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• Fourier analysis is carried out for the first time to study the convergence performance of CMFD, pCMFD, odCMFD and lpCMFD acceleration schemes for fixed-source neutron transport problem in two-dimensional geometry. • Detailed linearization and formulation of the Fourier analysis of CMFD, pCMFD, odCMFD and lpCMFD are derived. • CMFD becomes unstable when increasing the coarse mesh optical thickness for large scattering ratio. • lpCMFD is unconditional stable for the coarse mesh optical thickness up to 100 for the scattering ratios tested. • The observations from the 2D results are consistent with the 1D results in literature. Previous works have derived the Fourier analysis for different CMFD based neutron transport acceleration schemes for 1D problems to study the convergence performance, in which the recently developed lpCMFD scheme gives best convergence performance due to its employment of linear prolongation technique in flux updating that maintains continuous flux at the coarse mesh boundary. However, the theoretical convergence performance for variants of CMFD schemes in 2D neutron transport problems has not been previously investigated and is an important result that will complement other studies of neutron transport acceleration methods. This paper carries out the Fourier analysis of the CMFD, pCMFD, odCMFD and lpCMFD acceleration schemes for the fixed source neutron transport problem in two dimension, along with detailed linearization and formulation derivation. The Fourier analysis results show that compared to other schemes, the lpCMFD scheme has the lowest spectral radius for coarse mesh optical thickness larger than 1 and is unconditionally stable for scattering ratios c = 0.6, 0.8, 0.9 and 0.99 for a large range of coarse mesh optical thicknesses up to 100, which are consistent with 1D findings. It is also shown that there is almost no impact of the choice of level-symmetric S N quadrature set on the spectral radius. The results presented here can support the application of lpCMFD for fixed source neutron transport problems in 2D space for practical problems. Our research provides valuable insight to the field of acceleration methods in neutron transport. [ABSTRACT FROM AUTHOR]

• Abu Dhabi Polytechnic Reactor Simulator (ADPRES) is an open source reactor simulator which can be used and modified by everyone. • ADPRES solves static and time-dependent three-dimensional, multi-group neutron diffusion equation in Cartesian geometries. • ADPRES' input is straight forward and it has many features of today's modern reactor simulators. • ADPRES has been verified for both static and time-dependent reactor benchmarks. • ADPRES can be obtained from author's GitHub repository. Abu Dhabi Polytechnic Reactor Simulator (ADPRES) is an open source reactor simulator recently developed at Abu Dhabi Polytechnic. It solves static and time-dependent three-dimensional, multi-group neutron diffusion equation in Cartesian geometries. Fourth order standard Nodal Expansion Method (NEM) is implemented in ADPRES for spatial discretization. The time-dependent diffusion equation is numerically solved by fully-implicit scheme. ADPRES input is straight-forward and designed with minimal keywords. ADPRES also has most features of today's modern reactor simulators. In this paper, ADPRES solutions for several reactor benchmarks were verified. The first part of PWR MOX/UO2 core benchmark was used to verify ADPRES on static reactor problem. For transient reactor problems, LMW three-dimensional and NEACRP 3D PWR transient benchmarks were used. ADPRES solutions were generally accurate and consistent with the solutions from other computer codes. The source codes can be found in the author's GitHub repository. [ABSTRACT FROM AUTHOR]

