Annals of Nuclear Energy

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1 Annals of Nuclear Energy 37 (2010) Contents lists available at ScienceDirect Annals of Nuclear Energy journal homepage: Acceleration of source convergence in Monte Carlo k-eigenvalue problems via anchoring with a p-cmfd deterministic method Sunghwan Yun, Nam in Cho * Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon , Republic of Korea article info abstract Article history: Received 29 March 2010 Received in revised form 26 July 2010 Accepted 29 July 2010 Available online 21 August 2010 Keywords: Monte Carlo Fission source distribution p-cmfd method Heterogeneity Continuous-energy Anchoring The Monte Carlo method is widely used in neutron transport calculations, especially in complex geometry and continuous-energy problems. However, an extended application of the Monte Carlo method to large realistic eigenvalue problems remains a challenge due to its slow convergence and large fluctuations in the fission source distribution. In this paper, a deterministic partial current-based Coarse-Mesh Finite Difference (p-cmfd) method is proposed that achieves fast convergence in fission source distribution in Monte Carlo k-eigenvalue problems. In this method, the high-order Monte Carlo method provides homogenized and condensed cross section parameters while the low-order deterministic p-cmfd method provides anchoring of the fission source distribution. The proposed method is implemented in the MCNP5 code (version 1.30) and tested on realistic one- and two-dimensional heterogeneous continuous-energy large core problems, with encouraging results. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction * Corresponding author. Tel.: ; fax: address: nzcho@kaist.ac.kr (N.. Cho). The Monte Carlo method is widely used in particle transport calculations, especially in complex geometry and continuous-energy problems. However, an extended application of Monte Carlo method when used to large realistic eigenvalue problems such as whole-core problems remains a challenge due to the slow convergence and large fluctuations associated with the fission source distribution. The use of a small number of histories per generation can reduce the burden of slow convergence while occasionally showing undesirable fission source fluctuation even in active generations (Whitesides, 1971; Yamamoto et al., 2000; Blomquist et al., 2006). When using a large number of histories per generation, on the other hand, the slow convergence caused by a large dominance ratio is more problematic, because the required total number of histories will become very large. The difficulty caused by slow convergence was observed in deterministic criticality problems, and many acceleration methods have been proposed to counter this over the past several decades (Cho, 2005). The partial current-based Coarse-Mesh Finite Difference (p-cmfd) method (Cho et al., 2003a,b), one of these deterministic acceleration methods, is an improvement over the CMFD method (Smith and Rhodes, 2000), in that it is unconditionally stable in accelerating deterministic methods such as the discrete ordinates method and the methods that use the characteristics of transport calculations. This paper proposes an acceleration or speedup of the source convergence