Neutron Diffusion Problems Research Articles

PARALLEL MESHLESS RADIAL BASIS FUNCTION COLLOCATION METHOD FOR NEUTRON DIFFUSION PROBLEMS

The meshless global radial basis function (RBF) collocation method is widely used to model physical phenomena in science and engineering. The method produces highly accurate solutions with an exponential convergence rate. However, due to the global approximation structure of the method, dense node distributions lead to long computation times and hinder the applicability of the technique. In order to overcome this issue, this study proposes a parallel meshless global RBF collocation algorithm. The algorithm is applied to 2-D neutron diffusion problems. The multiquadric is used as the RBF. The algorithm is developed with Mathematica and eight virtual processors are used in calculations on a multicore computer with four physical cores. The method provides accurate numerical results in a stable manner. Parallel speedup increases with the number of processors up to five and seven processors for external and fission source problems, respectively. The speedup values are limited by the constrained resource sharing of the multicore computer’s memory. On the other hand, significant time savings are achieved with parallel computation. For the four-group fission source problem, when 4316 interpolation nodes are employed, the utilization of seven processors instead of sequential computation decreases the computation time of the meshless approach by 716 s.

Solution of Neutron Diffusion Problems by Discontinuous Galerkin Finite Element Method With Consideration of Discontinuity Factors

Abstract Neutronics calculations are the basis of reactor analysis and design. Finite element methods (FEM) have gained increasing attention in solving neutron transport problems for the rigorous mathematical formulation and the flexibility of handling complex geometric domains and boundary conditions. In order to reduce the computational errors caused by the homogenization of cross sections, this paper adopts the discontinuous Galerkin finite element method (DG-FEM) to solve the generalized eigenvalue problem formulated by the neutron diffusion theory and compensates the homogenization error by incorporating discontinuity factors. The results show that the discontinuous Galerkin finite element method can introduce the discontinuity factors with clear mathematical and physical meanings. The computational results of the discontinuous Galerkin finite element method are slightly better than those of the continuous Galerkin finite element method. However, the computation cost of the former is higher than that of the latter. Although good parallel efficiency can be achieved, the discontinuous Galerkin finite element method is not preferable for large-scale problems unless the effect of the discontinuity factors is significant.

A variational nodal formulation for multi-dimensional unstructured neutron diffusion problems

A variational nodal method (VNM) with unstructured-mesh is presented for solving steady-state and dynamic neutron diffusion equations. Orthogonal polynomials are employed for spatial discretization, and the stiffness confinement method (SCM) is implemented for temporal discretization. Coordinate transformation relations are derived to map unstructured triangular nodes to a standard node. Methods for constructing triangular prism space trial functions and identifying unique nodes are elaborated. Additionally, the partitioned matrix (PM) and generalized partitioned matrix (GPM) methods are proposed to accelerate the within-group and power iterations.Neutron diffusion problems with different fuel assembly geometries validate the method. With less than 5 pcm eigenvalue (keff) error and 1% relative power error, the accuracy is comparable to reference methods. In addition, a test case based on the kilowatt heat pipe reactor, KRUSTY, is created, simulated, and evaluated to illustrate the method's precision and geometrical flexibility. The Dodds problem with a step transient perturbation proves that the SCM allows for sufficiently accurate power predictions even with a large time-step of approximately 0.1 s. In addition, combining the PM and GPM results in a speedup ratio of 2–3.

