

Today, reactor core calculations are done by solving a few-group diffusion equation in a simplified core geometry ob-tained by assembly homogenization. The quantities of interestare the reaction rates and the core eigenvalue l and, usually,homogenized cross-sections are obtained by preserving trans-port reference reaction rates in a reference problem. Forassembly homogenization, the reference problem shouldclosely reproduce the state of the assembly in the reactorcore, but is usually chosen as that of an infinite lattice obtainedby replication of the basic assembly. A leakage model can bealso applied to obtain a critical (l¼ 1) assembly with conser-vative boundary conditions, although other reference situa-tions can be used in particular locations in the core, such asfor an assembly near the reflector. The construction of thereference problem is discussed in Section 2 where we advancetheoretical arguments in support of the infinite-lattice ap-proach combined with a leakage correction. A short discussionof the fundamental mode is included in Appendix for easy ofreference. In the following section we present the basic ideasof cross-section homogenization and analyze, in particular, thecase of anisotropic transfer cross-sections. The application ofequivalence theory to pin-by-pin assembly homogenizationis shortly reviewed in Section 4, where it is shown that foran infinite lattice, or for an assembly with no surface leakage,the equivalence problem degenerates and requires a supple-mentary constraint. Two familiar normalization techniques,flux conservation and Selengut’s normalization, are also dis-cussed in this section.

The remaining of the paper is consecrated to the black-boxhomogenization technique via current discontinuity factors(CDFs). The CDFs are introduced in Section 5 in order tocomply with black-box homogenization. The analysis isdone for an arbitrary reference situation with surface leakageand for a non-specified low-order operator, first in the caseof full-assembly homogenization and then for pin-by-pinhomogenization. In the presence of surface leakage there aretwo CDFs per surface, one for the incoming current and thesecond for the outgoing one. These discontinuity factors areequal only in the case of conservative boundary conditions.Also, for full-assembly homogenization, the CDFs are inde-pendent of the low-order operator when the reference problemhas conservative boundary conditions; the reason is, of course,that in this case the solution of the homogenized problem isanalytical (the flux is isotropic and constant.) For homogeniza-tion with surface leakage, and this is the case of pin-by-pinhomogenization, the CDFs have to be determined by solving afew source problems (with zero scattering) with the low-order

16 R. Sanchez / Progress in Nuclear Energy 51 (2009) 14e31

operator; note that, as opposed to equivalence theory, theconstruction of the CDFs is done via linear equations and doesnot involve iterations. The following three sections illustratespecific applications to three different low-order operators:diffusion, simplified PN (SPN) and quasi diffusion. For thesethree diffusion-like operators it is also possible to implementpartial current conservation via flux discontinuity factors(FDFs). There is an advantage in the use of FDFs in thatthey need a single value per surface, even in the presence ofsurface leakage. However, although the CDFs and the FDFsare identical for conservative boundary conditions, their appli-cation to core calculations will give different results, as shownin Section 6, where we also show that the FDFs are identicalto Smith’s discontinuity factors when the reference problem issolved with diffusion; it is only in this case that the familiardiscontinuity factors (Smith, 1986) preserve also the partialcurrents in the reference situation. At the end of this sectionwe give a numerical comparison between Smith’s discontinuitycoefficients, the FDFs and equivalence theory with Selengut’snormalization for a small 3� 3-assembly PWR motif with a cen-tral AIC assembly. Homogenization for SPN or quasi diffusioncore calculations is discussed in Sections 7 and 8. In each of thesesections, which begin with a short introduction to the basic modelequations, we present the particular form of CDFs and, then, in-troduce the equivalent formulation based on FDFs.

There are, of course, other homogenization techniquesbesides equivalence theory and discontinuity factors. As an ex-ample, we present in Section 9 a technique based on perturba-tion theory. Also, it is difficult to talk about homogenizationwithout discussing dehomogenization, i.e., the techniques thatare used to reconstruct detailed power distribution from the finalcore fluxes. For completeness, in Section 10 we briefly discusssome of the more popular dehomogenization methods. Conclu-sions and further comments are given in Section 11.

2. The reference problem

The reference problem should represent as close as possiblethe actual state of the assembly in the core, as described bytransport theory. But, most often assembly homogenizationis done using reference reaction rates obtained from an infinitelattice-transport calculation with critical leakage. In this sec-tion, we present some theoretical justifications in support ofthis choice for the reference problem.

The angular flux j(x) in an assembly, or in any subdomainD of the core, is given by the transport equation

ðU,VþSÞjcore ¼ qcore; x ˛X;

jcore ¼ j�; x ˛G�:ð1Þ

Here X ¼ xf ¼ ðE; r;UÞ; E ˛ E; r ˛ D; U ˛ 4pg is thephase space with energy domain E and geometric domain D,G� ¼ x; E ˛ E; r ˛ vD; U ˛ð2pÞ�g

�is the entering bound-

ary of X, j�(x) is the angular flux entering the assembly fromthe surrounding assemblies and

qcore ¼ Hjcore þ1

lcore

Pjcore

is the source density, where H and P stand for the scatteringand the production operators, and lcore is the core eigenvalue.

Eq. (1) defines the exact reference problem. However, theincoming boundary flux j� is not known and one is led todefine an approximated reference problem by introducing a re-alistic approximation for the boundary condition. A familiarapproximation is to homogenize the assembly with specularboundary conditions. This model is based on the assumptionthat an assembly surrounded by identical assemblies can beviewed, as a good approximation, as an infinite lattice. A the-oretical justification can be found in a seminal work by Larsen(1975), who proved that the angular flux in a core of largedimensions composed of identical assemblies can be written as

jcoreðxÞ ¼ f ðrÞjðxÞ þjeðxÞ: ð2Þ

In this expression f is the solution of a one-group diffusionequation with a symmetrical diffusion tensor and globallyhomogenized cross-sections (Larsen, 1975), j is the assemblyflux in an infinite lattice and je is a term of order e¼ assemblysize/core size. This result translates in clear mathematicalterms the physical intuition that, as the number of identicalassemblies in the core increases, the flux in the core gets themore and more close to f(r)j(x).

However, even when the original assemblies are identical,an infinite lattice may be a poor approximation for an assem-bly in a depleted or reloaded core, or for BWR cores andmodern PWR cores with MOX fuel, which contain assemblieswith markedly different neutronic properties. In this case, tohomogenize an assembly one may define a reference problemin a larger domain comprising the assembly and its immediateneighbors or a suitable homogeneous buffer obtained byhomogenization of the neighboring assemblies, or an assemblycolor-set. Nevertheless, even by today’s standards these calcu-lations are relatively expensive and one usually resorts to theinfinite-lattice reference problem.

Another caveat is that the calculation of an assembly in aninfinite lattice leads to an eigenvalue different from the coreeigenvalue and, therefore, the flux in the assembly can befar from the critical spectrum in the core. To constructa more realistic reference problem it is customary to adda leakage model so that the eigenvalue of the lattice calcula-tion is lcore¼ 1.

The leakage model also can be explained by resorting to as-ymptotic theory. By replacing the main flux in (2) in assemblyproblem (1) one finds that, to leading order in e, the assemblyflux obeys the transport equation:�

U$VþU$Vf

fþS

�j¼ q; x ˛X

with periodic boundary conditions. In this equation the macro-scopic function f plays the role of a critical parameter thatmust be adjusted so that the eigenvalue equals lcore.


By choosing a exponential mode (Benoist, 1964; Deniz,1986; Sanchez, 1995; Petrovic and Benoist, 1996),f(r)¼ exp(iB$r), the above equation becomes

ðU$Vþ iU$BþSÞj¼ q; x ˛X:

But now the transport equation contains the anisotropic cross-section iU$B and the angular flux j is a complex function sothat jcoreðxÞ ¼ R½expðiB$rÞjðxÞ�, where Rf indicates the realcomponent of f. This equation is very difficult to solve andsolutions have been only derived for low-order P1-like approx-imations (Petrovic and Benoist, 1996). Hence, for typical corecalculations, an approximate leakage model is defined byreplacing the anisotropy term iU$Bj by its angle averagevalue i(J$B/F)j, and by computing the latter from the flux-homogenized assembly so that i(J$B/F) w DB2, where D(E )is the leakage coefficient obtained from the flux-homogenizedassembly and B2 is the corresponding critical buckling. Thisleakage technique relies in the well-known fundamental-mode theory. For reference, a summary of this theory is givenin Appendix.

