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Annals of Nuclear Energy 35 (2008) 868–876

annals of

NUCLEAR ENERGY

Improvements of the reactivity devices modelingfor the advanced CANDU reactor

R. Le Tellier, G. Marleau *, M. Dahmani 1, A. Hebert

Institut de Genie Nucleaire, Ecole Polytechnique de Montreal, P.O. Box 6079, Station CV, Montreal, Quebec, Canada H3C 3A7

Received 8 June 2007; accepted 12 September 2007Available online 31 October 2007

Abstract

In the context of the ACR� (Advanced CANDU Reactor), 3D transport calculations are required in order to simulate the reactivitydevices located perpendicularly to the fuel channels. The computational scheme that is usually used for CANDU-6 and ACR reactors isbased on a simplified supercell geometry in which the fuel clusters and devices are replaced by annuli. Recently, an exact modeling of 3Dsupercell configurations was introduced within the framework of the ACR calculations. However, with such a model, fine meshingrequirements lead to problems that are very demanding in terms of computational resources.

In this paper, we present improvements introduced in the ACR context to reduce the cost of the 3D supercell calculations. Twoavenues of investigations are reported. First, the introduction of an accelerated characteristics method permits to reduce the compu-tational burden of such calculations involving a large number of regions. In addition, contrarily to CANDU-6 supercell configura-tions, the ACR 3D geometry is prismatic and consequently a special tracking procedure can be used. This approach introduces noapproximation and is significantly faster than the general 3D tracking technique. Thanks to these modifications in the computationalprocedure, 3D supercell calculations with a level of mesh discretization comparable to 2D cell configurations become affordable forindustrial applications.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Contrarily to light water reactors, the numerical simula-tion of CANDU reactors involves 3D transport calcula-tions in addition to the 2D depletion calculations for themodeling of the safety and control systems. Indeed, suchreactors contain vertical reactivity devices located betweenhorizontal fuel channels. The analysis of the devices pres-ent in the core such as zone control units (ZCU) and shut-off rods (SOR) has been usually performed by 3Dcalculations on a so-called supercell geometry that consistsin a single device surrounded by two fuel bundles. The pur-pose of these calculations is to generate incremental cross-sections that are superimposed onto the cell cross sections

0306-4549/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2007.09.011

* Corresponding author. Tel.: +1 514 340 4711; fax: +1 514 340 4192.E-mail address: guy.marleau@polymtl.ca (G. Marleau).

1 Present address: Atomic Energy of Canada Ltd., 2251, SpeakmanDrive, Mississauga, ON, Canada L5K 1B2.

in a reactor model, reflecting the effect of the device. In thepast, diffusion theory has first been used for such calcula-tions. Dastur and Buss (1983) proposed a simplified modelin which the perturbation of the flux introduced by thedevices is considered to be significant only in the moderatorregion and the diffusion equation is solved for this soleregion imposing boundary conditions (current-to-fluxratios) on the surface of the fuel and the device from 2Dtransport calculations. Donnelly et al. (1996) presented atwo-group diffusion methodology for the complete super-cell based on condensed and homogenized cross-sectionsmodified by a superhomogenization (SPH) technique(Hebert, 1993). Nowadays, 3D transport theory on a sim-plified geometry without condensation is commonly used(Roy et al., 1994). In this case, 2D transport calculationson a single cell are necessary to obtain cross-sectionshomogenized per ring of the fuel cluster because the fuelbundles are modeled as tubes in the 3D supercellcalculations.

R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876 869

The ACR�2 (Advanced CANDU Reactor), a genera-tion III+ reactor is under active development in Canada.Accordingly, there is a strong interest in improving themodeling of the reactivity devices in the specific contextof this reactor. DRAGON (Marleau et al., 2006b) is thelattice code that is currently used to perform such calcula-tions. The modeling of the ACR reactivity devices with thiscode was presented by Dahmani et al. (2008). Apart fromstudying the effect of burnup and enrichment on incremen-tal cross-sections, these authors introduced an exact mod-eling of the 3D supercell getting rid of the partialhomogenization of the fuel cluster. If this possibility washighlighted, a more complete study on the representationof the supercell geometry remained to be done. Moreover,the treatment of the exact 3D supercell is time consumingand improvements of the computational methods is neededin order to fully take advantage of this new model. Thepurpose of this paper is to present these new developments.