in Monte Carlo criticality calculations for heterogeneous continuous-energy problems based on the p-cmfd deterministic acceleration method. To achieve this, homogenized and condensed cross section parameters are obtained from the Monte Carlo (i.e., high-order transport) results, while a fission source distribution is provided by the p-cmfd (low-order diffusion-like deterministic) method that is defined by the homogenized and condensed cross section parameters. In previous studies by the authors (Cho et al., 2004a,b; Yun, 2005), the proposed method was demonstrated as feasible using script files and was tested on heterogeneous multigroup problems. A summary of those results are shown in the appendix. In the present paper, the p-cmfd acceleration method is applied in the inactive generations to accelerate slow convergence (reducing the required number of inactive generations) and then switched to the conventional Monte Carlo in active generations. The proposed method is implemented in the MCNP5 code (version 1.30) through the incorporation of a scattering cross section tally routine based on a collision estimator, a deterministic p-cmfd acceleration routine, and a modification to the fission source banking system. The new method is tested on several one- and two-dimensional but heterogeneous, continuous-energy problems /$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi: /j.anucene

2 1650 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) The p-cmfd acceleration method in Monte Carlo criticality problems 2.1. Description of the p-cmfd method in Monte Carlo problems We start with the following steady-state neutron transport equations: ~ rwð~r; ; ~ EÞþr t ð~r; EÞwð~r; ; ~ EÞ ¼ d ~ 0 de 0 r s ð~r; E 0! E; ~ 0 Þwð~r; ~ ~ 0 ; E 0 Þ þ 1 vðeþ d k eff 4p ~ 0 de 0 mr f ð~r; E 0 Þwð~r; ~ 0 ; E 0 Þ; ð1þ where the notations are standard. In a conventional method for the Monte Carlo eigenvalue problem, Eq. (1) is solved with power iteration as: ~ rw lþ1 ð~r; ; ~ EÞþr t ð~r; EÞw lþ1 ð~r; ; ~ EÞ ¼ d ~ 0 de 0 r s ð~r; E 0! E; ~ 0 Þw ~ lþ1 ð~r; ~ 0 ; E 0 Þ þ 1 vðeþ k l 4p Q l ð~rþ; ð2þ eff Q l ð~rþ d ~ 0 de 0 mr f ð~r; E 0 Þw l ð~r; ~ 0 ; E 0 Þ; where l is the generation index. In the p-cmfd acceleration method, Eq. (2) at the lth high-order Monte Carlo generation (or iteration) is rewritten with high-order iteration index l + 1/2 as: ~ rw lþ1=2 ð~r; ; ~ EÞþr t ð~r; EÞw lþ1=2 ð~r; ; ~ EÞ ¼ d ~ 0 de 0 r s ð~r; E 0! E; ~ 0 Þw ~ lþ1=2 ð~r; ~ 0 ; E 0 Þ þ 1 vðeþ k l 4p Q l ð~rþ: ð4þ eff After the high-order Monte Carlo calculation, deterministic p-cmfd parameter updating is performed with the following definitions: m;g 1 d~r de d w V ~ lþ1=2 ð~r; ; ~ EÞ; ð5þ m ~r2v m E2E g J þ;lþ1=2 ð~rþ 1 Mþ1=2;g dc de d j A ~ ~ ~njw lþ1=2 ð~r; ; ~ EÞ; Mþ1=2 ~r2a Mþ1=2 E2E g ~~n>0 ð6þ J ;lþ1=2 ð~rþ 1 Mþ1=2;g dc de d j A ~ ~ ~njw lþ1=2 ð~r; ; ~ EÞ; Mþ1=2 ~r2a Mþ1=2 E2E g ~~n<0 ð7þ t; ¼ 1 V M n2v M ~r2v M d~r E2E g de ð3þ d ~ r t;n ð~r; EÞw lþ1=2 ð~r; ~ ; EÞ; s; 0!g ¼ 1 V d~r de 0 M n2v ~r2v 0 de M M E 0 2E g 0 E2E g d ~ d ~ 0 r s;n ð~r; E 0! E; ~ 0! Þw ~ lþ1=2 ð~r; ~ 0 ; E 0 Þ; m f ; ¼ 1 d~r V M n2v ~r2v M M E2Eg de ð8þ ð9þ d ~ mð~r;eþr ð~r;eþw lþ1=2 ð~r; ~ ;EÞ; ð10þ ¼ 1 V M m2m m;g V m ; ð11þ where m is the fine-mesh cell index in the coarse-mesh cell M, n is the isotope index