With this approximate leakage model the final referenceproblem is�U$VþDB2þS

�j¼ q; x ˛X

with periodic boundary conditions.We end this section by pointing out that another way of

defining the reference problem, without need for a leakagemodel, could consist of using an albedo boundary conditionof the form

j�ðxÞwaðbjÞðxÞ; x ˛G�; ð3Þ

where a is a critical parameter, which serves to adjust theeigenvalue to that of the core, and b is a normalized albedooperator. The albedo can be specular or piecewise isotropic,or send back neutrons according to a given normalized spec-trum. The latter can be obtained by calculating the assemblyand its immediate surroundings with an approximate (forexample, specular) boundary condition.

3. Cross-section homogenization

In this section we discuss general properties of the homog-enization process. The first aim of homogenization is topreserve the reaction rates and the eigenvalue obtained fromthe transport solution of the reference problem. The result isa set of homogenized cross-sections {SI

G} that are constantover a set of macrogroups G;Gf ¼ 1;NEg and a set of macro-regions {I, I¼ 1, ND}. The associated reference reaction rates,eigenvalue and leakage are typically obtained from group col-lapsing and spatial homogenization of a fine-group, spatiallydetailed transport calculation or from a Monte Carlo calcula-tion (Ilas and Rahnema, 2003).

Homogenized cross-sections are generally obtained by in-troducing a spectral weight function. For a given macrogroupG and a macroregion I:

Sh ¼

ZG�I

ðSwÞðE;rÞdE drZG�I

wðE;rÞdE dr

¼ t

VIwI

; ð4Þ

where Sh is the homogenized total cross-section, S(E, r) is theheterogeneous cross-section and w(E, r) is a given weightfunction. Also, t is the total ‘reaction rate’ for cell G� I (mac-rogroup G and macroregion I ), VI is the volume of macrore-gion I and wI is the (energy-integrated) volume average ofthe weight function.

For example, volume averaging results from the choicew¼ 1, while flux averaging is obtained by using the referencetransport flux F:

Sh ¼ S¼ t

VIFI

ð5Þ

where t ¼R

G�IðSFÞðE; rÞdE dr is the total reaction rate overG� I and FI is the volume average of the reference flux. Afurther example is given by equivalence homogenization,where explicit use of the low-order flux Fh

I ,

Sh ¼ t

VIFhI

;

guarantees that the low-order operator will preserve reactionrates.

A general property of spectral weight homogenization isthat all other homogenized cross-sections in G� I can beobtained from the homogenized total cross-section Sh andthe reference reaction rates. Indeed, if Sx,r

h denotes thehomogenized cross-section for isotope x and reaction r, then

Shx;r ¼

tx;r

VIwI

¼ Shtx;r

t; ð6Þ

where tx, r is the reference reaction rate. Also, for an isotropictransfer cross-section (r¼ s),

�Sh

x;s

�G/G0

¼�Sh�G

�th

x;s

�G/G0

tG; ð7Þ

where G and G0 denote the departure and arrival macrogroups,respectively, and ðth

x;sÞG/G0 is the reference transfer rate.

Because global particle balance is fundamental to thehomogenization process we turn our attention to the globalbalance equations for the reference and the low-order opera-tors. For a given macrogroup G we haveZ

G�vD

dE dS$JðE;rÞ þX

I

SIVIFI ¼X

I

VIQI ð8Þ

andZG�vD

dE dS$JhðE; rÞ þX

I

ShI VIF

hI ¼

XI

VIQhI ; ð9Þ


where we have omitted the macrogroup index and used upperindex h to denote the low-order operator, vD is the assemblyboundary, the SI are the flux-homogenized cross-sectionsand FI and QI are the averaged flux and source over macrore-gion I, respectively. For an eigenvalue problem the source termon the right-hand-side of these equations contains the contri-bution from isotropic scattering and fission:

Q¼ HFþ 1

lPF;

QhI ¼ HhFh

I þ1

lhPhFh

I ;

where H and P are the transfer and production operators,respectively, and l is the eigenvalue.

The eigenvalue depends on global balance and it is clearfrom a comparison of Eqs. (8) and (9) that preservation ofmacrogroup and macroregion averaged reaction rates is notenough to ensure that the homogenized problem will havethe same eigenvalue, lh¼ l. For this to be true, one needsalso to preserve the overall reference leakage. This latter con-dition is automatically fulfilled, however, when the referenceproblem has conservative boundary conditions, so the leakagevanishes for both operators. As we shall see, conservativeboundary conditions entail a degeneracy of the homogeniza-tion technique in that there is an infinity of solutions permacrogroup.

3.1. Anisotropy of collision

We turn now to the problem of homogenization of aniso-tropic transfer cross-sections. Global balance does not explic-itly depend on the anisotropic cross-sections and, therefore,does not provide a guideline as to how homogenize thesecross-sections. Moreover, even when the reference transportcalculation includes anisotropy of scattering, it is alwayspossible to construct an isotropic low-order operator thatpreserves isotropic transfers (this is necessarily the case forthe diffusion operator.) Nevertheless, anisotropy effects canbe considered important to be incorporated in the low-orderoperator. To analyze this case we write global conservationfor the angular flux moments in macrogroup G:Z

G�vD

dE dS$JklðE;rÞ þZ

G�D

SakFklðE;rÞdE dr¼X

I

VIQkl;I

andZG�vD

dE dS$JhklðE;rÞ þ

XI

Shkl;IVIF

hkl;I ¼

XI

VIQhkl;I :

In these equations

FklðE; rÞ ¼Z

AklðUÞjðxÞdU;

JklðE; rÞ ¼Z

UAklðUÞjðxÞdU

are the angular flux and current moments with respect to the(real) spherical harmonic Akl, jlj � k, the Qkl are the angularmoments of the source and Sak¼S� fkSs, where fk ¼R 1�1dmPkðmÞPðmÞ=

R 1�1dmPðmÞ is the angular average of Legen-

dre polynomial PkðU$U0Þ with the scattering kernel PðU$U0Þ.Since fission is taken to be isotropic, the source terms Qkl

contain only out-of-group anisotropic contributions and, there-fore, the previous balance equations do not explicitly dependon the eigenvalue. Assuming that the anisotropic leakage ispreserved, one can enforce preservation of the anisotropicreaction rates by defining

Shkl ¼

tkl

VIFhkl;I

; ð10Þ

where tkl ¼R

G�IðSakFklÞðE; rÞdEdr and Fklh is the angular

flux moment as predicted by the low-order operator. Notethat the Skl

h depend on both k and l. This is unavoidable ifone wants to preserve the tkl for all l, but, if necessary, requiresminimal changes on typical PN and SN solvers.

A more cumbersome problem is that the weight functionFkl is not unconditionally positive and that, therefore, itsmacrogroup-integrated value may be close to zero, a fact thatcan result in unphysical values for the anisotropic cross-sections.

Our opinion is that, from the numerical point of view, useof a non positive weight function should be avoided at allcosts, and that, instead of Eq. (10), one should use flux weight-ing to define meaningful anisotropic cross-sections:

Shkl ¼ Skl ¼

tkl

VIFI

: ð11Þ

The drawback of this definition is that the anisotropic reac-tions rates cannot be preserved, except for the case of full-assembly homogenization. We shall come back to this pointin Section 7.

A particular case when the l-index degeneracy does notoccur is that for mode k¼ 1 in one-dimensional geometries,where the current vector has a constant direction. Here onecan define (Rimpault et al., 2005):

Sh1 ¼

ZG�I

Sa1JðE;rÞdEdr

VIJhI

: ð12Þ

However, because of the problem just discussed, to avoidunphysical values of the weight function JI one should choosemacrogroups were the current does not change of direction.For example, for a fuel cell, where the current exits the fuelat high energies and enters the fuel at intermediary and ther-mal energies, one should avoid using macrogroups overlyingboth energy domains.

4. Equivalence theory

Equivalence theory (Kavenoky, 1978; Kavenoky and He-bert, 1981; Sanchez et al., 1988; Hebert, 1993) is used forpin-by-pin homogenization for a reference problem with


conservative boundary conditions. Preservation of reactionrates over macrogroups and macroregions implies th

I ¼Sh

I VIFhI ¼ tI , where tI is the reference total reaction rate for

macrogroup G and macroregion I, so that for macrogroup G:

ShI ¼

tI

VIFhI

; I ¼ 1;ND: ð13Þ

Because the FhI depends on all the homogenized cross-sections

for macrogroup G, the above relations define a set of nonlinearequations for the SI

h. Note that (13) is of the general form of(4), with weight function w ¼ Fh

I , and that, therefore, all theother homogenized cross-sections, including the leakageterm DB2, can be obtained from SI

h as in Eqs. (6) and (7).In practice, nonlinear equations (13) are solved by itera-

tions using as starting values the flux-homogenized cross-sections in (5). A set of external iterations is done to convergethe spatial shape of the fission sources and of the outer scatter-ing contributions. Thus, within a given external iteration, thenonlinear problem is solved per macrogroup by iterativelyminimizing a functional of the type:

F�

S!h�¼X

I

1� th

I ðS!hÞtI

2

;

where S!h ¼ Sh

I ; I�

¼ 1;NDg is the set of homogenizedcross-sections for the macrogroup and th

I ðS!hÞ ¼ Sh

I VIFhI are

the macrogroup average reaction rates. This procedure offersthe advantage of solving a nonlinear problem with fewerunknowns, ND instead of the global number NE � ND. Eachmacrogroup nonlinear iteration uses the previous values forS!h to solve the reference problem with the low-order operatorand fixed external sources. This gives the macroregion averagefluxes Fh

I and the associated reaction rates tIh. At this point the

functional is checked and the iterations are halted if thedesired precision has been achieved.