First, when introducing this exact model, the number ofregions rapidly exceeds the practical limit of the collisionprobability method (CPM). As a consequence, the methodof characteristics (MoC) has to be used and the computa-tional efficiency is directly linked to MoC solver perfor-mances. Dahmani et al. (2008) considered parallelprocessing to reduce the computational burden while inthe present paper, we focus on the improvement of thesequential algorithm. Contrarily to CPM that has beenstandardized in DRAGON, MoC has been the object ofdifferent implementations over the last 10 years that stillcoexist in different versions of the code. Within the frame-work of 2D lattice calculations, the method of cyclic ofcharacteristics was introduced in the MOCC module(Roy, 1998; Roy, 1999) available in DRAGON sinceRelease 3.04. Later on, the MCI module was developedfor 3D non-cyclic characteristics (Wu and Roy, 2003b)based on a simple preconditioner, the self-collision rebal-ancing (SCR) (Wu and Roy, 2003a). This module wasthe starting point for investigations on parallel processingin a distributed memory environment for MoC (Dahmaniet al., 2003; Dahmani and Roy, 2005, 2007). The sequentialversion of this implementation is available in DRAGONRelease 3.05 (Marleau et al., 2006b) in the MCU moduleadding the capability of an inline tracking for MoC (Dah-mani et al., 2004). These implementations are limited tomultigroup flux calculations. Recently, with DRAGONVersion4 (Hebert, 2006; Marleau et al., 2006a), a MoCimplementation (referred as MCCG in the remainder ofthis paper) fully integrated in the computation channel ofa lattice calculation was released. MoC is available for sub-group self-shielding, multigroup flux with or without iso-tropic streaming effects and transport-to-transport SPHequivalence calculations (Le Tellier et al., 2006b). Both cyc-

2 A trademark of Atomic Energy of Canada Limited.

lic and non-cyclic methods are available in 2D while thenon-cyclic method is available for 3D calculations (Le Tel-lier et al., 2006a). In this implementation, special care wastaken for the sequential profiling and acceleration of MoC.Krylov iterative techniques were introduced for the solu-tion of the one-group transport equation (Le Tellier,2006) and the algebraic collapsing acceleration (ACA) orig-inally introduced by Suslov (2001) was improved andimplemented (Le Tellier and Hebert, 2007).

The modeling of ACR reactivity devices was discussedwithin the framework of DRAGON 3.05 (Dahmaniet al., 2008) and the MCU module was used both in itssequential version and its parallel version. If parallel com-puting was found efficient to reduce the CPU time, thenumerical convergence of SCR-preconditioned iterationsin MCU was found to be rather poor. As observed by LeTellier et al. (2006a) for CANDU-6 ZCU calculations,ACA method is more efficient and consequently, we areinterested in the present paper in comparing MCCG andMCU performances.

Supercell configurations are also geometrically simplerfor the ACR than for CANDU-6 reactors; indeed, bothSOR and ZCU are plates. As a consequence, a general3D tracking procedure is not mandatory and a 3D pris-matic approach can be used. This issue is discussed in Sec-tion 2 along with details on its implementation with MoC.Then, Section 3 presents the supercell configurations andthe methodology for the calculation of incremental cross-sections. Numerical results are discussed in Section 4 wherethe different solvers, tracking approaches and geometricalconfigurations are compared. Conclusions are finally givenin Section 5.

2. Computational methods

In this section, we focus on the ray-tracing procedurethat is the starting point for both CPM and MoC. We firstconsider the general 3D tracking procedure and then wediscuss the special case where the geometry can be extrudedfrom a 2D geometry (prismatic geometry) and give somedetails on our approach and the implementation of thistechnique with MoC in DRAGON.

2.1. Integral transport formalism

Integral methods are based on a tracking summationoperator that can be defined as

Sðf Þ ¼ 1

4p

Zð4pÞ

d2XZ

P~X

d2pf ð~T Þ; ð1Þ

where ~T is a so-called tracking line defined by a direction~X and a point~p in a normal 2D plane P~X. Such formalismappears both in the integration of collision probabilitiesfor CPM and in the integration for the region averagedscalar fluxes for MoC. In this context, the quantities thatthe line tracing process must calculate are the indices Nk

P

||

T

I

I

T

Ω

ψ

TΩ

θ

z

Ω

y

x

Fig. 1. 3D ray-tracing.