in the coarse-mesh cell M, V is the volume of the mesh cell, A is the area of the mesh surface, and ~n is the outnormal vector at the coarse-mesh cell surface M + 1/2. The low-order p-cmfd equations are now formulated for each coarse-mesh cell and group g by requiring the l + 1th iterate to satisfy the following balance relation: M 0 dc^n ~ J lþ1 MM 0 g ¼ v g k lþ1 eff where g 0 J lþ1 Mþ1=2;g ¼ Jþ;lþ1 Mþ1=2;g J þ;lþ1 Mþ1=2:g ¼ e D Mþ1=2;g J ;lþ1 Mþ1=2:g ¼ e DMþ1=2;g þ t; /lþ1 V M m f ; 0 0; V M þ g 0 J :lþ1 Mþ1=2;g ; Mþ1;g Mþ1;g /lþ1 2 /lþ1 2 ed Mþ1=2;g ¼ 2 ðd =D M ÞðD Mþ1;g =D Mþ1 Þ ; D =D M þ D Mþ1;g =D Mþ1 D ¼ 1 3 t; s; 0!g /lþ1 0V M; þ 2 b D þ Mþ1=2;g /lþ1 þ 2 b D Mþ1=2;g /lþ1 Mþ1;g ð12þ ð13þ ; ð14þ ; ð15þ ð16þ ; ð17þ bd Mþ1=2;g ¼ 2J ;lþ1=2 Mþ1=2;g þ e D Mþ1=2;g Mþ1;g 2 Mþ1=21=2;g /lþ1=2 : ð18þ The system of Eq. (12) for the entire problem provides the coarsemesh averaged scalar flux. The fine-mesh scalar flux distributions are then updated via m;g ¼ /lþ1=2 m;g ; for m 2 M: ð19þ Finally, the high-order Monte Carlo fission source term Q lþ1 ð~rþ at l + 1th generation is updated as follows, based on a weight concept used in heterogeneous problems (Yun, 2005): Q lþ1 ð~rþ ¼ de d mð~r; ~ EÞ ð~r; EÞw lþ1=2 ð~r; ; ~ EÞ n2v m Pg 0mrlþ1=2 f ; 0 m;g 0V m R~r2V m d~r R de R d mð~r; ~ EÞ P n2v m ð~r; EÞw lþ1=2 ð~r; ; ~ EÞ ¼ FSD lþ1=2 ð~rþ ^w lþ1 m ; for ~r 2 V m; m 2 M; ð20þ where FSD lþ1=2 ð~rþ de d ~ mð~r; EÞ n2v m ð~r; EÞw lþ1=2 ð~r; ~ ; EÞ; ð21þ P ^w lþ1 m g 0mrlþ1=2 f ;m;g 0 m;g 0V m R~r2V m d~r R de R d mð~r; ~ EÞ P n2v m ð~r; EÞw lþ1=2 ð~r; ; ~ EÞ for ~r 2 V m ; d~rq lþ1 ð~rþ ¼N; V ð22þ ð23þ and N is the number of histories per generation. The updated Q lþ1 ð~rþ is an accelerated fission source distribution, and it will anchor the high-order Monte Carlo procedure during inactive generations.

3 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) As an alternative to using Eqs. (19) and (20), the following coarse-mesh based p-cmfd distribution can be also used without the need of fine-mesh flux tally: Q lþ1 ð~rþ ¼ de d mð~r; ~ EÞ ð~r; EÞw lþ1=2 ð~r; ; ~ EÞ n2v m Pg 0mrlþ1=2 f ; 0 0V M R~r2V M d~r R de R d mð~r; ~ EÞ P n2v M ð~r; EÞw lþ1=2 ð~r; ; ~ EÞ ¼ FSD lþ1=2 ð~rþ ^w lþ1 M ; for ~r 2 V m: ð24þ 2.2. Cross section accumulation scheme In general, the variance of a homogenized and condensed cross section is known to be smaller than the variance of the reaction Score flux and partial current tally (Eqs. (5), (6), and (7)) No Start with initial guess One generation Monte Carlo calculation Score and make accumulated homogenized and condensed cross sections (Eq. (25)) Solve multi-group coarse-mesh p-cmfd low order eigenvalue problem Fission source update based on p-cmfd results End of inactive generation? Start conventional active Monte Carlo calculaiton End of whole procedure Fig. 1. Flow chart of the p-cmfd speedup method in the modified MCNP5 code. Yes rate (Larsen and Yang, 2008; Wolters et al., 2009), as it is divided by the stochastically estimated flux. However, this does not guarantee that the variance of a homogenized and condensed cross section is less than the variance of the neutron flux, especially in heterogeneous problems composed of multiple nuclides with considerable resonance. Thus, in this study, a cross section accumulation scheme is