For conservative boundary conditions, J¼ Jh¼ 0, the leak-age terms vanish and global balance (9) reduces toX

I

thI

�S!h�¼X

I

VIQhI ¼

XI

tI;

which implies thatP

I thI ðS!hÞ is preserved, regardless of the

values of the homogenized cross-sections. Therefore, the non-linear problem defined by the conditions th

I ðS!hÞ ¼ tI , I¼ 1,

ND, is degenerate and admits an infinite number of solutions.The reason for this degeneracy is that both operators, thereference and the low-order one, obey the same conservationrelation. Hence, in order to single out a unique solution, a sup-plementary condition has to be used per macrogroup. Usualnormalization conditions are the conservation of the assemblyaverage flux or Selengut’s normalization (Hebert and Mathon-niere, 1993). In the latter, the normalization consists of pre-serving the total transport incoming current, add the axialleakage and use the transverse leakage as incoming currenton the assembly boundary.

4.1. Full-assembly homogenization

In the particular case of full-assembly homogenization(ND¼ 1) with conservative boundary conditions, the total re-action rate is preserved, regardless of the value of the homog-enized cross-sections Sh. Therefore, one can choose any valuefor Sh. A meaningful physical choice is by flux weighting:

Sh ¼ S¼ t

VFð14Þ

where V is the assembly volume and F is the reference assem-bly average flux. This choice offers the supplementary advan-tage of preserving the assembly average flux,

Fh ¼ F: ð15Þ

Notice that this is equivalent to using as normalization condi-tion the conservation of the assembly average flux. As we shallsee, this homogenization is as the basis of the discontinuityfactor technique.

5. Black-box homogenization

In the past, discontinuity factors have been introduced topreserve boundary fluxes with full-assembly homogenization,conservative boundary conditions and diffusion as a low-orderoperator. The ground work was done by Koebke (1978, 1981)and refined later by Smith (1980, 1986). However, the point ofview adopted in the present work is to preserve the referenceboundary partial currents and not the reference boundaryfluxes.

In this section we present a general approach to the prob-lem of assembly homogenization where discontinuity factorsare used to enforce black-box homogenization, that is, to pre-serve the response of the assembly in the homogenizedreference situation. For a given incoming angular flux, theresponse of the assembly to its environment is given by theoutgoing angular flux. For the homogenized assembly onecan only expect to preserve average reference values and inour analysis we have chosen to preserve the boundary partialcurrents.

We shall consider the case of a general low-order operatorin a reference problem with arbitrary leakage. The theory willbe presented for full-assembly homogenization and, then, gen-eralized to pin-by-pin homogenization. The relation with theclassical discontinuity factors a la Smith will be discussedin the next section.

With reaction rates, sources and partial surface currentsprovided by the reference calculation, the basic idea behindthe black-box approach is to homogenize an assembly sothat, with homogenized sources Qh ¼ Q and incoming currentsJ�

h¼ J�, it reproduces the reference outgoing currentsJþ

h¼ Jþ, while preserving average reaction rates. By ‘homog-enized’ we understand here the numerical representation bythe low-order operator, or any other macroscopic compatibledescription, i.e., uniform current per side of the assembly.Also, when the low-order operator is transport, then ‘current’


may be substituted by ‘angular flux moment.’ Hence, thereference problem is defined by the reference reaction ratesand eigenvalue and also by the reference leakage, as givenfor example by the average partial currents for every side ofthe assembly.

However, with one homogenized cross-section per macro-region there are not enough degrees of freedom to simulta-neously preserve the macroregions reaction rates and thesurface incoming and outgoing currents. Thus, once thehomogenized cross-sections have been chosen to preservereaction rates, preservation of incoming and outgoing currentsneeds the introduction of current discontinuity factors (CDFs).The latter are then calculated as the ratio between the refer-ence value and the value produced by the low-order operatorin the reference problem, and taken to be valid in actualcore calculations. Note that we introduce discontinuity factorson the partial currents, not on the surface average scalar fluxes.It is only when the low-order operator is diffusion-like that onemay advantageously introduce discontinuity factors on thesurface average fluxes to implement the preservation of thepartial currents.

The physical basis of the CDFs will be illustrated for thecase of full-assembly homogenization with an arbitrary low-order operator and arbitrary boundary conditions (leakage),and generalized, then, to pin-by-pin homogenization.

5.1. Full-assembly homogenization

Jh-

Jh+

J+

J-

Fig. 1. Illustration for current discontinuity factors.

We consider full-assembly homogenization for an assemblywith general boundary conditions. Again, we base our discus-sion on the conservation relations per macrogroup for thereference and for the homogenized operators:Z

G�vD

dE dS$JðE;rÞ þSVF¼ VQ;

andZG�vD

dE dS$JhðE;rÞ þShVFh ¼ VQh;

where S is the flux-weighted total cross-section, V is theassembly volume and F and Q are the assembly averagereference flux and source, respectively.

We set the homogenized source to the reference value, Qh

¼ Q, and require to preserve the total reaction rate. This ispossible only if the total leakages are equal. Letting the anal-ysis of this condition for later, we see that the preservation ofthe total reaction rate allows one to arbitrarily define the aver-aged cross-section. We recall that the reason for this is the de-generacy of the equivalence problem. This degree of freedomallows us to use the flux-weighted cross-section, as in Eq. (14),so the assembly average flux is also preserved.

Remember that this is only possible under the conditionthat the total leakage is also preserved:Z

G�vD

dE dS$JhðE; rÞ ¼Z

G�vD

dE dS$JðE; rÞ: ð16Þ

However, it is easy to see that preservation of incoming andoutgoing partial currents is not in general possible withoutintroducing new degrees of freedom. Indeed, since the low-order operator is linear we must have a relation of the type:

J!hþ ¼ EVQhþ T J

!h�; ð17Þ

where J!h� denotes the exiting (þ) and entering currents (�)

for each side of the assembly and E and T, the escape andtransmission linear operators, which are solely determinedby the value of the homogenized cross-section and the discre-tization of the low-order operator. Therefore, with E and Tcomputed with the flux-weighted cross-section, the net currentis given by the following expression:

J!h ¼ J

!hþ � J!h� ¼ EVQh� ð1� TÞ J

!h�:

Because preservation of the eigenvalue requires the leakageof the low-order model to be equal to the reference leakage,J!h ¼ J

!, we must have

J!h� ¼ ð1� TÞ�1�EVQh � J

!� ð18Þ

with the corresponding exiting current given by Eq. (17):

J!hþ ¼ ð1� TÞ�1�

EVQh � T J!�

: ð19Þ

5.1.1. Current discontinuity factorsClearly, the values for J

!h� and J

!hþ in (18) and (19) are not

necessarily equal to the reference values J!� and J

!þ. This

situation is illustrated in Fig. 1. The figure shows that, in orderto recover the good values for J

!h� and J

!þ, one must intro-

duce discontinuity factors (CDFs) for the entering and exitingcurrents per side:

Jh� ¼ f �1

� J�

Jþ ¼ fþ Jhþ

�hJ� ¼ f�Jh

�; ð20Þ

where J�h and Jþ

h are the values of the low-order equation onthe ‘inside’ of the boundary and J� and Jþ are the values onthe ‘outside’ of the boundary; that is, the values that, in thelow-order core calculation, arrive from and are passed to theneighboring assemblies. Physically, the process is equivalentto adding a surface source that increases the exiting and the


entering partial currents, Jþh and J�, by the amounts ( fþ� 1)Jþ

h

and ( f ��1� 1)J�, respectively.

