Analytic Nodal Method Research Articles

An analytical nodal method for solving the SN transport equation in 2-D Cartesian geometry

A novel Analytical Nodal Discrete Ordinates (ANDO) method for the solution of the discrete ordinates (SN) neutron transport equation in Cartesian geometry with isotropic scattering and constant neutron source is presented. A nodal method approximates the multi-dimensional transport equation as a system of coupled one-dimensional transport equations along each coordinate axis through transverse integration. The resulting transverse-integrated equations are then solved separately using a one-dimensional solution technique. The one-dimensional solution utilized in the new ANDO method is based on a recent closed-form analytical solution in slab geometry (WANG and BYAMBAAKHUU, 2017). The novel solution approach allows for a compact closed-form solution on each computational cell. Results for constant and linear transverse leakage approximations are presented. The ANDO method possesses a number of favorable properties, demonstrated with numerical results, such as high accuracy, rapid convergence, asymptotic preserving, positivity preserving, near linear computational complexity, and local-hp adaptivity. The ANDO method can also be easily extended to higher-order transverse leakage approximations, to three-dimensional Cartesian geometry, and to multi-group.

Analytical nodal method for solution of neutron diffusion equation in polar coordinates

The well-known analytic nodal method approach as proposed in earlier work for Cartesian coordinates with constant transverse leakage approximation is adapted to solve the neutron diffusion equation in 2-d polar coordinates. The approach in present work incorporates the recently developed methodology to overcome the problem involved with the transverse integration in θ–direction. Unlike the earlier methods which solve for surface averaged fluxes, the node averaged fluxes are obtained in the present work. The advantage of using node average flux is that it allows the straightforward coupling with mesh centered schemes for thermal hydraulics or other physics. Furthermore, the conventional time marching schemes can be developed in a simplified manner using this approach. The solutions obtained from the current approach are compared with the available analytical and benchmark solutions. The eigenvalues obtained with current methodology is in good agreement with those obtained for analytical and internationally available benchmark solutions.

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.

An analytical nodal method for energy multi-group discrete ordinates transport calculations in two-dimensional rectangular geometry

A spectral nodal method for energy multi-group X, Y-geometry, discrete ordinates (SN) problems in non-multiplying medium is developed. This analytical coarse-mesh method is referred to as the multi-group spectral Green's function - constant nodal (SGF-CN) method. The SGF-CN method uses the multi-group SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. As the energy-group transfer scattering source terms are treated analytically in the offered method, the only approximations occur in the group transverse leakage terms. Numerical results are given to illustrate the method's accuracy for coarse-mesh calculations.

An analytical nodal method for energy multi-group discrete ordinates transport calculations in two-dimensional rectangular geometry

An analytical nodal method for energy multi-group discrete ordinates transport calculations in two-dimensional rectangular geometry

Solution of neutron diffusion equation in 2d polar (r,θ) coordinates using Nodal Integral Method

The nodal methods are significantly more accurate than the traditional methods such as finite difference method (FDM), finite element method (FEM) etc. However, these methods can be used only for the nodes of only a few limited shapes such as rectangular (in 2D) or cuboidal (in 3D). In this paper the approach for solving neutron diffusion equation in 2-d cylindrical polar geometry (r,θ) using nodal method is discussed. The analytic nodal method using transverse integration process is used to solve the 2-d neutron diffusion equation in polar coordinate. The problem involved with the transverse integration in θ-direction has been resolved by approximating the average of products by product of averages. This approximation leads to three different formulations of the scheme. A detailed study of the numerical error for source problems having analytical solutions is carried out. Using this analysis it has been shown that the error is second order for these problems irrespective of the formulation used. In addition to that the methodology is used to solve criticality problems for which analytical or benchmark solutions are available. The solution of the source problem shows that method maintains its accuracy and order even with approximations made to deal with transverse integration in θ-direction. The comparison of eigenvalues obtained with current methodology with those obtained analytically or available as benchmark show that the method is capable of accurately predicting the eigenvalues.

