Multigroup Neutron Transport Equation Research Articles

Development of multi-physics calculation method in lead cooled fast reactor code system MOSASAUR

A full-core three-dimensional multi-physics calculation method was researched based on the Lead cooled fast reactor (LFR) code system MOSASAUR, which is focused on both core steady-state and transient analyses of LFR. In the previous version of MOSASAUR, cross-sections generation module and core steady-state simulation module have been developed. In this research, the stiffness confinement method is employed to solve the time-dependent multi-group neutron transport equation to expand the capabilities of core transient analyses. And the thermal hydraulic calculation method is developed as thermal feedback module for core steady-state and transient analyses. Based on the fixed-point iteration scheme, the neutron module and thermal hydraulic module are coupled at each time step. The LMW benchmark and the problem based on ORAL 19-rod bundle are employed to verify the accuracy of transient calculation and thermal hydraulic module respectively. Finally, the multi-physics calculation method is utilized to simulate the transient process of the MicroURANUS core. The simulation results demonstrate the capability of the constructed multi-physics calculation framework to accurately simulate the transient behaviors of LMRs. Numerical results showed the good accuracy of the newly-developed multi-physics calculation module with MOSASAUR.

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

FEMFFUSION and its verification using the C5G7 benchmark

FEMFFUSION is an open source code that solves the multigroup neutron transport equation using the diffusion and the SPN approximations. This code uses the continuous Galerkin finite element method to be able to deal with any type of geometry, structured or unstructured, and problem dimension. To verify the program, a research study of the two-dimensional C5G7 benchmark has been performed, including the static, time-dependent and neutron noise computations. Numerical results include the effective multiplication factor, keff, the spatial neutron fluxes, and neutron power distributions for the static computations. The total neutron power evolution of a given transient and the neutron flux distribution evolution for the different energy groups. In addition, neutron noise frequency domain calculation were completed for perturbations of some cross-section in the reactor core. All computation have been performed using different SPN approximations. FEMFFUSION results are verified against the official OECD/NEA benchmark results.

Spherical Harmonics and Discontinuous Galerkin Finite Element Methods for the Three-Dimensional Neutron Transport Equation: Application to Core and Lattice Calculation

The spherical harmonics or PN method is intended to approximate the neutron angular flux by a linear combination of spherical harmonics of degree at most . In this work, the PN method is combined with the discontinuous Galerkin (DG) finite elements method and yield to a full discretization of the multigroup neutron transport equation. The employed method is able to handle all geometries describing the fuel elements without any simplification nor homogenization. Moreover, the use of the matrix assembly-free method avoids building large sparse matrices, which enables producing high-order solutions in a small computational time and less storage usage. The resulting transport solver, called NYMO, has a wide range of applications; it can be used for a core calculation as well as for a precise 281-group lattice calculation accounting for anisotropic scattering. To assess the accuracy of this numerical scheme, it is applied to a three-dimensional (3-D) reactor core and fuel assembly calculations. These calculations point out that the proposed PN -DG method is capable of producing precise solutions, while the developed solver is able to handle complex 3-D core and assembly geometries.

Asymptotic Diffusion Limits of the Multigroup Neutron Transport Equations

In this paper, the standard multigroup neutron diffusion equations are derived as an asymptotic approximation to the multigroup neutron transport equations. The asymptotic analysis employs a scaling that (1) is suggested by the multigroup neutron diffusion equations themselves and (2) generalizes the long-known asymptotic scaling for monoenergetic transport problems. Two other asymptotic scalings of the multigroup transport equations are also considered, both of which lead to a new “group-collapsed” (monoenergetic) “equilibrium” diffusion approximation. The standard multigroup and equilibrium diffusion approximations are shown to preserve certain nonasymptotic properties of the multigroup transport equations. Generalizations of the analyses in this paper, and possible practical applications, are discussed.