870 R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876

and path lengths Lk of the successive regions crossed by aneutron traveling on this line. With these notations, thereduced collision probability from region i to regionj 6¼ i can be written in 3D as

pij ¼1

RiRjV iS

Xk

di;Nk

Xh

dj;Nh 1� e�RiLk� �

e�sk;h 1� e�RjLh� � !

;

ð2Þwhile the scalar flux averaged over region i as integrated byMoC is given by

Ui ¼Qi

Riþ 1

RiV iS

Xk

djNkDk

!; ð3Þ

where Dk = in/j(k) � out/j(k) is the flux variation on seg-ment k of track ~T .

2.2. Ray-tracing techniques

In practice, Eq. (1) is evaluated by a numerical quadra-ture. In this paper, two 3D tracking approaches were testedto generate the set of tracking lines necessary to this evalu-ation. Both of them consist in a non-cyclic tracking andrequire that the reflection surfaces are treated as whiteboundaries.

First, the general 3D tracking procedure of DRAGONas implemented in EXCELT (Roy et al., 1989) and NXT(Marleau, 2006) modules was used. It is based on a com-bined 2D angular and 2D spatial numerical integration.The general procedure that is used for selecting therequired quadrature points is:

� Decompose the angular domain into a set of directionsand associate with each direction ~Xm a weight xX

m in sucha way that 1

4p

Rð4pÞ d

2X ¼P

mxXm~Xm ¼ 1.

� Select a 2D Cartesian plane P~Xmnormal to each direc-

tion and subdivide this plane into a uniform grid com-patible with a constant step quadrature therebygenerating a series of integration points ~pm;n with aunique weight xP

m;n for each direction ~Xm.

In this case, Eq. (1) can be directly written as

Sðf Þ ¼X

m

xXm

Xn

xPn;mf ð~T n;mÞ: ð4Þ

Concerning the angular treatment, EQN (see Carlson,1971) is the only quadrature available in the EXCELTtracking module. In NXT, symmetrical versions of theLegendre–Chebyshev (PCN) quadrature proposed byLongoni and Haghighat (2001) and the Legendre-trapezoi-dal (PTN) quadrature proposed by Sanchez et al. (2002) canalso be used. For a fixed order N, EQN, PCN and PTN gen-erate respectively NðNþ2Þ

8, 3NðNþ2Þ

8and 3N2

2discrete directions

per octant of the unit sphere.In addition, as mentioned earlier, both ZCU and SOR

devices in an ACR consist of plates and the 3D supercellgeometries can be extruded from a 2D geometry. The

corresponding axis will be arbitrarily chosen as ~ez. In thisspecial case, as the mesh is Cartesian and uniform along~ez, a so-called prismatic 3D tracking procedure can be used.First, restricting the angular quadrature to product quadr-atures, the angular integration is decomposed as

1

4p

Zð4pÞ

d2X ¼ 1

4p

Z 2p

0

dwZ 1

�1

dl; ð5Þ

where w is the azimuthal angle and l = cosh the polar co-sine. Then, as depicted in Fig. 1, the spatial integration isseparated into two parts, an integration along a line de-noted I^ in the x–y plane, perpendicular to ~X and an inte-gration along the line denoted Ik perpendicular to I^ and ~X.In this way, the tracking summation operator of Eq. (1)can be rewritten as

Sðf Þ ¼ 1

4p

Z 2p

0

dwZ

I?

dp?Z 1

�1

dlZ

Ik

dpkf ð~T Þ; ð6Þ

and the tracking procedure is decomposed into two steps.First, a 2D tracking �? ¼ f~T?g is generated by consid-

ering the first two parameters (w,p^) for a 2D geometryresulting from a projection of the 3D prismatic model.The 2D tracking procedure is similar to what was explainedin the 3D case and is performed by the same modules. Atrapezoidal quadrature TN where N is the number of angles2]0,p/2[, denoted ðwm;x

wmÞ is selected for w tracking angle

and a constant step quadrature ðp?m;n;x?m;nÞ is used for the1D spatial integration along I^ normal to ~X?.

