Development of Discrete Ordinates SN Code in Three-Dimensional (X,Y, Z) Geometry for Shielding Design

A discrete ordinates transport code ENSEMBLE in (X, Y, Z) geometry has been developed for the purpose of shielding calculations in three-dimensional geometry. The code has some superior features, compared with THREETRAN which is the only code of the same kind so far developed. That is, the code can treat higher order anisotropic scattering and employs a coarse mesh rebalancing method. Moreover it has a negative flux fix-up routine using a variable weight diamond difference equation scheme and has a ray-effect fix-up option using a fictitious source based on S_N→P_N-1 conversion technique. Formulations for these advanced features in three-dimensional space have been derived.

As the demonstration of the capabilities of the code, several numerical analyses and an analysis of an annular duct streaming experiment in JRR-4 at Japan Atomic Energy Research Institute, have been performed.

As a result of these analyses, confirmation has been obtained for the prospect of applicability of ENSEMBLE to practical shielding design.     

Least-squares finite-element SN method for solving three-dimensional transport equation

A discrete ordinates finite-element method for solving three-dimensional first-order neutron transport equation is proposed using a least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation. Different from using the standard Galerkin variation applying to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. The approach allows a continuous finite-element. To eliminate the discontinuity of the angular flux on the fixed flux boundary in the spherical harmonics method, the equation is discretized using the discrete ordinates method for angular dependency. A three-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results demonstrate the accuracy and feasibility of the method.     

A New Mixed Method with Finite Difference and Finite Element Method for Neutron Diffusion Calculation

A new method for obtaining three-dimensional neutron flux distribution in a reactor has been developed by taking into account the fact that the X-Y planar geometry is generally complex but the geometry along Z-axis is simple. In this method, the finite element method is applied to the X-Y plane calculation and the finite difference method to the Z-axis. For solving a three-dimensional neutron diffusion equation, these two methods are iterated successively until a consistency of the leakage coefficients is attained between the two. The present method is embodied as a computer program FEDM for FACOM M200 computer. With this program, a three-dimensional diffusion calculation was performed for comparing some numerical results with those by a conventional standard computer code ADC. The comparison has shown that they agree well with each other. Computing time required for this problem by the FEDM was shorter than that by the ADC for obtaining same accuracy on the eigenvalue. To indicate usefulness of this method, a demonstration calculation for a reactor with a complex geometry was performed, which was a difficult case to calculate with a conventional finite difference code.     

Sub-cell balanced nodal expansion methods using S4 eigenfunctions for multi-group SN transport problems in slab geometry

A highly accurate S₄ eigenfunction-based nodal method has been developed to solve multi-group discrete ordinate neutral particle transport problems with a linearly anisotropic scattering in slab geometry. The new method solves the even-parity form of discrete ordinates transport equation with an arbitrary S_N order angular quadrature using two sub-cell balance equations and the S₄ eigenfunctions of within-group transport equation. The four eigenfunctions from S₄ approximation have been chosen as basis functions for the spatial expansion of the angular flux in each mesh. The constant and cubic polynomial approximations are adopted for the scattering source terms from other energy groups and fission source. A nodal method using the conventional polynomial expansion and the sub-cell balances was also developed to be used for demonstrating the high accuracy of the new methods. Using the new methods, a multi-group eigenvalue problem has been solved as well as fixed source problems. The numerical test results of one-group problem show that the new method has third-order accuracy as mesh size is finely refined and it has much higher accuracies for large meshes than the diamond differencing method and the nodal method using sub-cell balances and polynomial expansion of angular flux. For multi-group problems including eigenvalue problem, it was demonstrated that the new method using the cubic polynomial approximation of the sources could produce very accurate solutions even with large mesh sizes.     

Twodant solutions for the 2-D C5G7 MOX benchmark

The C5G7 MOX benchmark was proposed to test the ability of commercial transport codes to treat reactor core problems without spatial homogenization. The benchmark requires solutions in the form of normalized pin powers as well as the eigenvalue. In the work, the two-dimensional benchmark calculation using the TWODANT code within the DANTSYS code package has been performed with proper spatial and angular approximations. The TWODANT code solves the multigroup discrete ordinates form of the Boltzmann transport equation in two-dimensional geometry. The calculation results show a good agreement in comparison with the reference solution obtained from a seven-group MCNP calculation. In addition, sensitivity studies on mesh and angular refinements have been performed to produce a higher quality solution. In the results, it is found that in the TWODANT calculation the spatial approximation in a staircase form of the circular fuel pin with relatively high quadrature order of S_N is a viable method for solving the2-D C5G7 benchmark.     