Application of stiffness confinement method within variational nodal method for solving time-dependent neutron transport equation

The stiffness confinement method (SCM) is implemented in the variational nodal method (VNM) framework to solve time-dependent multi-group neutron transport equations. With frequency-transform, the time-dependent neutron equation is decoupled from the precursor equation and transformed into a dynamic eigenvalue problem which can be solved with the in-house VNM code VITAS. An improved time-source iteration method within the VNM-SCM framework is proposed to accelerate constructing response matrices and reduces memory usage. Based on the VNM-SCM scheme, the kinetics module of VITAS is developed for the Cartesian and hexagonal-z geometry and verified by several benchmark problems.Numerical results demonstrate that VNM-SCM can perform accurate and efficient transient calculations with large time-step sizes. For the two-dimensional TWIGL benchmark, the VNM-SCM produces a maximum core power error of -0.57% with the time-step size of 20 ms. For the three-dimensional LMW benchmark, the VNM-SCM results in a maximum core power error of 0.95% with the time-step size of 2 s. The VNM-SCM also yields an accurate neutron flux prediction for the hexagonal-z geometry problem. In addition, the effect of the angular expansion is investigated systematically. Examinations with the three-dimensional LMW benchmark display an error reduction in the core power error from 0.131% to 0.011% from P1 to P3 interfacial angular expansion, with a substantial computing time increase of 6.14 hours. The study indicates that the high-order angular expansion significantly increases the computing time but contributes little to the accuracy of the core power. Besides, it was found that the initial steady-state calculation dominates the shape error of the flux in the transient calculation.

Development of a cross section self-shielding method in arbitrary geometry

We are developing a neutron transport calculation code based on the Iso-Geometric Analysis (IGA) method that can resolve the multi-group neutron transport equation for any arbitrary spatial domain. The energy variable is treated with multi-group theory. Therefore, self-shielding calculations become necessary. We have decided to adopt the subgroup method because this method does not rely on any specific geometrical domain and should be used in combination with the IGA method. Some basic theory of the IGA method is given to introduce the concept of basis function and NURBS for the representation of the spatial domain. Validation of the implementation of the subgroup method for IGA and SN transport calculations is presented. The result is that the combination of SN transport theory, IGA and the subgroup method gives correctly shielded cross-sections. Thus, effective multi-group cross-sections can be generated for any arbitrary spatial domain.

A mesh-free treatment for even parity neutron transport equation

Common family of numerical methods such as finite element and finite difference involve the discretization of domain into regular grids or meshes. Mesh generation and re-meshing process, however, for obtaining required accuracy are complicated and time-consuming. Recently, Mesh Free (MFree) methods have been developed to overcome the drawbacks of such conventional numerical methods. In the present work, a novel approach is developed for solving multi-group neutron transport equation in two-dimensional X-Y geometry based on a mesh free scheme using the Radial Point Interpolation Method (RPIM). The K+ principle which is an extremum variational principle for solving even parity neutron transport equation is our choice to implement RPIM. The directional dependence of even-parity angular flux is expanded by the spherical harmonic polynomials which leads to PN method. The multi-quadrics radial basis function is used to construct the RPIM shape functions for spatial approximation of angular flux to obtain the discretized MFree weak-form of neutron transport equation. As the RPIM shape functions possess the Kronecker delta function property, essential boundary conditions is enforced as efficiently as in the finite element method (FEM). The obtained results are compared with some mesh-based methods such as conventional finite element method to evaluate the performance of the presented approach. It is demonstrated that the proposed method is robust, stable, reliable and efficient for treatment of neutron transport.

OpenNTP: Implementation of the S[formula omitted] method in cartesian 2D geometry and the CP method in cylindrical and spherical 1D geometry

This paper presents the implementation of the discrete ordinates method (SN) in 2D cartesian geometry and the collision probability method (CP) in cylindrical and spherical 1D geometry in OpenNTP code (Open Neutron Transport Package). This code is a pedagogical tool for computer analysis of nuclear reactors. Its main features are as follows: a free software with an open source, it solves the neutron transport equation to a few steady-state groups on a grid structured in one or two spatial dimensions with an isotropic and anisotropic dispersion source, and any new calculation method and algorithm would be easy to implement in this proposed code. Also, the code offers the possibility to calculate the main parameters of nuclear reactors such as the multiplication factor and the distribution of the scalar and angular neutron fluxes. Additional parameters, like the reaction rates, the pin power distribution and the boundary currents are also calculated. Moreover, a graphical user interface written in Python 3 programming language has been developed to simplify the use of OpenNTP. Some applications of the OpenNTP code have been compared on the one hand with the Monte Carlo OpenMC and MCNP6.1 codes and the WIMSD-5B lattice transport code, on the other hand. Numerical results are given to illustrate the accuracy of the OpenNTP code. Program summaryProgram Title: OpenNTP, version 1.2CPC Library link to program files:https://doi.org/10.17632/gm2b6799j6.1Developer’s repository link:https://github.com/mohamedlahdour/OpenNTPLicensing provisions: GPLv2Programming language: fortran90 and Python 3External routines/libraries: NumPy, Matplotlib, PyQt5, f2pyNature of problem: Solving the steady-state multigroup neutron transport equation by different methods in one, or two spatial dimensions.Solution method: the characteristic method (MOC), the discrete ordinate (SN) method and the collision probability method (CP)

