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Progress in Nuclear Energy 85 (2015) 640e647

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Progress in Nuclear Energy

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Partitioned-Matrix acceleration to the Fission-Source iteration of theVariational Nodal Method

Yunzhao Li a, *, Yongping Wang a, Boning Liang a, Wei Shen a, b

a School of Nuclear Science and Technology, Xi'an Jiaotong University, Chinab Canadian Nuclear Safety Commission, Canada

a r t i c l e i n f o

Article history:Received 1 June 2015Received in revised form27 July 2015Accepted 3 August 2015Available online 3 September 2015

Keywords:Variational Nodal MethodNeutron-diffusion calculationPWRPartitioned-Matrix accelerationFission-Source iteration

* Corresponding author. School of Nuclear Science aUniversity, 28 West Xianning Road, Xi'an, Shaanxi 71

E-mail address: yunzhao@mail.xjtu.edu.cn (Y. Li).

http://dx.doi.org/10.1016/j.pnucene.2015.08.0010149-1970/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

The Variational Nodal Method (VNM) expands the nodal volumetric flux and surface partial current intothe sums of orthogonal basis functions without using the transverse integration technique. The exclusionof the transverse integration provides a number of advantages for employing the VNM in PressurizedWater Reactor (PWR) core simulation. The orthogonality of those basis functions guarantees the con-servation of neutron balance regardless of the expansion orders, providing an opportunity to acceleratethe computationally expensive full-order iteration by using cheap low-order sweeping with high-ordermoments fixed. This was named as the Partitioned-Matrix (PM) technique in the legacy VNM codeVARIANT, and was applied to the within-group (WG) iteration. It is very effective for neutron-transportcalculation, but less effective for neutron-diffusion mainly due to the reduced number of high-orderpartial current moments. In this paper, we extend the PM technique to the Fission-Source (FS) itera-tion to accelerate the flux convergence by using low-order flux moments also. From the macroscopicacceleration point of view, it converges the fission- and scattering-source distributions by usingcomputationally cheap low-order iteration faster than the original full-order sweeping. Based on ournew VNM code VIOLET, considering the fact that the discontinuity factor used for preserving neutronleakage rates during spatial homogenization slows down the nodal iteration convergence, numericaltests were carried out for two typical PWR problems respectively without and with discontinuity factors.By analyzing both the computational effort in terms of FLOP (FLoating-point OPeration) and computingtime, the following conclusions have been demonstrated. The legacy PM technique for WG iteration canprovide an acceleration ratio of about 2 for the PWR core neutron-diffusion calculation with or withoutusing discontinuity factors, while the one for FS iteration itself can accelerate by a factor of about 3 whichis higher. By accelerating both the WG and FS iteration simultaneously, the acceleration ratio is about 4for both the two PWR problems. In addition, by extending the PM technique from the WG iteration to theFS iteration, the neutron-diffusion calculation of the VNM can be accelerated very effectively with almostno extra storage or implementation cost to the existing computer code.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The Variational Nodal Method (VNM) (Lewis and Miller, 1984;Carrico et al., 1992) was first developed by Northwestern Univer-sity and Argonne National Laboratory (ANL) to solve the multi-group steady-state neutron-diffusion and -transport equations forreactor core calculations. It uses a variational principle for the even-

nd Technology, Xi'an Jiaotong0049, China.

parity form of the Boltzmann transport equation. In this variationalprinciple, the odd-parity Lagrange multipliers along the nodal in-terfaces guarantee neutron conservation for each node. The clas-sical Ritz procedure is employed by using orthogonal polynomialsin space and spherical harmonics in angle. Nodal response matricesare then formed for the volumetric flux moments and surfacepartial current moments. The VARIANT code (Palmiotti et al., 1995),developed at ANL inmid 90s was the first production code based onVNM. It has been employed for fast reactor routinely designing bothin ANL such as the REBUS code (Toppel, March 1983) and in Europesuch as the ERANOS code (Doriath et al., 1994). In 2007, a newversion of the VARIANT code named NODAL was developed in ANL

Y. Li et al. / Progress in Nuclear Energy 85 (2015) 640e647 641

as one of the solvers in the UNìC package (Palmiotti et al., 2007; Liet al., 2015). In 2011, it has also been implemented into the INSTANTcode in Idaho National Laboratory (INL) (Wang et al., 2011).