Surrogate modeling for neutron diffusion problems based on conservative physics-informed neural networks with boundary conditions enforcement

Application of physics-informed neural network(PINN) on neutron diffusion equation, which is of great engineering significance for reactor physics field, has not received much attention yet. Meanwhile, the non-smoothness of solution for neutron diffusion equation brings difficulties for PINN’s application. Therefore, we introduce the conservative PINN(cPINN) which develops PINN for each sub-domain and considers additional conservation law along the sub-domains’ interfaces, to solve heterogeneous neutron diffusion problems. Specifically, we develop PINNs on each sub-domain which has same material property and set equality constraints for neutron flux and neutron current on adjacent sub-domains. Furthermore, we propose a neural network constructing method to ensure that PINN/cPINN predictions strictly conform to three types of boundary conditions(BCs) involved in neutron diffusion problems. The results of numerical examples demonstrate that cPINN can solve heterogeneous neutron diffusion problems with non-smooth solutions and the proposed BC-imposed method can help to improve the cPINN performance on complex heterogeneous problems.

Higher Order U N Method for the Solution of the Neutron Diffusion Problem

The first application of the U N (Chebyshev polynomials of the second kind) method with higher order approximations is performed to solve the neutron diffusion problem in a slab reactor. The moments of equations are carried out by solving neutron transport equation using first the conventional spherical harmonics (P N) and then the U N method. These differential equations with constant coefficients are then solved together to obtain the diffusion equation corresponding to related approximation. The roots of the diffusion equation are estimated to calculate the diffusion lengths of the neutrons for various values of c, the number of secondary neutrons per collision. Numerical results obtained by the present method with its easily executable equations are tabulated with the ones already existing in literature. A good accordance is observed between them. Better results are also obtained than the conventional P N method for certain values of c.

Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary

In this paper, the artificial neural networks (ANN) based deep learning (DL) techniques were developed to solve the neutron diffusion problems for the continuous neutron flux distribution without domain discretization in advance. Due to its mesh-free property, the DL solution can easily be extended to complicated geometries. Two specific realizations of DL methods with different boundary treatments are developed and compared for accuracy and efficiency, including the boundary independent method (BIM) and boundary dependent method (BDM). The performance comparison on analytic benchmark indicates BDM being the preferred DL method. Novel constructions of trial function are proposed to generalize the application of BDM. For a more in-depth understanding of the BDM on diffusion problems, the influence of important hyper-parameters is further investigated. Numerical results indicate that the accuracy of BDM can reach hundreds of times higher than that of BIM on diffusion problems. This work can provide a new perspective for applying the DL method to nuclear reactor calculations.

A fully adaptive nonintrusive reduced-order modelling approach for parametrized time-dependent problems

We present an approach to build a reduced-order model for nonlinear, time-dependent, parametrized partial differential equations in a nonintrusive manner. The approach is based on combining proper orthogonal decomposition (POD) with a Smolyak hierarchical interpolation model for the POD coefficients. The sampling of the high-fidelity model to generate the snapshots is based on a locally adaptive sparse grid method. The novelty of the work is in the adaptive sampling of time, which is treated as an additional parameter. The goal is to have a robust and efficient sampling strategy that minimizes the risk of overlooking important dynamics of the system while disregarding snapshots at times when the dynamics are not contributing to the construction of the reduced model. The developed algorithm was tested on three numerical tests. The first was an advection problem parametrized with a five-dimensional space. The second was a lid-driven cavity test, and the last was a neutron diffusion problem in a subcritical nuclear reactor with 11 parameters. In all tests, the algorithm was able to detect and include more snapshots in important transient windows, which produced accurate and efficient representations of the high-fidelity models.

Fast solution of neutron diffusion problem with movement of control rods

Accurate and fast-running reduced order neutronics models are expected in many challenging applications such as multi-physics coupling analysis and data assimilations of reactors. The emphasis of the present study is to investigate an appropriate order reduction method for treating the movement of control rods. The empirical interpolation method is adopted to perform an affine decomposition to the spatial distribution of the cross sections with respect to the positions of control rods. The method can well recover the abrupt change of the cross sections near the end of control rods. With incorporation of the decomposed components, accurate and very efficient reduced order models can be constructed for the neutron diffusion problem with one-step or two-step movements of control rods.