• A set of new expansion functions are developed for solving transport problems in hex-z geometry. • The method eliminates need for mapping hex geometry to Cartesian geometry. • New method was implemented into the co arse m esh radiation t ransport code COMET. • Transport solutions are more accurate with the new method than the mapping method. Some of the radiation transport solvers that are based on expansion function methods in hexagonal geometry generally use projection techniques that transform the spatial dependence of the angular flux in hexagonal geometry to Cartesian geometry by mapping and then expanding in terms of expansion functions (e.g., Legendre polynomials). However, this approach usually requires a relatively higher-order expansion in space for accurate representation of the spatial distribution of the flux. To overcome this limitation, a set of consistent spatial expansion functions have been developed in hexagonal geometry in this paper. These functions satisfy the orthogonality condition on hexagonal surfaces and guarantee the conservation of currents after expansion regardless of the truncation order. The latter is an important property for preservation of neutron balance. The new spatial expansion method has been implemented into the co arse m esh radiation t ransport (COMET) code and tested in both the single assembly and the full core Advanced High Temperature Reactor (AHTR) benchmark problems. The eigenvalues and assembly-averaged and stripe-wise fission density distributions predicted by the new method with 2nd order spatial expansion functions were found to exhibit similar accuracy as those computed by the current mapping method with 4th order spatial expansion functions. This indicates that the angular fluxes in AHTRs can be predominantly represented by a low order expansion if the new spatial expansion functions are used. As a result, to maintain the same accuracy, the computational efficiency of COMET is improved significantly (2–5 times) by using the new spatial expansion method. [ABSTRACT FROM AUTHOR]

• Fourier analysis is carried out for the first time to study the convergence performance of CMFD and lpCMFD for 2D k-eigenvalue neutron transport problem. • Detailed linearization and formulation of the Fourier analysis of CMFD and lpCMFD are derived. • CMFD becomes unstable when increasing the coarse mesh optical thickness. • lpCMFD is unconditional stable for the coarse mesh optical thickness up to 100 for the scattering ratios tested. • Increasing the inner source iteration number reduces the spectral radius up to an asymptotic value for both the CMFD and lpCMFD schemes. The recently developed lpCMFD scheme has been shown to be unconditionally stable and more effective than CMFD scheme for 2D fixed source neutron transport problems by using Fourier analysis. Previous works have derived the Fourier analysis to study the convergence performance of the CMFD acceleration scheme for 1D k -eigenvalue neutron transport problem, but the study has not yet been performed for 2D k -eigenvalue neutron transport problem, which we believe is an important work that complements other studies of neutron transport acceleration methods. In this paper, we extend the Fourier analysis of CMFD and lpCMFD for k -eigenvalue neutron transport problems to 2D space, along with the detailed linearization and formulation derivation. The Fourier analysis results show that for both CMFD and lpCMFD, increasing the number of inner iterations per outer iteration can reduce the spectral radius up to an asymptotic value of the spectral radius. CMFD is found to be unstable at higher coarse mesh optical thickness while lpCMFD is unconditionally stable for scattering ratios c = 0.3, 0.6, 0.9, 0.99 and for a large range of coarse mesh optical thickness up to 100. The results presented here can support the application of lpCMFD for k -eigenvalue problems in 2D space for realistic neutronic simulations. [ABSTRACT FROM AUTHOR]

Accurate implementation of reflective Boundary Condition (B.C.) in the P N method for both forward and adjoint analysis in one, two and three dimensional geometries is investigated in detail. Slab, spherical and cylindrical geometries are covered for 1D space, while a comprehensive procedure for general 3D Cartesian geometry is explained which encapsulates 2D (XY) geometry as a special case. It is shown that the reflective B.C. in the P N method ends to a series of moment couplings on the mirror surface which should be treated as an essential B.C. While these constraints are derived for the conventional P N method, they are applicable for the even parity P N approach, as well. Necessary guideline for imposing reflective constraints on the final system of linear equations in eigenvalue search or fixed source problems is also described. We finally verify the proposed approach by several test cases at the end. [ABSTRACT FROM AUTHOR]

To solve neutron transport problems with void regions accurately and efficiently, the legacy discrete nodal transport method (DNTM) was improved in three-dimensional Cartesian geometry. Firstly, the efficient nodal-equivalent finite difference (NEFD) algorithm was modified, named M-NEFD, to directly treat the zero total cross section in the denominator for void nodes. Furthermore, an angle projection discrete-ordinates (APSN) method was proposed to combine with the NEFD algorithm, named APSN-NEFD, to treat the void regions without sweeping the void nodes one by one. Thirdly, the angular sweeps within one octant of both M-NEFD and APSN-NEFD were parallelized using OpenMP. Based on these improvements, a discrete-ordinates nodal transport code named NECP-HONESTY has been developed in three-dimensional Cartesian geometry. To test the performance of the method and the code, a fission-source problem with two small void regions and a fixed-source problem with one large void region were presented in this paper by comparing with the Monte Carlo method. It has been demonstrated by the numerical results that the improved DNTM can provide accurate eigenvalue and scalar flux, no matter directly sweeping the void nodes with M-NEFD or skipping the void regions with APSN. In addition, compared with M-NEFD, APSN-NEFD can efficiently reduce both computing time and storage requirement for problem with large void regions. [ABSTRACT FROM AUTHOR]

