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.     

Two-dimensional nodal transport method for triangular geometry

The advanced nodal method for solving the multi-group neutron transport equation in two-dimensional triangular geometry is developed. To apply the transverse integration procedure, an arbitrary triangular node is transformed into a regular triangular node using coordinate transformation. The angular distributions of intra-node neutron fluxes and its transverse-leakage are represented by the S_N quadrature set. The spatial distributions of neutron flux and source in the regular triangle are given approximately by an orthogonal quadratic polynomial, and the spatial expansion of transverse-leakage is approximated by a second-order polynomial. To establish a stable and efficient iterative scheme, the improved nodal-equivalent finite difference algorithm is used. The results for several benchmark problems demonstrate the higher capability of the method to yield the accurate results in significantly smaller computing times than those required by the standard finite difference method and the finite element spherical-harmonics method.     

Finite element spherical harmonics (PN) solutions of the three-dimensional Takeda benchmark problems

A set of multi-group eigenvalue (K_eff) benchmark problems in three-dimensional homogenised reactor core configurations have been solved using the deterministic finite element transport theory code EVENT and the Monte Carlo code MCNP4C. The principal aim of this work is to qualify numerical methods and algorithms implemented in EVENT. The benchmark problems were compiled and published by the Nuclear Data Agency (OECD/NEACRP) and represent three-dimensional realistic reactor cores which provide a framework in which computer codes employing different numerical methods can be tested. This is an important step that ought to be taken (in our view) before any code system can be confidently applied to sensitive problems in nuclear criticality and reactor core calculations. This paper presents EVENT diffusion theory (P₁) approximation to the neutron transport equation and spherical harmonics transport theory solutions (P₃–P₉) to three benchmark problems with comparison against the widely used and accepted Monte Carlo code MCNP4C. In most cases, discrete ordinates transport theory (S_N) solutions which are already available and published have also been presented. The effective multiplication factors (K_eff) obtained from transport theory EVENT calculations using an adequate spatial mesh and spherical harmonics approximation to represent the angular flux for all benchmark problems have been estimated within 0.1% (100 pcm) of the MCNP4C predictions. All EVENT predictions were within the three standard deviation uncertainty of the MCNP4C predictions. Regionwise and pointwise multi-group neutron scalar fluxes have also been calculated using the EVENT code and compared against MCNP4C predictions with satisfactory agreements. As a result of this study, it is shown that multi-group reactor core/criticality problems can be accurately solved using the three-dimensional deterministic finite element spherical harmonics code EVENT.     

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.     

Effect of Polynomial Expansion Order of Intranode Flux Treatment in Nodal SN Transport Calculation Code NSHEX for Large-Size Fast Power Reactor Core Analysis

The nodal discrete ordinates (S_N) transport calculation code for three-dimensional hexagonal geometry NSHEX treats intranode flux distribution using a polynomial series and considers the angular dependence of flux by the SN method. For the improvement of calculation accuracy of NSHEX for practical use to large-size fast reactor plants, the maximum order of the polynomial series is extended from two to six. In order to check the effect of the polynomial expansion order, NSHEX is applied to the intermediate-size fast power reactor core “Monju” and the large-size one “Super Phenix,” including various control rod insertion conditions. From the application, it is found that extension of the polynomial expansion order is effective especially for the large-size core “Super Phenix” under the control-rod-inserted condition.     

A closed-form solution for the two-dimensional transport equation by the LTSN nodal method in the energy range of Compton effect

In the present work we report on a closed-form solution for the two-dimensional Compton transport equation by the LTS_N nodal method in the energy range of Compton effect. The solution is determined using the LTS_N nodal approach for homogeneous and heterogeneous rectangular domains, assuming the Klein–Nishina scattering kernel and a multi-group model. The solution is obtained by two one-dimensional S_N equation systems resulting from integrating out one of the orthogonal variables of the S_N equations in the rectangular domain. The leakage angular fluxes are approximated by exponential forms, which allows to determine a closed-form solution for the photons transport equation. The angular flux and the parameters of the medium are used for the calculation of the absorbed energy in rectangular domains with different dimensions and compositions. In this study, only the absorbed energy by Compton effect is considered. We present numerical simulations and comparisons with results obtained by using the simulation platform GEANT4 (version 9.1) with its low energy libraries.     

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.     