The exclusion of the transverse integration in the VNM providesseveral advantages (Lawrence, 1986; Wagner, 1989). Firstly, theVNM expands the volume flux by using basis functions whichusually are orthogonal polynomials. Once obtained those fluxmoments, continuous flux profile within each node can be ob-tained, leaving no need for pin power reconstruction which usuallyintroduces more approximations. Secondly, for adjoint flux calcu-lation usually employed in transient simulation, the VNM canguarantee that the corresponding mathematical adjoint flux isexactly the same with the physical one. Thirdly, it is possible toextend the homogeneous VNM to heterogeneous VNM which cantreat heterogeneous cross section distribution within each node(Smith et al., 2003; Li et al., 2014; Wang et al.). Fourthly, the VNMemploys the Pnmethod for angular variable withinwhich neutron-diffusion equation is equivalent to the P1 approximation, enablingthis method can to be consistently extended to neutron-transportcalculation. Thus, recently a new VNM code named VIOLET hasbeen developed at Xi'an Jiaotong University (XJTU) for thermalreactor such as PressurizedWater Reactor (PWR) neutron-diffusionsimulation.

The numerical process of the VNM contains three levels ofiteration. The outermost is the Fission-Source (FS) iteration (alsotermed as the outer iteration in literatures) based upon the PowerMethod (Lewis and Miller, 1984). At each FS iteration, just in case ifup-scattering shows up, the multi-group (MG) flux system is solvedby using the legacy Gauss-Seidel (GS) algorithm. Only one sweepover the energy groups is required if there is no up-scattering. Foreach group, the within-group (WG) response matrix system issolved by using the Red-Black Gauss-Seidel (RBGS) algorithm(Palmiotti et al., 1995). It is the so-called WG iteration (typicallytermed as the inner iteration in literatures).

Traditionally, the VARIANTcode employs the Partitioned-Matrix(PM) technique to accelerate the WG iteration. Before each full-order partial current moments iteration, a number of low-orderpartial current moments iterations are carried out with the high-order ones fixed. Usually, only one full-order sweep is carried outfor each energy group within each MG iteration. The PM techniqueperforms very well in transport cases due to the large number ofhigh-order moments. However, the effect is less effective in diffu-sion because there are fewer high-order moments to eliminate.Though other techniques or algorithms such as the Krylov (Saad,2003; Saad and Schultz, 1986) ones including CG (ConjugateGradient) (Wang et al., 2011) and GMRES (Generalized MinimalResidual Method) (Wang et al., 2011; Lewis et al., 2013; Li et al.,2012) have also been proposed and tested, they usually requiremore memory due to the storage of orthogonal vectors. In addition,these algorithms usually require preconditioners to be compatiblewith the PM accelerated WG RBGS iteration, making the codesystem much more complicated.

The rest of this paper is organized as following. Section 2 de-scribes the theory of the VNM including its iteration process, thePM technique and its implementations to the both the WG and FSiterations of the VNM, Section 3 assesses the PM techniquenumerically by using two typical PWR problems respectively withand without discontinuity factors. Section 4 summarizes the con-clusions and discussions.

2. Theoretical formulation

After introducing the VNM response matrices and the iterationprocess, the PM techniques for both the WG and FS iterations aredescribed in detail. The computing efforts of applying these

response matrices are evaluated and summarized based on theseformulas.

