A PWR ASSEMBLY COMPUTATIONAL SCHEME BASED ON THE DRAGON V4 LATTICE CODE

Transcription

1 Supercomputing in Nuclear Applications (M&C + SNA 2007) Monterey, California, April 15-19, 2007, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2007) A PWR ASSEMBLY COMPUTATIONAL SCHEME BASED ON THE DRAGON V4 LATTICE CODE R. Le Tellier and A. Hébert École Polytechnique de Montréal C.P succ. Centre-Ville, Montréal Qc. CANADA H3C 3A7 romain.le-tellier@polymtl.ca; alain.hebert@polymtl.ca ABSTRACT This paper describes the validation of a two-level computational scheme dedicated to PWR assembly calculations within the framework of the DRAGON v4 lattice code. The first level uses the interface current method for both the subgroup-based self-shielding and main flux calculation, with a 172-group XMAS energy discretization. The second level uses the method of characteristics preconditioned with the algebraic collapsing acceleration technique, posterior to the condensation of the material properties with a representative weighting flux obtained from the first level. This two-level scheme is compared to both a reference single-level computational scheme with the method of characteristics and to a Monte-Carlo calculation on a zero-burnup benchmark. The numerical results obtained with both twoand one-level computational schemes are in very good agreement with the Monte-Carlo calculation. Moreover, the deviation of the two-level scheme with respect to the one-level scheme is assessed when depletion is considered and is found to be small. Key Words: Method of Characteristics, DRAGON v4, PWR Assembly, Computational Scheme, Depletion 1. INTRODUCTION This paper is related to the validation studies of the DRAGON v4 lattice code[1, 2]. We propose to study computational schemes based on the method of characteristics[3] (MOC) and dedicated to pressurized water reactor (PWR) assembly calculations. The validation is done with respect to the continuous energy Monte-Carlo code TRIPOLI4 version 4.3[4]. Contrarily to the benchmarking of both MOC and different self-shielding models presented in [5], the present study is oriented towards the definition of a production computational scheme for PWR assemblies within the DRAGON environment. Consequently, besides the use of a direct one-level MOC scheme, we discuss the implementation of a two-level computational scheme as proposed in [6, 7]. The idea is to reduce the MOC CPU time by decreasing the number of groups for which the calculation is performed; to do so, a prior calculation with an interface current method is performed in order to condensate the macroscopic cross section to a lower number of energy groups. The usage of an SPH equivalence[8] between these two calculations is also investigated. This paper is focused on the definition and validation of such a computational scheme by comparison with TRIPOLI4 on a zero burnup PWR-UOX assembly and comparison with a single level scheme for the depletion of this assembly. The details on the implementation of the method of characteristics within DRAGON can be found elsewhere[9]. The assembly modeling is described in Sect. 2. The validation at zero burnup of both the one-level and two-level schemes is given in Sect. 3 while Sect. 4 presents the results of the two-level scheme when considering a depletion calculation. Finally, our conclusions are given in Sect. 5. Throughout the presentation of the different components of the computational scheme, CPU times are given so that they can be compared between the different parts of the calculation.

2 R. Le Tellier and A. Hébert 2. DESCRIPTION OF THE UOX ASSEMBLY AND METHODOLOGY The PWR assembly is fueled with 1.8 % enriched UO 2 fuel pellets and corresponds to production assembly calculations as reported in [10] for commercial PWR reactors. The lattice geometry is depicted in Fig. 1, colored according to the mixture index used in the assembly modeling. Each fuel rod is split in terms of resonant mixtures into 4 rings representing (inner to outer) respectively 50 %, 30 %, 15 % and 5 % of the rod volume according to the recommendation in [11] in order to treat correctly the spatial distribution of the resonant absorption of 238 U and the concentration changes of actinides and fission products when burnup is considered. A total of 20 fuel mixtures is used in order to differentiate the depleting mixtures according to their position with respect to a guide tube or instrumentation tube[11]. The geometry is treated according to its 1/8 assembly symmetry. Figure 1. PWR Assembly (colored per mixture) Isotopic cross sections are represented with an XMAS (172 groups) library in the DRAGON format, built from JEF-2.2 with NJOY99[12] and the Dragr module[13]. For the validation procedure, the library contains only the isotopes of the benchmark processed at the exact temperatures found in the benchmark. The PENDF files obtained in building this library were used for TRIPOLI4 calculations. Our TRIPOLI4 runs are not using probability tables and, consequently, cannot represent unresolved self-shielding effects. In order to be consistent, we have disabled the self-shielding treatment in DRAGON for all group indices lower than 45, corresponding to an energy greater than kev (i.e. corresponding to the unresolved 2/13

