A nodal SN transport method for three-dimensional triangular-z geometry

A new nodal S_N transport method has been developed to perform accurate transport calculation in three-dimensional triangular-z geometry, where arbitrary triangles are transformed into regular triangles via a coordinate transformation. The transverse integration procedure is applied to treat the neutron transport equation in the regular triangle. The neutron angular distributions of intra-node fluxes are represented using the S_N quadrature set, and the spatial distributions of neutron fluxes and sources are approximated by a quadratic polynomial. The nodal-equivalent finite difference algorithm for 3D triangular geometry is applied to establish a stable and efficient iterative scheme. The present method was tested on four 3D Takeda benchmark problems published by the nuclear data agency (NEACRP), in which the first three problems are in XYZ geometry and the last one is in hexagonal-z geometry. The results of the present method agree well with those of the reference Monte-Carlo calculation method, the difference in k_eff being less than 0.1%. This shows that multi-group reactor core/criticality problems can be accurately and effectively solved using the present method.     

中子输运方程的三角形节块SN方法研究

利用面积坐标思想,将任意三角形变换为正三角形,使用横向积分方法对正三角形节块进行处理.节块内横向积分通量、中子源的空间分布使用新的正交二次多项式近似;横向泄漏项的空间分布使用二阶多项式近似;中子通量和横向泄漏的角度通过离散纵坐标(SN)求积组离散.采用节块平衡有限差分方法建立稳定有效的迭代方案;编制了二维三角形节块SN输运计算程序(DNTR),对一系列基准题进行了验证.结果表明,本方法在同等计算精度下比细网差分程序(DOT4.2)快5～7倍,在同等计算精度和相同节块尺寸下比矩形离散节块输运方法(DNTM)快1～3倍,但DNTR程序可应用于非结构几何区域问题,具有DNTM等其它结构化节块SN程序无可比拟的优势.     

Application of Finite Element Method to Two-Dimensional Multi-Group Neutron Transport Equation in Cylindrical Geometry

The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries.

Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S _n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies.     

Application of Hexagonal Element Scheme in Finite Element Method to Three-Dimensional Diffusion Problem of Fast Reactors

The finite element method is applied in Galerkin-type approximation to three-dimensional neutron diffusion equations of fast reactors. A hexagonal element scheme is adopted for treating the hexagonal lattice which is typical for fast reactors. The validity of the scheme is verified by applying the scheme as well as alternative schemes to the neutron diffusion calculation of a gas-cooled fast reactor of actual scale. The computed results are compared with corresponding values obtained using the currently applied triangular-element and also with conventional finite difference schemes.

The hexagonal finite element scheme is found to yield a reasonable solution to the problem taken up here, with some merit in terms of saving in computing time, but the resulting multiplication factor differs by 1% and the flux by 9% compared with the triangular mesh finite difference scheme. The finite element method, even in triangular element scheme, would appear to incur error in inadmissible amount and which could not be easily eliminated by refining the nodes.     

The Davidson method as an alternative to power iterations for criticality calculations

The Davidson method is implemented within the neutron transport core solver parafish to solve k-eigenvalue criticality transport problems. The parafish solver is based on domain decomposition. It uses spherical harmonics (P_N method) for angular discretization, and non-conforming finite elements for spatial discretization. The Davidson method is compared to the traditional power iteration method in this context.     

三维六角形节块多群中子扩散程序NDHEX

本文介绍用DIF3D (NOD)求解二、三维六角形几何系统下中子扩散方程的理论模型及数值计算方法。六角形节块内的中子通量密度分布采用高次多项式近似表示,最后导出通量矩方程及偏流的响应矩阵方程。应用粗网再平衡和渐近源外推方法加速收敛。参考此方法编制了计算程序NDHEX,并对一些六角形基准问题进行了计算。结果表明:NDHEX的计算结果与DIF3D(NOD)的计算结果符合很好;与差分程序相比,具有更高的精度与计算效率。它可用于快堆计算。     

用于求解细网SP3中子输运方程的两节块方法的精度与效率分析

采用两节块方法求解细网3阶简化球谐函数(SP3)中子输运方程,该方法只对零阶角通量密度的拉普拉斯算子进行节块法处理,对应的零阶通量密度采用2阶展开,横向泄漏采用零阶近似;以此方法开发了适用于细网全堆输运计算的CORCA-PIN程序,该程序同时集成了细网有限差分方法。验证算例采用KAIST 3A基准问题及扩展三维问题。数值结果表明,采用栅元1×1划分的两节块法具有可接受的计算精度,而计算时间只有相同精度的细网有限差分方法的11%。因此,本文提出的两节块方法适用于细网SP3中子输运方程计算。     

