Multigroup Neutron Diffusion Equation Research Articles

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.

New 3D Diffusion Code Based on The Nodal Polynomial Expansion

Nodal Expansion Method (NEM) is widely employed in the neutronic design of nuclear reactors core. The main reason is its higher computational performance and efficiency when compared with the conventional fine mesh difference method or finite element method. The NEM diffusion calculation uses coarse spatial meshes and the size can lie in assembly scale. This is the key for the computational efficiency and high accuracy calculations. The NEM consists mainly of basis polynomial function expansion for each nodal direction. The Nodal Expansion Method (NEM), as proposed by Finnemann in 1977, has been used to solve the multigroup neutron diffusion equations in three-dimensional (3D) rectangular geometry. The weighted residual technique has been applied to determine the higher order coupling coefficients. Resulting from the combination of nodal and finite element methods, NEM provides rigorously accurate equations obtained by integrating the neutron balance equation. The two group coarse mesh 3D IAEA benchmark has been simulated by NEM3D-1A using different nodes sizes. The VENTURE

ADAPTIVE TIME-STEP CONTROL FOR THE MODAL METHOD TO INTEGRATE THE MULTIGROUP NEUTRON DIFFUSION EQUATION

The distribution of the power inside a reactor core can be described by the time dependent multigroup neutron diffusion equation. One of the approaches to integrate this time-dependent equation is the modal method, that assumes that the solution can be described by the sum of amplitude function multiplied by shape functions of modes. These shape functions can be computed by solving a _-modes problems. The modal method has a great interest when the distribution of the power cannot be well approximated by only one shape function, mainly, when local perturbations are applied during the transient. Usually, the shape functions of the modal methods are updated for the time-dependent equations with a constant time-step size to obtain accurate results. In this work, we propose a modal methodology with an adaptive control time-step to update the eigenfunctions associated with the modes. This algorithm improves efficiency because of time is not spent solving the systems to a level of accuracy beyond relevance and reduces the step size if they detect a numerical instability. Step size controllers require an error estimation. Different error estimations are considered and analyzed in a benchmark problem with a out of phase local perturbation.

Virtual element methods for the spatial discretisation of the multigroup neutron diffusion equation on polygonal meshes with applications to nuclear reactor physics

The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretisation methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. This paper presents the first application of VEM to the field of nuclear reactor physics, specifically to the steady-state, multigroup, neutron diffusion equation (NDE). In this paper the theoretical convergence rates of the CG-VEM are verified using the Method of Manufactured Solutions (MMS) for a reaction-diffusion problem in the presence of both highly distorted and non-convex elements and also in the presence of discontinuous material data. Finally, numerical results for the 2D IAEA and the 2D C5G7 industrial nuclear reactor physics benchmarks are presented using both block-Cartesian and general polygonal meshes.

Nodal Integral method for multi-group neutron diffusion equation in three dimensional cylindrical coordinate system

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.

Development and verification of open reactor simulator ADPRES

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.

Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

The meshless radial basis function collocation method is an efficient numerical technique for solving partial differential equations. The multiquadric is the most widely utilized radial function for this purpose; but it contains a shape parameter, which has a significant effect on the performance of the method. In this study, the meshless collocation method employing multiquadric as the radial function with optimum shape parameters is applied to the numerical solution of the multigroup neutron diffusion equation. The optimization of the shape parameter is performed by minimizing the Madych-Nelson function. One external and two fission source problems are solved to investigate the performance of the method. The results show that the meshless collocation method with optimized shape parameters yield a high level of accuracy with an exponential convergence rate.

Iteration Methods with Multigrid in Energy for Eigenvalue Neutron Diffusion Problems

In this paper we present nonlinear multilevel methods with multiple grids in energy for solving the k-eigenvalue problem for multigroup neutron diffusion equations. We develop multigrid-in-energy algorithms based on a nonlinear projection operator and several advanced prolongation operators. The evaluation of the eigenvalue is performed in the space with smallest dimensionality by solving the effective one-group diffusion problem. We consider two-dimensional Cartesian geometry. The multilevel methods are formulated in discrete form for the second-order finite volume discretization of the diffusion equation. The homogenization in energy is based on a spatially consistent discretization of the group diffusion equations on coarse grids in energy. We present numerical results of model reactor-physics problems with 44 energy groups. They demonstrate performance and main properties of the proposed iterative methods with multigrid in energy.