3 A PWR Assembly Computational Scheme Based on the DRAGON v4 Lattice Code resonance domain). The self-shielding treatment of the assembly is based on a subgroup approach where the statistical model is used along with physical probability tables computed by fitting dilution-dependent cross sections[14]. In this context, the transport solver is based on a UP 1 interface current method which considers the cell interface currents to be uniform and angularly represented by a DP 1 expansion i.e. a linear anisotropic half-range spherical harmonics expansion. The flux-current UP 1 system is solved by an iterative approach rather than an algebraic elimination of the current unknowns for both the self-shielding and multigroup flux calculations. As discussed in Sect , this has a direct impact on the CPU time. With such a solver it is a common practice to merge different cells considering that they share the same flux in order to reduce the CPU time. In this study, both the 8-cell grouping and the 12-cell grouping depicted in Fig. 2 were considered. They were created by differentiating the cells according to their position with respect to a guide tube or instrumentation tube; the cells orientation was chosen accordingly. R 8 R 12 Figure 2. Two Different Cell Grouping for the UP 1 Solver Moreover, for the self-shielding calculation, one can also merge different resonant mixtures; starting from the 8-cell grouping, two different configurations were tested: R + 8 in which all the resonant mixtures are merged into one for all the resonant isotopes except 238 U; R ++ 8 in which, beside the R + 8 merging, the resonant mixtures for 238 U are merged between the different cells, only the distinction in four different layers within a pincell is kept. For the MOC flux calculation, different geometry discretizations were considered using the NXT: tracking module that was introduced in the release 3.05 of DRAGON[15]. This module can analyze 2-D and 3-D assemblies of pin cells and CANDU clusters with a non-uniform mesh. The geometrical configurations we used for this PWR assembly are denoted C i [1,5] and are depicted in Fig. 3. For the parametric study, we present results in which both the geometry and the tracking parameters were varied. We denote N a the number of angles [0,π/2] for the azimuthal quadrature, d the uniform track density in cm 1 and N p the 3/13

4 R. Le Tellier and A. Hébert number of angles [0, π/2] for the polar quadrature derived from the optimization procedure proposed in [16]. C 1 C 2 C 3 C 4 C 5 Figure 3. Five Different Discretizations of the PWR Assembly For comparison purpose, the output reaction rates are condensed into a four-group structure compatible with the different energy meshes we used in this study. It is presented in Table I and is based on the decomposition into fast, resonant (unresolved/resolved) and thermal regions. In this study, we focus on the fission rate in group 4, the radiative capture rate in groups 2 and 3 and finally, the total interaction rate in group 1. Table I. Macro Energy Groups for Condensation group Energy Interval (ev) 1 ] [ 2 ] [ 3 ] [ 4 ] [ We present the results in terms of the difference in k eff i.e k eff = k eff k eff ref. and the average ( ǫ) and maximum (ǫmax) differences on macroscopic reaction rates (τ) per fuel cell. These differences are defined as ǫmax = max i ( τi τ ref. i τ ref. i ), (1) 4/13