introduced during inactive generations to provide more stable deterministic results: x; 1 l l l 0 ¼1 Pn2V M R ~r2v M d~r R E2E g de R d r ~ x;n ð~r; EÞw l0þ1=2 ð~r; ; ~ 1 EÞ R ~r2v M d~r R E2E g de R d w ~ l0þ1=2 ð~r; ; ~ A; EÞ ð25þ for all cross section types. A schematic computational flow is shown in Fig Numerical results 3.1. One-dimensional continuous-energy small core problem The first test problem is a one-dimensional continuous-energy small core problem. Its configuration is shown in Fig. 2 and its material compositions are shown in Table 1. In this problem, 2.52 cm p-cmfd fine-mesh, 5.04 cm p-cmfd coarse-mesh, and histories per generation are used. The p-cmfd low-order calculation is performed using one-group condensed cross sections with a 10 8 error criterion. As the p-cmfd acceleration calculation can accelerate the source convergence, 50 inactive generations are used in the p-cmfd accelerated case while 100 inactive genera- Table 1 Material composition of one-dimensional continuous-energy small core problem. Materials Isotopes a Weight fraction Isotopes a Weight fraction Water (1.0 g/cm 3 ) H E 01 B E 04 O E 01 B E w/o UO2 fuel (10.40 g/cm 3 ) 8.7 w/o MO fuel (10.40 g/cm 3 ) a ENDF-B/VI.6 libraries are used. U E 01 U E 02 U E 05 O E 01 U E 05 Pu E 03 U E 03 Pu E 02 U E 01 Pu E 02 Am E 04 Pu E 03 O E 01 Pu E 03 Fig. 2. Configurations of one-dimensional continuous-energy small core problem.

4 1652 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) tions are used in the conventional Monte Carlo calculation. The initial fission source starts at the center of each fuel region within a p-cmfd fine-mesh cell. The p-cmfd acceleration calculation is made during inactive generations while the conventional Monte Carlo procedure is done during active generations. To ascertain the source convergence, the Shannon entropy H(S) is used, that is defined as HðSÞ ¼ Ns where s¼1 p s ln 2 ðp s Þ; number of fission source particles in bin s p s number of fission source particles in all bins ; N S ¼ number of total bins; ð26þ ð27þ ð28þ Fig. 3. Shannon entropies of one-dimensional continuous-energy small core problem. with each p-cmfd fine-mesh cell selected as a Shannon entropy bin (Brown, 2006). The results are shown in Fig. 3. The reference values are generated by averaging 30 independent conventional Monte Carlo batch runs with histories per generation, 100 inactive generations, and 300 active generations. The p-cmfd accelerated case without cross section accumulation shows much more fluctuation in fission source distribution compared to that noted in the conventional Monte Carlo case during inactive generations owing to the fluctuations of the cross section as shown in Fig. 4. In the cross section accumulation scheme, on the other hand, a more stable and reliable fission source distribution is obtained, as shown in Fig. 3, outweighing the effect of the bias in the cross sections originated by the cross section accumulation process. To estimate the actual variance of the p-cmfd acceleration method during inactive generations, 30 independent batch runs were made. From these results, the average Shannon entropies and the average eigenvalues are shown in Figs. 5 and 6, respec- Fig. 4. Cross sections in one-dimensional continuous-energy small core problem.