Axial SPN and Radial MOC Coupled Whole Core Transport Calculation

The Simplified P_N (SP_N) method is applied to the axial solution of the two-dimensional (2-D) method of characteristics (MOC) solution based whole core transport calculation. A sub-plane scheme and the nodal expansion method (NEM) are employed for the solution of the one-dimensional (1-D) SP_N equations involving a radial transverse leakage. The SP_N solver replaces the axial diffusion solver of the DeCART direct whole core transport code to provide more accurate, transport theory based axial solutions. In the sub-plane scheme, the radial equivalent homogenization parameters generated by the local MOC for a thick plane are assigned to the multiple finer planes in the subsequent global three-dimensional (3-D) coarse mesh finite difference (CMFD) calculation in which the NEM is employed for the axial solution. The sub-plane scheme induces a much less nodal error while having little impact on the axial leakage representation of the radial MOC calculation. The performance of the sub-plane scheme and SP_N nodal transport solver is examined by solving a set of demonstrative problems and the C5G7MOX 3-D extension benchmark problems. It is shown in the demonstrative problems that the nodal error reaching upto 1,400 pcm in a rodded case is reduced to 10pcm by introducing 10 sub-planes per MOC plane and the transport error is reduced from about 150pcm to 10pcm by using SP₃. Also it is observed, in the C5G7MOX rodded configuration B problem, that the eigenvalues and pin power errors of 180 pcm and 2.2% of the 10 sub-planes diffusion case are reduced to 40 pcm and 1.4%, respectively, for SP₃ with only about a 15% increase in the computing time. It is shown that the SP₅ case gives very similar results to the SP₃ case.     

A nodal collocation method for the calculation of the lambda modes of the PL equations

P_L equations are classical approximations to the neutron transport equation admitting a diffusive form. Using this property, a nodal collocation method is developed for the P_L approximations, which is based on the expansion of the flux in terms of orthonormal Legendre polynomials. This method approximates the differential lambda modes problem by an algebraic eigenvalue problem from which the fundamental and the subcritical modes of the system can be calculated. To test the performance of this method, two problems have been considered, a homogeneous slab, which admits an analytical solution, and a seven-region slab corresponding to a more realistic problem.     

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.     

Transport Calculations of MOX and UO2 Pin Cells by the Method of Characteristics

A new transport theory code for two-dimensional calculations of both square and hexagonal fuel lattices by the method of characteristics has been developed. The ray tracing procedure is based on the macroband method, which permits more accurate spatial integration in comparison to the equidistant method of tracing. The neutron source within each region is approximated by a linear function and linearly anisotropic scattering can be optionally accounted for. Efficient new techniques for both azimuthal and polar integration are presented. The spatial discretization problem in case of P ₁-scattering has been studied. Detailed analyses show that the P ¹-scattering in case of regular infinite array of fuel cells is significant, especially for MOX fuel, while the transport correction is inadequate in case of real geometry multi-group calculations. Finally, the complicated nature of the angular flux in MOX and UO₂ fuel cells is demonstrated.     

An exponential spectral nodal method for one-speed X,Y-geometry deep penetration discrete ordinates problems

Presented here is an exponential spectral nodal method applied to deep penetration X,Y-geometry heterogeneous neutron transport problems in the discrete ordinates (S_N) formulation. This numerical method uses the spectral Green's function (SGF) scheme for solving the one-dimensional transverse-integrated S_N exponential nodal equations with no spatial truncation error. Based on the physics of deep penetration problems, we approximate the transverse leakage terms by exponential functions. We show in two numerical experiments that the SGF-exponential nodal method (SGF-ExpN) generates very accurate results when compared to the conventional transport nodal methods for coarse-mesh deep penetration S_N problems, specially in highly absorbing media.     

Analytical reconstruction scheme for the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates transport model in slab geometry

Coarse-mesh numerical methods are very efficient in the sense that they generate accurate results in short computational time, as the number of floating point operations generally decrease, as a result of the reduced number of mesh points. On the other hand, they generate numerical solutions that do not give detailed information on the problem solution profile, as the grid points can be located considerably away from each other. In this paper we describe two steps for the analytical reconstruction of the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates (S_N) transport model in slab geometry. The first step of the algorithm is based on the analytical reconstruction of the coarse-mesh solution within each discretization cell of the grid set up on the spatial domain. The second step is based on the angular reconstruction of the discrete ordinates solution between two contiguous ordinates of the angular quadrature set used in the S_N model. Numerical results are given so we can illustrate the accuracy of the two reconstruction techniques, as described in this paper.     