5 A PWR Assembly Computational Scheme Based on the DRAGON v4 Lattice Code ǫ = 1 τ i τ ref. i V i Vtot i τ ref.. (2) i TRIPOLI4 calculations were carried out with 20 millions histories. The standard deviation on the k eff is 25 pcm while it is about 0.1 % for the fission rate in group 1, 0.2 % for the capture rates in group 2 and 3 and 0.05 % for the total interaction rate in group VALIDATION AT ZERO BURNUP WITH RESPECT TO TRIPOLI4 In this section, we study the zero-burnup benchmark which was used in order to design and validate the computational schemes. A direct 172-group MOC computational scheme is first investigated and starting from this modeling, a two-level computational scheme is then introduced One-Level Computational Scheme Parametric Study We give here some details on the parametric study regarding the geometry, the tracking and the scattering anisotropy as well as the cell and mixture grouping option for the self-shielding calculation. In Table II, we present the geometry discretization and tracking refinements for a cyclic tracking. We clearly see that the meshing can be largely coarsened without noticeably deteriorating the results. The most influential parameter is the number of cyclic azimuthal angles and consequently, configuration C 1 with N a = 20, d = 10.0 cm 1 and N p = 2 was selected. It is denoted Cspec in the remaining of the paper. Table II. Effect of Geometry Discretization and Tracking Refinement Conf. k eff ǫ (ǫmax) (%) (N a - d - N p ) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr. 1 C 5 ( ) (-0.03) 0.02 (0.03) 0.00 (-0.01) 0.12 (-0.12) C 4 ( ) (0.04) 0.03 (0.03) 0.01 (-0.01) 0.12 (-0.12) C 4 ( ) (0.04) 0.03 (0.05) 0.01 (-0.01) 0.23 (-0.24) C 3 ( ) (0.07) 0.04 (0.05) 0.01 (-0.01) 0.23 (-0.25) C 2 ( ) (0.08) 0.05 (0.11) 0.01 (-0.02) 0.23 (-0.25) C 1 ( ) (0.10) 0.05 (0.11) 0.01 (-0.02) 0.23 (-0.25) C 1 ( ) (0.12) 0.12 (0.15) 0.00 (-0.00) 0.68 (-0.74) The reference is configuration C 5 with N a = 24, d = cm 1 and N p = 3. Beside, on such an assembly, one can question the necessity of using a cyclic tracking with specular reflective boundary conditions. Thus, in Table III, a non-cyclic tracking with white boundary conditions 5/13

6 R. Le Tellier and A. Hébert has been considered for treating the C 1 geometrical configuration with N p = 2. We clearly see that for N a = 20 and d = 10.0 cm 1 the results are comparable to what was obtained with the cyclic tracking. As a non-cyclic tracking is time efficient, this configuration denoted C iso was also considered in the remaining of the study. Table III. Usage of a Non-Cyclic Tracking N a d k eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr (-0.13) 0.03 (0.07) 0.03 (0.19) 0.06 (0.11) (-0.16) 0.04 (0.06) 0.03 (0.20) 0.06 (0.13) (-0.09) 0.05 (0.14) 0.03 (0.22) 0.22 (-0.25) (0.11) 0.06 (0.17) 0.04 (0.29) 0.53 (-1.08) The reference is the same as in Table II. Then, we have tested the influence of the scattering treatment on this benchmark. Configuration Cspec was used and the results are reported in Table IV where P0 corresponds to an APOLLO-type transport correction for isotropic scattering[1]. As we can see, a P 1 expansion is sufficient to achieve convergence in this case. Moreover, the performances of the P0 scattering treatment are fully acceptable. In the context of the development of a computational scheme, this option was kept as it is time efficient. Table IV. Effect of the Scattering Anisotropy Treatment Order k eff ǫ (ǫmax) (%) L (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr. 1 P (-0.60) 0.15 (0.73) 0.20 (0.60) 0.08 (0.14) P (0.13) 0.13 (-0.23) 0.06 (0.12) 0.18 (-0.20) P (-0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) The reference is the P 3 scattering treatment. Finally, to conclude this parametric study for the one-level scheme, we have examined the effects of region and mixture grouping for the self-shielding calculation with the UP 1 solver. Configurations R 8, R + 8 and R ++ 8 are compared to R 12 in Table V. We clearly see that the only noticeable deterioration in the results appears when mixtures for 238 U are merged. In the remaining of the study, R + 8 configuration was used. Table V. Effect of Cell and Mixture Grouping in the Self-Shielding Process Conf. k eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr. 1 R (0.00) 0.04 (0.12) 0.01 (0.02) 0.00 (-0.00) R (0.00) 0.04 (0.12) 0.01 (0.03) 0.00 (-0.00) R (-0.05) 0.74 (-1.12) 0.18 (-0.26) 0.01 (-0.02) The reference is the R 12 configuration. 6/13