The CDFs f� and fþ introduce a discontinuity for the partialcurrents at the boundary so that the ‘outside’ values are closeto those in the core situation. They are computed in the refer-ence situation and assumed to be valid, to first order, in theactual core calculation:

f� ¼J�Jh�

��ref

: ð21Þ

Notice that in the actual core calculation, the currents enteringtwo neighboring assemblies, A and B, are related by the dis-continuity conditions:

ðf�ÞA��Jh��

A¼ ðfþÞB�

�Jhþ�

B;

ðf�ÞB��Jh��

B¼ ðfþÞA�

�Jhþ�

A; ð22Þ

as illustrated in Fig. 2.

5.1.2. Periodic lattice homogenizationIf the reference situation is a periodic lattice, both the ref-

erence and the low-order boundary currents vanish and wehave Jh

�jref ¼ Jhþjref and J�jref ¼ Jþjref . Therefore the CDFs

for a given assembly are identical,

f� ¼ f ¼ J�Jh�

��ref

; ð23Þ

and the core discontinuity conditions become

fA��Jh��

A¼ fB�

�Jhþ�

B;

fB ��Jh��

B¼ fA �

�Jhþ�

A: ð24Þ

The present case clearly illustrates our discussion in the prece-dent section: with J¼ 0 the solution of Eq. (18) is independentof the reference value J� and depends only on the internalsources and on the flux-homogenized cross-section.

To obtain the value J�h, needed for the calculation of the

CDFs in (23), we realize that, for global ensemble homogeni-zation with conservative boundary conditions, the homoge-nized low-order equation is that for a homogeneous infinitemedium. Hence, the angular flux is constant and we haveJ�

h¼pjh¼Fh/4 and, since the ensemble average flux is pre-served, Fh ¼ F, we obtain

J -h J- J+ J+

h

assemblyA

assemblyB

=

Fig. 2. Case of core calculation. The two adjacent assemblies have been sep-

arated for clarity.

f ¼ 4J�

F

��ref

; ð25Þ

where F is the reference assembly average flux. This showsthat the CDFs are independent of the low-order equation.But this is only true for the case of global assemblyhomogenization.

5.2. Comment

We note that a different possibility for full-assembly ho-mogenization with conservative boundary conditions wouldbe to select a value of the homogenized cross-section that pre-serves the reference global partial currents. This can be doneby fixing the value of the homogenized assembly averageflux to

Fh ¼ 4J�;

where J� are the reference partial currents averaged over theentire surface of the assembly. The value of the homogenizedcross-section follows from reaction rate conservation:

Sh ¼ t

4VJ�:

This is Hebert’s Selengut normalization for the case of full-assembly homogenization (Hebert and Mathonniere, 1993).With this value for Sh we lose the attractive feature of preserv-ing the assembly average flux but, on the other hand, we obtaina black-box homogenization with unit CDFs for all sides ofthe assembly. However, this procedure should be used onlywhen the original assembly has the proper symmetries thatassure that the partial currents per side are identical.

5.3. Pin-by-pin homogenization

We extend the concept of current discontinuity factors tothe case of pin-by-pin homogenization. In this case, one canintroduce CDFs at the pin level to ensure that the low-orderoperator will reproduce pin reaction rates in the referencesituation, as given by the transport reference assembly calcu-lation. Notice that in the reference calculation the pin reactionrates account for the physical exchanges between pins in theassembly, as opposed to the separate homogenization ofeach pin in the assembly. A consequence of our approach isthat the average pin flux will also be preserved.

Let I denote one of the pins in the assembly or, for that mat-ter, any macroregion to be homogenized. Then the cross-section for pin I is determined by direct flux weighting, asin Eqs. (14) and (15), but CDFs are introduced at the pin’sboundary to account for the reference neutron exchangesbetween the pin and the neighboring pins or, if one or moresides of the pin lay on the boundary of the assembly, for thereference exchanges at the boundary.

Clearly, and except for an assembly with zero leakage com-posed of identical pins, the pins will not have zero current attheir boundaries. In this case the CDFs have to be determined


by the general formula (21), where the reference currents J�and Jþ are those on the surface of the pin and the J�

h and Jþh

have to be calculated using the exact low-order operator repre-sentation in Eqs. (18) and (19) with J equal to the referencecurrent on the boundary of the pin.

Thus, because the low-order flux in the pin is not anymoreflat, the low-order operator is needed to compute the CDFs perpin. Therefore, the CDFs will depend on the numerical discre-tization of the low-order operator. As we discuss next, thecalculation of these factors entails the use of the low-orderoperator, with the same discretization as the one that will beused in the final core calculation, and with no scattering. Aseparate calculation has to be done for each side of the cellwith a unit current entering the side, zero currents enteringthe other sides of the cell and no internal sources. Thesecalculations yield the transmission matrix T in Eq. (17), whilea last calculation, with no incoming currents and the flat refer-ence source Qh ¼ Q, provides the escape vector EVQh. Finally,the desired values for J�

h and Jþh follow from Eqs. (18) and

(19). All these calculations have to be done for each pin inthe assembly and for each macrogroup.

Although having a natural appeal, especially if the confi-dence in the representativeness of the reference situation ishigh, this approach demands the extraction of local currentsat the pin level from the reference assembly transport calcula-tion, a number of low-order calculations for each pin and eachmacrogroup and a substantial storage for the pins’ current dis-continuity factors.

6. Diffusion as low-order operator

In diffusion theory the partial boundary currents are

Jh� ¼

Fhbd

4� Jh

2; ð26Þ

where the net current Jh¼ Jþh� J�

h is measured in the exitingdirection.

In order to preserve Jh¼ J in the homogenization process,we must use the boundary condition:

�Dhnþ$VFh��bd¼ J;

where Dh is the diffusion coefficient and nþ is the outwardnormal at the boundary and, then, obtain the partial currentsfrom Eq. (26) with Jh¼ J.

Hence, in the reference situation the CDFs in Eq. (21) are

f� ¼J�

Fhbd

4� J

2

��ref

: ð27Þ

Notice that the value of the diffusion coefficient is not givenby the homogenization process, and that one is free to adoptany conventional definition for Dh.

6.1. Periodic lattice homogenization

For a periodic lattice, with Jþ¼ J�, we have

f� ¼ f ¼ 4J�

Fhbd

��ref

ð28Þ

which, for the case of full-assembly homogenization, becomesEq. (25).

Finally, when the reference operator is also the diffusionoperator, Eq. (25) yields the well-known formula:

f ¼ Fbd

F

��ref

: ð29Þ

6.2. Homogenization by flux discontinuity factors

The precedent discussion concerns the straightforward ap-plication of CDFs to the case when the low-order operator isdiffusion. There is, however, a better way to implement currentdiscontinuity factors for diffusion. Indeed, Eq. (26) shows that,if the current is imposed, Jh¼ J, and if we want to preserve J�and Jþ, we may introduce a single flux discontinuity factor(FDF) on the boundary flux and write

Jh� ¼

f Fhbd

4� Jh

2; ð30Þ

where f is determined in the reference situation with J�h¼ J�:

f ¼ 2J� þ Jþ

Fhbd

��ref

: ð31Þ

Recently, a similar formula has been proposed for homogeni-zation of non-multiplying materials such as the baffle/reflectordomain (Tahara and Sekimoto, 2002; Mittag et al., 2003).Notice that Eq. (30) preserves the total reference currentJh¼ Jþ

h� J�h¼ Jþ� J� ¼ J and, therefore, reaction rate con-

servation. For periodic lattice homogenization the FDF isequal to the CDF given in Eq. (28). Thus, for full-assemblyhomogenization with periodic boundary conditions Eq. (31)gives Eq. (25), and, only when the reference calculation isdiffusion, (29).

Formula (30) implies that for the diffusion core calculationone must use current continuity and a discontinuity conditionfor the interface flux:�

Jhbd

�A¼�Jh

bd

�B;

fA

�Fh

bd

�A¼ fB

�Fh

bd

�B:

ð32Þ

As compared with the general formula for the CDFs in (20),the FDFs have the advantage of requiring a single coefficientper surface for the correction, even for problems with bound-ary leakage. Moreover, even though both formulations pre-serve the reference partial currents for the referenceproblem, their impact on the final core calculation is different,as a comparison of Eqs. (22) and (32) shows. Consider, for


example, the frequent case of homogenization with conserva-tive boundary conditions for which the flux and current dis-continuity factors are identical. For this case the core inter-assembly conditions given by the general formulation in Eq.(24) can be written as

fA

�Jh

bd

�A¼ fB

�Jh

bd

�B;

fA

�Fh

bd

�A¼ fB

�Fh

bd

�:

to be compared with the corresponding conditions in (32),which are to be used with the formulation based on fluxdiscontinuity factors.

6.2.1. CommentsThe flux discontinuity factors introduced in Eq. (30) and

defined by Eq. (31) are not to be confused with the classicaldiscontinuity factors a la Smith (Smith, 1986). The formerhave been introduced as another way to enforce the preserva-tion of the reference boundary partial currents when the low-order operator is diffusion, while the latter were introducedto preserve reference flux boundary values for diffusion.

