In this paper, a discontinuous Bubnov-Galerkin isogeometric analysis discrete ordinate (DBG-IGA-S-N) form of the subgroup projection and resonance spectrum expansion (RSE) methods within the DRAGON5 code is presented. The DBG-IGA-S-N method uses non-uniform rational B-spline (NURBS) basis functions to spatially discretize the neutron transport equation. The NURBS basis functions, which are used to represent both the solution field and the geometry, are able to exactly represent pin cell geometries even on the coarsest spatial representation. This is not possible with traditional Lagrangian finite elements. This is the first instance of a DBG-IGA-S-N discretization of the RSE method. We compared the numerical accuracy of the DBG-IGA-S-N method to the existing collision probability (CP) method within DRAGON5 against Serpent and CP ultrafine-group reference solutions for a series of uranium-oxide and mixed-oxide Rowlands pins. In DRAGON5, the CP method is frequently used for resonance self-shielding calculations. We use it in this work as a deterministic method standard for comparison to assess DBG-IGA-S-N performance. We found that our implementation of the DBG-IGA-S-N method achieved similar numerical accuracy compared to the CP method overall. The infinite multiplication factor errors were no greater than 107 pcm for the uranium-oxide cases and 152 pcm for the mixed-oxide cases with a Serpent reference, with a variation of about 10 pcm between DBG-IGA-S-N and CP. For the CP ultrafine-group method reference cases, the infinite multiplication factor errors did not surpass 108 pcm and 127 pcm for the uranium-oxide and mixed-oxide cases, respectively. In addition, DBG-IGA-S-N had lower infinite multiplication factor numerical errors than CP in all cases, in one case by 21 pcm. We also found that the RSE method had higher errors in the multiplication factors and reaction rates than the subgroup projection method for both the uranium-oxide and mixed-oxide cases when using the CP ultrafine-group method reference. Future work will investigate the relative performance of the DBG-IGA-S-N for larger, more complex cases, where strengths of the method, such as its scalability and ability to model exact geometries and anisotropic scatter, may make it desirable for production resonance self-shielding calculations.

A novel approach to modeling the response of a homogenized region has been applied for reflector calculations. This method was developed and tested in the DRAGON5/DONJON5 version 5 environment coupled to the BRISINGR nodal diffusion solver and the open-source NSSF module.This method leads to the introduction of a discontinuity matrix, called the $\mathbb F$F matrix, which contains parameters that are analogous to discontinuity factors (DFs). The matrix relaxes the flux continuity equation between nodes by introducing new free parameters. This work introduces the linear equivalent reflector method, which proceeds in two steps: (1) The $\mathbb F$F matrix is calculated from the results of a simplified one-dimensional or two-dimensional (2D) fuel-reflector geometry calculated in transport theory, and (2) the matrix is introduced into full-core calculations in diffusion theory using the nodal expansion method (NEM) or the analytic nodal method (ANM).In this case of the first step, two-group response matrices are obtained so as to preserve the net currents and reaction rates of the reference transport calculation over the macrogeometry with two different sets of cross sections. The calculation of the $\mathbb F$F matrix is straightforward, noniterative, and extendable to any number of groups.This approach was validated for a 2D small-core representation of a pressurized water reactor with different fuel loadings and a steel reflector. The calculations show that standard NEM and ANM codes are capable of handling such matrices as input, while giving results comparable to reflectors modeled by DFs. These results show that the approach presented in this paper may be a suitable option for calculating reflector parameters for nodal diffusion calculations.