COSINE软件包组件参数计算程序LATC中SN输运模块的开发与初步验证

本文论述了组件参数计算程序LATC的离散纵标（S_N）输运模块的理论模型。通过对基准问题的校验,验证了自主开发的组件参数计算程序LATC中基于一维、二维S_N理论及扩散综合加速收敛方法的输运模块。结果表明,LATC组件参数计算程序的S_N输运模块与基准解吻合良好,初步验证了LATC组件参数计算程序的S_N输运模块的正确性。     

扩散综合加速收敛方法在离散节块输运方法中的应用

本文介绍了一、二维直角坐标的离散节块输运方法中采用扩散综合加速收敛的理论、方法和计算公式，并给出了一些数值结果，以了解其加速效果。     

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.     

Parallelization of Monte Carlo Code MCACE for Shielding Analysis and Estimation of Its Efficiency by Simulator

An improved coarse-mesh discrete ordinates method has been developed for three-dimensional hexagonal transport calculations of high-conversion light water reactors and fast reactors. This method employs a new weighted diamond difference approximation which is obtained by using the neutron balance equations in divided submeshes. The weight is a function of neutron direction and scaler flux, and this method can be easily incorporated into conventional discrete ordinates transport codes.

The present method was applied to hexagonal fuel assembly calculations of high-conversion reactor and fast reactor core calculations, and the results were compared with those of Monte- Carlo calculations. The values of kefi and power distributions agreed with each other within 0.5 and 3%, respectively, verifying accuracy of the present improved coarse-mesh discrete ordinates transport calculation method.     

A least-squares finite-element Sn method for solving first-order neutron transport equation

A discrete ordinates finite-element method for solving the two-dimensional first-order neutron transport equation is derived using the least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation which contains the inversion of the cross-section. Different from using the standard Galerkin variation to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. To eliminate the discontinuity of the angular flux on the vacuum boundary in the spherical harmonics method, the angle variable is discretized by the discrete ordinates method. A two-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results verified the validity of this method.     

Unstructured partial- and net-current based coarse mesh finite difference acceleration applied to the extended step characteristics method in NEWT

The NEWT (NEW Transport algorithm) code is a multi-group discrete ordinates neutral-particle transport code with flexible meshing capabilities. This code employs the Extended Step Characteristic spatial discretization approach using arbitrary polygonal mesh cells. Until recently, the coarse mesh finite difference acceleration scheme in NEWT for fission source iteration has been available only for rectangular domain boundaries because of the limitation to rectangular coarse meshes. Therefore no acceleration scheme has been available for triangular or hexagonal problem boundaries. A conventional and a new partial-current based coarse mesh finite difference acceleration schemes with unstructured coarse meshes have been implemented within NEWT to support any form of domain boundaries. The computational results show that the new acceleration schemes works well, with performance often improved over the earlier two-level rectangular approach.     

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.     

强散射介质中子输运计算的角度相关附加再平衡加速算法

利用最小二乘有限元离散坐标方法,对一阶中子输运方程进行离散求解,给出了基于非结构网格的角度相关附加再平衡加速算法,采用附加修正量的办法达到再平衡的原理加速计算过程,同时也给出了其外推算法。将算法应用到强散射介质中子输运方程的计算中,一些基准问题的数值结果表明,计算速度可加速到原来的1.5～2倍。     

A comparison of finite elements and discrete ordinate methods for one-dimensional multigroup problems

The finite element formulation for the multigroup neutron transport equation is utilized to solve both shielding and eigenvalue problems in one dimension. In particular the variational principles formulated by Ackroyd are used. Results obtained for a three-group constant source and a five-group eigenvalue problem show the accuracy and efficiency of the finite element method in comparison with the discrete ordinates finite difference method.     