• VITAS is a multi-purpose simulation code for handling steady-state and time-dependent neutron transport problems. • VITAS incorporates models inside a unified variational nodal framework based on the second-order transport equation. • VITAS is compatible with problems using Cartesian, hexagonal, triangular, and Cartesian with finite element geometries. • The space-angle truncation is extensively examined to demonstrate the asymptotic convergence with p- and h-refinement. • Results indicate that VITAS has the potential to become a versatile toolset for solving various nuclear reactor problems. Existing neutronic whole-core simulation codes are problem-oriented. With the advancement of computer technology and the development of complicated micro-nuclear reactors, there is a pressing need to create solution methods with broad applicability and the ability to accommodate more complex, high-fidelity, unstructured geometries. This necessity motivates the development of a neutron transport code that can adapt to different types of nuclear reactors. This work describes the code design and theoretical model of VITAS, a multi-purpose simulation code for handling steady-state and time-dependent neutron transport problems. This code incorporates algorithms and models inside a unified variational nodal framework based on the second-order neutron transport equation. The most recent version of VITAS employs the isotropic scattering approximation and permits arbitrary spatial and angular refinement. VITAS is compatible with Cartesian geometry, hexagonal geometry, triangular geometry, and Cartesian geometry with finite element subdivisions in the radial direction (Cartesian-FE). The three-dimensional modeling can be performed by axially extruding the two-dimensional mesh. Users can choose the mesh type and expansion order on-demand for optimal computational efficiency. In this paper, VITAS is validated using benchmark problems, i.e., the TAKEDA3 benchmark and the transient LMW benchmark with Cartesian homogenized geometry, the C5G7 and C5G7-TD exercises with Cartesian-FE geometry, the TAKEDA4 benchmark and the Buckner and Stewart benchmark with hexagonal-z homogenized geometry, and the Dodds benchmark with cylindrical-z geometry described by triangular-z unstructured meshes. The space-angle truncation is extensively examined to demonstrate the asymptotic convergence behavior with p- and h-refinement. The results indicate that VITAS has the potential to become a versatile toolset for solving a variety of nuclear reactor problems. [ABSTRACT FROM AUTHOR]

The multigroup time-integro-differential equation of the neutron diffusion kinetics (IDE-NDK) was solved numerically in 3D Cartesian geometry with the use of the basic-progressive polynomial approximation (BP n ) using the finite difference method (FDM) for the spatial discretization. Applications involving ramp and instantaneous change of the thermal removal or fission macroscopic cross sections were used to assess the accuracy of the BP 2 algorithm. Two reactor models were used: a homogeneous and the 3D-TWIGL reactor. The BP 2 algorithm showed good accuracy when it is compared to the results of other codes. Also the static neutron diffusion equation was solved numerically with the Lagrange interpolation polynomial to assess the K eff accuracy of the FDM used for the steady state problem. In some applications calculations were performed as function of the time integration step and mesh size. Extrapolations to infinitesimal mesh size were performed in some cases. [ABSTRACT FROM AUTHOR]

This work presents three efficient higher order analytical nodal methods for the numerical solution of a two-dimensional multigroup neutron diffusion equation in Cartesian geometry based on the use of the successive polynomial-weighted transverse integrations technique to convert a one-group diffusion equation to a system of coupled one-dimensional ordinary differential equations. These equations are then solved analytically over each homogenized cell after adequate approximations of the resulting effective sources after transversal integrations. Coupling between the approximate transverse flux-moments is achieved by imposing uniqueness constraint on their moments values. Adjacent elements are coupled by enforcing continuity conditions on the flux and current moments at interfaces cells. The weighted cell-balance equations and current-continuity conditions are then used to derive the discrete equations. These methods are applied for solving numerically various 2D benchmark problems and theirs performances discussed. Numerical results demonstrates more efficiency for the third higher order analytical nodal method for which the alone unknowns considered are the transverse flux moments on the interfaces of the homogenized elements. [ABSTRACT FROM AUTHOR]