Then, the 3D tracks ~T can be constructed on-the-flyfrom �^ by a simple 2D tracking procedure in each planeð~T?;~ezÞ. This second tracking procedure is simplified dueto the fact that the intersection of the domain with a planeð~T?;~ezÞ consists only in a Cartesian grid. In our implemen-tation, Ik is decomposed into two parts: Ik;Z and Ik;L for thetracks incoming in the domain through bottom/top andlateral surfaces, respectively, as depicted in Fig. 2. Thisdecomposition is not a requirement but it simplifies the

z

p

p

T T

T T_ _

I

I

+L Z

+

Z

||,L

||,Z

||,L

||,Z

L

T T

Fig. 2. Inline 3D track construction for a prismatic geometry.

R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876 871

algorithm when considering symmetry boundary condi-tions with respect to x–y outer surfaces. EXCELT orNXT treatment for these symmetries consists in unfoldingthe geometry and merging adequately the regions and sur-faces. For the prismatic tracking, we chose to take directlyinto account these symmetries by reflecting the trajectorieswhen they encounter a symmetry plane. With thisapproach, symmetry conditions can be applied on boththe bottom and top x–y boundaries at the same time.Moreover, this treatment is compatible with anisotropicscattering contrarily to the geometry unfolding. In addi-tion, tracks corresponding to +l and �l are constructedduring a single sweep of these two intervals. This approachcorresponds to Eq. (6) rewritten as

Sðf Þ ¼ 1

4p

Z 2p

0

dwZ

I?

dp?Z 1

0

dlZ

Ik;Z

dpk;Z f ~T Zþ

� �þ f ~T Z

�� �� �

þZ

Ik;L

dpk;L f ~T Lþ

� �þ f ~T L

�� �� �!

; ð7Þ

additional notations being defined in Fig. 2. Consideringan angular quadrature lu;x

lu

� �for l 2]0,1[ and constant

step quadratures pk;Lu;v ;xk;Lu;v

� �and pk;Zu;v ;x

k;Zu;v

� �for the spatial

integration along Ik;L and Ik;Z , respectively, Eq. (7) becomes

Sðf Þ ¼X

m

xwm

Xn

x?n;mX

u

xlu

Xv

xk;Zu;v f ~T Zþ;u;v

� ��

þf ~T Z�;u;v

� ��þX

v

xk;Lu;v f ~T Lþ;u;v

� �þ f ~T L

�;u;v

� �� �!:

ð8Þ

In this paper, a Gauss–Legendre quadrature PN where N isthe number of points 2]0,1[ was used for lu;x

lu

� �when a

prismatic tracking is considered.

2.3. Prismatic tracking with MoC in DRAGON

This prismatic tracking capability has recently beenimplemented in the characteristics solver MCCG of DRA-GON Version4. The code was modified at two levels. First,to be coherent with DRAGON environment and minimizethe modification in the user’s computational schemes, a 3Dprismatic geometry is defined as any other 3D geometry;the prismatic treatment is only an option at the trackingmodule level. EXCELT and NXT modules perform theanalysis of the 3D geometry; when the prismatic option isenabled, tests are performed to ensure that the geometryis prismatic and the analysis of the corresponding 2D pro-jected geometry is created along with a region mappingindex between the 2D geometry and the different x–y

planes of the initial 3D geometry. This analysis is then usedby EXCELT or NXT tracking routines to produce the 2Dtracking file. MCCG was then modified to incorporate thein-line construction of the 3D tracks from the 2D trackingfile when a prismatic tracking procedure is requested. Allthe options available in the MCCG module were madecompatible with this special tracking. In particular, theACA preconditioning technique as used in this work wasmodified according to this prismatic tracking approach.ACA is based on a collapsing hypothesis with respect tothe tracking summation operator S. It leads to a sparsecorrective system the order of which is the same as the ini-tial transport problem. In practice, in 3D, ACA overheadper group is about 12 times the system order in terms ofthe number of real scalars to be stored. An optimized stor-ing and solving strategy presented by Le Tellier and Hebert(2007) is used. Moreover, by introducing a track mergingtechnique (Wu and Roy, 2003a), a two-step algebraic col-lapsing process was obtained: the acceleration is improvedwhile the time required to construct the corrective system isdrastically reduced. This merging process was optimized inthe prismatic case by introducing two levels of track merg-ing: a first one is applied to the 2D tracks and a subsequentone is used for the 3D tracks (constructed from a 2Dmerged track).