Stability analysis of predictor-corrector schemes for coupling neutronics and depletion

Numerical instability resulting from coupling isotopic depletion and neutronics is widely reported, but little work has been done to interrogate the phenomenon theoretically. This work extends the pioneering analysis developed by Densmore et al., who examined the stability of a monoenergetic neutron diffusion problem coupled with a simple depletion system through the explicit Euler scheme. Their Fourier stability analysis is applied to the predictor-corrector scheme and its iterated and implicit variants. The results reproduce behaviour described in previous reports of numerical instability and its solution, and develop a future avenue for diagnosing the stability of burn-up problems.

Application of proper generalized decomposition to multigroup neutron diffusion eigenvalue calculations

In this paper, proper generalized decomposition (PGD) is utilized to reduce the computational burden of evaluating multigroup neutron diffusion eigenvalue problems. PGD is a reduced order modeling technique that seeks a separated representation to a multi-dimensional variable. In this application, each multigroup flux is sought as a finite sum of separable one-dimensional functions. This representation can significantly reduce the burden of evaluating multi-dimensional linear systems. Therefore, we use PGD to evaluate the linear systems within the power iteration process of the eigenvalue problem. In this paper, we discuss our implementation of PGD to these eigenvalue systems including a derivation of PGD operators for multigroup neutron diffusion problems with standard power iteration and power iteration accelerated with adaptive Wielandt shift. To illustrate PGD’s effectiveness, we apply our implementation to eigenvalue problems ranging from homogeneous to highly heterogeneous geometries with one-,two-, and four-group material properties. With comparison to full-order model evaluation with MOOSE, we find that the effectiveness of PGD is problem dependent. PGD always out performs the full-order model with close to homogeneous problems, but performs more poorly with more realistic reactor problems.

A unified space–time fully-transversed general nodal expansion method for multidimensional and multigroup steady and transient nuclear reactor neutronics problems

A new unified space–time fully-transversed general nodal expansion method (STFT_GNEM) is developed to solve the multidimensional and multigroup steady and transient neutron diffusion problems by making full use of the transverse-integration technique and the respective advantages of the two popular nodal methods, the analytical nodal method (ANM) and the nodal expansion method (NEM). Specifically, the developed STFT_GNEM can easily obtain the ANM’s local analytical solutions of all the transverse-integrated equations in the spatial coordinate directions based on the basis function expansion framework of the traditional NEM. The temporal operators are appropriately discretized using the transverse-integration procedure in the same way as the spatial operators and then the transverse-integrated equations in the temporal direction are also analytically solved so that both the steady and transient nuclear reactor neutronics problems can be easily integrated into a unified nodal discretization formulation. Besides, the STFT_GNEM code can be easily developed from the existing NEM code. In this paper, the TWIGL reactor kinetics problem, the LRA super prompt-critical transient problem and the three dimensional transient LMW problem are analyzed to test the numerical properties of STFT_GNEM. Numerical results show that the STFT_GNEM code can track the reference solutions very well even on the coarse meshes and for large time-step sizes, which show the effectiveness and the potential of STFT_GNEM for steady and transient nuclear reactor neutronics problems.

A new moving-mesh Finite Volume Method for the efficient solution of two-dimensional neutron diffusion equation using gradient variations of reactor power

Abstract A new moving-mesh Finite Volume Method (FVM) for the efficient solution of the two-dimensional neutron diffusion equation is introduced. Many other moving-mesh methods developed to solve the neutron diffusion problems use a relatively large number of sophisticated mathematical equations, and so suffer from a significant complexity of mathematical calculations. In this study, the proposed method is formulated based on simple mathematical algebraic equations that enable an efficient mesh movement and CV deformation for using in practical nuclear reactor applications. Accordingly, a computational framework relying on a new moving-mesh FVM is introduced to efficiently distribute the meshes and deform the CVs in regions with high gradient variations of reactor power. These regions of interest are very important in the neutronic assessment of the nuclear reactors and accordingly, a higher accuracy of the power densities is required to be obtained. The accuracy, execution time and finally visual comparison of the proposed method comprehensively investigated and discussed for three different benchmark problems. The results all indicated a higher accuracy of the proposed method in comparison with the conventional fixed-mesh FVM.