5 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) tively, while the behavior of the flux at cm region versus number of generations is shown in Fig. 7. Both the Shannon entropies and the eigenvalues converge much faster (by more than a factor of two) compared to those in the conventional Monte Carlo method. The batch averaged eigenvalue of the p-cmfd acceleration with cross section accumulation is (relative standard deviation is %), while the eigenvalue of the conventional Monte Carlo is (relative standard deviation is %), over 30 independent batch runs. To assess the bias effect incurred by the cross section accumulation scheme, the averaged relative difference in the eigenvalue over the batch runs is defined as ðaveraged relative difference of the eigenvalueþ 1 J k j k eff ;p-cmfd eff ;reference ; ð29þ J k eff ;reference j¼1 where J represents total number of independent batch runs, k j eff ;p-cmfd is the mean eigenvalue of the p-cmfd accelerated method with cross section accumulation at each single run, and k eff,reference is the batch averaged eigenvalue of the conventional Monte Carlo method. The averaged relative difference in the eigenvalue, which is defined in Eq. (29), is % and it is smaller than the standard deviation of the conventional Monte Carlo ( %). The averaged relative difference of the flux at cm Fig. 7. The flux at cm vs number of generations in one-dimensional continuous-energy small core problem. is %, while the relative standard deviation of the conventional Monte Carlo is %. Thus, we can say that the p-cmfd acceleration method with cross section accumulation provides results in good agreement with those of the conventional Monte Carlo method in both of the eigenvalue and spatial flux distribution. The observed bias effect in cross section accumulation is negligible for the problem tested. The computing times during the inactive generations are shown in Table 2. Since more tally information is required for p-cmfd calculation in inactive generations, 55% more calculation time is required in the same number of inactive generations. However, the required number of inactive generations is reduced by the p- CMFD acceleration, and thus the total computation burden is decreased by 30% in the inactive generation calculation Continuous-energy large core problem with a two-dimensional configuration Fig. 5. Batch-averaged Shannon entropies of one-dimensional continuous-energy small core problem. The second test problem is a continuous-energy eight assembly core problem including two-dimensional configurations of a fuel rod ( cm in radius) and cladding ( cm in radius), as shown in Fig. 8. The cladding material properties are shown in Table 3. One fuel pin cell (1.26 cm 1.26 cm) was used as a p- CMFD fine-mesh cell and four fuel pin cells (2.52 cm 2.52 cm) were used as a p-cmfd coarse-mesh cell. The p-cmfd low-order calculation was performed using one-group condensed and homogenized cross sections with a 10 7 error criterion and the cross section accumulation scheme. A million (10 6 ) histories per generation were used with 50 inactive generations in the p-cmfd accelerated case and 200 inactive generations in the conventional Monte Carlo calculation. The p-cmfd acceleration calculation was done during the inactive generations, and the strategy was switched to the conventional Monte Carlo procedure in the active generations. Fig. 6. Batch-averaged k eff problem. of one-dimensional continuous-energy small core Table 2 Comparison of computing times in inactive generations for the first test problem. Method Monte Carlo anchoring with p-cmfd Conventional Monte Carlo (MCNP5) a Number of inactive generations 6 Core 2 Duo Dual-Core Processor E8400 CPUs. Computing time a [s] Speedup

6 1654 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) Fig. 8. Configurations of continuous-energy eight assembly problem. Table 3 Material composition of cladding material a Materials Isotopes a Weight fraction ircaloy-4 cladding (6.489 g/cm 3 ) ENDF-B/VI.6 libraries are used. Isotopes a Weight fraction Cr E 05 Fe E 05 Cr E 04 Fe E 03 Cr E 05 Fe E 05 Cr E 05 Fe E 06 r E 01 Sn E 02 Batch averaged k eff Conventional MCNP5 p-cmfd accelerated MCNP5 (-section accumulation) Reference Generations Fig. 10. Batch-averaged k eff over five independent batch runs. The computing times during the inactive generations are shown in Table 4. In the same number of inactive generations, 120% more calculation time is required. However, with the reduced number of inactive generations, the total computation burden is reduced by 50% in the inactive generation calculation. 4. Conclusions Fig. 9. Batch-averaged Shannon entropies. In this paper, an acceleration or speedup of the source convergence in a Monte Carlo criticality calculation via anchoring with a The averaged Shannon entropies (a p-cmfd fine-mesh cell was used as a Shannon entropy bin) and the averaged eigenvalues over five independent batch runs are shown in Figs. 9 and 10, respectively. The Shannon entropies converge around 25 inactive generations in the p-cmfd accelerated case, whereas 200 inactive generations are required in the conventional Monte Carlo case. The eigenvalue of the p-cmfd accelerated case converged within 20 inactive generations and the eigenvalue of the conventional Monte Carlo converged with 120 inactive generations. Table 4 Comparison of computing times in inactive generations for the second test problem. Method Monte Carlo anchoring with p-cmfd Conventional Monte Carlo (MCNP5) a Number of inactive generations 6 Core 2 Duo Dual-Core Processor E8400 CPUs. Computing time a [s] Speedup factor 50 11, , ,422

7 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) p-cmfd deterministic acceleration method is proposed. In this novel method, the high-order Monte Carlo method provides homogenized and condensed cross section parameters while the low-order deterministic p-cmfd method provides an accelerated fission source distribution that anchors the high-order Monte Carlo fission source distribution. A p-cmfd acceleration calculation is performed during inactive generations (reducing the required number of inactive generations) while the conventional Monte Carlo procedure is utilized during active generations. Fig. A.1. Computational flow of MCNP/p-CMFD scheme. Fig. A.3. Configurations of a two-dimensional multigroup small core problem. Fig. A.2. Computational flow of improved MCNP/p-CMFD scheme.

