Analytic Function Expansion Nodal Research Articles

Time-dependent neutronics model of nodal neutronics program Ants

Ants is a nodal neutronics program developed at VTT since 2017. Ants solution method for the nodal diffusion equation is based on the function expansion nodal method (FENM) and analytic function expansion nodal (AFEN) method. Ants supports rectangular, hexagonal and triangular nodal geometries. The solution methodology of Ants extended to solve time-dependent problems is presented in this work. The performance of Ants is presented for all three geometry options in a series of established two- and three-dimensional numerical time-dependent benchmarks for both rectangular and hexagonal lattices. The Ants results agree with other published nodal program results, especially if the problem is not sensitive to control rod cusping effects or a fine enough axial mesh is used.

A response matrix method for the refined Analytic Function Expansion Nodal (AFEN) method in the two-dimensional hexagonal geometry and its numerical performance

In order to improve calculational efficiency of the CAPP code in the analysis of the hexagonal reactor core, we have tried to implement a refined AFEN method with transverse gradient basis functions and interface flux moments in the hexagonal geometry. The numerical scheme for the refined AFEN method adopted here is the response matrix method that uses the interface partial currents as nodal unknowns instead of the interface fluxes used in the original AFEN method. Since the response matrix method is single-node based, it has good properties such as good calculational efficiency and parallel computing affinity. Because a refined AFEN method equivalent nonlinear FDM response matrix method tried first could not provide a numerically stable solution, a direct formulation of the refined AFEN response matrix were developed. To show the numerical performance of this response matrix method against the original AFEN method, the numerical error analyses were performed for several benchmark problems including the VVER-440 LWR benchmark problem and the MHTGR-350 HTGR benchmark problem. The results showed a more than three times speedup in computing time for the LWR and HTGR benchmark problems due to good convergence and excellent calculational efficiency of the refined AFEN response matrix method.

Analytic function expansion nodal (AFEN) method for solving multigroup neutron simplified P3 (SP3) equations in hexagonal-z geometry

In this paper, analytic function expansion nodal (AFEN) method was developed to solve multi-group and multi-dimensional neutron simplified P3 (SP3) equation in reactor cores with hexagonal fuel assembly. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group and Legendre moment. Fourteen boundary conditions has 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. The code takes 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 the two and three dimensional IAEA benchmark problem. The 3D IAEA problem has been calculated for three different cases. The numerical results demonstrate that the SP3 AFEN method exhibits better accuracy compared to diffusion method especially when the control materials are inserted in the core.

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.

New analytic function expansion nodal (AFEN) method for solving multigroup neutron simplified P3 (SP3) equations

In this study, a new analytic function expansion nodal (AFEN) method was developed to solve multi-group and three dimensional neutron simplified P3 equations (SP3) in reactor cores with rectangular fuel assemblies. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group and moment. The nodes are coupled through the surface averaged partial currents at each nodal interface. Thus, six boundary conditions at each group and Legendre moments have been considered. Coarse group rebalancing (CGR) method was applied to increase the speed of code calculations. The code takes few-groups cross sections produced by a lattice code such as WIMS and calculates the effective multiplication factor, zeroth and second moments of the flux in multi-group energy, reactivity, and the relative power density at each fuel assembly. The numerical results for different benchmark problems demonstrate that solution of SP3 equations by our AFEN method improves both effective multiplication factor (keff) and power distribution compared to our AFEN diffusion method, especially in heterogeneous geometry and mixed-oxide (MOX) fuel problems.

Analytic Function Expansion Nodal Method for Multigroup Diffusion Equations in Cylindrical (r,θ,z) Geometry

A coarse-mesh nodal method in cylindrical (r, θ,z) geometry, e.g., of pebble bed reactors, based on the analytic function expansion nodal (AFEN) methodology, is described in this paper. Two unique features are (a) no use of transverse integration—allowing a nodal scheme in (r, θ,z) geometry—and (b) nodal solution expressed in terms of analytic basis functions—leading to high accuracy and readily available reconstruction of homogeneous flux distributions. Additional features of multigroup formulation, two methods of void region treatment, and coarse-group-rebalance acceleration are implemented in the TOPS code and tested on several benchmark problems, including the Organisation for Economic Co-operation and Development/Nuclear Energy Agency PBMR-400 Benchmark Problem. The TOPS results are in excellent agreement with those of the VENTURE code, using significantly less computer time.