A solution to the singularity problem in the meshless method for neutron diffusion equation

In recent years, the moving least square (MLS) approximation has been widely used for numerical analysis of scattered data. Many numerical methods for solving partial differential equations are developed based on the MLS approximation.The MLS method is ideal for problems where the geometry of the domain is complex. An example of this is the neutron diffusion calculation. The collocation method is known that it has fast convergence speed, however, it is unstable and non-robust in many cases. In particular, it is the reason resulting in the fluctuation of the solution at the position that is on overlapping boundary region. This drawback can be dealt with the meshless local Petrov-Galerkin weak form for nodes on the interface boundary. In this paper, a combination of the weak form of the meshless local Petrov-Galerkin (MLPG) and the collocation method are applied to solve the neutron diffusion equation. Trial functions employed in the weak form of MLPG are constructed via the MLS method. In most MLPG methods numerical integration is required. The exception to this is the case of the collocation method in which the test functions are Dirac delta functions. Therefore, if the shape function is not constructed precisely enough, within an acceptable error range, the second order derivative of the shape function will lack accuracy. One of the reasons for the inaccuracy of the shape function is singularity problems occurring in the process of constructing the shape function. In this study, a solution is introduced to eliminate the singularity problem, allowing us to obtain the derivative of the shape function with sufficient precision. Finally, neutron diffusion problems are implemented in a combination of the weak-form of the meshless local Petrov-Galerkin (MLPG) and the collocation method is used to evaluate the efficiency and accuracy.

High-order lattice Boltzmann method for multi-group neutron diffusion solution

The detailed simulation of neutron diffusion problem plays an important role in neutronics, wherein a high-order method that can be easily implemented is highly desirable, since this would likely reduce the difficulty of application of these computational techniques. In this report, a high-order lattice Boltzmann method (HLBM) is presented to solve multi-group neutron diffusion problems in both transient and eigenvalue situations. The lattice Boltzmann equation is established for the neutron diffusion problem, and the corresponding variables are determined via high-order Chapman-Enskog expansion. Both the truncation errors and the numerical results indicate that the HLBM can be used to simulate the neutron diffusion problem, with an accuracy that is much higher than that of the conventional lattice Boltzmann method. This approach may form the basis for an efficient and accurate technique for large-scale engineering, and the numerical accuracy of the proposed model can be continuously increased by adopting higher-order truncation while preserving the current level of calculation efficiency.

Mathematical foundation of the neutron diffusion problem for a reflected nuclear reactor

The analytical approach to the neutron kinetics of a reflected multiplying system constitutes a challenging problem in mathematical physics and provides useful results for a full physical understanding of the multiplication process in non-homogeneous media, opening the possibility to establish reliable benchmarks. In this work, the basic mathematical features of the neutron diffusion operator in a reflected configuration are investigated. The completeness of the eigenstates is discussed and proven. The case of the infinite reflector is taken into consideration, highlighting its peculiarities that lead to the appearance of a continuum spectrum. The limitation of the critical eigenfunctions is also discussed. The analysis is carried out at first disregarding the presence of delayed emissions. The work is then extended to include also in the model the presence of delayed neutron precursors.