3. Reactivity device simulation

The fuel bundle geometry of the ACR consists in a 43pins cluster geometry. The central poisoned pin is largerthat the three outer rings of fuel pins. Two types of bundlesthat differ only by their Uranium enrichment are present inthe ACR reactor. A so-called supercell consists in a singlevertical reactivity device inserted between two fuel channelsas depicted in Fig. 3 for the case of a zone control unit(ZCU). In general, reflective boundary conditions areapplied on the external surfaces in the x, y and z directions.

3.1. Geometrical modeling

Here, we focus our attention on the geometry modeling.Fresh fuel with the lowest enrichment is considered

Right bundleLeft bundle

Fig. 3. Projections in the x–y (bottom) and x–z (top) planes of the 3DZCU supercell geometry.

Left FuelBundle

Right FuelBundle

Fig. 4. Coarse discretization of the ZCU supercell.

872 R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876

throughout our study. Two discretized configurations ofthe supercell will be analyzed; they are presented in Figs.4 and 5 and are referred to as ‘‘coarse’’ and ‘‘fine’’ modelsin the remainder of the paper. They were derived from aprevious study for an earlier design of the ACR (Dahmaniet al., 2006). For the fuel bundles within the supercell, threegeometries, as depicted in Fig. 6, were used. The first onecalled ‘‘annular model’’ consist in an homogenization ofthe fuel cluster into four annuli. The main problem hereis to select the flux distribution that will be used to producethe cross-sections associated with the annular fuel. Assum-ing that the impact of the reactivity devices on the flux dis-tribution inside the fuel is relatively small, a 2D cellcalculation in the exact cluster geometry should providean adequate flux distribution for the homogenization pro-cess. In order to preserve as much information as possible,no energy condensation is considered. This is the standardprocedure used for CANDU-6 reactors (Roy et al., 1994).Two other fuel geometries that consist in the exact clustergeometry were considered: the ‘‘coarse cluster model’’ issimilar to the geometry used for self-shielding calculationwhile the ‘‘fine cluster model’’ corresponds to the geomet-rical modeling for the multigroup flux calculation in 2D celldepletion calculations (Dahmani et al., 2008). The differentconfigurations used in the remainder of the paper are sum-marized in Table 1 along with the number of regions andexternal surfaces they generate. While configurations based

on the annular model were treated with the EXCELTtracking module of DRAGON, the cluster models wereanalyzed and tracked with the NXT module.

3.2. Incremental cross-sections evaluation

The procedure for evaluating the incremental cross-sec-tions associated with the devices involves three successive3D transport calculations. The first calculation, which isused to obtain the average supercell few group cross-sections in the absence of the reactivity device (RG

A;x) mustbe performed using a model where the device and the guidetube are replaced with moderator. Assuming that the mul-tigroup cross-section Rg

i;x of type x associated with a regionof volume Vi sees a flux /g

i , then RGA;x is obtained using

RGA;x ¼

Pg2G

Pi

V i/gi R

gi;xP

g2G

Pi

V i/gi

ð9Þ

The second step consist in inserting the guide tube andrepeating the transport calculation to obtain a new fluxdistribution. The resulting homogenized and condensedcross-sections RG

T ;x, computed using an equation similar toEq. (9) include the effect of the guide tube, weighted by

Left FuelBundle

Right FuelBundle

Fig. 5. Fine discretization of the ZCU supercell.

Table 1Description of the geometrical configurations

Configurations Supercell model Fuel model Nra Ns

b

Cc-a Coarse Annular 168 202Cf-a Fine Annular 2400 1040Cf-c Fine Coarse cluster 3040 1104Cf-f Fine Fine cluster 6440 1444

a Number of regions.b Number of external surfaces.

R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876 873

the new flux. Finally, a third transport calculation is per-formed where the guide tube is present and the controlplate is fully inserted and occupies a full lattice pitch, thecorresponding homogenized and condensed cross-sectionsbeing denoted RG

TþD;x. Three types of incremental cross-sec-tions can then be defined

DRGT ;x ¼ RG

T ;x � RGA;x; ð10Þ

DRGD;x ¼ RG

TþD;x � RGT ;x; ð11Þ

DRGTþD;x ¼ RG

TþD;x � RGA;x: ð12Þ

Annular Coarse cl

Fig. 6. Fuel bundle m

4. Numerical results

In this section, we first evaluate the performances of thedifferent transport solvers and tracking options using thesupercell geometries described in Table 1. These differentgeometrical configurations are then compared in terms ofthe accuracy for the evaluation of the incremental cross-sections.