The introduction of discontinuity factors in the literature,which has been done only for the case of full-assembly ho-mogenization with conservative boundary conditions, is basedin a different argument. The idea is to introduce a discontinuityfactor to ‘recover’ the good reference flux at the boundary. In-deed, as illustrated in Fig. 3 for the simple case of an interfacebetween two homogenous media, the reference transport fluxis continuous across the boundary, whereas the asymptoticboundary fluxes must be discontinuous to assure the correctflux levels on each medium.

Thus, one may multiply the low-order boundary flux timesa discontinuity factor to get the ‘correct’ transport flux at theboundary:

f Fhbd

��ref¼ Fbdjref/f ¼ Fbd

F

��ref

; ð33Þ

having recognized that Fhbdjref ¼ Fhjref ¼ Fjref . This entails the

use of interface conditions (32) in the core calculation.

left

right

Φbd

Φbd

Φbd

Fig. 3. The effect of the discontinuity factor is to ensure the good low-order

average flux in two neighboring assemblies (shown here by the dashed lines.)

The continuous curve represents the collapsed transport flux.

However, this definition does not respect the ‘black-box’ ap-proach in that it does not preserve the reference incomingand outgoing boundary currents, except if the reference oper-ator is also diffusion.

6.3. Numerical example for full-assemblyhomogenization

We present here a comparison between three differenthomogenization techniques: Smith’s discontinuity coefficients(DCs), black-box flux discontinuity factors (FDFs) and equiv-alence theory with Selengut normalization. We recall that thelatter constructs homogenized cross-sections that ensure unitDCs. We have considered a small 3� 3-assembly motif withspecular boundary conditions comprising a central AIC assem-bly surrounded by eight UOX assemblies. These calculationsare aimed to analyze the impact of FDFs in a simple interfaceproblem. Reference results for the two 3� 3 motifs have beenobtained from a 281-group lattice APOLLO2 calculation ina detailed geometric mesh (Fig. 4) with the method of charac-teristics. The same number of groups and detailed mesh werealso used to separately homogenize the UOX and AIC assem-blies in an infinite lattice with critical buckling. This was re-peated for the three different homogenization techniques.The homogenized cross-sections and surface discontinuity fac-tors (DCs, FDFs or unity factors for Selengut’s normalization)were then used to run a diffusion calculation with the codeCRONOS2.

The homogenization step and the core calculations weredone for two and eight groups. The latter comprise four fastgroups and four thermal groups. The thermal group for thetwo-group partition contains the lowest three thermal groupsof the eight-group partition. In Table 1 the results are com-pared to the corresponding collapsed results from the transportreference calculation in terms of the relative errors for theassembly absorption (Sa) and production (nSf), 1� tdiff/tref,and the reactivity, r¼ 1/keff

ref� 1/keffdiff, with normalization of

the diffusion and reference calculations to the same productionrate. Equivalence theory with Selengut normalization gave thebest results in reaction rates for the harder AIC problem, but itis much less precise in reactivity for the two-group problem.For this problem, the FDF technique gives better results thanthe classical discontinuity factors, reducing the errors inabsorption and production by 10% for the two-group calcula-tion and by 25% for the eight groups calculation, where trans-port effects are more prominent.

A comparison of FDFs and Smith’s DCs for the two and theeight-group homogenizations is given in Table 2 for the AICand UOX assemblies. Strong assembly absorption makes theaverage flux smaller than the boundary flux and results inDCs greater than one. The values of FDFs and DCs followthe same trends: for the AIC assembly they exceed one forall groups because the current predominantly moves particlestowards the interior of the assembly where the absorption ishigher, resulting in an anisotropy of the boundary fluxes whichincreases with decreasing energies because of the increasingabsorption; for the UOX assembly the fast currents flow

Fig. 4. Computing mesh for the 3� 3-assembly motif (1/8 symmetry).

Table 2

Flux discontinuity coefficients for the two assemblies in the AICeUOX

problem

AIC UOX

DC FDF DC FDF

Two groups 1.039 1.035 0.992 0.998

1.392 1.341 1.032 1.004

Eight groups 1.031 1.031 0.973 0.996

1.002 1.019 0.972 0.995

1.017 1.018 0.997 1.000

1.073 1.050 1.020 1.002

1.139 1.112 1.002 0.993

1.185 1.152 1.007 0.995

1.285 1.241 1.016 0.994

1.462 1.407 1.041 1.010

DC¼ Smith’s flux discontinuity coefficient, FDF¼ flux discontinuity factor.


outwards, as shown by the values of the DCs and FDFs beingsmaller than one, while for thermal groups the currents flowinward because of the absorption. Next, we consider the ratiobetween the FDF and the DC coefficients. For specular bound-ary conditions the angular flux at the assembly boundary, withunit normal n, can be written as

jðxÞ ¼ 1

4pFðE;rÞ þjeðxÞ; r ˛vD; ð34Þ

where je(x)¼ je(Sx), S is the specular symmetry SU ¼U� ðU$nÞn and

Rð2pÞþ

jeðxÞdU ¼ 0. Since je changes ofsign over (2p)þ one can view Eq. (34) as a perturbation aroundan isotropic angular flux. Then, with the help of Eqs. (25), (33)and (34) we have

fFDF

fDC

¼ 4JþFbd

��ref

¼ 1þ 4

*Zð2pÞþ

jeðxÞjU$njdU

+Fbd

;

Table 1

3� 3 Motif with central AIC assembly. Relative errors in assembly absorption

(ta), production (tp) and reactivity (r) from Smith’s discontinuity coefficients

(DCs), flux discontinuity factor (FDF) and equivalence theory with Selengut

normalization (S )

Assembly Two groups Eight groups

DC FDF S DC FDF S

ta (%) UOX corner 1.52 1.50 1.28 0.56 0.51 0.51

UOX side �0.55 �0.63 �0.86 �0.06 �0.13 �0.57

AIC �7.29 �6.59 �5.23 �2.60 �1.79 �0.81

tp (%) UOX corner 1.42 1.40 1.34 0.39 0.34 0.55

UOX side �0.29 �0.37 �0.46 0.12 0.04 �0.23

AIC �8.33 �7.64 �6.60 �3.62 �2.77 �2.40

r (pcm) 56 40 114 �104 �123 �32.51

where <$> represents the average over a macrogroup andover the side of the assembly. For je¼ 0 the angular flux isisotropic at the boundary and the coefficients are equal,fFDF¼ fDC. For all other cases, the ratio fFDF/fDC depends onthe type of anisotropy of j: if this flux increases with m ¼jU$nj we have fFDF> fDC while, for j decreasing with m,fFDF< fDC. The values displayed in Table 2 for the eight-groupcase show that fFDF T fDC for fast groups, where the angularflux is higher in directions close to m¼ 1, and fFDF< fDC forthermal groups, where diffusion in water makes the fluxmore prominent in directions with small values of m. Thevalues of the FDFs are within a few percent of those for theDCs, reaching 6% for thermal groups in the AIC assembly.

Because the present interface between the spectral codeAPOLLO2 and the diffusion code CRONOS2 does not includediscontinuity factors, we were in the impossibility to run largecore calculations at this time. However, the results obtained forthe small 3A3-assembly motif show that the FDFs behavealike the classical Smith’s discontinuity coefficients with a po-tential to get better results in situations were transport effectsare more important, as evidenced by the eight-group AICcalculation.

6.4. Pin-by-pin homogenization

For diffusion as low-order operator, the treatment of pin-by-pin homogenization with CDFs or FDFs is done alongthe general lines discussed earlier. For every cell one needsto solve the diffusion equation with no scattering; first withno internal source and with a unit entering current on eachside and zero in the other sides, and second with zero enteringcurrents and a unit source.

7. Simplified PN as low-order operator

Recently, the simplified PN (SPN) equations have been putto use for three-dimensional core calculations in productioncodes (Lewis and Palmiotti, 1997; Baudron and Lautard,2007). In this section we generalize the notion of current dis-continuity factors for this case.


We start with a brief discussion of the SPN equations. In thispresentation we shall follow the asymptotic formulation ofPomraning (1993). A different asymptotic analysis has beenconducted by Larsen et al. (1993) who derived the SPN equa-tions as a higher-order approximation of transport with diffu-sion being the leading-order equation.