Application of the response matrix method to BWR lattice analysis

A lattice calculation code RESPLA has been developed for light-water reactor lattices on the basis of the response matrix method treating the heterogeneity in pin cells. The spatial dependency of neutron flux distribution along each cell boundary is taken into account by dividing the cell boundary into several subsurfaces and the anisotropy of neutron angular distribution is considered up to the P₁ component by using a relation between the P₀ and P₁ components. The RESPLA code has been applied to BWR lattice calculations and the calculational results have been compared with those obtained by the S_n method and the collision probability method. It has been found that the present response matrix method has the same accuracy as the collision probability method with fine spatial meshes and the error caused by the use of coarse meshes is much smaller than that by the collision probability method. Furthermore, the required computing time is smaller by about a factor of five than that in the collision probability method.     

Detected-neutron multiplication factor measured by neutron source multiplication method

An alternative definition of neutron multiplication factor measured by the neutron source multiplication (NSM) method is newly proposed. This newly defined neutron multiplication factor, k_det, is derived on the basis of neutron detection process in a subcritical system with an external neutron source. The definition of k_det is expressed as a ratio of total number of detected fission-neutrons to total number of detected all neutrons. In this paper, a heuristic derivation of k_det is presented, and another interpretation of k_det is explained by using the detector importance function. Based on the idea of k_det, the measurement principle of NSM method is reinterpreted, and the correction factors in the NSM method are clarified. In order to verify our proposed NSM method, numerical analysis of the NSM method is carried out. The numerical results suggest that target neutron multiplication factors of the NSM method can be well estimated even without any corrections by putting a neutron detector where the effective neutron multiplication factor k_eff is well approximated by k_det.     

A new spectral nodal method based on Larsen’s extended diamond approach

In this article, we describe a new spectral nodal method for solving discrete ordinates (S_N) neutron transport problems with anisotropic scattering for arbitrary order N of angular quadrature. The key to our new spectral nodal method is a consistent derivation of nonstandard auxiliary equations that relate angular neutron fluxes only in the upwind directions. These nonstandard equations are angularly coupled extensions of very basic auxiliary equations proposed by Edward W. Larsen in his extended diamond scheme of solving S₂ problems in the presence of scattering and free from spatial truncation error. The resulting method here is also free from spatial truncation error and, in contrast to previously developed spectral nodal methods, it is compatible with an efficient use of iteration on the scattering source and is free from the storage of cell-edge angular fluxes.     

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry. Part I: Modeling and verification for time-independent solution

This paper describes a novel method based on using cellular neural networks (CNN) coupled with spherical harmonics method (P_N) to solve the time-independent neutron transport equation in x–y geometry. To achieve this, an equivalent electrical circuit based on second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. We use the CNN model to simulate spatial response of scalar flux distribution in the steady state condition for different order of spherical harmonics approximations. The accuracy, stability, and capabilities of CNN model are examined in 2D Cartesian geometry for fixed source and criticality problems.     

Analysis of an a posteriori error estimator for the transport equation with SN and discontinuous Galerkin discretizations

We present an error estimator for the S_N neutron transport equation discretized with an arbitrary high-order discontinuous Galerkin method. As a starting point, the estimator is obtained for conforming Cartesian meshes with a uniform polynomial order for the trial space then adapted to deal with non-conforming meshes and a variable polynomial order. Some numerical tests illustrate the properties of the estimator and its limitations. Finally, a simple shielding benchmark is analyzed in order to show the relevance of the estimator in an adaptive process.     

Application of the TN method to critical slab problem for one-speed neutrons with forward and backward scattering

The critical slab problem which includes isotropic forward and backward scattering has been studied in one-speed neutron transport equation using first kind of Chebyshev polynomials. The critical half-thicknesses are computed for different degrees of c and forward and backward scattering with Mark and Marshak boundary conditions in the uniform finite slab. It is shown that T_N method gives accurate results in one-dimensional geometry and the results are agreement P_N approximation.     

A spherical harmonics—Finite element discretization of the self-adjoint angular flux neutron transport equation

The spherical harmonics (P_N) method is widely used in solving the neutron transport equation, but it has some disadvantages. One of them comes from the complexity of the P_N equations. Another one comes from the difficulty of dealing with the vacuum boundary condition exactly. In this paper, the P_N method is applied to the self-adjoint angular flux (SAAF) neutron transport equation and a set of P_N moments equations coupled with each other are obtained. An iterative method is utilized to decouple them and solve them moment by moment. The corresponding vacuum boundary condition is derived based on the Marshak boundary condition. The spatial variables are discretized on unstructured-meshes by use of the finite element method (FEM). The numerical results of several problems demonstrate that this method can provide high precision results and avoid the ray effect, which appears in the discrete ordinate (S_N) method, with relatively high computational efficiency.     