Development of a new analytic function expansion nodal code, HexDANM, for solving the neutron diffusion equation in hexagonal-Z geometry

Abstract In this paper, we developed a new approach of analytic function expansion nodal (AFEN) method to solve the multi-group and multi-dimensional neutron diffusion equation in reactor cores with hexagonal fuel assembly. This method represents a multidimensional intra nodal flux distribution in terms of analytic basis functions at any points in the node. New types of boundary conditions have been considered that constrain the intranodal flux distributions in the hexagonal-z node, which include twelve radial surface-averaged partial currents and two axial surface-averaged partial currents. We utilized the coarse group rebalancing (CGR) method to increase the speed of code calculations. The computer code takes a few-groups cross sections produced by a lattice code and calculates the effective multiplication factor (keff ), flux in multi-group energy, reactivity, and the relative power density at each fuel assembly. Finally, the solution accuracy is tested for two well-known benchmark problems. The numerical results demonstrate that the new AFEN method is an accurate method for calculating keff and power density distribution in hexagonal-z geometries.

Three higher order analytical nodal methods for multigroup neutron diffusion equations

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.

Convergence analysis of two-node CMFD method for two-group neutron diffusion eigenvalue problem

In this paper, the nonlinear coarse-mesh finite difference method with two-node local problem (CMFD2N) is proven to be unconditionally stable for neutron diffusion eigenvalue problems. The explicit current correction factor (CCF) is derived based on the two-node analytic nodal method (ANM2N), and a Fourier stability analysis is applied to the linearized algorithm. It is shown that the analytic convergence rate obtained by the Fourier analysis compares very well with the numerically measured convergence rate. It is also shown that the theoretical convergence rate is only governed by the converged second harmonic buckling and the mesh size. It is also noted that the convergence rate of the CCF of the CMFD2N algorithm is dependent on the mesh size, but not on the total problem size. This is contrary to expectation for eigenvalue problem. The novel points of this paper are the analytical derivation of the convergence rate of the CMFD2N algorithm for eigenvalue problem, and the convergence analysis based on the analytic derivations.

Two group, three dimensional diffusion theory coupled with single heated channel model: A study based on xenon transient simulation of Tehran research reactor

In this work xenon transient analysis of the Tehran research reactor (TRR) core from hot full power without poisoning to xenon saturation is carried out using neutronic and thermal hydraulic coupling. A complete two group, three dimensional neutronic model coupled with one dimensional single phase heat transfer model has been developed and applied to power distribution calculations. We used the WIMS-D5 cell calculation code in order to find a linear relation between important parameters (fuel temperature, moderator temperature and density, and xenon concentration) and group constants of different fuel assemblies. In this way, WIMS running will be bypassed at each time step. We developed and used the ANOMOS code that solves three dimensional diffusion equations using analytic nodal method by transverse leakage approximation in order to find the effective multiplication factor as well as flux and power distributions. Also, we developed and used a SHC code that solves the conservation equations for one dimensional axial homogeneous downward flow through channel using single heated channel model to find the assembly-wise fuel temperature, moderator temperature and density profiles versus generated power. The results have a good agreement with the safety analysis report of the TRR.

Comparison of the calculated neutron noise using finite differences and the Analytical Nodal Method

In this paper, a comparison of the calculated neutron noise, i.e. the fluctuation of the neutron flux around its average value assuming that all processes are stationary, is conducted, where the neutron noise is calculated using finite differences alone and with finite differences where the Analytical Nodal Method is used to correct the neutron currents, respectively. It is seen that the lower the frequency of the noise source, the larger difference between the two solutions. The main conclusion from this work is that the gain of calculating the neutron noise using the more sophisticated Analytical Nodal Method compared to the increase of the corresponding computational burden is too little to motivate the use of the ANM.

Neutron noise calculations using the Analytical Nodal Method and comparisons with analytical solutions

In this study, the neutron noise, i.e. the stationary fluctuations of the neutron flux around its mean value, is calculated in 2-group diffusion theory using the Analytical Nodal Method. A brief description of the calculation of the static flux is also included. The static solution is benchmarked against a reference solution in the case of a homogeneous core. The same calculational scheme for the neutron noise as for the static flux is used. As a dynamical benchmark, the calculated neutron noise for a 2D fully homogeneous reactor is compared with the analytical solution of a centered noise source at different frequencies. The numerical solution is also benchmarked to an off-centered source where the analytical solution is determined using the power reactor approximation, extended to two energy groups. The results of the benchmarks are that the numerical calculations using ANM accurately match the analytical solutions.