• High-order diamond differencing is proposed for the solution of the Boltzmann Fokker-Planck (BFP) equation. • The BFP equation is used to represent the transport of electrons. • A continuous slowing down termwith stopping power coefficient is introduced. • Soft and catastrophic energy domains are used for representing collisional and radiative scattering. 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. [ABSTRACT FROM AUTHOR]

Highlights: [•] Coupled neutronic/thermal–hydraulic simulations on hybrid Cartesian geometries. [•] Analysis of discrepancies between full pin-by-pin and hybrid solutions. [•] Extension of COBAYA3/SUBCHANFLOW coupling scheme within the NURESIM platform. [•] Code-to-code comparison done to verify the accuracy of the coupled hybrid scheme. [•] Local safety parameters studied for pin-by-pin, hybrid, and nodal based core solutions. [Copyright &y& Elsevier]

Highlights: [•] The 2D-static neutron diffusion equation was solved with Lagrange polynomial (LAP). [•] No variational principle was used in the development of the LAP algorithms. [•] The progressive polynomial approximation was used in the LAP algorithms. [•] The LAP algorithms showed better accuracy than a finite difference method. [ABSTRACT FROM AUTHOR]

Abstract: The multi-group time-integro-differential equations of the neutron diffusion kinetics (IDE-NDK) was solved numerically in 2D Cartesian geometry with the use of the basic-progressive polynomial approximation (BP n ). Two applications were computed: a ramp, and an instantaneous change of the thermal removal macroscopic cross sections of the driver material of the 2D-TWGL benchmark problems. The BP2 algorithm showed good accuracy when compared with the results of other codes. BP n did not show the undesirable oscillations that appeared in other codes. [Copyright &y& Elsevier]

Abstract: In this work, a solution for a two-dimensional neutron transport problem, in cartesian geometry, is proposed, on the basis of nodal schemes. In this context, one-dimensional equations are generated by an integration process of the multidimensional problem. Here, the integration is performed for the whole domain such that no iterative procedure between nodes is needed. The ADO method is used to develop analytical discrete ordinates solution for the one-dimensional integrated equations, such that final solutions are analytical in terms of the spatial variables. The ADO approach along with a level symmetric quadrature scheme, lead to a significant order reduction of the associated eigenvalues problems. Relations between the averaged fluxes and the unknown fluxes at the boundary are introduced as the usually needed, in nodal schemes, auxiliary equations. Numerical results are presented and compared with test problems. [Copyright &y& Elsevier]

• We develop the 3D and cylindrical version of CNEM_JFNK based on NEM and JFNK. • The efficient row-column-iteration-based preconditioners are developed. • CNEM_JFNK is successfully applied to complicated HTR pebble bed core models. • Numerical results show CNEM _JFNK obtains the good accuracy and efficiency. A cylindrical version of Jaocbian-free Newton Krylov nodal expansion methods in cylindrical coordinates (CyliNEM_JFNK) are successfully developed to solve the three-dimensional and multigroup k -eigenvalue reactor problems in cylindrical geometry. Unlike nodal methods in Cartesian geometry, CyliNEM_JFNK adopts the special polynomial-expansion approximation and some separate treatments to solve the transverse-integrated equations in each direction and it is not a simple and trivial extension from Cartesian geometry to cylindrical geometry. To reduce the computational cost and the scale of residual systems constructed in the CyliNEM_JFNK as much as possible, the outgoing interface partial currents, node-averaged fluxes, third and fourth expansion coefficients are only chosen as the final-solving variables to obtain the residual functions by eliminating all the other variables including incoming interface partial currents, first and second expansion coefficients, source and leakage moments. Then the efficient physics-based preconditioners are successfully developed by making full use of the efficient Row-Column iteration and original power iteration framework. The cylindrical version of 3D LWR IAEA benchmark problem and a series of three-dimensional high temperature gas-cooled pebble bed reactor models are chosen to verify the numerical accuracy and computational efficiency of the CyliNEM_JFNK method. Numerical solutions show that the proposed CyliNEM_JFNK method can track the reference solutions very well and obtain higher computational efficiency than the CyliNEM method with efficient Row-Column iteration in three-dimensional cylindrical coordinates, which indicates the advantages of the developed CyliNEM_JFNK method that can serve as a potential nodal diffusion analysis tool for three-dimensional cylindrical advanced reactor problems. [ABSTRACT FROM AUTHOR]