7 A PWR Assembly Computational Scheme Based on the DRAGON v4 Lattice Code Comparison with respect to TRIPOLI4 The comparison of both Cspec and C iso configurations with respect to TRIPOLI4 is reported in Table VI. We clearly see that wathever the tracking type is, there is a good agreement between DRAGON and TRIPOLI4 : the k eff is predicted within 60 pcm while the maximum error per cell on the thermal fission rate is lower than 0.5 %. Table VI. DRAGON - TRIPOLI4 Comparison for the One-Level Scheme Conf. k eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr. 1 Cspec (-0.28) 0.35 (-1.25) 0.17 (-0.47) 0.14 (0.33) C iso (-0.40) 0.37 (-1.21) 0.18 (-0.47) 0.14 (0.50) TRIPOLI4 Reference: k eff = (σ = 25 pcm) Two-Level Computational Scheme We now consider the two-level scheme. The MOC calculation is performed with the two previous configurations Cspec and C iso but on a few group energy structure. In this paper, we focus on the usage of the 20-group structure reported in [11] Cell Grouping for the First-Level Flux Calculation Concerning the cell grouping, both R 8 and R 12 configurations were tested along with a configuration without any grouping. In Table VII, the comparison of these two-level calculations with respect to a MOC 172-group reference is presented for Cspec tracking parameters. We clearly see that grouping the cells has not a strong impact. In practice, when solving the flux-current UP 1 system with an iterative approach, the computational time for this first-level flux calculation is about 5, 2, 1 s. for the configuration without grouping, R 12 and R 8 respectively. Table VII. Cell Grouping Effect for the First-Level Flux Calculation Conf. k eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr (-0.11) 0.30 (0.80) 0.00 (-0.01) 0.01 (0.02) R (-0.11) 0.33 (0.86) 0.00 (0.01) 0.01 (0.02) R (-0.11) 0.34 (0.89) 0.00 (-0.01) 0.01 (0.02) The reference is the MOC 172-group calculation. A major drawback of this grouping strategy is that it is assembly-dependent; in cases where the grouping pattern becomes difficult to select (as for assemblies with poisoned rods), one can decide to use a configuration without merging as the CPU time is not drastically affected. One can note that this is only true when the UP 1 system is solved iteratively; when an algrebraic elimination of the current unknowns is used, the CPU time largely increases when no cell-grouping is used, by a factor of about 30. 7/13