A Comparison Between Recent Advances in Cylindrical Nodal Diffusion Methods

The resurgence of high temperature reactor (HTR) technology has prompted the development and application of modern calculation methodologies, many of which are already utilized in the existing power reactor industry, to HTR designs. To this end, the use of nodal diffusion methods for full core neutronic analysis is once again considered for both their performance and accuracy advantages. Recently a number of different approaches to two-dimensional and 3D multigroup cylindrical nodal diffusion methods were proposed by various institutions for use in HTR and, specifically, pebble-bed modular reactor (PBMR) calculations. In this regard, we may mention the NEM code from the Pennsylvania State University based on the nodal expansion method and the OSCAR-4 code from NECSA, utilizing a conformal mapping approach to the analytic nodal method. In this work we will compare these two approaches in terms of accuracy and performance. Representative problems, selected to test the methods thoroughly, were devised and based on both a modified version of the PBMR 400 MW benchmark problem and a “cylindrisized” version of the IAEA two-group problem. The comparative results between OSCAR-4 and NEM are given, focusing on global reactivity estimation, as well as power and flux errors as compared with reference finite-difference solutions. These results indicate that both OSCAR-4 and NEM recover the global reference solution for the IAEA problem and show power errors, which are generally acceptable for nodal methods. For the PBMR problem the accuracy is similar, but some convergence difficulties are experienced at the outer boundaries of the system due to the very large dimensions of the reflector (when compared with typical water-moderated reactors). For both codes a significant performance increase was found, as compared with finite-difference calculations, which is the method currently employed by the PBMR (Pty) Ltd. In conclusion it seems that nodal methods have potential for use in the HTR analysis and, specifically, the PBMR calculational arena, although cylindrical geometry based nodal methods will have to develop toward maturity before becoming the industry standard.

Multi-group unified nodal method with two-group coarse-mesh finite difference formulation

The one-node kernels of the unified nodal method (UNM) which were originally developed for two-group (2 G) problems are extended to solve multi-group (MG) problems within the framework of the 2 G coarse-mesh finite difference (CMFD) formulation. The analytic nodal method (ANM) kernel of UNM is reformulated for the MG application by adopting the Padé approximation to avoid the similarity transform required to diagonalize the G × G buckling matrix. In addition, a one-node semi-analytic nodal method (SANM) kernel which is considered adequate for multi-group calculations is also integrated into the UNM formulation by expressing it in the form consistent with the other UNM kernels. As an efficient global solution framework, the 2 G CMFD formulation with dynamic group condensation and prolongation is established and the performance of the various MG kernels is examined using various static and transient benchmark problems. It turns out that the SANM kernel is the best one for MG problems not only because it retains accuracy comparable to MGANM with a shorter computing time but also because its accuracy or its convergence does not depend on the eigenvalue range of the buckling matrix of the system. The 2 G CMFD formulation with MG one-node UNM kernels turns out to be very effective in that it conveniently accelerates the MG source iteration.

A simplified presentation of the multigroup analytic nodal method in 2-D Cartesian geometry

The nodal diffusion algorithms used in many production reactor simulation codes are originating from a common ancestry developed in the 1970s, the analytic nodal method (ANM) of the QUANDRY code. However, this original presentation of the ANM is complex and makes difficult the calculation of the nodal coupling matrices. Moreover, QUANDRY is limited to two-energy groups and its generalization to more groups appears laborious. We are presenting a simplified implementation of the ANM requiring only limited programming work. This formulation is consistent with the initial QUANDRY implementation and is easily generalizable to arbitrary G-group problems. A Matlab script is provided to highlight the simplicity of our presentation. For the sake of clarity, our implementation is limited to G-group, 2-D Cartesian geometry.

The analytic nodal method in cylindrical geometry

Nodal diffusion methods have been used extensively in nuclear reactor calculations, specifically for their performance advantage, but also for their superior accuracy. More specifically, the Analytic Nodal Method (ANM), utilising the transverse integration principle, has been applied to numerous reactor problems with much success. In this work, a nodal diffusion method is developed for cylindrical geometry. Application of this method to three-dimensional (3D) cylindrical geometry has never been satisfactorily addressed and we propose a solution which entails the use of conformal mapping. A set of 1D-equations with an adjusted, geometrically dependent, inhomogeneous source, is obtained. This work describes the development of the method and associated test code, as well as its application to realistic reactor problems. Numerical results are given for the PBMR-400 MW benchmark problem, as well as for a “cylindrisized” version of the well-known 3D LWR IAEA benchmark. Results highlight the improved accuracy and performance over finite-difference core solutions and investigate the applicability of nodal methods to 3D PBMR type problems. Results indicate that cylindrical nodal methods definitely have a place within PBMR applications, yielding performance advantage factors of 10 and 20 for 2D and 3D calculations, respectively, and advantage factors of the order of 1000 in the case of the LWR problem.