Open source modeling of VVER-type reactors could become a medium-term objective in Eastern Europe. As the deterministic code DRAGON5 could meet such a need, we confronted DRAGON5 against a stochastic reference code, SERPENT2. Our validation comprises 7 cells and 4 assemblies from the Khmelnitsky-2 reactor in Ukraine, within a wide range of heterogeneity levels in fuel composition. Two calculation schemes have been developed and compared. The first, the ALAMOS scheme, is highly discretized in energy and spatial resolution, while the second, the REL2005-like scheme, is calculated in two levels (one highly discretized in energy and the other highly discretized in space). In the majority of cases studied, both schemes offer satisfactory accuracy (e.g. less than 300 pcm in k eff ), although there are difficulties related to energy deposition with gadolinium-poisoned fuel. While showing significantly poorer results than the ALAMOS scheme, the REL2005-like scheme offers lower computation times and major avenues for improvement remain to be explored. This work offers a first step towards the simulation of VVER-type reactors in DRAGON5, and paves the way for full-core simulations.

A monotone finite-difference scheme is proposed for the multidimensional angular Fokker-Planck operator in discrete ordinates calculations. It is compatible with nonorthogonal quadrature sets on the unit sphere and preserves the null space, the zeroth, and the first three angular moments of the analytical angular Fokker-Planck operator. Numerical results demonstrate that this discretization, combined with the suitable Galerkin quadrature method, eliminates spurious oscillations in the flux solution related to highly anisotropic scattering.

For many contemporary applications, ionizing radiation transport plays a pivotal role, requiring an accurate assessment of its impact on the exposed environment. While Monte Carlo simulations are widely considered the gold standard for accurate general-purpose coupled transport of photons, electrons and positrons in matter, discrete ordinates algorithms provide a viable alternative. This work consolidates cross-section models for coupled photon-electron-positron transport and provides the methodology to generate the data required by the multigroup Boltzmann Fokker-Planck transport equation and by energy and charge deposition formulas. It includes elastic, collisional and radiative inelastic interactions of leptons, annihilation of positrons, Compton scattering, Rayleigh scattering, photoelectric effect, pair production as well as fluorescence and Auger electron production from relaxation cascades following ionization. Comparative analyses of energy deposition in water, aluminum, and gold are conducted for incident beams of 1 MeV, 10 MeV, and 100 MeV electrons and photons, and juxtaposed against Monte Carlo reference calculations. While disparities of a few percent are typical, higher deviation can be observed due to discretization or physical model limitations. Energy spectrums per particle type at varying depths in the medium are also contrasted with Monte Carlo calculations to discern limitations in the current implementation and to propose potential avenues for enhancing the presented models. Energy and charge deposition calculations are also compared to experimental measurements. The cross-section production and transport algorithms are implemented in an open-source Julia package, Radiant.jl.

The resonance spectrum expansion (RSE) self-shielding method was recently proposed by Nagoya and Osaka universities as a powerful alternative to existing approaches. First investigations of the RSE at Polytechnique Montreal show that it can effectively replace the actual subgroup method used for production calculations in DRAGON5. The Japanese implementation of the RSE method is limited to a solution of the Boltzmann transport equation (BTE) with the method of characteristics. We are proposing a new implementation of the RSE method compatible with various types of solutions for the BTE, including the collision probability and the interface current methods. We based our validation study on a subset made up of eight Rowlands pin cell benchmark cases. The absorption rates obtained after self-shielding are compared with exact values obtained using an elastic slowing-down calculation where each resonance is modeled individually in the resolved energy domain. Validation of Rowlands benchmark with effective multiplication factor calculations was also conducted with respect of the SERPENT2 Monte Carlo code. It is shown that the RSE method is compatible with both advanced and legacy energy meshes and performs slightly better than the production subgroup methods actually used.

A multi-group cross-section library is fundamental for deterministic lattice physics calculations. Most existing multi-group cross-section libraries are customized for particular computer codes, as well as for particular types of nuclear reactors. This paper presents an HDF5-format multi-group cross-section library named XPZLIB. XPZLIB was produced using a self-developed XPZR module integrated into the NJOY2016 code, and an in-house PyNjoy2022 system was developed for auto-processing. XPZLIB contains detailed data content and well-organized data structures that are user- and developer-friendly. Three typical XPZLIBs with different numbers of energy groups, nuclides, and depletion reaction types were released via the Tsinghua cloud website. Furthermore, the applicability of the released XPZLIBs was investigated using HTGR and PWR lattice calculations, which can provide guidance for applying XPZLIB under different scenarios.