2.1. The Variational Nodal Method

After the multi-group approximation for the energy variableand the P1 approximation for the angular variable, isotropic scat-tering with transport correction and isotropic fission, the neutron-transport equation becomes the Multi-Group neutron-diffusionequation together with its albedo boundary condition:

8>>><>>>:

VJg þPr;gFg¼ Sg

13VFg þ P

tr;gJg¼0

; g ¼ 1 � G (1)

Sg ¼Xg0sg

Ps;gg0

Fg0 þ 1k

Xg0

Fgg0Fg0 (2)

Fg � 2JTgng ¼ bg;g$�Fg þ 2JTgng

�; r2Gg (3)

where common symbols are used as in literature (Lewis and Miller,1984), Jg and ng are column vectors. The variational principle (Lewisand Miller, 1984) turns out to be:

FhFg; Jg

i¼

Xv

FvhFg ; Jg

i(4)

FvhFg ; Jg

i¼

ZvdV

n�3Str;g

��1VFgVFg þ Sr;gF

2g � 2FgSg

oþ 2

�Xg

Zg

Fg$JTgngdG

(5)

For each energy group and each node, the volumetric flux andsource are expanded by using basis functions:

FgðrÞ ¼XI

i¼1

fi;gfiðrÞ ¼ f T4g ; r2v (6)

SgðrÞ ¼XIi¼1

si;gfiðrÞ ¼ f Tsg ; r2v (7)

And the surface net outgoing current is expanded as

JTgðrÞng ¼XKk¼1

jg;k;ghg;kðrÞ ¼ hTgjg;g r2Gg (8)

where fi(r) and hg,k(r) are orthogonal polynomial basis functionsrespectively on nodal volume v and surface Gg, I and K are thenumber of expansion terms, 4g, sg, jg,g, f and hg are column vectorscontaining the corresponding moments and functions.

Nodal response matrices can be formed (Lewis and Miller, 1984;Carrico et al., 1992; Palmiotti et al., 1995; Wang et al., 2011) byfirstly substituting the expansions in Eqs. (6)e(8) into the func-tional in Eq. (5), the boundary conditions in Eq. (3) and the source inEq. (2) and then requiring the functional to be stable in terms of 4g

and jg respectively:

Fig. 1. The VNM iteration process.

Y. Li et al. / Progress in Nuclear Energy 85 (2015) 640e647642

sg ¼Xg0sg

Sgg04g0 þ 1k

Xg0

Fgg04g0 (9)

jþg ¼ RgP5IK jþg þ Bgsg (10)

4g ¼ Hgsg � Cg�IN$NS

�P�5IK j

þg (11)

where П stands for the spatial connection matrix containing thealbedo outer boundary condition and the continuity inner bound-ary condition, IX stands for an X� X identitymatrix,N is the numberof nodes, NS is the number of surfaces per node, 5 refers to tensorproduct between the two matrices, the surface partial currentvector is defined as

j±g ¼ 14MT4g±

12jg (12)

jg ¼hjT1;g / jTg;g /

iT(13)

The response matrices R, B, H, C and M, depends on nodal ge-ometry, material and the volumetic and surface expansion orders,are block diagonal over spatial nodes and can be found elsewhere(Palmiotti et al., 1995). Once the discontinuity factors (DF) show up,this inner boundary condition can be obtained by imposing netcurrent continuity and flux discontinuity:

j�g;g ¼ 2fg0

fg0 þ fgjþg0;g þ

fg0 � fgfg0 þ fg

jþg;g (14)

fg and fg0 are the discontinuity factors (DF) of the two adjacentnodes g and g0 obtained by using the legacy Generalized Equiva-lence Theory (GET) (Smith, 1986, 1980).

The appearance of the fission source makes the above equationan eigenvalue problem. The legacy Power Method (Lewis andMiller, 1984; Xie and Deng, 2005) is employed and referred asFission-Source (FS) iteration here. Within each FS iteration, the fluxhas to be updated with fission source fixed by employing the legacyGauss-Seidel algorithm. This iteration is termed as multi-group(MG) iteration. It is noticeable that only one MG iteration wouldbe required if there is no up-scattering which is usually the case forthe PWR core diffusion calculation. Within each MG iteration, theflux vector is updated one group after another, which means thereis an sweeping from the 1st energy group to the G'th energy group.For each energy group, a) the source is constructed by using Eq. (9),b) the current is iteratively updated as in Eq. (10) by employing theRed-Black Gauss-Seidel (RBGS) algorithm, c) the flux is constructedby using the neutron balance equation in Eq. (11).