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.

AFEN method and its solutions of the hexagonal three-dimensional VVER-1000 benchmark problem

The unique features of the analytic function expansion nodal (AFEN) method in hexagonal- z geometry are described. The COREDAX code implementing the AFEN method is verified testing on the VVER-440 benchmark problem and a “simplified” VVER-1000 benchmark problem. The COREDAX code then applied to the original VVER-1000 benchmark problem exercise 2 (HZP case and HP case) provides very good results in comparison with those of other benchmark participants.

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.

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.

The Analytic Function Expansion Nodal (AFEN) Method with Half-Interface Averaged Fluxes in Mixed Geometry Nodes for Analysis of Pebble Bed Modular Reactor (PBMR) Cores

The analytic function expansion nodal (AFEN) method has been successfully applied to the rectangular and hexagonal geometries in the cartesian coordinates system. In this paper, we extended the AFEN method to the cylindrical geometry in the R-Z coordinates for the analysis of pebble bed modular reactors (PBMRs). To treat the mixed geometry of rectangular and triangular nodes appearing in the lower periphery of the reactors, we used half-interface averaged fluxes as nodal unknowns. Numerical results obtained attest to their accuracy and applicability to practical problems.

The Analytic Function Expansion Nodal (AFEN) Method with Half-Interface Averaged Fluxes in Mixed Geometry Nodes for Analysis of Pebble Bed Modular Reactor (PBMR) Cores

The analytic function expansion nodal (AFEN) method has been successfully applied to the rectangular and hexagonal geometries in the cartesian coordinates system. In this paper, we extended the AFEN method to the cylindrical geometry in the R-Z coordinates for the analysis of pebble bed modular reactors (PBMRs). To treat the mixed geometry of rectangular and triangular nodes appearing in the lower periphery of the reactors, we used half-interface averaged fluxes as nodal unknowns. Numerical results obtained attest to their accuracy and applicability to practical problems.

Kinetics Calculation Under Space-Dependent Feedback in Analytic Function Expansion Nodal Method via Solution Decomposition and Galerkin Scheme

To develop kinetics calculational capability of the analytic function expansion nodal methodology for space-dependent feedback problems, a novel method with the time-dependent solution decomposed into an analytic part and a polynomial correction part is proposed. The analytic part consists of the analytic solutions of the “quasi-static” diffusion equation and the polynomial part is determined by applying a Galerkin scheme. The results tested on several benchmark problems (two-dimensional and three-dimensional) show that 1 node/assembly calculation and a large time-step size can be used for high accuracy. The new feedback calculation method removes almost all the errors induced from space-dependent feedback. Also, it is shown that the coarse group rebalance acceleration scheme and conventional techniques for kinetics calculation (exponential transformation for time variable and bilinear weighting for control rod cusping problem) can be easily incorporated into the method.

Unified Nodal Method Formulation for Analytic Function Expansion Nodal Method Solution to Two-Group Diffusion Equations in Rectangular Geometry

The analytic function expansion nodal (AFEN) method formulation for the solution to two-group diffusion equations in rectangular geometry is reformulated in the principle of the unified nodal method (UNM) formulation. Except for the corner point neutron balance equations, the nodal coupling relations of the reformulated AFEN method are shown to resemble exactly those of the nodal expansion method (NEM) so that they not only can be easily incorporated into the existing NEM production codes but also can enable one to make the most of the well-established numerical solution schemes including the nonlinear coarse-mesh finite difference (CMFD) schemes for speedy AFEN method calculations. A one-node CMFD scheme for the speedy AFEN calculations of the UNM formulation is newly proposed. The effectiveness of the one-node scheme is compared with that of the two-node CMFD scheme in terms of UNM solutions to the International Atomic Energy Agency and Organization for Economic Cooperation and Development L336 neutronics benchmark problems. Advantages of the UNM formulation for the AFEN method calculations over the original AFEN method formulation are discussed.