A Highly Parallel Multilevel Newton--Krylov--Schwarz Method with Subspace-Based Coarsening and Partition-Based Balancing for the Multigroup Neutron Transport Equation on Three-Dimensional Unstructured Meshes

The multigroup neutron transport equation is crucial for studying the motion of neutrons and their interaction with materials. Numerical simulation of the multigroup neutron transport equation is computationally challenging because the equation is defined on a high-dimensional phase space, the computational spatial domain is complex, and the materials are heterogeneous. A scalable parallel solver is required to address such a challenge. In this paper, we study a highly parallel Newton--Krylov--Schwarz (NKS) method consisting of a Newton-based eigenvalue solver, a Krylov subspace method, and a novel multilevel Schwarz preconditioner. The multilevel method is one of the most popular preconditioners for accelerating neutron transport calculations, but the construction of coarse spaces can be expensive and often unscalable when a large number of processors is used. We propose a novel matrix coarsening algorithm in which a multilevel hierarchy is constructed using a single-component matrix instead of the full matrix of the neutron transport equation. This new coarsening algorithm is referred to as “subspace-based coarsening.” Above 8,000 processors, we show a 13x enhancement in multilevel preconditioner setup time when using the subspace-based coarsening method. A partition-based balancing strategy is studied to enhance the parallel efficiency of the NKS algorithm by equalizing the work for each processor. A hierarchical mesh partitioning algorithm is employed to generate a large number of submeshes while minimizing off-node communication. We demonstrate that the proposed algorithm is scalable with more than 10,000 processors for a realistic application on three-dimensional unstructured meshes with a few billion degrees of freedom. Neutron transport calculations using the improved NKS algorithm are twice as fast as those based on the unmodified NKS solver when over 8,000 processors are employed.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: I. Effects of Imprecisely Known Microscopic Total and Capture Cross Sections

The subcritical polyethylene-reflected plutonium (PERP) metal fundamental physics benchmark, which is included in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook, has been selected to serve as a paradigm illustrative reactor physics system for the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) that was developed by Cacuci. The 2nd-ASAM enables the exhaustive deterministic computation of the exact values of the 1st-order and 2nd-order sensitivities of a system response to the parameters underlying the respective system. The PERP benchmark is numerically modeled in this work by using the deterministic multigroup neutron transport equation discretized in the spatial and angular independent variables. Thus, the numerical model of the PERP benchmark developed includes the following imprecisely known uncertain parameters: 180 group-averaged total microscopic cross sections, 21,600 group-averaged scattering microscopic cross sections, 120 fission process parameters, 60 fission spectrum parameters, 10 parameters describing the experiment’s nuclear sources, and six isotopic number densities. Thus, the numerical simulation model for the PERP benchmark comprises 21,976 uncertain parameters, which implies that, for any response of interest, there are a total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to the model parameters. Computing these sensitivities exactly represents the largest sensitivity analysis endeavor ever carried out in the field of reactor physics. Only 241,483,276 are distinct from each other, and some of these turned out to be zero due to the symmetry of the 2nd-order sensitivities. The numerical results for all of these sensitivities, together with discussions of their major impacts, will be presented in a sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”. This work is the first in this sequence, presenting formulas of general use for neutron transport problems, along with the numerical results that were produced by these formulas for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic total cross sections. For comparison, this work also presents formulas of general use and numerical results for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic capture cross sections. It has been widely believed hitherto that, for reactor physics systems modeled by the neutron transport or diffusion equations, the second-order sensitivities are all much smaller than the first-order ones. However, contrary to this widely held belief, the numerical results that were obtained in this work prove, for the first time ever, that many of the 2nd-order sensitivities are much larger than the corresponding 1st-order ones, so their effects can become much larger than the corresponding effects stemming from the 1st-order sensitivities. For example, the 2nd-order sensitivities of the PERP leakage response cause the expected value of this response to be significantly larger than the corresponding computed value. The importance of the 2nd-order sensitivities increases as the relative standard deviations for the cross sections increase. For the extreme case of fully correlated cross sections, for example, neglecting the 2nd-order sensitivities would cause an error as large as 2000% in the expected value of the leakage response and up to 6000% in the variance of the leakage response. The significant effects of the mixed 2nd-order sensitivities underscore the need for reliable values for the correlations that might exist among the total cross sections, which are unavailable at this time. The 2nd-order sensitivities with respect to the total cross sections also cause the response distribution to be skewed towards positive values relative to the expected value. Hence, neglecting the 2nd-order sensitivities could potentially cause very large non-conservative errors by under-reporting of the response variance and expected value.