The iteration process of VNM is shown in Fig. 1. It has to bepointed out that there is an Absolute Error Tolerance (AET) and aRelative Error Tolerance (RET) used as the iterative vector conver-gence criteria. Taking vector v as an example, the AET εv and RET evare as following:

rðiÞ ¼��vðiÞ � vði�1Þ��

2��vðiÞ��2 (15)

rðiÞ < εv (16)

rðiÞ

rð1Þ< ev: (17)

2.2. The Partitioned-Matrix acceleration for within-group iteration

Because of the hierarchical system of orthogonal polynomials inspace, a low-order system can closely resemble the original full-order system by maintaining neutron balance. Thus, traditionallythe Partitioned-Matrix (PM) technique is applied to the within-group (WG) iteration.

Partitioning the surface partial current vector:

jþg ¼��

jþ0g

�T �jþ1g

�T �T(18)

where 0 and 1 denote the flat and non-flat surface momentsrespectively. Correspondingly, the WG iteration formulas can alsobe partitioned

"jþ0g

jþ1g

#¼

"R00g R01

g

R10g R11

g

#"P5I1j

þ0g

P5IK�1jþ1g

#þ"B0g

B1g

#sg (19)

As in VARIANT, the WG iteration can be done only for the flat-moment system with non-flat ones fixed

jþ0g ¼

hR00g R01

g

i" P5I1jþ0g

P5IK�1jþ1g

#þ B0

gsg (20)

Once the flat-moment system converged, a full sweep can becarried out to update the all of the moments including the non-flatones. The VNM iteration process is shown in Fig. 2.

By comparing the Eqs. (19) and (20), it can be found that the PMtechnique is designed to save numbers of updates on the non-flatpartial current moments by requiring a number of more updateson the flat ones.

Y. Li et al. / Progress in Nuclear Energy 85 (2015) 640e647 643

2.3. The Partitioned-Matrix acceleration for Fission-Source iteration

Due to the fact that the PM acceleration forWG iteration reducescomputing times by smaller percentages than when used in higherorder transport calculations, we extend the PM technique to the FSiteration in this paper.

Partitioning the surface partial current and the volume fluxvectors:

jþg ¼� �

jþag

�T �jþbg

�T �T(21)

4g ¼� �

4lg

�T �4hg

�T �T(22)

where a and b denote the low (K1) and high-order (K2) surfacemoments, and l and h denote the low (I1) and high-order (I2)volume moments. Correspondingly, the iteration formulas can bepartitioned

"jþag

jþbg

#¼

"Raag Rab

g

Rbag Rbb

g

#"P5IK1j

þag

P5IK2jþbg

#þ"Bag

Bbg

#sg (23)

"4lg

4hg

#¼

"Hl

g

Hhg

#sg �

"Clag Clb

g

Chag Chb

g

#" �IN$NS

�P�5IK1j

þag�

IN$NS�P

�5IK2j

þbg

#(24)

Before each full-order sweeping, a number of low-ordersweeping is carried out with high-order moments fixed.

Fig. 2. The VNM iteration process with partitioned-matrix acceleration for WGiteration.

jþag ¼

hRaag Rab

g

i"P5IK1jþag

P5IK2jþbg

#þ Ba

gsg (25)

4lg ¼ Hl

gsg �hClag Clb

g

i" �IN$NS�P

�5IK1j

þag�

IN$NS�P

�5IK2j

þbg

#(26)

By comparing the Eqs. (23) and (24) with Eqs. (25) and (26), itcan be found that the PM technique for FS iteration is designed tosave numbers of updates on the high-order currents and flux mo-ments by requiring a number of more updates on low-order cur-rents and flux moments.

The VNM iteration process is shown in Fig. 3. It can be found thatthe total computing effort in FLOP is mainly contributed by foursteps: (1) the source construction or the applications of the scat-tering and fission matrices; (2) the nodal response or the applica-tions of the response matrices B, R, H and C; (3) the specification ofnodal incoming partial current if discontinuity factors show up; (4)iterative error estimations.