4.1. Solvers comparison

To compare the different solvers, the coarse discretiza-tion of the supercell and the annular fuel model were con-sidered in order to limit the number of regions. Thisconfiguration is denoted Cc-a. The two characteristics sol-ver MCU and MCCG described in Section 2.3 were usedalong with CPM. Two approaches are reported for CPM:the so-called inline CPM where the tracks are generatedon-the-fly when constructing the CP matrices and the stan-dard CPM where the tracking file is generated once and forall. For MCU and MCCG, the tracks are also computedprior to the flux calculation and stored in a tracking file.Note that the MCU solver has also an inline capabilitybut it considerably increases the CPU time (Dahmaniet al., 2008) and was not used in this study.

From the point of view of the multigroup iterativescheme, both MCU and MCCG are based on a Jacobischeme as they integrate vectorially (over the energygroups) the flux upon the tracking. This approach is timeefficient when one considers tracking lines stored in a bin-ary file since it limits the number of file accesses. For stan-dard CPM, the CP matrices are integrated vectorially upon

uster Fine cluster

odeled geometries.

874 R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876

the tracking file for the same reason. CPM is based on aGauss–Seidel scheme and calculates the flux one-groupafter the other starting from the fastest one. All the solversuse a multigroup rebalancing and a variational accelerationof the power iterations (Marleau et al., 2006b).

Two sets of parameters were used for the general 3Dtracking procedure with the EXCELT module: an EQ16

angular quadrature with a track density of 100 tracks/cm2 (see Table 2) and an EQ8 angular quadrature witha track density of 20 tracks/cm2 (see Table 3). The resultsin terms of k1 for the three calculations clearly demon-strate the coherence of the four solvers. It is worth notingthat MCU and MCCG, although being independentMoC implementations, give the same answer within0.01 mk. When comparing CPM to MoC, we observethat the difference is 0.02 mk for the fine tracking and0.06 mk for the coarser tracking. In 3D, MoC andCPM differences come directly from the approach usedto ensure the neutron conservation. While MoC usesnumerically integrated surfaces to apply the isotropicboundary conditions, CPM renormalizes the CP matricesin order to ensure the conservation laws. When the track-ing is coarsened, the difference between both treatments isenhanced. If one considers an EQ2 angular quadraturewith a track density of 10 tracks/cm2, this differencereaches 0.5 mk.

Tables 2 and 3 also include t, the total CPU time andNiter, the total number of multigroup iterations for thethree calculations. With such a low number of regions,standard CPM is the fastest, about two times faster thaninline CPM or MCCG. Concerning the MoC solvers,MCU is slower than MCCG because of the greater numberof iterations needed to achieve convergence: even with alow number of regions, ACA preconditioned approach in

Table 2Solvers comparison on Cc-a configuration (EQ16 – 100.0 tracks/cm2)

Solver CPM Inline CPM MCU MCCG

kA1 1.12019 1.12019 1.12017 1.12017

kT1 1.11746 1.11746 1.11745 1.11745

kTþD1 1.04179 1.04179 1.04177 1.04177

ta (h) 21.1 49.0 81.8 40.5Niter

b 44 45 85 33

a CPU time.b Total number of multigroup iterations.

Table 3Solvers comparison on Cc-a configuration (EQ8 – 20.0 tracks/cm2)

Solver CPM Inline CPM MCU MCCG

kA1 1.12032 1.12032 1.12026 1.12026

kT1 1.11760 1.11760 1.11754 1.11754

kTþD1 1.04165 1.04165 1.04159 1.04159

t (h) 1.3 2.8 4.8 2.8Niter 45 44 82 33

MCCG is about two times faster than the SCR based strat-egy in MCU. As the geometry is refined and the number ofregions becomes large, we clearly see the advantage ofusing the MCCG implementation of the characteristicsmethod for these supercell calculations. This comparisonconfirms the conclusions of (Le Tellier et al., 2006a) onCANDU-6 ZCU configurations. In the next sections,MCCG is used for all the calculations.

Note that within the framework of MCU, when con-sidering parallel computing, at the worst, a speed-up of25 can be reached with 64 processors. Consequently, therewould be a strong interest in combining MCCG sequen-tial acceleration strategy and MCU parallel processing.Moreover, as discussed by Le Tellier (2006), the ACA-based strategy could benefit from a distributed memoryenvironment by applying ACA hypothesis separately forthe tracking part of each processor: the communicationoverhead induced by the ACA preconditioner could lar-gely be overcome by the improvement of the syntheticpreconditioner.