8 1656 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) The method was tested on continuous-energy problems including realistic mixed isotopes and a heterogeneous geometry of core problems. However, significant fluctuation of the deterministic p- CMFD fission source distribution occurred due to the fluctuation of the cross sections. This phenomenon was likely caused by the highly heterogeneous structure at fine-mesh cells, which were composed of multiple nuclides with considerable resonance. Thus, a cross section accumulation scheme to run during the inactive generations was suggested for more stable deterministic results, as the bias effect in the cross section is much less than the fluctuation with a conventional fission source. The results were noticeably improved. For the problems tested, the eigenvalue converges very rapidly, within approximately 30 inactive generations. In contrast, the conventional MCNP5 code requires inactive generations. The numerical results demonstrated that the p- CMFD deterministic acceleration method works with continuousenergy, heterogeneous k-eigenvalue Monte Carlo problems quite well. This is particularly true for large core problems. An alternative possibility of the application of the p-cmfd method in the Monte Carlo eigenvalue calculation was tested by the authors (Yun and Cho, 2009a,b). The p-cmfd method used in both inactive and active generations, with anchoring factor a, was tested in two-group loosely-coupled rod Monte Carlo eigenvalue problems and showed less fluctuation in the fission source distributions than the conventional Monte Carlo method. Similar results (i.e., better performance in its real variance) by the p-cmfd method were reported in a simple two-group piece-wise homogeneous problem; however, its degraded performance was also observed in more realistic continuous-energy, heterogeneous, multiple-nuclide problems (Yun and Cho, 2010). It is suspected that the degraded results (larger real variance than that of the conventional Monte Carlo) in such realistic problems could have originated from the homogenized and condensed cross sections. It may be possible that the variance of the generated cross sections can be significantly larger than the variance of the scalar flux in complex problems, leading to poor performance of the loworder p-cmfd method. This aspect warrants further studies, such as the use of dynamic anchoring factor suggested in (Cho, 2010). Appendix A A.1. The MCNP/p-CMFD scheme (script file) In a previous study by the authors entitled MCNP/p-CMFD Rebalance (Cho et al., 2004a), the Monte Carlo k-eigenvalue problem was decomposed into a fixed-source Monte Carlo problem (for a whole-core) and an eigenvalue coarse-mesh deterministic problem. In a particular generation (inactive and active) of the Monte Carlo calculation, the problem is posed as a fixed-source problem in which the source is provided by the fission neutron distribution Q lþ1 ð~rþ given as Eq. (A.1): Fig. A.4. k eff as functions of generation and number of histories (N) per generation (no generations skipped).