3. Numerical results

Based on the formulations of the Variational Nodal Method, acode named VIOLET has been developed in Xi'an Jiaotong Univer-sity for Pressurized Water Reactor core neutron-diffusion calcula-tion. And the PM technique has been implemented to accelerate the

Fig. 3. The VNM iteration process with partitioned-matrix acceleration for FS iteration.

Table 1Cross sections for the PWR problem without discontinuity factor.

Material g Dg/cmPg

a/cm�1 n

Pgf /cm

�1 P21a /cm�1

1 1 1.5 0.01 0 0.022 0.4 0.08 0.135

2 1 1.5 0.01 0 0.022 0.4 0.085 0.135

3 1 1.5 0.01 0 0.022 0.4 0.13 0.135

4 1 2.0 0 0 0.042 0.3 0.01 0

5 1 2.0 0 0 0.042 0.3 0.055 0

Y. Li et al. / Progress in Nuclear Energy 85 (2015) 640e647644

FS iteration. Neutron diffusion calculation usually is employed forassembly or quarter assembly homogenized reactor core calcula-tion. During the homogenization process, the neutron leakage ratesduring spatial homogenization can be preserved by using Gener-alized Equivalence Theory with discontinuity factor (Smith, 1986,1980) or other homogenization techniques without discontinuityfactor, such as superhomogenization (H�ebert, 2009). Consideringthe fact that the application of discontinuity factor slows the nodaliteration convergence (Zika and Downar, 1993), two typical testproblems without and without discontinuity factor respectivelyhave been chosen to evaluate and analyze the code and the algo-rithms. During these tests for PM technique, all other accelerationtechniques such as the multi-level iteration optimization (Li et al.,2013) in VIOLET are turned off. These calculations were per-formed using a 32 bit Intel® Core™ i7-2600 CPU @3.40 GHz withWindows 7 operating system and 3 GB memory.

3.1. PWR problem without discontinuity factor

As shown in Fig. 4, the first test problem is a quarter coreproblem representing a typical PWR operating state. There are 5types of fuel assemblies named as A-E, containing 5 different ma-terials tagged by numbers 1 to 5 as listed in Table 1. Each type ofassembly has its own axial composition as shown in Fig. 4. Thereare partially and fully inserted control rods in assemblies E and Crespectively. Assembly D is the reflector. The mesh sizes are thesame as the assembly size in radial direction, and 20 cm each in theaxial direction. Totally, there are 1311 spatial nodes with 6 surfacesper node.

The volumetric flux is represented by a 5th order polynomial inradial and 7th order polynomial axially, while the partial current oneach interface is represented by a 2nd order polynomial. This leadsto 58 degrees of freedoms (DOFs) per node, and 6 DOFs for eachnode surface. The convergence criteria for keff and fission sourcewere set as ek¼ 1.0� 10�5 and ef¼ 1.0� 10�4 respectively, with themaximum number of FS iterations set to be 500. Due to the absenceof up-scattering, there is no MG iteration. For the WG iteration ofeach energy group, a maximum iteration number Igmax ¼ 100, anAET εWG ¼ 1.0 � 10�12 and a RET eWG ¼ 1.0 � 10�3 were employedfor the partial current update. It is worth to point out that thenumerical results before and after acceleration are the same withthe convergence criteria, which means the difference betweenthem are less than 1.0 � 10�4.

For this problem, computing effort in FLOP for the full-orderresponse matrices B, R, H and C are respectively 2088, 1296, 3364and 2088. After 244 FS iterations with 9044 WG iterations, theresults converged to the right answer. For the PM acceleration, the

Fig. 4. The configuration for the PWR pr

number of low-order current moments K1 is 1 (flat), while first 4(linear in three dimensional spaces) volumetric flux moments areconsidered as the low-order ones. The number of low-order FS it-erations before each full-order one is determined by the reductionof keff error. The low-order improvement is taken as achieved oncethe relative error of keff (ek) is reduced by 50%. In this case, it turnedout that only 20 full-order FS iterations (containing 802 full-orderWG iterations) together with 272 low-order ones (containing8553 low-order WG iterations) are required to reach the sameresults.