Analytic Function Expansion Nodal (AFEN) Method in Hexagonal-Z Three-Dimensional Geometry for Neutron Diffusion Calculation

The analytic function expansion nodal (AFEN) method in hexagonal-z geometry is described focusing on its unique features, including the use of node-interface flux moments. Multigroup extension based on matrix function theory and coarse group rebalance (CGR) acceleration are also described. The COREDAX code implementing the AFEN method is verified testing on the VVER-440 benchmark problem, a “simplified” VVER-1000 benchmark problem, and the SNR-300 benchmark problem.

Temporal Adaptive Three-Grid Correction Method for Transient Nonlinear Nodal Calculations

As an efficiency enhancement numerical scheme of transient nonlinear nodal calculations, a three-grid correction scheme (3GCS) using a modified W cycle based on three grid structures of three-dimensional (3-D) four-node-per-assembly (4N/A), 3-D 1N/A, and two-dimensional (2-D) 1N/A is developed. Its computational efficiency is compared with a single-grid biconjugate gradient stabilized (BICGSTAB) iteration scheme in popular use in terms of 3-D 4N/A nonlinear analytical nodal method solutions to Nuclear Energy Agency Committee on Reactor Physics pressurized water reactor rod ejection benchmark problems. It is shown that in computational efficiency, the 3GCS excels the BICGSTAB iteration method using preconditioners such as Jacobi, incomplete lower and upper (ILU), and 3-D block incomplete lower and upper (BILU3D) preconditioners. It is also shown that coarse-grid residual equations based on the 3-D 1N/A grid structure can predict temporal truncation errors as accurately as the 3-D 4N/A fine-grid residual equation but with considerably less overhead computing time for variable time-step size control calculations by a step doubling method. In addition, incorporation of preconditioners into the 3GCS is shown to enhance further efficiency of the nonpreconditioned 3GCS. From these results, it is concluded that the temporal adaptive 3GCS employing coarse-grid residual equations for temporal step-size control as well as the preconditioner like the BILU3D can provide a very efficient iterative solution scheme for transient nonlinear nodal calculations.

Unified Nodal Method for Solution to the Space-Time Kinetics Problems

The two popular transverse integrated nodal methods (TINMs), the nodal expansion method (NEM) and analytical nodal method (ANM), and the analytic function expansion nodal (AFEN) method are integrated into a single unified nodal formulation for the space-time kinetics calculations in rectangular core geometry. In particular, the nodal coupling equations of the conventional ANM and AFEN method are reformulated by the matrix function theory based on the unified nodal method (UNM) principle for the solution to the transient two-group neutronics benchmark problems. The difference between the two transient AFEN formulations by the UNM and the conventional AFEN principles is pointed out. The performance of the UNM formulation is examined in terms of the solutions to the transient light water reactor benchmark problems such as the Nuclear Energy Agency Committee on Reactor Physics pressurized water reactor rod ejection kinetics benchmark problems. Through comparison of several nodal computational options by the UNM formulation, it is shown that one node-per-fuel assembly (N/A) calculations by the AFEN method are superior to those by the NEM and the ANM, but that 4 N/A calculations by the AFEN method are not better than those by ANM, in prediction accuracy at the sacrifice of the computational time. The advantages of the transient UNM formulation over the conventional TINM and AFEN method formulations are discussed.

Super-Nodal Methods for Space-Time Kinetics

A Super-Nodal method is developed to improve computational efficiency of core simulations for three-dimensional (3-D) core neutronics models. Computational performance of the neutronics model is increased by reducing the number of spatial nodes used in the core modeling. The Super-Nodal method reduces the errors associated with the use of coarse nodes in the analyses by providing a new set of cross sections and discontinuity factors for the new nodalization. These so-called homogenization parameters are obtained by employing a consistent collapsing technique.During this research a new type of singularity, namely, “fundamental mode singularity,” is addressed in the analytical nodal method solution. The “coordinate shifting” approach is developed as a method to address this singularity. Also, the “buckling shifting” approach is developed as an alternative to address the “zero buckling singularity.” In the course of addressing the treatment of these singularities, an effort was made to provide better and more robust results from the Super-Nodal method by developing several new methods for determining the collapsed diffusion coefficient. A simple error analysis based on the relative residual in the 3-D few-group diffusion equation at the fine mesh level is also introduced in this work.