8 R. Le Tellier and A. Hébert In the remaining of this study, the R 8 configuration was kept for the first-level flux calculation SPH Equivalence Another interesting point when using a two-level scheme is the usage of an SPH equivalence when condensing the macroscopic cross sections. Although this type of equivalence is considered as mandatory when a spatial homogenization takes place (as explained in [11] for S n calculations), its usefulness when only an energetic condensation is at stake is unclear. For example, the 26-group energy mesh used in [6] for BWR-MOX calculations was partly introduced in order too avoid using any equivalence procedure in the condensation process. The results on this PWR-UOX assembly presented in Table VIII confirm that this equivalence is not efficient and can be avoided. Note that, if found advantageous, such an equivalence only takes 1 s. Table VIII. Effect of the SPH Equivalence k Conf. eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr. 1 No Eq (-0.11) 0.34 (0.89) 0.00 (-0.01) 0.01 (0.02) Cspec Eq (-0.61) 0.31 (-0.85) 0.07 (0.14) 0.03 (0.12) No Eq (-0.10) 0.33 (0.86) 0.00 (-0.01) 0.01 (0.03) C iso Eq (-0.60) 0.31 (-0.89) 0.07 (0.14) 0.03 (0.12) The reference is the MOC 172-group calculation in the considered configuration Comparison with respect to TRIPOLI4 Finally, we present in Table IX the results of the two tracking configurations with this two-level approach in comparison with TRIPOLI4. Table IX. DRAGON - TRIPOLI4 Comparison for the Two-Level Scheme Conf. k eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Total gr. 1 Cspec (-0.39) 0.64 (1.73) 0.17 (-0.48) 0.14 (0.35) C iso (-0.42) 0.63 (1.77) 0.18 (-0.46) 0.14 (0.52) TRIPOLI4 Reference: k eff = (σ = 25 pcm). With the two-level scheme without equivalence, we can see that we obtain a prediction of the k eff within 100 pcm when compared to TRIPOLI4: indeed, on one hand, there is +50 pcm between the one-level DRAGON calculation and TRIPOLI4 and, on the other hand, -50 pcm between the one-level and two-level schemes. As observed in Tables VII and VIII, the deterioration of the reaction rates when using this two-level scheme is localized in the resolved resonance energy region. Beside, a cell-by-cell comparison of the thermal fission rate is given in Fig. X for the fuel cells of a 1/8 assembly. We clearly see that the agreement between DRAGON and TRIPOLI4 is good: the maximum error is about 0.4 %. 8/13

9 A PWR Assembly Computational Scheme Based on the DRAGON v4 Lattice Code Table X. Thermal Fission Rate Comparison MOC Computational Time Details In this section, we give some CPU time details regarding the MOC implementation we used for these two computational schemes. This presentation is limited to the Cspec tracking parameters. The CPU time of both 172- and 20-group MOC calculations are presented in Table XI with and without the Algebraic Collapsing Acceleration (ACA) based strategy described in [9]. Table XI. MOC CPU Time Details 172 gr. 20 gr. a b c d e Acceleration CPU time (s.) Option Assembly d Flux Calc. Total Nout a N b track N c calc None Two-step ACA Two-step ACA (Init.) 2 1 e None Two-step ACA (Init.) Nout is the number of outer iterations, N track is the number of tracking accesses, N calc = N g (i) where N g (i) is the number of groups processed at iteration i, i Assembly corresponds to the ACA matrices construction prior to the MOC flux calculation, The multigroup ACA technique is used and an ACA-simplified transport calculation is performed to initialize the multigroup fluxes. 9/13

10 R. Le Tellier and A. Hébert We obtain a speed-up of 2.2 (resp. 2.4) with ACA for the 20-group (resp. 172-group) calculation, the usage of ACA to initialize the fluxes providing an additional 10 % reduction of the CPU time. Considering the self-shielding, UP 1 and tracking CPU times, the 20-group calculation provides an overall speed-up of about when compared to the 172-group calculation for the C spec tracking parameters. The usage of a non-cyclic tracking (C iso configuration) helps further reducing the total time to = 32 s. 4. DEPLETION CALCULATION In this last section, we present a comparison of the two-level computational schemes with respect to the one-level scheme when considering the depletion of this PWR-UOX assembly from 0 to 60 GWd/t. Such a comparison is mandatory to fully validate the two-level scheme. For this study, the C iso tracking parameters were used. 70 burnup steps were selected according to [11]. A library produced in the DRAGON format as explained in Sect. 2 containing 268 isotopes without any pseudo-fission product was used. This constant fuel power depletion calculation is performed with a burnup computed using the energy released in the complete geometry Two-Level Scheme Accuracy The evolution of the difference in k eff between the two-level and one-level schemes is shown in Fig. 4 with or without an SPH equivalence in the two-level scheme. A comparison in terms of reaction rates is provided in Table XII for the last burnup step. 40 k eff (pcm) with 172 gr. as reference Δ gr. No Eq. 20 gr. Eq Burnup (GWd/ton) Figure 4. Evolution of the Two-level Schemes Discrepancy in k eff The two-level scheme deviation k eff between 0 and 60 GWd/t is limited to 50 k eff 30 (resp. 50 k eff 0) in pcm without (resp. with) an equivalence. The relative discrepancies in the reaction rates introduced by the two-level scheme are rather stable as one can notice by comparing Tables VIII and XII. The thermal fission rate is the most affected and the maximum relative difference with respect to the one-level scheme goes from % to % when the fuel is depleted. 10/13