9 Q lþ1 ð~rþ Q lþ1 m ¼ m2m g 0 m f ; 0 m;g 0V m: S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) ða:1þ coupled. Hence, the scheme was limited to multigroup problems and constant source updating within the fine-mesh region. This coupled procedure continues until a deterministic solution converges. Fig. A.1 shows a schematic of the computational flow. This MCNP/p-CMFD scheme was implemented in a script file in which the MCNP5 code and the deterministic p-cmfd code were A.2. The improved MCNP/p-CMFD scheme (script file) The MCNP/p-CMFD scheme was refined in an unpublished study by the authors entitled Improved MCNP/p-CMFD Scheme (Yun, 2005). In this scheme, the high-order Monte Carlo calculation was solved as a one-generation k-eigenvalue problem with a loworder p-cmfd fission source distribution and the k eff value, in which the low-order p-cmfd fission source distribution was transformed into weights and written on a modified srctp file. Thus, in the improved MCNP/p-CMFD scheme, the bias caused by the piecewise constant fission source was eliminated by preserving the information of the high-order fission source distribution within each fine-mesh cell. A schematic of the computational flow is shown in Fig. A.2. The improved MCNP/p-CMFD scheme was tested on multigroup heterogeneous problems. Recently, a similar weight adjustment scheme based on the CMFD deterministic acceleration method was described and tested on multigroup piece-wise homogeneous problems (Lee et al., 2009). A.3. Numerical results of MCNP/p-CMFD and improved MCNP/p-CMFD schemes Fig. A.5. k eff as functions of generation and number of histories (N) per generation (first 10 generations skipped in MCNP/p-CMFD and first 200 generations skipped in MCNP). Fig. A.6. k eff of the improved MCNP/p-CFMD scheme as functions of generation (initial five generations are skipped). The MCNP/p-CMFD scheme was tested on two-dimensional multigroup small core problems. The boundary conditions were all reflective, and seven energy-group cross sections from a benchmark study (Lewis et al., 2001) were used. Details of the configuration of the problem are described in Fig. A.3. Figs. A.4 and A.5 show that k eff in the MCNP/p-CMFD scheme converges very rapidly and exhibits only slight oscillation as the number of generations increases, whereas k eff in the conventional MCNP scheme converges slowly with irregular oscillations. This may be due to the fact that the rebalanced fission source distribution in the MCNP/p-CMFD scheme reaches equilibrium quickly. As shown in Fig. A.5, there was a visible bias associated with the k eff value in the MCNP/p-CMFD scheme. As an alternative use of the scheme, the solid green line in Fig. A.5 shows the result of a switching calculation in which the first 20 generations were done with the MCNP/p-CMFD scheme and then switching to the conventional MCNP for up to 300 generations. The result was very close to the conventional MCNP calculation alone while omitting the first 200 generations. Thus, the MCNP/p-CMFD scheme used in a switching mode can facilitate the inclusion of all generations with none skipped. The resulting values of k eff of the improved MCNP/p-CMFD scheme versus various histories per generation are shown in Table A.1 Sensitivity of k eff on particle histories and generations a. Total number of particles k eff % error in k eff Total number of particles k eff % error in k eff MCNP/p-CMFD (100 b / ) c ( ) d (100/ ) ( ) (100/10 4 ) ( ) (100/ ) ( ) (100/10 5 ) ( ) (500/ ) ( ) Total number of particles k eff % error e Improved MCNP/p-CMFD 10 6 (100/10 4 ) MCNP (500/ ) ( ) (500/ ) ( ) Ref. a b c d e 200 inactive generations in conventional MCNP and 10 inactive generations in MCNP/p-CMFD and improved MCNP/p-CMFD. Number of generations. Number of histories per generation. Standard deviation. Relative difference of k eff via reference MCNP.