Richardson Extrapolation-Based Verification Method of Scientific Calculation Program without the Oracles: A Case Study

For the verification test of some scientific calculation programs, different comparison methods are commonly applied to ensure the correctness of the computations. However, it is difficult to verify whether the testing output is correct, because the oracles which include the expected output are not always available or too hard to get. For this reason, the authors focus on using the Richardson Extrapolation to estimate the convergences of the numerical solution on different levels of mesh refinement. These numerical convergence properties can be applied to verification test, without the need for giving the oracles. In the present study, the authors take the program test of the multigroup neutron diffusion equations as a study case and propose the Richardson Extrapolation-based verification method. Three verification criterions are obtained based on our approach. In addition, a test experiment is conducted demonstrating the validity of our theoretical results.

DEVELOPMENT OF OPEN REACTOR SIMULATOR: ADPRES

Abu Dhabi Polytechnic Reactor Simulator (ADPRES) is an open reactor simulator designed particularly as a learning and research tool for nuclear engineering students. ADPRES is written in pure FORTRAN 90 language from scratch. It solves static and transient three-dimensional, multi-group neutron diffusion equation in Cartesian geometries. Standard fourth order Nodal Expansion Method (NEM) is implemented in ADPRES. Currently, the simple adiabatic approach is used for transient problems without thermal-hydraulics feedbacks. It had been tested on several well-known benchmarks to evaluate its accuracy: IAEA-3D, DVP BWR, for static problems and LMW transient benchmark for time-dependent problem. The results showed that ADPRES results agree very well against reference solutions. The ADPRES’ source codes can be found in the author’s GitHub repository.

A Reduced Order Model for the Fast Predictions of Reactivity and Neutron Distributions within Reactor Cores

A new Reduced Order Model (ROM) is developed for solving the neutron eigenvalue problem for the fast and accurate prediction and simulation of the neutron flux within light water reactor cores. The method of Proper Orthogonal Decomposition is employed to form the ROM which uses snapshots obtained from a full order model based on the finite element discretisation of the spatial dependence of the multi-group neutron diffusion equation. We detail how the temperature variation and control rod adjustments can be efficiently integrated into the model and their influence then accurately predicted within the model's solution. This is particularly important as the former poses the non-linearities that form within the neutron transport equations, whilst the later can easily form challenging fluxes for the ROM to resolve. <br/> The ROM is used to predict eigenvalues for studying the reactivity of a PWR assembly under varying operating conditions, with varying temperatures, control rod settings and fuel compositions. It is able to accurately resolve reactivity under these different scenarios and provide detailed solutions of the energy dependent neutron scalar flux distributions. Importantly the run times to produce these solutions are drastically reduced, amounting to several orders of magnitude in comparison to a full order model using methods such as the finite element method. Thus it allows extremely fast solutions to a large array of calculations that may be required in scoping calculations, uncertainty quantification, and time dependent reactor transient analysis.

Multi-level coarse mesh finite difference acceleration with local two-node nodal expansion method

A computationally efficient and effective multi-level coarse mesh finite difference (CMFD) acceleration is proposed using the local two-node nodal expansion method for solving the three-dimensional multi-group neutron diffusion equation. The multi-level CMFD acceleration method consists of two essential features: (1) a new one-group (1G) CMFD linear system is established by using cross sections, flux and current information from the multi-group (MG) CMFD to accelerate the MG CMFD calculation, (2) an adaptive Wielandt shift method is proposed to accelerate the inverse power iteration of 1G CMFD in order to provide an accurate estimate of the eigenvalue at the beginning of the iteration for both 1G and MG CMFD linear system. Additionally, a nodal discontinuity factor and a diffusion coefficient correction factor are defined to achieve equivalence of the 1G and MG CMFD system. The accuracy and acceleration performance of multi-level CMFD are examined for a variety of well-known multi-group benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level CMFD acceleration method is efficient, particularly for the larger, multi-group systems.