The basic idea of SPN is that at any location there is a localreference frame (xl, yl, zl) in which, to first order, the angularflux is a function of the form j(zl, m), where m ¼ ez$U and ez

are the direction of the local zl axis (Pomraning, 1993):

jðxÞ ¼ jðE; zl;mÞ þjeðxÞ; ð35Þ

where je is a term of order e� 1.In these conditions, to lowest order in e the local flux obeys

the slab-geometry transport equation:

ðmvzlþSÞj¼ Ss

2

Xk�0

ð2kþ 1ÞfkPkðmÞfkðE; zlÞ þQðE; zl;mÞ;

where Pk is the Legendre polynomial of order k and

fkðE; zlÞ ¼Z 1

�1

PkðmÞjðE; zl;mÞdm:

Notice that the definition of the fk implies a flux expansion ofthe form

jðE; zl;mÞ ¼1

2

Xk�0

ð2kþ 1ÞPkðmÞfkðE; zlÞ ð36Þ

and that a similar expansion can be assumed for the source.In terms of these expansions the PN formulation of the slab-

geometry transport equation reads

akvzlfkþ1þ bkvzl

fk�1 þSakfk ¼ Qk; 0� k � N; ð37Þ

where f�1¼ fNþ1¼ 0, ak¼ (kþ 1)/(2kþ 1), bk¼ 1� ak, Qk

accounts for external transfers and, for k¼ 0, fission and

Sak ¼ S� fkSs: ð38ÞThe SPN equations are then obtained by writing Eq. (37) in

the global coordinates x ¼ ðE; r;UÞ. This amounts to replac-ing the even flux moments as fkðE; zlÞ/fkðE; rÞ and theodd flux moments by vectors, fkðE; zlÞ/ f

!kðE; rÞ, and,

accordingly, the vzlby the gradient or the divergence (Pomran-

ing, 1993):

akV$ f!

kþ1þ bkV$ f!

k�1 þSakfk ¼ Qk; k even;

akVfkþ1þ bkVfk�1þSak f!

k ¼ Q!

k; k oddð39Þ

with 0� k� N, N odd, and where now f!

kðE; rÞ is a vector fork odd.

The SPN approximation in (35) implies that, to lowest orderin e, the flux j(x) is invariant by rotations around the local axiszl, and so are the SPN equations (39). Consequently, to lowestorder in e, Vfk for k even and f

!k for k odd are in the direction

of the local direction ez, which entails that, to the same order, theangular flux is given by expansion (36) with the replacements

fkðE; zlÞ/

fkðE;rÞ; k even;

ezðrÞ$ f!

kðE;rÞ; k odd:ð40Þ

To derive interface and boundary conditions for the SPN equa-tions one assumes that at interfaces and boundaries the localreference frame has axis ez in the direction of the normaln(r) to the interface or boundary surface. Therefore we canuse expansion (36) with (40) by simply replacing ez(r) byn(r) in the latter.

Since at an interface between two materials flux continuityentails the continuity of the local flux expansion coefficientsfk, the interface conditions for SPN are

fkðE;r�Þ ¼ fkðE;rþÞ; k even;

nðrÞ$ f!

kðE;r�Þ ¼ nðrÞ$ f!

kðE;rþÞ; k oddð41Þ

with r� ¼ lime/0 ðr� enÞ.At an external surface the boundary condition is given by

the entering flux. In the PN method one may use Mark orMarshak boundary conditions. Marshak conditions amountto the preservation of the odd angular moments over the enter-ing directions:Z 0

�1

PkðmÞjðE; zl;mÞdm¼Z 0

�1

PkðmÞjinðE; zl;mÞdm; k odd;

ð42Þ

where jin(E, zl, m) is the given incoming flux. To express theseboundary conditions in terms of the flux angular moments atthe boundary we need to compute the projection of expansion(36) into a half space. We shall express these boundary condi-tions in terms of the SPN currents:

Jh�;kðE; rÞ ¼

Z�

PkðmÞjðE; zl;mÞdm

¼Fh

bd;kðE;rÞ4

� Jhk ðE; rÞ

2; k odd; ð43Þ

where we have taking into account the orthogonality andparity properties of the Legendre polynomials,

RðþÞh

R 10,R

ð�ÞhR 0

�1 and we have defined

Fhbd;kðE; rÞ ¼ 4

XN�1

k0 even

Akk0fhbd;k0 ðE; rÞ; Jh

k ¼ nðrÞ$ f!

kðE; rÞ

with Akk0 ¼ ðð2k0 þ 1Þ=2ÞR 1

0PkðmÞPk0 ðmÞdm.Thus, boundary conditions (42) can be written as

Jh�;kðE; rÞ ¼

�Jh�;k

�inðE;rÞ; k odd; ð44Þ

where (J�h

,k)in is computed with the entering flux jin. Theboundary condition typical of a periodic lattice is that of spec-ular reflection and therefore the boundary angular flux is evenin m. In terms of the angular flux moments this condition


entails that the odd moments must vanish at the surface. Thus,we are left with

nðrÞ$ f!

kðE;rÞ ¼ 0; k odd: ð45Þ

We end this brief discussion of the SPN equations by notingthat for N¼ 1, the SPN equations with isotropic scatteringand sources and boundary condition (44) reduce to the diffu-sion equation with F¼ f0, J ¼ f

!1 and D¼ 1/(3S).

7.1. Homogenization by current discontinuity factors

As opposed to diffusion, the SPN operator gives a deeperdescription of the angular flux and we shall use the black-box approach to respect ‘higher’ angular moments of theboundary fluxes. In transport theory, entering and exitingcurrents ought to be viewed as incoming and outgoing angularfluxes. For the SPN case these are equivalent to the incomingand outgoing currents J�

h,k for k odd.

To take into account anisotropy of collision the homogeni-zation has to provide homogenized values for the anisotropycross-sections in (38). For reasons discussed earlier, thesecross-sections will be calculated here by flux weighting withthe reference flux F:

Shak ¼ Sak ¼

ZG�I

ðSakfkÞðE;rÞdE drZG�I

FðE;rÞdE dr

¼ tak

VIFI

; k even; ð46Þ

where F¼ f0. A problem is the calculation of the referencerates tak from the transport solution. The reason is that SPN

is not truly a coherent approximation of transport and thatthe transport reference flux does not necessarily have a localreference frame in which the angular flux is of the formj(E, zl, m). A solution is to take as local z direction that definedby the transport current (Pomraning, 1993):

ezðrÞ ¼JðE;rÞJðE;rÞ/fkðE;rÞ ¼

1

2p

ZPkðez$UÞjðxÞdU:

The anisotropic cross-sections for k odd are analogous todiffusion coefficients and can be homogenized in a similarway. This likeness is evidenced by eliminating the odd vectorcomponents in Eq. (39) to write the even-parity form of theSPN equations:

�akakþ1V$1

Sa;kþ1

Vfkþ2 � bkbk�1V$1

Sa;k�1

Vfk�2

��

gkV$1

Sa;kþ1

þ gk�1V$1

Sa;k�1

�Vfk þSakfk

¼ Qk � akV$1

Sa;kþ1

Q!

kþ1� bkV$1

Sa;k�1

Q!

k�1; k even;

where gk¼ akbkþ1.We only consider the case of full-assembly homogenization

with non-zero leakage. The extension to pin-by-pin

homogenization is done in the way discussed in Section 5.By introducing current discontinuity factors,

f�;k ¼J�;kJh�;k

��ref

; k odd;

one assures that leakage is preserved and therefore can useflux-weight homogenization as in (46) so that fh

k ¼ F for keven. As discussed in Section 5, the J�

h,k have to be computed

by using the boundary conditions:

Jhk ¼ Jkjref

with the average reference sources Qk and Q!

k.Finally, for a periodic assembly we have a single CDF per

current:

fk ¼J�;kJh�;k

��ref

; k odd;

where the J�,k values are obtained from the reference transportcalculation and, accounting by boundary condition (45),Jh�;kjref ¼

PN�1k0 even Akk0f

hbd;k0 jref .

Moreover, for full-assembly homogenization, the SPN fluxis constant and it is given by the infinite, homogeneousmedium solution of (39), fh

bd;k ¼ fhk ¼ Qh

k=Shak ¼ F for k

even. Hence, the CDFs are independent of the discretizationof the low-order operator:

fk ¼ ak

J�;k

F

��ref

; k odd;

where ak ¼ 1=PN�1

k0 even Akk0 . Notice that for N¼ 1 we recoverthe diffusion value ak¼ 4.