The methods presented in this paper solve the Simplified Spherical Harmonics approximation to the multidimensional neutron transport equation. 1D, 2D and 3D systems were modeled with Cartesian geometry using the finite difference method to discretize the spatial variables. The method is able to simulate any energy group discretization, including up-scattering terms. The Krylov Shur method was used to calculate the solution of the steady-state equation by solving a generalized eigenvalue problem. This methodology has the capability to calculate any number of eigenfunctions. A formulation review of the Simplified Spherical Harmonics is explained in this work, as well as, a study of the boundary conditions for different approaches of the finite difference method. The results calculated by this methodology are compared with the discrete ordinates and diffusion approximation methods, all of them, using the same spatial discretization in order to show the different accuracy of each method without influence of the method used for discretizing the spatial variable. The results show the validity of each method for different benchmark problems. [ABSTRACT FROM AUTHOR]

• Neutron diffusion for cylindrical geometry is solved by Nodal Integral Method. • Unlike Cartesian, each direction is different making extension to 3D difficult. • Method is used for multi-group neutron diffusion equation. • Method is applied to solve both the source and criticality benchmark problems. • The results are in good agreement with those obtained with other approaches. Due to their high accuracy, the nodal methods are quite extensively used for solving neutron diffusion and transport equations. However, their use is mainly restricted to the geometries which can be mapped by rectangular (cuboidal in 3D) or hexagonal cells. For several Generation IV reactors the equations are solved in cylindrical geometries, needing fine meshing near the boundaries if traditional nodal methods are used. Therefore, a Nodal Integral Method (NIM) for one group neutron diffusion in 2D polar coordinates was developed recently. Here, the method is extended to multi-group neutron diffusion in 3D cylindrical coordinates. Unlike the Cartesian geometries, each direction needs separate treatment in case of the cylindrical geometries. Therefore, the extension is neither trivial nor straightforward. The developed method is tested by solving three different problems for which analytical or benchmark solutions exist. It is observed that the methodology preserves its high accuracy for multi-group 3D problems. [ABSTRACT FROM AUTHOR]

• A new 3D discrete-ordinates method in a modular geometrical modelling. • The linear short characteristics as a linear Galerkin projection of the integral equation. • A practical non-overlapping domain decomposition method for complicated and memory-consuming problems. • A stable Coarse-Mesh Finite Difference accelerates the Parllel Block Jacobi iteration of the DDM. • A 60% parallel efficiency has been measured up to 0(1000) cores. We present in this paper a discrete-ordinates transport method to perform 3D PWR transport simulations. The numerical technique takes profit of the Cartesian modular construction of the PWR geometry. The spatial mesh of the proposed method is composed of Heterogeneous Cartesian Cells (HCC). HCCs are basic geometrical patterns delimited by a box and having an arbitrary number of locally-extruded heterogeneous regions. The source is spatially expanded by piece-wise linear approximation in each region. The faces of the box, composing the boundary of the HCC, are discretized with a uniform Cartesian mesh. This surface sub-mesh is the support of a piece-wise linear representation of the interface angular flux. The linear expansion of the sources allows for a considerable reduction of the number of regions. Because of the Cartesian nature of the geometry, the method uses the effective spatial sweeping based on progression by front. Results on three-dimensional core simulations show accurate power distribution while minimizing the number of degrees of freedom. Since parallel computing is mandatory to access high-fidelity flux distributions in a reasonable amount of time, a domain decomposition algorithm is applied at each power iteration. Also, because of the slow convergence of the parallel block-Jacobi iterations, outer power iterations are accelerated by the Coarse-Mesh Finite Difference method (CMFD). In the present work, both transport and outer CMFD share the same domain splitting configuration and perform local multigroup source iterations to stabilize the scattering source. This allows for non-intrusive coarse-grained hybrid parallelism based on MPI/OpenMP directives. A set of 3D benchmarks will be proposed to access the accuracy and the parallel efficiency of the implementation. [ABSTRACT FROM AUTHOR]