The Analytic Function Expansion Nodal Method Refined with Transverse Gradient Basis Functions and Interface Flux Moments

A refinement of the analytic function expansion nodal (AFEN) method is described. By increasing the number of flux expansion terms in the way that the original basis functions are combined with the transverse-direction linear functions, the refined AFEN method can describe the flux shape in the nodes more accurately, since the added flux expansion terms still satisfy the diffusion equation. The additional nodal unknowns introduced are the interface flux moments, and the additional constraints required are provided by the continuity conditions of the interface flux moments and the interface current moments. Also presented is an algebraically exact method for removing the numerical singularity that can occur in any analytic nodal method when the core contains nearly no-net-leakage nodes. The refined AFEN method was tested on the Organization for Economic Cooperation and Development (OECD)-L336 mixed-oxide benchmark problem in rectangular geometry, and the VVER-440 benchmark problem and a nearly no-net-leakage node embedded core problem, both in hexagonal geometry. The results show that the method improves not only the accuracy in predicting the flux distribution but also the computing time, and that it can replace the corner-point fluxes with the interface flux moments without accuracy degradation, unless the problem consists of strongly dissimilar nodes. The possibility of excluding the corner-point fluxes increases the flexibility in implementing this method into the existing codes that do not have the corner-point flux scheme and may make it fit better for the nonlinear scheme based on two-node problems.

Acceleration of the Analytic Function Expansion Nodal Method by Two-Factor Two-Node Nonlinear Iteration

A nonlinear iterative scheme is developed to reduce the computing time of the Analytic Function Expansion Nodal (AFEN) method and is applied to three test problems, including a mixed-oxide fuel problem. The new nonlinear scheme is based on solving two-node problems and using two nonlinear correction factors at every interface instead of one factor, as in the conventional scheme. The use of two correction factors provides higher-order accurate interface fluxes as well as currents, which are used as the boundary conditions of the two-node problem. The numerical results show that this new nonlinear scheme reproduces the same solution as that of the original AFEN method and that the computing time is significantly reduced in comparison with the original AFEN method.

Higher order polynomial expansion nodal method for hexagonal core neutronics analysis

A higher-order polynomial expansion nodal(PEN) method is newly formulated as a means to improve the accuracy of the conventional PEN method solutions to multi-group diffusion equations in hexagonal core geometry. The new method is applied to solving various hexagonal core neutronics benchmark problems. The computational accuracy of the higher order PEN method is then compared with that of the conventional PEN method, the analytic function expansion nodal (AFEN) method, and the ANC-H method. It is demonstrated that the higher order PEN method improves the accuracy of the conventional PEN method and that it compares very well with the other nodal methods like the AFEN and ANC-H methods in accuracy.

Extension of Analytic Function Expansion Nodal Method to Multigroup Problems in Hexagonal-Z Geometry

The analytic function expansion nodal (AFEN) method has been successfully applied to two-group neutron diffusion problems. However, the current AFEN method cannot treat complex eigen-modes, which appear in the general multigroup equations. The AFEN method is extended such that complex eigenmodes are treated within the framework of the original AFEN method for any type of geometry. Also, a suite of new nodal codes based on the extended AFEN theory is developed for hexagonal-z geometry and applied to several benchmark problems. Numerical results obtained attest to their accuracy and applicability to practical problems.

Analytic Function Expansion Nodal Method for Hexagonal Geometry

A new hexagonal nodal method that directly solves the multidimensional diffusion equation without the transverse integration procedure is described. The new method expands the homogeneous flux dist...