Basic Characteristics of Centrifuges, (III)

A method of numerically integrating the Navier-Stokes equations is presented for axisymmetric compressible flows. A modified Newton's method is employed to determine the steady motion of gas in a rotating cylinder without the use of a time-consuming marching process with respect to time. A suitable form of the finite difference equations gives a computationally-stable integration with reasonable representation of the spatial characteristics of the flow. The method includes a Gaussian elimination procedure which consists of the transformation of the Jacobian matrix to a triangular matrix followed by the backward substitution. By using an auxiliary constant matrix algorithm, the method gives the solution within reasonably acceptable computation time.

As an example of the method, some features of solutions are presented for the steady flow of UF₆ gas in the centrifuges which have the openings for feed and withdrawal on the end plates.     

Space asymptotic method for the study of neutron propagation

Space asymptotic theory is shown to be a suitable model for the study of pulsed experiments in neutron multiplying systems. After a short revisitation of the basic aspects of space asymptotic theory applied on the Laplace transformed one-group transport equation, the full solution is derived. It is shown how results are exact in representing localized pulse propagation in the first portion of the transient, until the boundary is reached by the neutron signal, since it propagates with a finite velocity. Approximate models are then derived starting from the exact formulation and the B_N method is used to account for anisotropy effects. Numerical results are presented for one-dimensional systems, discussing the physical phenomena and noting the distortions introduced by approximate models, which may then turn out to be inadequate for the simulation of realistic pulsed experiments situations.     

Calculation of extrapolation distances by low order Chebyshev polynomial approximation of transport equation in slab geometry

In this study, the problem of extrapolated end point has been studied in one-speed neutron transport equation with isotropic scattering by using the Chebyshev polynomial approximation which is called T_N method. Assuming neutrons of one speed, extrapolated end point are calculated for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. It is shown that low order T_N method gives very good results of low order spherical harmonics approximation and diffusion theory for extrapolation of the flux of neutrons leaking from the medium. We present an alternative method which is similar to P₁ method to calculate the extrapolation distances z₀. Moreover, we prefer new solution of transport equation in one-dimensional slab geometry.     

Higher order analytical nodal methods in response-matrix formulation for the multigroup neutron diffusion equations

A higher analytical nodal method for the multigroup neutron diffusion equations, based on the transverse integration procedure, is presented. The discrete 1D equations are cast with the interface partial current techniques in response matrix formalism. The remaining Legendre coefficients of the transverse leakage moment are determined exactly in terms of the different neutron flux moments order in the reference node. In the weighted balance equations, the transverse leakage moments are linearly written in terms of the partial currents, facial and centered fluxes moments. The self-consistent is guaranteed. Furthermore, as the order k increase the neutronic balance in each node and the copulate between the adjacent cell are reinforced. The convergence order in L²-norm is of O(h^k+3−δ_k0) under smooth assumptions. The efficacy of the method is showed for 2D-PWR, 2D-IAEA LWR and 2D-LMFBR benchmark problems.     

A multigroup finite-element solution of the neutron transport equation—I: X-Y geometry

The result of extending a variational finite element method of solving the neutron transport equation, to energy dependence, is reported. Detailed results are given, in the form of tables and graphs, of P₁ and higher-order transport solutions to a number of benchmark problems in X-Y geometry. The accuracy and flexibility of the method are demonstrated. Some suggestions are made for the future development of the computer implementation of the method.     

Evaluation of dominant time-eigenvalues of neutron transport equation by Meyer’s sub-space iterations

The aim of this paper is to explore the use of Meyer’s sub-space iteration (SSI) method for the evaluation of dominant prompt time-eigenvalues of the neutron transport equation. The integro-differential form of the transport equation is considered. The SSI method is known to be an efficient technique to find the dominant eigenvalues of a non-symmetric matrix. It has been earlier used for eigenvalue problems in neutron diffusion theory. However, it does not seem to be tried in the transport theory case. Here, the use of SSI has been tested in transport theory for some 1-D mono-energetic homogeneous and heterogeneous benchmark problems. The space variable is discretised by finite differencing while neutron directions are discretised by discrete ordinates (S_n-) method. The SSI method needs frequent multiplication of the relevant matrix operator with vectors. As known from earlier works in this area, this can be achieved in terms of external source calculations for which a 1-D programme was developed and used. With the availability of more versatile S_n-method codes, it may perhaps be possible to extend use of SSI to more realistic cases.