4.2. Use of a prismatic tracking

In this section, we consider the fine discretization of thesupercell along with the annular fuel model (Cf-a configura-tion) to compare the general 3D tracking with the prismatictracking approach. Results are reported in Table 4 in termsof the reactivity effects

DqT ¼ 1000� 1

kA1� 1

kT1

!; ð13Þ

DqD ¼ 1000� 1

kT1� 1

kTþD1

� ; ð14Þ

and the incremental cross-sections for the device DRgD;x.

Prismatic tracking cases are denoted Tn · Pm as they usea product quadrature composed of a trapezoidal quadra-ture for the azimuthal angle and a Gauss–Legendre quad-rature for the polar angle.

Table 4Comparison of general and prismatic tracking based calculations on Cf-a

configuration with MCCG

Ref. value Relative difference (%)

Tracking EQ16 T6 · P6 T4 · P3 T6 · P3 T4 · P6

Parameters 100.0 10.0 · 10.0 5.0 · 5.0 10.0 · 5.0 5.0 · 10.0

DqT �2.21 �0.01 �0.08 �0.10 �0.11DqD �61.40 �0.04 0.11 0.08 �0.02

DR1D;t 1.58e�03 0.03 0.21 0.16 0.07

DR1D;a 3.46e�05 0.03 0.23 0.13 0.08

DR1!1D;s0 1.60e�03 0.02 0.22 0.16 0.08

DR2!1D;s0 1.45e�05 �0.02 0.27 0.15 0.10

DR2D;t 1.62e�03 0.09 0.63 0.56 0.16

DR2D;a 7.27e�04 �0.03 0.23 0.13 0.08

DR1!2D;s0 �5.46e�05 �0.12 0.01 �0.01 0.01

DR2!2D;s0 8.77e�04 0.20 0.98 0.94 0.23

t (h) 100.4 28.6 2.6 8.4 9.5

Table 5Comparison of different tracking parameters on Cf-c configuration usingNXT and MCCG

Ref. value Relative difference (%)

Tracking EQ16 T6 · P6 PT4 EQ12 PC8

Parameters 100.0 10.0 · 10.0 100.0 100.0 100.0

DqT �2.14 0.09 �0.25 0.07 �0.05DqD �61.91 0.09 0.37 0.09 0.34

DR1D;t 1.58e�03 0.04 0.14 0.05 0.11

DR1D;a 3.46e�05 0.03 0.28 0.06 0.15

DR1!1D;s0 1.60e�03 0.03 0.13 0.04 0.11

DR2!1D;s0 1.44e�05 0.01 0.66 0.10 0.39

DR2D;t 1.58e�03 �0.03 0.84 0.12 0.49

DR2D;a 7.25e�04 �0.02 0.65 0.10 0.38

DR1!2D;s0 �5.48e�05 �0.07 �0.06 �0.02 �0.05

DR2!2D;s0 8.41e�04 �0.04 1.00 0.14 0.59

t (h) 148.6 39.7 97.1 87.1 107.2

Table 6Comparison of the different geometrical configurations using MCCG anda prismatic tracking

Ref. value Relative difference (%)