An Approximate Calculation Method of the λ, ω p and ω d Eigenvalue Problem of the Group Diffusion Equation

The approximate solutions of the, λ, ω _p and ω _d eigenvalue problems of the group-diffusion equation for a multi-region reactor are obtained by expanding neutron fluxes into finite numbers of eigenfunctions satisfying the Helmholtz equation and the boundary condition at the extrapolated boundary of the reactor. The original eigenvalue problem is reduced to that of an asymmetric real matrix for the vector whose components constitute the expansion coefficients. For the numerical calculation of the real matrix thus derived, to determine the higher λ, ω _p and ω _d modes, the QR iteration method based on numerically stable unitary transformation, in combination with inverse iteration is effective in saving computation time.

The λ, ω _p and ω _d modes obtained by the above method are expressed by a linear combination of a comparatively small number of simple elementary functions, and are thus of high practical value in the numerical calculation of higher order perturbations and for examining snace-deDendent reactor dynamics.     

A high-order discontinuous Galerkin method for the SN transport equations on 2D unstructured triangular meshes

This paper presents high-order numerical solutions to the _SN$S_N$ transport equation on unstructured triangular meshes using a Discontinuous Galerkin Finite Element Method (DGFEM). Hierarchical basis functions, up to order 4, are used for the spatial representation of the solution. Numerical results are provided for source-driven and eigenvalue problems. Convergence rates (as a function of the mesh size and CPU time) are discussed.     

Analysis of Cavity Effect on Space- and Time-Dependent Fast and Thermal Neutron Energy Spectra

The effects of the presence of a central cavity on the space- and time-dependent neutron energy spectra in both thermal and fast neutron systems are analyzed theoretically with use made of the multi-group one-dimensional time-dependent S_n method. The thermal neutron field is also analyzed for the case of a fundamental time eigenvalue problem with the time-dependent P₁ approximation. The cavity radius is variable, and the system radius for graphite is 120 cm and for the other materials 7 cm.

From the analysis of the time-dependent S_n calculations in the non-multiplying systems of polythene, light water and graphite, cavity heating is the dominant effect for the slowing-down spectrum in the initial period following fast neutron burst, and when the slowing-down spectrum comes into the thermal energy region, cavity heating shifts to cavity cooling. In the multiplying system of ²³⁵U, cavity cooling also takes place as the spectrum approaches equilibrium after the fast neutron burst is injected.

The mechanism of cavity cooling is explained analytically for the case of thermal neutron field to illustrate its physical aspects, using the time-dependent P₁ approximation. An example is given for the case of light water.     

Wavelet-based angular dependence analysis of the heterogeneous calculation on MOX fuel lattice

A wavelet-based transport method is developed to satisfy the high order angular approximation, which has been proved to be necessary in the heterogeneous calculation of MOX fuel lattice. Based on the new angular discretization scheme, the angular dependence of flux is analysed to find out the origin of complicated angular anisotropy and its effects on the heterogeneous calculation. Both of the geometric and neutronic effects are investigated quantitatively to find out the angular dependence in heterogeneous calculations. Comparisons between the traditional S_N angular discretization scheme and wavelet-based scheme are analysed to indicate the challenges brought from the MOX fuel lattice heterogeneous calculation. An effective solution is given by using wavelets in the angular discretization of neutron transport equation. Improvements of high order angular approximation are suggested.     

An improved three-dimensional wavelet-based method for solving the first-order Boltzmann transport equation

A new angular discretization scheme based on the Daubechies’ wavelets has been developed in recent studies. A decoupled S_N and wavelet expansion method was proposed. This paper discusses the limitations and improvements of this decoupled scheme. The scaling function, instead of the wavelet function, is applied as the basis function. It significantly improved the efficiency and computational stability. A new series of wavelets on the interval are applied instead of the ‘wrapped wavelets’, which eliminate the edge effect in the angular subdomain scheme. Based on the improvements, a wavelet-based neutron transport code package WAVTRAN is developed and the previous work is extended to the three-dimensional calculation and anisotropic scattering calculation. Numerical results demonstrate that the improvements are effective. Further investigations demonstrate that the wavelet-based angular discretization scheme is more powerful than the traditional ones in some highly anisotropic angular flux problems.     

Multi-group pin power reconstruction method based on colorset form functions

A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-analytic basis functions,and a colorset based form function generating method is proposed,which can accurately model the spectral interaction occurring at assembly interface.To demonstrate its accuracy and applicability to realistic problems,the new method is tested against two benchmark problems,including a mixed-oxide fuel problem.The results show that the new method is comparable in accuracy to fine-mesh methods.     

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 (2G) problems are extended to solve multi-group (MG) problems within the framework of the 2G 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 2G 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 2G CMFD formulation with MG one-node UNM kernels turns out to be very effective in that it conveniently accelerates the MG source iteration.