As for the case of diffusion, it is possible to associate fluxdiscontinuity factors with the even flux components and to usecontinuity of the odd components at the interfaces. Thus, Eq.(43) is replaced by

Jh�;k ¼

fkFhbd;k

4� Jh

k

2; k odd:

Black-box homogenization implies the simultaneous conser-vation of the total and partial currents for all odd k orders.Therefore we replace Jk

h in the precedent expression with thereference values Jk and, from conservation of incoming andoutgoing currents in the reference situation, we obtain

fk ¼ 2J�;k þ Jþ;k

Fhbd;k

��ref

; k odd: ð47Þ

For a periodic lattice we have Jþ,k¼ J�,k. Moreover, for globalassembly homogenization Fh

bd;k ¼ F=ak . Hence the FDFs areindependent of the low-order discretization. Calculation fora truly heterogeneous homogenization will require to solve


SPN source problems in order to calculate the low-operatorflux moments fh

k0 produced by the lattice boundary conditionsJk

h¼ 0 with source Qh and with fixed eigenvalue equal to thereference eigenvalue.

8. Quasidiffusion as low-order operator

Quasidiffussion (QD) is essentially an improved diffusiontheory with a diffusion tensor that incorporates transport cor-rections. It was earlier introduced as an acceleration methodfor transport iterations (Anistratov and Gol’din, 1993; Miftenand Larsen, 1993) and, only recently, has been considered asa potential tool for core calculations (Anistratov, 2002; Nesand Palmer, 2002; Hiruta et al., 2005; Hiruta and Anistratov,2006). As a final example of the black-box homogenizationtechnique we consider here the use of current discontinuityfactors when QD is used as a low-order operator.

The derivation of the QD equations is based on the first twoangular moments of the transport equation (Miften and Larsen,1993):

V$JþSa0F¼ Q0;

V$j2þSa1J¼Q1;ð48Þ

where j2 is the symmetric tensor j2ðE; rÞ ¼R

UUjðxÞdU.Next, by defining the transport-dependent Eddington tensor,

EðE; rÞ ¼

ZUUjðxÞdUZ

jðxÞdU

¼ j2ðE;rÞFðE;rÞ ; ð49Þ

the second equation in (48) provides a Fick’s law:

J¼� 1

Sa1

V$ðEFÞ þ 1

Sa1

Q1:

Finally, replacement of the Fick’s law into the balance equa-tion yields the QD equation:��V$

1

Sa1

V$EþSa0

�F¼ Q0�V$

1

Sa1

Q1: ð50Þ

In a similar way, a transport-dependent boundary Eddingtonfactor B is introduced to derive the boundary conditions interms of the transport partial currents:

J� ¼Zð2pÞ�

jU$nþjj dU¼ZðjU$nþj �U$nþÞj dU

¼ BFbd

4� J

2; ð51Þ

where nþ(r) is the outward normal to the surface,J¼ J$nþ¼ Jþ� J� and

BðE;rÞ ¼ 2

ZjU$nþjjðxÞdUZ

jðxÞdU

¼ 2JþðE;rÞ þ J�ðE;rÞ

FbdðE; rÞ: ð52Þ

8.1. Homogenization by current discontinuity factors

As previously, we consider only the case of full-assemblyhomogenization with arbitrary leakage. As usual, the homog-enization of Sa0 is obtained from flux weighting, so the assem-bly averaged flux is preserved. Also, Sa1 is homogenized asa diffusion coefficient. It remains to homogenize the Edding-ton tensor that incorporates the effects of flux anisotropy.This tensor can be homogenized over the entire macroregion:

Eh ¼

ZG�I�ð4pÞ

jðxÞUUdE dr dUZG�I�ð4pÞ

jðxÞdE dr dU

:

Current discontinuity factors are defined by Eq. (21) and, forthe present case, they can be written as

f� ¼J�

BFhbd

4� J

2

��ref

¼ 2J�

ðJþ þ J�ÞFh

bd

Fbd

� J

��ref

;

where Fbdh has to be determined from the solution of the low-

order QD equation with boundary conditions:

Jh ¼ Jjref :

with conservative boundary conditions the solution of the low-order QD equation is a constant flux and the above formulagives an unique coefficient independent of the low-orderoperator:

f� ¼ f ¼ Fbd

Fhbd

��ref

¼ Fbd

F

��ref

: ð53Þ


As for the case of the diffusion equation we introduce anFDF for the partial currents,

Jh� ¼ fBFh

bd

4� Jh

2;

and calculate it in the reference problem:

f ¼ 2Jþ þ J�

BFhbd

��ref

¼ Fbd

Fhbd

��ref

;

which, for full-assembly homogenization, gives Eq. (53). No-tice that the FDF is always independent of the discretization ofthe QD operator, even in the presence of leakage. The reason


is that the boundary coefficient B ¼ 2ðJþ þ J�Þ=Fbdjref is alsocomputed in the reference situation.

9. A case study: homogenization via perturbation theory

As an example we present here a different homogenizationtechnique based on a variational approach. We consider a gen-eral reference problem with sources and mixed boundary con-ditions comprising an incoming angular flux and an arbitraryalbedo operator, and write the reference and low-order equa-tions as

Bj¼ S; x ˛X;

j¼ bjþj0; x ˛G�ð54Þ

and

Bhjh ¼ Sh; x ˛X;

jh ¼ Bhjhþj0h; x ˛G�;

ð55Þ

where B ¼ SþU$V� H � ð1=lÞP, S and j0 are an externalsource and an incoming angular flux, respectively, and b is analbedo operator.

The idea is to consider the low-order equation as a perturba-tion of the reference one and use the duality relation (Sanchez,1998):

ðj;SÞ þ hj;j0i�¼ ðS;jÞ þ�j0;j

�þ ð56Þ

where

ðf ;gÞ ¼Z

X

fg dx; h f ;gi�¼Z

G�

fgdbx

are the volume and surface scalar products, dx ¼ dE dr dU

and dbx ¼ jU$njdE dS dU.Also, in Eq. (56) j* is the adjoint flux solution of the

equation

Bj ¼ S; x ˛X;

j ¼ bj þj0; x ˛Gþ;

where B ¼ S�U$V� H � ð1=lÞP, H* and P* are theadjoints of H and P with respect to the volume scalar product,respectively, and b* is the adjoint of b: <b*f,g>þ¼<f, bg>�.

In order to use Eq. (56) with Eq. (55) we write the latterequation with the unperturbed operators as in Eq. (54) butwith now S¼ Shþ dBjh and j0¼ j0

hþdbjh, wheredB¼ B� Bh and db¼ b� bh. We shall consider only thecase of a critical assembly so that j0¼ j0*¼ S*¼ 0, thenfrom Eq. (56) we obtain�j;dBjh

�¼ 0;

where we have assumed that homogenization preserves thealbedo, db¼ 0, and does not add any sources, Sh¼ 0.

This equation provides a single condition for homogeniza-tion. In order to define the homogenized cross-sections permacrogroup and macroregion we request that the homogeniza-tion conserves the eigenvalue, writing explicitly the aboveequation,

XG;I

�j;dSjh

�G

IþXG;I

�j;dHjh

�G

I� 1

lcore

XG;I

�j;dPjh

�G

I¼ 0;

where ðf ; gÞGI ¼R

G�I�ð4pÞfg dx, and cancel each contributionseparately. For example, for the total cross-section we have(j*, dSjh)I

G¼ 0 which yields

�Sh�G

I¼ ðS;wÞ

GI

ð1;wÞGI

with the weight function w¼ j*jh, which is of the generaltype of Eq. (4). The other cross-sections, transfers and fissiongive similar expressions. Note that the weight depends on thelow-order solution and that, therefore, the homogenization isnonlinear.

We note that the use of duality is only possible if the low-order operator is also the transport operator. This type ofvariational homogenization has been applied to homogenizecontrol rods for fast reactors (Rowlands and Eaton, 1978).

10. Dehomogenization

The ultimate aim of reactor core calculations is to obtainspatially detailed reaction rates and power distributions and,in particular, to be able to predict the assembly power peak.Thus, once the core calculation has been done, one needsa method to reconstruct the detailed behavior of the solutionwithin every assembly. This is a typical inverse problemthat, because of the lack of exact information on the localbehavior in the actual core calculation, is ill posed andrequires a regularization technique for its solution. Most ofthe reconstruction techniques in use fall in one of the fourcategories that we present hereafter.

10.1. Factorization

Because the only information on the detailed solution is theassembly transport solution in the reference homogenizationproblem (typically an infinite lattice calculation), the simplerreconstruction technique is by factorization. Let F(r) denoteany quantity of interest (the power, a reaction rate or the mac-rogroup flux) in a given assembly. From the reference trans-port one has a representation of the form

FrefðrÞ ¼X

k0F ref

k0 f refk0 ðrÞ;

where typically, the f kref are piecewise constant functions. On

the other hand, the core solution for the same assembly is


FcoreðrÞ ¼X

k

Fcorek f core

k ðrÞ;

where most often, the f kcore are polynomials.