11 A PWR Assembly Computational Scheme Based on the DRAGON v4 Lattice Code Table XII. Two-level Schemes Reaction Rates Discrepancy at 60 GWd/t Conf. k eff ǫ (ǫmax) (%) (pcm) Fission gr. 4 Capture gr. 3 Capture gr. 2 Totale gr. 1 No Eq (-0.41) 0.31 (0.79) 0.01 (-0.03) 0.01 (-0.03) Eq (-0.77) 0.30 (-0.82) 0.07 (0.15) 0.03 (0.14) The reference is the MOC 172-group calculation. In Table XIII, we gives the relative difference in some actinides concentration between the two-level and one-level schemes at 60 GWd/t. We see that these differences are within 1 % without equivalence. With an equivalence, the results are improved and the differences are within 0.5 %. Unfortunately, for a PWR three-zone MOX assembly, a similar calculation (not reported in this paper) has shown that the equivalence leads to a large discrepancy in the thermal fission rate. In such a context, the only way to improve the prediction of the actinides concentration without deteriorating the reaction rates is to increase the number of groups for the second level calculation. For this purpose, the 26-group structure used in [6] on BWR-MOX assemblies and further discussed in [7] is a good candidate. Table XIII. Two-level Schemes Discrepancy in Actinides Concentrations at 60 GWd/t Isotope Relative Error in Concentration (%) Relative Error in Concentration (%) Isotope No Eq. Eq. No Eq. Eq. 234 U Pu U Am U Am U Cm Np Cm Pu Cm Pu Cm Pu Cm Pu The reference is the MOC 172-group calculation Computational Time Details Finally, we present a detailed comparison of the computational cost for the one-level scheme and two-level scheme without equivalence on this depletion calculation. The cumulative times spent in the different components of the lattice code are reported in Table XIV. We see that for such a depletion calculation, the two-level scheme provides an overall speed-up of about 6.8. It is interesting to note that for the one-level scheme, the MOC calculations represent 95 % of the CPU time while for the two-level scheme, it is reduced to 38 %: the condensation from 172 to 20 groups is the second major time consumer with 28 % while the edition of the results i.e. the calculation of the condensed reaction rates at each step represents 11 %. This is directly related to the large number of tracked isotopic 11/13

12 R. Le Tellier and A. Hébert concentrations (6768) in the depletion; with such an optimized computational scheme, the impact of the number of isotopes in the library on the overall computational time becomes important. Table XIV. Depletion Computational Cost Details for the Different Schemes CPU time in s. One-level Two-level (20 gr. No Eq.) Library Access 77 (0.59 %) 77 (4.05 %) Self-Shielding 110 (0.83 %) 110 (5.79 %) UP 1 Flux Calc (3.69 %) Condensation (27.91 %) Tracking 4 (0.03 %) 4 (0.21 %) MOC Assembly 1718 (13.22 %) 177 (9.32 %) Flux Calc (81.85 %) 532 (28.01 %) Depletion 167 (1.28 %) 186 (9.79 %) Edition 285 (2.19 %) 213 (11.22 %) Total CONCLUSIONS A study dedicated to the definition of computational schemes for PWR assemblies was presented in the framework of DRAGON v4 validation studies. These schemes are based on a subgroup-based self-shielding approach and on the method of characteristics for the main flux calculation. Besides a direct MOC calculation, a computational scheme for industrial applications is based on a two-level approach. In this case, the MOC calculation is performed posterior to the condensation of the material properties in a few groups using a weighting flux obtained by an interface current calculation, leading to a speed-up factor of 6.8 for a complete depletion calculation. At zero burnup, it was compared to a Monte-Carlo calculation and was found in very good agreement: the k eff is predicted within 100 pcm while the maximum error on the thermal fission per cell is less than 0.5%. When depletion is considered, the deviation between this two-level scheme and a direct MOC calculation was assessed, it is limited to 80 pcm in k eff and 1 % in the concentration of major actinides. REFERENCES [1] G. Marleau, A. Hébert and R. Roy, A User Guide for DRAGON Version4, Report IGE-294, Institut de Génie Nucléaire, École Polytechnique de Montréal, Montréal (2006). [2] A. Hébert, Towards DRAGON Version4, Workshop at Int. Mtg. on the Physics of Fuel Cycles and Advanced Nuclear Systems: Advances in Nuclear Analysis and Simulation PHYSOR 2006, Vancouver (2006). [3] J. R. Askew, A Characteristics Formulation of the Neutron Transport Equation in Complicated Geometries, Report AAEW-M 1108, United Kingdom Atomic Energy Establishment, Winfrith (1972). [4] J. P. Both and Y. Peneliau, The Monte Carlo Code TRIPOLI-4 and its First Benchmark Interpretations, Proc. of Int. Conf. on the Physics of Reactors PHYSOR 1996, Mito (1996). 12/13