This work investigates reactor model biases and their consequences in nuclear scenario simulations. Usually, the models for Pressurized Water Reactors are based on infinite 2D assembly depletion simulations, but recent work has shown the importance of 3D complete core simulation for uncertainty reduction. The consideration of a whole core leads to new reactor parameters in the simulations that may bring additional biases. The fuel temperature distribution is one of them, and previous work considered isothermal reactors, leading to probable uncertainties in spent fuel inventory at reactor discharge. To quantify those biases and their propagation in a full scenario simulation, new advanced reactor models have been developed, based on neutronics and thermal-hydraulics couplings at the core level performed with DONJON5. Results show that the plutonium isotopic quality of spent fuel is biased for an isothermal core, with values systematically higher than for multi-physics calculations. In order to propagate those discrepancies in fuel cycle simulations that involve plutonium recycling in PWR MOX fuels, the coupling between CLASS and DONJON was renewed in order to add new fuel parameters such as the fuel temperature in the core burn-up simulation. A new methodology for data interpolation from lattice calculation has been implemented that allows acceptable computational time for DONJON5 calculations that are done within the fuel cycle simulation performed by CLASS. Comparison between isothermal and multi-physics reactor models for advanced scenario simulations performed with CLASS shows that the isothermal hypothesis leads to biases up to 10% for plutonium inventory in the UOX spent fuel stockpile, comparable with biases associated with other reactor parameters such as the loading pattern.

A calculation module is developed for testing and validating the improved nodal equivalence techniques of reflectors for full-core nodal calculations. This module, BRISINGR, is a new implementation of the nodal expansion method developed by Delft University of Technology and Framatome, and has been inserted into the version 5 environment of Polytechnique Montr & eacute;al, providing a fast prototyping setup used to assess the impact of different nodal equivalence approaches.We focus our investigations on an open-source implementation of the legacy equivalence technique Baff-Refl originating from the SCIENCE platform at Framatome. The proposed improvements to Baff-Refl are twofold: modification of the nodal equivalence procedure and modification of the reflector diffusion coefficients. We review the Nodal Expansion Method (NEM) and Analytical Nodal Method (ANM) for reflector calculations, the discontinuity factor (DF) renormalization, the DF decorrelation, the albedo calculation, and the procedure for obtaining few-group reflector diffusion coefficients from fine-group leakage coefficients.Our validation tests focus on the accuracy of the average nodal power of the fuel region in the downstream full-core calculation. A benchmark set of four two-dimensional 9 $$ \times $$x 9 core configurations with Evolutionary Power Reactor-type assemblies with either steel or water reflectors was used for validation. The results on the core impact of the reflector model show that the Inscatter model for the calculation of diffusion coefficients improves the accuracy of the full-core power in all benchmark configurations. DF renormalization is another studied aspect of this paper, and has been shown to provide notable improvements. Actually, renormalization to assembly DFs provides better results than renormalization to 1, which is itself more preferable than none for accuracy. Finally, calculating reflector constants with ANM is shown to have no conclusive improvement over NEM.

The Blackshaw-Murray[1] elastic kernel represents the effect of neutron up-scattering caused by thermal motion of target nuclei and resonance elastic scattering on the multigroup scattering matrices. A first implementation of this model was proposed by Ouisloumen and Sanchez and made available in proprietary cross-sections libraries of the PARAGON and APOLLO lattice codes.[2] Later, the same technique was reimplemented as module RESK in the NECP-Atlas cross-section generating code by J. Xu, T. Zu and L. Cao.[3, 4] The proposed reskr module in NJOY-2012 and NJOY-2016 is based on the implementation of module RESK in the NECP-Atlas cross-section generating code.[5] The reskr module is released under the BSD Open-Source license.