A useful reconstruction formula consist on using thefactorization

FRðrÞ ¼ FrefðrÞX

k

FRk f core

k ðrÞ

and determine the coefficients FkR so that the reconstructed

quantity preserves the behavior of the core solution:�f corek ;FR

�¼�f corek ;Fcore

�;

where

ðf ;gÞ ¼Z

assembly

ðfgÞðrÞdr: ð57Þ

10.2. Transport reconstruction

The drawback with the preceding approach is that the ref-erence transport solution has been obtained for the referenceproblem and, therefore, does not satisfy core inter-assemblyconditions. In order to avoid this approximation one canrecompute the assembly solution by solving the transportproblem (1) with the eigenvalue obtained from the core solu-tion and with incoming fluxes renormalized to the incomingcurrents from the core solution. For a transport group g con-tained in macrogroup G:

jg�ðxÞ ¼

½jg�ðxÞ�

ref�JG�ðxÞ

�ref

�JG�ðxÞ

�core; g ˛G; ð58Þ

where [j�g]ref is the flux from the reference transport solution

and [J�G]core is the current entering the assembly from the core

solution.Clearly, this transport calculation can be done with a num-

ber of groups lesser than that used for the reference calcula-tion, and all quantities of interest can be recovered from thesolution. However, the shape of the entering angular flux isdetermined from that of the reference solution.

10.3. Multiscale techniques

The idea here is to incorporate detailed spatial behavior inthe core solution by introducing extra degrees of freedom, soas to preserve other spatial moments of the reference solution(Koebke, 1981, 1985; Nichita and Rahnema, 2003; Hiruta andAnistratov, 2006), and use the factorization technique for thefinal reconstruction.

A general approach based on the multiscale methodology isto introduce a set of modified low-order expansion functions:

bf Gk ðrÞ ¼

�FrefðrÞ

�Gf core

k ðrÞ;

where the f kcore are typical expansion functions for the core

calculation and the [Fref(r)]G are the transport reference fluxescollapsed in macrogroup G. This type of basis functions havebeen used elsewhere for rebalancing acceleration for core cal-culations (Lautard et al., 1991) as well as in reactor kinetics(Chauvet et al., 2007).

In the frequent case when the low-order equation,

BcoreF¼ Q;

is discretized via a projective technique, the discretized equa-tions are obtained by introducing the expansion FGðrÞwP

kbFG

kbf G

k ðrÞ in the previous equation and by projecting overevery expansion function:X

k

�bf Gk ;B

corebf Gk

�bFGk ¼

�bf Gk ;Q

G�;

where ( f,g) is the scalar product in Eq. (57).

10.4. Dynamic homogenization

A better, albeit more expensive, way to obtain the detailedspatial behavior from the reactor core calculation is by dynam-ically homogenizing the assemblies accounting by the coreinter-assembly exchanges. Here, one starts the core calculationwith lattice-homogenized assembly cross-sections and everyfew iterations rehomogenizes the assemblies accounting forthe core interchanges, as in (58). The homogenization can bedone dynamically by homogenizing every assembly using theactual value of the core eigenvalue and the incoming currents(Aragones and Ahnert, 1986; Mondot and Sanchez, 2003), bypre-tabulating the homogenized cross-sections in terms of theeigenvalue and macrogroup-normalized incoming fluxes(Rahnema and Nichita, 1997), via a full core transport calcula-tion (Joo et al., 2002) or by introducing suitable approxima-tions on the assembly boundary fluxes (Smith, 1994).

This technique gives simultaneously the converged coresolution and the transport fluxes in the assemblies and, there-fore, does not need a reconstruction calculation.

11. Conclusions

There are still two main approximations in today’sapproach to reactor core design: the calculation of multigroupselfshielded cross-sections and the homogenization techniquesused in the two-steps lattice-transport to diffusion core calcu-lation. In spite of the continuous increase in computing power,routine 3D transport core calculations without need for self-shielding models are still far in the future. The trend in self-shielding seems to be an increase on the number of groupsthat allows for the use of simplified models in one or two-di-mensional geometries. On the other hand, access to large par-allel computers with thousands of processors has open thedoor to detailed assembly transport calculations and, surelyin the near future, to full core transport reference calculationswithout need for homogenization. However, a full transportcore calculation will require the solution of a discretized


problem with w1012 unknowns and, even with large parallelcomputers, routine design calculations based on large parallelcomputers will not be the accessible to the industry for years.

More realistically, one expects that, in a first time, increas-ingly available computer power would permit routine 3Dtransport core calculations with pin-by-pin homogenizedassemblies and dozens of groups. This raises the problem oftransport-to-transport homogenization. At present, simplifiedtransport (SPN) and quasidiffusion are being considered aspotential candidates for core calculations. Also, to diminishthe error introduced by the usual lattice homogenization, theassembly environment should be somewhat incorporated intothe homogenization process and this necessitates homogeniza-tion techniques for assemblies with boundary leakage. This isclearly the case for small research reactors or for applicationsto PBMRs for fuel near the inner and outer reflectors.A related problem is that of homogenization of the assembliesnear the reflector, where the impact of the reflector could betaken into account by a representative albedo.

In this work we have used the black-box paradigm to intro-duce current discontinuity factors (CDFs) for an arbitrary low-order operator in the presence of boundary leakage. The CDFsallow to preserve average reaction rates and average boundaryfluxes, as represented by partial currents, in a given reference sit-uation. We have also extend the notion of CDFs to pin-by-pinhomogenization (a problem where homogenization is done inthe presence of surface leakage) and show that the CDFs dependon the discretization of the low-order operator, but can be com-puted from the solution of a few low-order linear problems.

For applications to diffusion-like low-order operators, it isadvantageous to introduce flux discontinuity factors (FDFs)which, while preserving the partial reference currents, requireonly one value per side of the assembly. Among, this type oflow-order operators we have considered in detail diffusion,SPN and quasidiffusion. In the case of diffusion with full-as-sembly homogenization without boundary leakage, the FDFsequal the traditional Smith discontinuity coefficients (DCs)only in the case when the reference calculation is also diffu-sion. Preliminary tests calculations for a PWR 3� 3-assemblymotif with a central AIC assembly show that the FDFs give thesame precision than Smith’s DCs, while taking better intoaccount transport effects.

A last, necessary step after the core calculation is the recon-struction of detailed power distribution, in particular for theprediction of the local power peak. For completeness, wehave included a discussion of the prominent reconstructiontechniques: factorization, transport reconstruction, multiscalemethods and dynamic homogenization.

We hope that this note will help clarify some of the conceptsinvolved in the homogenization process that, while waiting forthe panacea of detailed 3D full transport core calculations, willstill be applied for some time for reactor core calculations.

Acknowledgements

Many thanks are due to generations of students that sat inmy ‘homogenization’ lecture. The 3� 3-assembly PWR motif

calculations were part of Cedric Reux’s master’s work. I thankhim and his advisor, Laure Mondelain, for making the resultsavailable to me. Thanks are also due to F. Dolci for herdetailed analysis of the results.

Appendix. Fundamental mode

In an infinite, homogeneous medium the cross-sectionsdepend only on the energy E and, therefore, the ratio betweentwo angular flux moments is also a sole function of the energy.The implication is that the solution of the transport equationobeys Fick’s law.

Because of invariance by translation one may introduceFourier modes of the form

jcoreðxÞ ¼ R�eiB$rjðU;EÞ

�;

where jðU;EÞ satisfies the transport equation

ðSþ iU$BÞj¼ Hjþ 1

lPj: ð59Þ

Because of the invariance by rotations around B¼ BeB, whereB is the module of B and eB is the unit vector in direction B,one has jðU;EÞ ¼ jBðm;EÞ with m ¼ U$eB. The correspond-ing scalar flux and current are FB(E ) and J(E )¼ JB(E )eB. Thecritical condition, l¼ 1, is defined by the critical buckling B2

that can be negative, zero or positive. For B2> 0 the real com-ponent of the flux is even in m, while the imaginary componentis odd in m. Hence, the solution of (59) can be normalized sothat the even angular flux moments are real and the odd onesare purely imaginary. On the other hand, for B2< 0 one cannormalize jB(m, E ) so it is real. The critical spectrumFB(E ), called the fundamental mode, is a basic characteristicof a multiplying homogeneous medium.

The critical core flux obeys the Fick’s law Jcoreðr;EÞ ¼�DBVFcoreðr;EÞ with the leakage coefficient

DBðEÞ ¼ iJBðEÞ

BFBðEÞ:

This coefficient and the critical buckling are computed fromthe BN solution of Eq. (59) for a given anisotropy of collision(Sanchez et al., 1988).

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Assembly homogenization techniques for core calculations

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