10 1658 S. Yun, N.. Cho / Annals of Nuclear Energy 37 (2010) Fig. A.6. The biasing effect in k eff was significantly reduced by conserving the information of the fission source distributions within each p-cmfd fine-mesh cell. Table A.1 shows the sensitivity of k eff with respect to the number of particle histories per generation. Although the improved MCNP/p-CMFD scheme showed smaller biases in the k eff values, stronger dependency on the number of particle histories per generation was observed. References Blomquist, R.N., Armishaw, M., Hanlon, D., Smith, N., Naito, Y., Yang, J., Mioshi, Y., Yamamoto, T., Jacquet, O., Miss, J., Source Convergence in Criticality Safety Analyses, NEA Report No. 5431, Nuclear Energy Agency/Organization for Economic Co-operation and Development. < pubs/2006>. Brown, F.B., On the use of Shannon entropy of the fission distribution for assessing convergence of Monte Carlo criticality calculations. PHYSOR-2006, ANS Topical Meeting on Reactor Physics Organized and Hosted by the Canadian Nuclear Society, Vancouver, BC, Canada (CD-ROM). Cho, N.., Fundamentals and recent developments of reactor physics methods. Nucl. Eng. Techol. 37, 25. < php?jid=jk > (formerly Journal of the Korean Nuclear Society). Cho, N.., Monte Carlo anchoring method vs global nodal diffusion/local Monte Carlo iterations in reactor core calculations. PHYSOR 2010 Technical Workshop on Transport Methods for Reactor Core Calculations, Pittsburgh, Pennsylvania, USA. Cho, N.., Lee, G.S., Park, C.J., 2003a. Partial current-based CMFD acceleration of the 2D/1D fusion method for 3D whole-core transport calculations. Trans. Am. Nucl. Soc. 88, 594. Cho, N.., Lee, G.S., Park, C.J., 2003b. On a new acceleration method for 3D wholecore transport calculations. Annual Meeting of the Atomic Energy Society of Japan, March 2003, Japan. Cho, N.., Yun, S., Lee, K.T., Lee, G.S., 2004a. Speedup of Monte Carlo k-eigenvalue calculations via p-cmfd rebalance. Trans. Am. Nucl. Soc. 90, 550. Cho, N.., Yun, S., Lee, K.T., and Lee, G.S., 2004b. An adjoint p-cmfd scheme for Monte Carlo k-eigenvalue calculations, Proceedings of the Korean Nuclear Society Spring Meeting, Gyeongju, Korea (CD-ROM). Larsen, E.W., Yang, J., A functional Monte Carlo method for k-eigenvalue problems. Nucl. Sci. Eng. 159, 107. Lee, M.J., Joo, H.G., Lee, D., Smith, K., A feasibility study of CMFD acceleration in Monte Carlo eigenvalue calculation. In: Proceedings of the Korean Nuclear Society Autumn Meeting, Gyeongju, Korea (CD-ROM). Lewis, E. E., et al., Expert group on 3-D radiation transport benchmarks summary of meeting on C5G7MO benchmark. NEA/NSC/DOC (2001)17. Smith, K.S., Rhodes III, J.D., CASMO characteristics method for twodimensional PWR and BWR core calculation. Trans. Am. Nucl. Soc. 83, 294. Whitesides, E.G., A difficulty in computing the k-effective of the world. Trans. Am. Nucl. Soc. 14, 680. Wolters, E.R., Larsen, E.W., Martin, W.R., A hybrid Monte Carlo-S2 method for preserving neutron transport effect. In: International Conference on Advances in Mathematics, Computational Methods, and Reactor Physics (M&C 2009), Saratoga Springs, New York, May 3 7, 2009 (CD-ROM). Yamamoto, T., Nakamura, T., Miyoshi, Y., Fission source convergence of Monte Carlo criticality calculations in weakly coupled fissile arrays. J. Nucl. Sci. Technol. 37, 41. Yun, S., A study on the bias in improved MCNP/p-CMFD method. NurapT Lab Seminar, Korea Advanced Institute of Science and Technology, November 8, 2005 (unpublished). Yun, S., Cho, N.., 2009a. Monte Carlo anchoring method for loosely-coupled k- eigenvalue problems. In: International Conference on Advances in Mathematics, Computational Methods, and Reactor Physics (M&C 2009), Saratoga Springs, New York, May 3 7, 2009 (CD-ROM). Yun, S., Cho, N.., 2009b. Monte Carlo anchoring method for asymmetric looselycoupled k-eigenvalue problems. Annual Meeting of AESJ (AESJ 2009), Tokyo, Japan, March (CD-ROM). Yun, S., Cho, N.., Refinement of Monte Carlo anchoring method tested on continuous-energy loosely-coupled fissile problems. PHYSOR 2010 Advances in Reactor Physics to Power the Nuclear Renaissance, Pittsburgh, Pennsylvania, USA (CD-ROM).