Pyramid finite elements for discontinuous and continuous discretizations of the neutron diffusion equation with applications to reactor physics

When using unstructured mesh finite element methods for neutron diffusion problems, hexahedral elements are in most cases the most computationally efficient and accurate for a prescribed number of degrees of freedom. However, it is not always practical to create a finite element mesh consisting entirely of hexahedral elements, particularly when modelling complex geometries, making it necessary to use tetrahedral elements to mesh more geometrically complex regions. In order to avoid hanging nodes, wedge or pyramid elements can be used in order to connect hexahedral and tetrahedral elements, but it was not until 2010 that a study by Bergot established a method of developing correct higher order basis functions for pyramid elements. This paper analyses the performance of first and second-order pyramid elements created using the Bergot method within continuous and discontinuous finite element discretisations of the neutron diffusion equation. These elements are then analysed for their performance using a number of reactor physics benchmarks. The accuracy of solutions using pyramid elements both alone and in a mixed element mesh is shown to be similar to that of meshes using the more standard element types. In addition, convergence rate analysis shows that, while problems discretized with pyramids do not converge as well as those with hexahedra, the pyramids display better convergence properties than tetrahedra.

Fast solution of neutron diffusion problem by reduced basis finite element method

For the important applications which need carry out many times of neutron diffusion calculations such as the fuel depletion analysis and the neutronics-thermohydraulics coupling analysis, fast and accurate solutions of the neutron diffusion equation are demanding but necessary. In the present work, the certified reduced basis finite element method is proposed and implemented to solve the generalized eigenvalue problems of neutron diffusion with variable cross sections. The order reduced model is built upon high-fidelity finite element approximations during the offline stage. During the online stage, both the keff and the spatical distribution of neutron flux can be obtained very efficiently for any given set of cross sections. Numerical tests show that a speedup of around 1100 is achieved for the IAEA two-dimensional PWR benchmark problem and a speedup of around 3400 is achieved for the three-dimensional counterpart with the fission cross-sections, the absorption cross-sections and the scattering cross-sections treated as parameters.

Stability analysis of nonlinear two-grid method for multigroup neutron diffusion problems

We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.

P-multigrid expansion of hybrid multilevel solvers for discontinuous Galerkin finite element discrete ordinate (DG-FEM-SN) diffusion synthetic acceleration (DSA) of radiation transport algorithms

Effective preconditioning of neutron diffusion problems is necessary for the development of efficient DSA schemes for neutron transport problems. This paper uses P-multigrid techniques to expand two preconditioners designed to solve the MIP diffusion neutron diffusion equation with a discontinuous Galerkin (DG-FEM) framework using first-order elements. These preconditioners are based on projecting the first-order DG-FEM formulation to either a linear continuous or a constant discontinuous FEM system. The P-multigrid expansion allows the preconditioners to be applied to problems discretised with second and higher-order elements. The preconditioning algorithms are defined in the form of both a V-cycle and W-cycle and applied to solve challenging neutron diffusion problems. In addition a hybrid preconditioner using P-multigrid and AMG without a constant or continuous coarsening is used. Their performance is measured against a computationally efficient standard algebraic multigrid preconditioner. The results obtained demonstrate that all preconditioners studied in this paper provide good convergence with the continuous method generally being the most computationally efficient. In terms of memory requirements the preconditioners studied significantly outperform the AMG.

New aspects in the implementation of the quasi-static method for the solution of neutron diffusion problems in the framework of a nodal method

The ability to accurately model the dynamic behaviour of the neutron distribution in a nuclear system is a fundamental aspect of reactor design and safety assessment. Due to the heavy computational burden associated to the direct time inversion of the full model, the quasi-static method has become a standard approach to the numerical solution of the nuclear reactor dynamic equations on the full phase space. The present paper is opened by an introductory critical review of the basics of the quasi-static scheme for the general neutron kinetic problem. Afterwards, the implementation of the quasi-static method in the context of a three-dimensional nodal diffusion theory model in hexagonal-z geometry is described, including some peculiar aspects of the adjoint nodal equations and the explicit formulation of the quasi-static nodal equations. The presentation includes the discussion of different formulations of the quasi-static technique. The results presented illustrate the features of the various formulations, highlighting the corresponding advantages and drawbacks. An adaptive procedure for the selection of the time interval between shape recalculations is also presented, showing its usefulness in practical applications.