Configurations Cf-f Cf-c Cf-a Cc-a

kA1 1.14190 �0.528 �1.898 �1.881

DqT �2.28 �6.14 �3.04 �4.65DqD �67.30 �7.93 �8.80 �3.45

DR1D;t 1.59e�03 �0.36 �0.34 0.61

DR1D;a 3.48e�05 �0.61 �0.55 0.35

DR1!1D;s0 1.61e�03 �0.34 �0.35 0.61

DR2!1D;s0 1.45e�05 �0.44 0.04 6.71

DR2D;t 1.63e�03 �3.35 �0.87 6.20

DR2D;a 7.36e�04 �1.53 �1.23 5.72

DR1!2D;s0 �5.48e�05 �0.12 �0.45 0.54

DR2!2D;s0 8.84e�04 �4.93 �0.60 6.58

t (h) 54.5 39.7 28.6 9.6Niter 56 51 51 33

R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876 875

First of all, if we compare the EQ16 calculation with theT6 · P6 case, we see that the results are very close both interms of reactivity and incremental cross-sections. Theyexhibit the same level of tracking refinement: both of themgenerate 36 tracking directions per octant and correspondto a density of 100 tracks/cm2 (in the case of the prismatictracking, a linear density of 10 tracks/cm was selected forthe 2D tracking while the inline 3D track construction usesa density of 10.0 tracks/cm). When comparing the CPUtimes, the interest in using a prismatic tracking is clear, itreduces the computational time by a factor of about 3.5while the total number of multigroup iterations remainsthe same, namely 51. In terms of storage, the 3D trackingfile generated by the general procedure is about 15 Gbwhile the 2D tracking file used by the prismatic approachis less than 1 Mb. Note that the ACA-based accelerationstrategy is improved when considering a prismatic trackingbecause the time spent in building the ACA matrices is lar-gely reduced. For example, for the EQ16 (resp. T6 · P6)tracking option, the ratio between the ACA building timeand the time spent for a characteristics iteration is 1.95(resp. 0.16). This is directly due to the use of a track merg-ing technique when building the ACA corrective system asexplained in Section 2.3.

In Table 4, we also report three prismatic configurationswith a coarser tracking. It is interesting to note when com-paring T6 · P3 and T4 · P6 configurations with T6 · P6 andT4 · P3 that the 2D tracking can be coarsened withoutdegrading the accuracy. The results are more sensitive tothe parameters of the inline 3D track construction becauseof the fine meshing along the z axis. The prismatic trackingis advantageous in this case as it inherently offers a decou-pling between the integration parameters for (w,p^) on theone hand and ðl; pkÞ on the other hand.

4.3. Comparison of different quadratures

In this section, the Cf-c configuration which exactly rep-resents the fuel cluster is used with the NXT tracking mod-ule. In this case, the three angular quadratures described inSection 2.2 can be used to generate the 3D tracks. Compar-ison between a prismatic tracking and different 3D generaltracking is presented in Table 5. We see that the prismatictracking results are totally coherent with the standardtracking results.

4.4. Comparison of the different geometrical configurations

We now focus on the comparison of the results for thefour geometries Cf-f , Cf-c, Cf-a and Cc-a. A prismatic track-ing (T6 · P6 angular quadrature and 10.0 · 10.0 trackdensity) was used for all the calculations. From Table 6,we clearly see that noticeable differences are observedfor the different geometrical configurations. What is par-ticularly interesting is the good performances of Cf-a ascompared to Cf-c: it seems preferable to use an annularrepresentation of the fuel with cross-sections homogenized

with a correct reference flux than to use a coarse meshexact fuel geometry. Moreover, if we consider the 2Ddepletion calculation on a single cell, k1 for fresh fuelis 1.14185, only 0.05 mk lower than the Cf-f result. Theintroduction of the prismatic tracking procedure permitsto use the geometrical configuration of the fuel bundleretained in the 2D depletion calculation for the supercellcalculation with a reasonable computational time: Cf-f

with a prismatic tracking is about two times faster thanCf-a with a standard tracking. In this way, the computa-tional procedure for the ACR reactor is improved andsimplified thanks to the coherence reached between the2D and 3D transport calculations.

5. Conclusion

In this paper, we have presented improvements intro-duced in the ACR context to reduce the cost of 3D supercellcalculations necessary to the reactivity device modeling.First, the introduction of an optimized characteristics

876 R. Le Tellier et al. / Annals of Nuclear Energy 35 (2008) 868–876

method has permitted to reduce the computational burdenof such calculations that involve a large number of regions.In addition, contrarily to CANDU-6 supercell configura-tions, because the ACR 3D geometry is prismatic, a specialtracking procedure can be introduced. This approachinduces no approximation and is shown to be significantlyfaster than the general 3D tracking procedure while requir-ing a much smaller tracking file. Finally, 3D supercell calcu-lations on the exact ACR geometry without any priorhomogenization become affordable when considering amesh discretization comparable to 2D cell configurations.This simplifies the ACR computational procedure andremoves some of the hypothesis usually considered in thereactivity device modeling of such reactors. Moreover, con-sidering a coarser tracking, isotopic depletion calculationsfor 3D prismatic geometries may become practical withthese developments.

Acknowledgements

This work was supported in part by the Natural Scienceand Engineering Research Council of Canada and byAtomic Energy of Canada Limited.

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