13 A PWR Assembly Computational Scheme Based on the DRAGON v4 Lattice Code [5] R. Le Tellier and A. Hébert, Benchmarking of the Characteristics Method Combined with Advanced Self-Shielding Models on BWR-MOX Assemblies, Proc. of Int. Mtg. on the Physics of Fuel Cycles and Advanced Nuclear Systems: Advances in Nuclear Analysis and Simulation PHYSOR 2006, Vancouver (2006). [6] A. Santamarina, N. Hfaiedh, R. Le Tellier, V. Marotte, S. Misu, A. Sargeni, C. Vaglio-Gaudard and I. Zmijarevic, Advanced Neutronics Tools for BWR Design Calculations, Proc. of Int. Conf. on Nuclear Engineering ICONE 14, Miami (2006). [7] J. F. Vidal D. Bernard, O. Litaize, A. Santamarina and C. Vaglio-Gaudard, New Modelling of LWR Assemblies Using the APOLLO2 Code Package, this meeting. [8] A. Hébert, A Consistent Technique for the Pin-by-Pin Homogenization of a Pressurized Water Reactor Assembly, Nuclear Science and Engineering, 113, pp (1993). [9] R. Le Tellier and A. Hébert, An Improved Algebraic Collapsing Acceleration with General Boundary Conditions for the Characteristics Method, accepted for publication in Nuclear Science and Engineering. [10] A. Hébert and M. Coste, Computing Moment-Based Probability Tables for Self-Shielding Calculations in Lattice Codes, Nuclear Science and Engineering, 142, pp (2002). [11] A. Santamarina, C. Collignon and C. Garat, French Calculation Schemes for Light Water Reactor Analysis, Proc of Int. Mtg. on the Physics of Fuel Cycles and Advanced Nuclear Systems PHYSOR 2004, Chicago (2004). [12] R. E. MacFarlane and D. W. Muir, NJOY99.0 Code System for Producing Pointwise and Multigroup Neutron and Photon Cross Sections from ENDF/B Data, Report PSR-480/NJOY99.0, Los Alamos National Laboratory, Los Alamos (2000). [13] A. Hébert and H. Saygin, Development of DRAGR for the Formatting of DRAGON Cross Section Libraries, Seminar on NJOY-91 and THEMIS for the Processing of Evaluated Nuclear Data Files, Saclay (1992). [14] A. Hébert, The Ribon Extended Self-Shielding Model, Nuclear Science and Engineering, 151, pp.1-24 (2005). [15] G. Marleau, A. Hébert and R. Roy, A Users Guide for DRAGON 3.05, Report IGE-174 Rev. 6, Institut de Génie Nucléaire, École Polytechnique de Montréal, Montréal (2006). [16] A. Leonard and C. T. McDaniel, Optimal Polar Angles and Weights for the Characteristics Method, Transactions of the American Nuclear Society, Vol. 73, pp (1995). 13/13