A fully coupled two‐level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations

SummaryThe multigroup neutron diffusion equations (an approximation of the neutron transport equation) are widely used for studying the motion of neutrons and their interactions with stationary background materials. Solving the multigroup neutron diffusion equations is challenging because the unknowns are tightly coupled through scattering and fission events, and solutions with high spatial resolution of full reactor cores in multiphysics environments are frequently required. In this paper, we focus on the development of a scalable, parallel preconditioner for solving the system of equations arising from the finite‐element discretization of the multigroup neutron diffusion equations in space and an implicit finite‐difference scheme in time. The parallel preconditioner (here referred to as the “fully coupled Schwarz preconditioner”) is constructed by monolithically applying the overlapping domain decomposition method together with a smoothed aggregation‐based coarse space to the coupled system. Our approach is different from the traditional block Gauss–Seidel sweep method that applies the preconditioner from the fast group to the thermal group sequentially, and we demonstrate that it provides significant improvements in terms of both the number of iterations required and the total compute time for a system of equations with millions of unknowns on a large supercomputer.

Multigroup neutron diffusion equation with the finite volume method in reactors using MOX fuels

ABSTRACTThe use of mixed oxide (MOX) fuel to partially fill the cores of commercial light water reactors (LWRs) gives rise to a reduction of the radioactive waste and production of more energy. However, the use of MOX fuels in LWRs changes the physics characteristics of the reactor core, since the variation with energy of the cross sections for the plutonium isotopes is more complex than for the uranium isotopes. Although the neutron diffusion theory could be applied to reactors using MOX fuels, more emphasis on treatment of the energy discretization should be placed. This energy discretization could be typically 4–8 energy groups, instead of the standard 2-energy group approach. In this work, the authors developed a finite volume method for discretizing the general multigroup neutron diffusion equation. This method solves the eigenvalue problem by using Krylov projection methods, in which the size of the vectors used for building the Krylov subspace does not depend on the number of energy groups, but it can solve the multigroup formulation with upscattering and fission production terms in several energy groups. The method was applied to MOX reactors for its validation.

Development of a Nodal Expansion Method based space-time kinetics code for hexagonal geometry

A two and three dimensional, multigroup, neutron space-time kinetics theory based computer code IQSHEX (Improved Quasi Static method based kinetics code for HEXagonal geometry) is developed in order to carry out reactivity initiated transient in large size thermal (i.e. neutronically loosely coupled reactor core) and fast breeder reactors with hexagonal lattice. For such reactors, neutron flux is considered as function of both space and time and neutron space-time kinetics equations are solved for obtaining the dependence of flux on space and time. This code uses Improved Quasi Static (IQS) method to solve the equations. In this method, it is assumed that shape function changes slowly relative to amplitude function and product of these two functions gives the complete behavior of neutron flux both in space- and time. Multigroup neutron diffusion equation is solved over macro (or larger) time step by advanced Nodal Expansion Method (NEM) to find out shape function, and point kinetics equations are solved over micro (or smaller) time step by Piecewise Constant Approximation (PCA) method to find out amplitude function. The code has been validated against several international benchmark problems with and without temperature feedback. Results corroborate well with that of the benchmark problems.

Development of a GPU-accelerated 3D neutron dynamics code for PB-FHR

Pebble bed fluoride-salt cooled high temperature reactor (PB-FHR) is a kind of novel nuclear energy systems, combining advanced techniques such as coated particle fuel and molten-salt coolant to achieve better safety and economy performance. The compact core of a PB-FHR is geometrically complicated due to the randomly-packed pebble bed with fuel motion in a complex core chamber, which yields a highly detailed core-modeling for accurate neutron dynamics analysis. Three-dimensional fine mesh finite volume method is expected to achieve better accuracy because of its strong geometry adaptability in detailed core-modeling, but is much more time-consuming compared to other methods such as coarse mesh nodal method. A GPU-accelerated 3D fine mesh neutron dynamics code (GAND) is developed, using conjugate gradient method (CG) to solve the time-dependent multi-group neutron diffusion equations in r-z-θ coordinates. The GAND code is verified by a cylindrical reactor benchmark with good agreement, and is preliminarily applied to a PB-FHR core in both static and transient analysis. Speed-up ratio and other performance of GAND code using different equation preconditioners is studied, a best speed-up ratio of 21.65 has been achieved using Neumann polynomials-preconditioned CG.