As listed in Table 2, without the PM acceleration, 35-second CPUtime is needed by the entire iteration process to complete 21 � 109

FLOP. Among those FLOP, there are 16 � 109 FLOP (about 76%)required by the WG iteration. In contrast, only 8-second CPU timewould be needed to carry out 3 � 109 FLOP when PM technique isapplied. The CPU time for full-order iteration is reduced to about 4 swhile introducing about 4 s for extra low-order iterations. Fig. 5shows the convergences of keff and fission source with andwithout employing of the PM technique. It can be found that theerrors would rebound after each full-order sweep, but the low-order sweeps reduces it again quickly. And for the first full-ordersweeps, less number of low-order sweeps would be requiredsince the error of source distribution is large. Once the error ofsource distributing becomes small, more low-order iterations aregoing to be required to reduce the error of keff by 50%. In addition,non-monotonic convergence is observed in the accelerated case,which is usual for coarse mesh accelerations.

3.2. PWR problem with discontinuity factor

As shown in Fig. 6, the second test problem is also a quarter coreproblem referring to a typical PWR operating state. Comparing tothe first problem, there are several facts making it much more

oblem without discontinuity factor.

Table 2Computing efforts for the PWR problem without discontinuity factor.

Total Full-order FS excluding WG Full-order WG Low-order FS excluding WG Low-order WG

Without PM Time (s) 34.7 8.9 25.8 e e

FLOP (109) 20.7 4.9 15.8 e e

With PM Time (s) 8.3 1.5 2.3 1.1 3.4FLOP (109) 2.6 0.4 1.4 0.3 0.5

Y. Li et al. / Progress in Nuclear Energy 85 (2015) 640e647 645

realistic. Firstly, there are three batches of full assemblies includingnew ones and the ones experienced 1 and 2 cycles already. Sec-ondly, there are numbers of burnable poisons (Gd) rod in theseassemblies. The numbers Gd rods per assembly are among 0, 8, 20and 24. Thirdly, all control rods are out of the core, but the boronconcentration is 1830 ppm (the corresponding critical boron con-centration is 1953 ppm). Fourthly, each assembly is divided into 4nodes in radial direction and 20 cm per node in axial. Totally, thereare 3978 spatial nodes with 6 surfaces per node. The cross sectionsare provided by using the CASMO-4 (University release) (StudsvikScandpower, 2009).

Fig. 5. Fission-Source iteration processes of the PWR problem without discontinuityfactor.

Fig. 6. The configuration for the PWR problem with discontinuity factor.

The volumetric flux is represented by a 7th order polynomialradially 5th order polynomial axially, while the surface partialcurrent is represented as a 3rd order polynomial. It leads to 71 DOFsper node and 10 DOFs per nodal surface. The convergence criteriaare the same with the first problem. Correspondingly, computingeffort in FLOP for the full-order response matrices B, R, H and C arerespectively 4260, 3600, 5041 and 4260. After 386 FS iterationswith 11,177 WG iterations, the results converged to the rightanswer. For the PM acceleration, the number low-order currentmoment K1 is 1 (flat), while 1 (flat) volumetric flux moments isconsidered as the low-order ones. The number of low-order FS it-erations before each full-order one is determined by the reductionof keff error. The low-order improvement is achieved once therelative error of keff (ek) is reduced by 50%. In this case, only 37 full-order FS iterations (containing 2156 full-order WG iterations)together with 309 low-order ones (containing 8311 low-order WGiterations) are required to reach the same results (the differencebetween the numerical results before and after acceleration is lessthan 1.0 � 10�4).