Abstract Legacy nuclear-reactor Boltzmann solvers start clinical deployment as an alternative to Monte Carlo (MC) codes and Fermi–Eyges semiemprical models in radiation oncology treatment planning. Today’s certified clinical solvers are limited to photon beams. In this paper, ELECTR, a state-of-the-art multigroup electron cross sections generation module in NJOY is presented and validated against Lockwood’s calorimetric measurements, EGS-nrc and GEANT-4 for 1–20 MeV unidirectional electron beams. The nuclear-reactor DRAGON-5 solver is upgraded to access the library and solve the Boltzmann–Fokker–Planck (BFP) equation. A variety of heterogeneous radiotherapy and radiosurgery phantom configurations were used for validation purpose. Case studies include a thorax benchmark, that of a typical breast Intra-Operative Radiotherapy and a high-heterogeneity patient-like benchmark. For all beams, $$100\%$$ 100 % of the water voxels satisfied the American Association of Physicists in Medicine accuracy criterion for a BFP-MC dose error below $$2\%$$ 2 % . At least, $$97.0\%$$ 97.0 % of adipose, muscle, bone, lung, tumor and breast voxels satisfied the $$2\%$$ 2 % criterion. The average BFP-MC relative error was about $$0.56\%$$ 0.56 % for all voxels, beams and materials combined. By irradiating homogeneous slabs from $$Z=1$$ Z = 1 (hydrogen) to $$Z=99$$ Z = 99 (einsteinium), we reported performance and defects of the CEPXS mode [US. Sandia National Lab., SAND-89-1685] in ELECTR for the entire periodic table. For all Lockwood’s benchmarks, NJOY-DRAGON dose predictions are within the experimental data precision for $$98\%$$ 98 % of voxels.

Fully coupled photon-electron-positron computational schemes are needed in a broad range of domains, from radiotherapy and radiosurgery treatment planning to nuclear-reactor safety analysis. Industrial deployment of such schemes remains problematic because of the prohibitively time-consuming charged-particle-transport aspects of both Monte Carlo algorithms and Boltzmann-Fokker-Planck solvers. In this paper, a partially coupled scheme based on reactor-physics models for secondary photons (RPSPs) is presented. RPSPs include thick-target-bremsstrahlung, annihilation, and full atomic cascade models, recovering long-range effects of both electrons and positrons. The RPSP-based scheme is compared with state-of-the-art fully coupled and uncoupled schemes in medical, nuclear, and multidisciplinary contexts. Within tissues, the average discrepancies from the fully coupled scheme are about 0.5%, with a maximum error below 1.3% at high energy. The CPU time of the coupled schemes is reduced by 58 and 78 times for 3-and 6-MeV beams, respectively. The discrepancies are also within the accuracy of the cross sections for 4Be, 6C, 13Al, 26Fe, 29Cu, 32Ge, 47Ag, 79Au, 82Pb, and 92U slabs. The same discrepancies are about 0.1%, 0.4% and 0.8%, respectively, in the reactor's shielding, vessel and fuel slabs. A power peaking fac-tor reduction of 4.35 times from the state-of-the-art uncoupled scheme is demonstrated for uranium oxide and mixed plutonium-uranium oxide slabs.

This paper presents the implementation of high-order diamond differencing schemes, specifically the DD1 and DD2 schemes which are 4th- and 6th-order accurate respectively, to handle the spatial discretization of the Boltzmann Fokker-Planck equation in 1D and 2D Cartesian geometries. While being as computationally expensive as linear/quadratic discontinuous Galerkin schemes, the DD1/DD2 schemes are respectively an order more accurate. The energy deposition calculations presented in this work show that, besides providing correction to the oscillations and a reduced propensity to yield negative fluxes than the classical diamond difference closure relation, they tend to perform better than discontinuous Galerkin schemes, even in regions with abrupt variations of the solution due to rapid energy loss, to interface implying high-Z variations and to domain borders. (C) 2022 The Authors. Published by Elsevier Ltd.