• Linear interpolation method is developed for lpCMFD in hexgonal geometry. • The proposed barycentric interpolation technique is demonstrated to be effective. • CMFD becomes unstable at large coarse mesh size. • pCMFD has slower or comparable performance compared to lpCMFD. The recently developed lpCMFD acceleration scheme has been shown to be unconditionally stable and effective for S N neutron transport problems in Cartesian rectangular geometry. However, studies of lpCMFD performance for multi-energy group k -eigenvalue neutron transport problem in hexagonal geometries have not been previously investigated. In this paper, linear interpolation method on general convex polygon coarse mesh geometries is developed for lpCMFD in multi-energy group k -eigenvalue neutron transport problems. This method is tested on two hexagonal type reactor benchmark problems discretized with an equilateral triangular fine mesh for S N and different sizes of equilateral triangular and a mixed hexagonal-triangle coarse mesh grid for low order diffusion equation. It was found that lpCMFD was stable and effective for the benchmark problems investigated here with different coarse mesh grids, while CMFD became unstable when coarse mesh size increases. pCMFD was found to be stable in all configurations but was generally slower or as effective as lpCMFD and CMFD. The barycentric interpolation technique developed in this paper for lpCMFD in 2D hexagonal geometry is demonstrated to be effective and can be applied in realistic practical problems. [ABSTRACT FROM AUTHOR]

• The solution of the 1D and 2D Neutron Transport Equation provide neutron distributions. • Discrete Ordinates method for the angular discretization is applied. • Finite Difference Method for the spatial discretization is applied. • Krylov-Schur method is applied to solve the eigenvalue problem. • Multiple eigenvalues and eigenfunctions can be calculated. The method explained in this paper solves the steady-state of the neutron transport equation for 1D and 2D systems modeled with Cartesian geometry, by using the Discrete Ordinates method S N for the angular discretization and the Finite Difference Method for the spatial discretization. The method applies the multi-group approach for any energy discretization, including upscattering terms. The method solves the steady-state equation by solving a generalized eigenvalue problem by means of a Krylov-Schur method. One of the main advantages of the method is the capability to calculate multiple eigenfunctions. The Discrete Ordinates methodology is used for the angular discretization, which uses a simple formulation involving the angles and direction cosines. The spatial discretization with Finite Difference Method is selected for its simplicity. The method is validated with several one-dimensional benchmark problems and four two dimensional benchmark problems. The results show good agreement with respect to the reference results for all the cases studied. [ABSTRACT FROM AUTHOR]

• The one-group 2D discrete ordinates adjoint equations are solved. • Explicit solutions in terms of the spatial variables are given. • Explicit expressions are derived for the detector responses. • Absorption rates are estimated from the adjoint discrete ordinates problems. A solution to the one-group discrete ordinates approximation of the adjoint to the transport equation in two-dimensional Cartesian geometry is developed in this work. Starting from an integration process over a region (node) of the domain, the model is reduced to one-dimensional transverse-integrated equations in the x and y variables which are then solved by the Analytical Discrete Ordinates Method (ADO Method). Particular attention is given to the effect of non-homogeneous boundary conditions in the formulation. Explicit expressions are derived and used for evaluating, in a fast way, absorption rates of internal detectors from the adjoint formulation. Excellent agreement was found between numerical results obtained for the absorption rates evaluated from the forward and adjoint problems. [ABSTRACT FROM AUTHOR]