As in Table 3, the total computational efforts required by theoriginal VNM iteration is 210� 109 FLOP, and correspondingly 442 sof CPU time. Again, about 76% effort was spent on the WG itera-tions. With the PM acceleration, the total computing time isreduced to 120 s to complete 40� 109 FLOP. The computing effort infull-order FS iteration was reduced to 37 � 109 FLOP while 3 � 109

extra FLOP are added in the low-order FS iterations. The FS iterationis shown in Fig. 7, indicates that the same acceleration performanceof the PM technique can be achieved for the PWR core when thediscontinuity factors are presented.

4. Conclusion and discussion

The legacy Partitioned-Matrix (PM) technique traditionallydeveloped for the within-group (WG) iteration acceleration is nowextended to accelerate the Fission-Source (FS) iteration of theVariational Nodal Method (VNM) for PWR core neutron-diffusioncalculation. Based on our new VNM code VIOLET, numerical testwere carried out for two typical PWR problems with and withoutusing discontinuity factors. By analyzing both the computationaleffort in terms of FLOP (FLoating-point OPeration) and thecomputing time, it has been demonstrated that the PM techniquecan provide an acceleration ratio of about 4 for the test problems.By extending the PM technique from the WG iteration to the FSiteration, the neutron-diffusion calculation can be accelerated veryeffectively with almost no extra storage or implementation cost tothe existing computer code. The PM technique roots in theorthogonality of the basis functions. Considering the fact that thePM technique performs very well in accelerating the WG iterationfor the VNM neutron-transport calculation, its extension from WGiteration to FS iteration is also expected to work well for the VNMneutron-transport calculation.

Based on the idea of macroscopic acceleration, the PM techniqueis employed to accelerate the FS iteration of the VNM. The generalterm macroscopic acceleration used here represents all the accel-eration techniques that uses coarse-mesh or low-order expansionmethods in space, angle or energy to accelerate the original fine-

Table 3Computing efforts for the PWR problem with discontinuity factor.

Total Full-order FS excluding WG Full-order WG Low-order FS excluding WG Low-order WG

Without PM Time (s) 442.2 103.2 339.0 e e

FLOP (109) 210.8 42.7 168.1 e e

With PM Time (s) 119.7 17.8 64.8 11.5 25.6FLOP (109) 40.0 4.1 32.4 1.7 1.8

Fig. 7. Fission-Source iteration processes of the PWR problem with discontinuityfactor.

Y. Li et al. / Progress in Nuclear Energy 85 (2015) 640e647646

mesh or high-order methods, such as the Coarse Mesh Rebalance(CMR) (Lewis and Miller, 1984; Xie and Deng, 2005), Coarse MeshFinite Differencing (CMFD) (Smith and Rhodes, 2002; Joo et al.,2002; Zhong et al., 2008), Diffusion Synthetic Acceleration (DSA)(Koph, 1963; Larsen, 1982; Mccoy and Larsen, 1982; Azmy et al.,1985), Non-Linear Iteration technique (NLI) (Smith, 1983; Liao,2002) and other coarse mesh acceleration methods (Li, 2013;Tatsumi and Yamamoto, 2003). There are two features in thesemethods. Firstly, they use an efficient coarse-mesh calculation toaccelerate a fine-mesh calculation which is expensive in bothcomputing efforts and storage. Secondly, since the coarse-meshmethod is not as accurate as the fine-mesh calculation, thecoarse-mesh calculation has to be corrected by the correspondingfine-mesh iterative result. There are different ways to implementthe correction. One can correct the diffusion coefficient as in CMRand DSA, or add a new term as in CMFD, or define a discontinuityfactor as in EFEN method (Li, 2013). The philosophy behind theseCMAmethods is to catch the source profile during iteration as earlyas possible. From this point of view, the PM technique is one ofthose CMA methods. It uses the low-order spatial expansion mo-ments corrected by adding the high-order moments to balance thesource distribution quickly. By extending the PM technique fromthe WG iteration to the FS iteration, the neutron-diffusion calcu-lation of the VNM can be accelerated very effectively with almostno extra storage and implementation costs to the existing code.

Acknowledgments

This work is supported by the National Natural Science Foun-dation of China (No. 11305123).

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