Calculation of [formula omitted] modes of the multi-group neutron transport equation using the discrete ordinates and Finite Difference Method

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 SN 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.

NtkFoam: An OpenFOAM based neutron transport kinetics solver for nuclear reactor simulation

Due to the complexity of detailed nuclear reactor numerical simulation, especially the complicated geometry and multi-physics coupling properties of the advanced reactor, a nuclear reactor kinetics solver for nuclear reactor engineering and design needs to be developed. Based on the open source C++ software OpenFOAM, this work establishes a neutron transport kinetics solver, namely ntkFoam, for the nuclear reactor kinetics simulation, from the governing equations to the detailed implementations. The coupling between multi-group neutron transport equations and the delayed neutron precursor balance equations are considered with using the finite volume method. By introducing the inverse power method to the OpenFOAM, the k-eigenvalue problems are calculated, and by coupling the neutron transport and delayed neutron precursor calculations using the Euler implicit scheme, the transient kinetics problems are simulated. Numerical results show that the proposed ntkFoam can simulate the multi-group neutron transport kinetics problem accurately and flexibility, and both the regular and irregular mesh configurations can be adopted. This work can provide some new perspectives and foundations to nuclear reactor coupling calculations.

Fully meshless solution of the one-dimensional multigroup neutron transport equation with the radial basis function collocation method

In this paper a fully meshless method is proposed for the numerical solution of the one-dimensional multigroup neutron transport equation with anisotropic scattering. Both first-order and even-parity forms of the transport equation are studied. The radial basis function collocation method is chosen for the spatial treatment, and Legendre polynomials are used to approximate the angular variable. The selection of the Legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables. Multiquadric is utilized as the radial function. Seven problems are considered to evaluate the performance of the method. The results show that the method converges exponentially, and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains. For the one-group isotropic benchmark problem, discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided, and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates-finite element scheme when the shape parameter is properly chosen.

A parallel unified gas kinetic scheme for three-dimensional multi-group neutron transport

In this paper, a parallel unified gas kinetic scheme (UGKS) is developed to simulate 3D multi-group neutron transport. The physical processes of neutron absorption, scattering, fission are included in the scheme. In order to design a multiscale method for capturing the physics in different regimes, the scheme is composed of solving coupled equations of microscopic neutron transport and macroscopic moment equation. As a result, a time accurate numerical fluxes for the neutron transport can be uniquely formulated and used in the construction of the multiscale method. The isotropic scattering neutron transport is studied in this paper, and it can be proved mathematically that this extended UGKS for multi-group neutron transport equation is an asymptotic preserving scheme. The scheme can be used effectively to capture the diffusion limit even with the mesh size being much larger than the neutron's transport characteristic scale, i.e., the so-called mean free path. Thus, the computation efficiency is greatly improved in comparison with the conventional single-scale scheme. In order to further speed up the computation, a parallelized UGKS has been developed for 3D neutron transport simulation as well. The newly developed scheme is verified through many numerical tests and shows high accuracy and strong robustness in the simulations under a large variation of transport condition.