Isogeometric analysis for the multigroup neutron diffusion equation with applications in reactor physics

Isogeometric Analysis (IGA) has been applied to heterogeneous reactor physics problems using the multigroup neutron diffusion equation. IGA uses a computer-aided design (CAD) description of the geometry commonly built from Non-Uniform Rational B-Splines (NURBS), which can exactly represent complicated curved shapes such as circles and cylinders, common features in reactor design. This work has focused on comparing IGA to finite element analysis (FEA) for heterogeneous reactor physics problems, including the OECD/NEA C5G7 LWR benchmark. The exact geometry and increased basis function continuity contribute to the accuracy of IGA and an improvement over comparable FEA calculations has been observed.

Extension of the GeN-Foam neutronic solver to SP3 analysis and application to the CROCUS experimental reactor

The Laboratory for Reactor Physics and Systems Behaviour at the PSI and at the EPFL has been developing since 2013 a multi-physics platform for coupled reactor analysis named GeN-Foam. The developed tool includes a solver for the eigenvalue and transient solution of multi-group neutron diffusion equations. Although frequently used in reactor analysis, the diffusion theory shows some limitations for core configurations involving strong anisotropies, which is the case for the CROCUS research reactor at the EPFL. The use of an SP3 approximation to neutron transport can often lead to visible improvements in a code predictive capabilities, especially for one-directional anisotropies, with acceptable added computational cost vs diffusion. Following some modelling issues for the CROCUS reactor, and in order to improve the GeN-Foam modelling capabilities, the GeN-Foam diffusion solver has been extended to allow for SP3 analyses. The present paper describes such extension and a preliminary verification using a mini-core PWR benchmark. The newly developed solver is then applied to the analysis of the CROCUS experimental reactor and results are compared to Monte Carlo calculations, as well as to the results of the diffusion solver.

Sensitivity Analysis of the Galerkin Finite Element Method Neutron Diffusion Solver to the Shape of the Elements

The purpose of the present study is the presentation of the appropriate element and shape function in the solution of the neutron diffusion equation in two-dimensional (2D) geometries. To this end, the multigroup neutron diffusion equation is solved using the Galerkin finite element method in both rectangular and hexagonal reactor cores. The spatial discretization of the equation is performed using unstructured triangular and quadrilateral finite elements. Calculations are performed using both linear and quadratic approximations of shape function in the Galerkin finite element method, based on which results are compared. Using the power iteration method, the neutron flux distributions with the corresponding eigenvalue are obtained. The results are then validated against the valid results for IAEA-2D and BIBLIS-2D benchmark problems. To investigate the dependency of the results to the type and number of the elements, and shape function order, a sensitivity analysis of the calculations to the mentioned parameters is performed. It is shown that the triangular elements and second order of the shape function in each element give the best results in comparison to the other states.

Numerical analysis for multi-group neutron-diffusion equation using Radial Point Interpolation Method (RPIM)

A mesh-free method is introduced to overcome the drawbacks (e.g., mesh generation and connectivity definition between the meshes) of mesh-based (nodal) methods such as the finite-element method and finite-difference method. In particular, the Point Interpolation Method (PIM) using a radial basis function is employed in the numerical analysis for the multi-group neutron-diffusion equation. The benchmark calculations are performed for the 2D homogeneous and heterogeneous problems, and the Multiquadrics (MQ) and Gaussian (EXP) functions are employed to analyze the effect of the radial basis function on the numerical solution. Additionally, the effect of the dimensionless shape parameter in those functions on the calculation accuracy is evaluated. According to the results, the radial PIM (RPIM) can provide a highly accurate solution for the multiplication eigenvalue and the neutron flux distribution, and the numerical solution with the MQ radial basis function exhibits the stable accuracy with respect to the reference solutions compared with the other solution. The dimensionless shape parameter directly affects the calculation accuracy and computing time. Values between 1.87 and 3.0 for the benchmark problems considered in this study lead to the most accurate solution. The difference between the analytical and numerical results for the neutron flux is significantly increased in the edge of the problem geometry, even though the maximum difference is lower than 4%. This phenomenon seems to arise from the derivative boundary condition at (x,0) and (0,y) positions, and it may be necessary to introduce additional strategy (e.g., the method using fictitious points and Hermite-type collocation method) in order to solve this problem. On the basis of these results, it is expected that the mesh-free method including the RPIM can be sufficiently employed in numerical analysis for the multi-group neutron-diffusion equation and can be considered as an alternative numerical approach to overcome the drawbacks of existing nodal methods.