Development and implementation of an integral variational nodal method to the hexagonal geometry nuclear reactors

This paper presents recent progress of the development and implementation of the integral transport variational nodal method (VNM) to the neutronics modelling of hexagonal geometry nuclear reactors. In this study, the three-dimensional multi-group neutron transport equation is solved with the variational nodal method utilizing an integral form of the with-in node angular flux distribution to generate response matrices more efficiently. The angular flux distributions on hexagonal nodal interfaces are approximated with even-parity spherical harmonics. The quasi-reflected interface condition (QRIC) method is employed to reduce the number of interfacial angular terms to save computational costs. Parallelization algorithms are accounted to accelerate the response matrix formation process. The newly-implemented methods are examined against the TAKEDA-4 benchmark cases, presenting superior accuracy and efficiency than the standard VNM approach. With the P7 angular approximation, the joint performance of the integral method and the QRIC method yields a remarkable computational time gain of 33.0. Furthermore, an 11-core parallelization to the response matrix formation process results in another 88% reduction in the total computation time.

NTP-ERSN: A new package for solving the multigroup neutron transport equation in a slab geometry

The package, called NTP-ERSN (N eutron T ransport P ackage from the R adiations and N uclear S ystems G roup), is an open-source code written in FORTRAN90 for a pedagogical purpose to solve the steady-state multigroup neutron transport equation. This package is based on three classical methods, namely the collision probability (CP) method, the discrete ordinates (SN) method and the method of characteristics (MOC). These methods are employed to obtain scalar and angular flux distributions in homogeneous and heterogeneous slab geometry with isotropic and anisotropic scattering. The source code algorithms are very simple to be comprehensive by engineering students. In addition, NTP-ERSN is a simple framework to add and test new algorithms. On the other hand, a graphical user interface written in Python programing language has been developed to simplify the use of NTP-ERSN. Numerical results are given to illustrate the NTP-ERSN code's accuracy. Finally, the present software can be useful as an academic tool for teaching reactor physics. It is freely available for download on GitHub (https://github.com/mohamedlahdour/NTP-ERSN).

Stabilization of multi-group neutron transport with transport-corrected cross-sections

Many deterministic neutron transport solvers rely on the source iteration method to solve the multi-group neutron transport equation. Often, these solvers rely on transport corrected cross-sections for accurate prediction of the neutron flux distribution. Transport corrected within-group scattering cross-sections can become negative, and if these negative cross-sections are large in magnitude, source iteration can become unstable and fail to converge for certain cases. In this study, we present evidence of this convergence issue for the method of characteristics (MOC) on full-core PWR problems with common transport correction schemes. A theoretical discussion is presented to illustrate the reason for the convergence issues. Previously established stabilization methods are compared with a newly proposed stabilization method. Results show that the new stabilization method allows for faster convergence than previous techniques. In addition, the effect of Coarse Mesh Finite Difference (CMFD) acceleration on stability is analyzed, showing that CMFD acceleration with full-group structure can overcome the convergence issues, but a stabilization technique is necessary for convergence when a condensed group structure is used.

An improved variational nodal method for the solution of the three-dimensional steady-state multi-group neutron transport equation

Abstract An improved variational nodal method is presented for the solution of the three-dimensional (3D) steady-state multi-group neutron transport equation. The variational functional is constructed that reproduces the even parity neutron transport equation with isotropic scattering. 3D orthogonal polynomials are used to approximate the spatial flux distribution within the nodes and across the nodal interfaces. The angular discretization utilizes a 3D even-parity integral method within the nodes, and standard spherical harmonics (P N ) on the interfaces. The generalized partitioned matrix (GPM) acceleration is derived and performed to speed up outer iterations of the transport formulation. Examined against three sets of TAKEDA benchmark cases, the integral method exhibits superior accuracy and efficiency than the standard VNM approach. In addition, the GPM method presents remarkable acceleration to outer iterations when tested on the 3D PWR problem. The joint employment of the GPM method and the PM method yields a gain of over 20 in the response matrix solution time.

Evaluation of the Effect of Linear Scattering Anisotropy and Boundary Shape on the Calculation of K ∞ for VVER-1000 Cells with Mixed Uranium-Plutonium Power-Grade Fuel

The method of surface pseudo sources for solving the multigroup neutron transport equation with scattering anisotropy and different boundary conditions by the operator splitting method was developed previously. The method was implemented in the RATsIYa option of the WIMS-SH-2.0 complex. The impact of scattering anisotropy and the form of the boundary conditions on K ∞ of VVER-1000 reactor cells with uranium-plutonium power-grade fuel was investigated. The calculations showed that the impact of each of these two effects is about 0.5%; they have different signs, as a result of which they largely compensate one another.