Effect of Lithium on Pressure of Hydrogen in Liquid Sodium

Abstract

Neutron spectra in a cylindrical straight duct and in bent ducts with angles of 30°, 60° and 90° have been measured by the multiple foil activation and thermoluminescence dosimetry methods. Two-dimensional discrete ordinates and three-dimensional Monte Carlo calculations are executed, and the results are compared with the measurements. The flow rate at the duct entrance calculated by the DOT3.5 code is underestimated by approximately 30%, due to a conversion of the core and reflector geometry from XY to RZ geometry. The fast neutron flux in the ducts is underestimated by 20% by the MORSE-SGC/S code due to a too coarse angular mesh of the source, which does not properly represent the actual angular distribution of the fast flux, which is highly peaked forwardly into the ducts. The thermal neutron flux was overestimated by the Monte Carlo calculation. A method is proposed to calculate the angular distribution of the flow rate at the duct entrance and to calculate the source strength and the angular distribution of the flow rate at the entrance of the second leg of the duct. The results are compared with those of the transport calculations. Generally, the agreement is quite satisfactory.     

A Two-Dimensional Benchmark Experiment for Neutron Transport in Water

For the purpose of providing standard data for checking two-dimensional neutron penetration calculations, fast neutron spectra as well as thermal and epithermal neutron fluxes were measured over a two-dimensional (R, Z) space in water shield using an activation method. Threshold reaction rates were converted to fast neutron scalar flux spectra with the aid of the SAND-II code. These results agree within a factor of 2 with the calculations by a two-dimensional discrete ordinates code PALLAS-2D. Thermal and epithermal neutron fluxes obtained with the Westcott's method agree quite well with the calculated values by the PALLAS-2D code in which the diffusion equation was adopted for dealing with low energy neutrons to reduce the computing time. All experimental results are given in the absolute values.     

三维离散纵标方法在堆内构件释热率计算中的初步应用

以美国H.B.Robinson-2#机组反应堆压力容器(RPV)基准实验的参数为输入数据,采用三维离散纵标方法程序(TORT)计算压力辐照监督管处中子能谱及典型核素的活度值。计算得到的辐照监督管处中子能谱与基准实验结果趋势一致、吻合较好;典型核素活度的计算值与测量值之比(C/M)为1.04±0.04。用TORT对福建宁德核电站堆内构件释热率分布进行初步计算,并与蒙特卡罗方法(MCNP)的计算结果相比较,两种方法的结果表现出良好的一致性。最后对TORT程序应用于堆内释热率计算进行讨论。     

基于离散纵标法与蒙特卡罗方法的三维耦合程序开发

辐射屏蔽设计是核装置工程设计的核心内容之一。单一的离散纵标法(比如SN)或蒙特卡罗方法(MC)在大型核装置屏蔽计算分析方面均存在一定限制。为了满足大型复杂核装置精确辐射屏蔽计算要求,本文实现了三维SN-MC耦合方法,并发展了相应的三维耦合程序系统。该程序结合了SN方法解决深穿透问题的优势和MC方法模拟复杂几何的长处,克服两种方法的缺点,为保证屏蔽系统优化设计的质量提供有力的技术支持。采用接口程序和MC自定义源抽样程序将SN计算得到的粒子角注量率转换为MC计算所需的源粒子信息,为下一步MC计算提供源项,实现三维SN-MC耦合输运计算。采用MC、SN、SN-MC耦合三种方法对直角坐标系和圆柱坐标系下的测试例题进行了计算比较分析。计算结果吻合良好,初步证明了所开发的三维SN-MC耦合程序的正确性。     

Fender: A finite element code for the solution of the diffusion equation in shield design applications

Diffusion theory remains an important method of calculation for shield design. Using adjusted coefficients the method provides an inexpensive solution of adequate accuracy for survey and optimization studies in two- and three-dimensional geometries. Solved in the adjoint mode, the method provides an estimate of the importance function which may be used for the acceleration of generalized-geometry Monte Carlo calculations.A number of computer codes exist to solve the diffusion equation by a finite difference approximation in one-, two- and three-dimensions. The mesh systems used in such codes usually impose restrictions on the accuracy of representation of shields with complicated geometries.The computer code FENDER solves the diffusion equation for neutron or gamma transport using the finite element technique. At present the code is written for a two-dimensional problem in which the geometry is specified as an array of triangular or rectangular elements. This permits a good representation to be made of shields containing curved surfaces. The variation of the calculated particle fluxes within an element is assumed to be quadratic.FENDER may take details of the element structure from an external mesh generating package but also contains a semi-automatic mesh generating routine for use as a stand-alone code. Multigroup diffusion parameters may be either input directly or generated from material compositions. The code is capable of handling problems with at least 1000 elements which is roughly equivalent in size and attenuation to 10,000 finite difference meshes. A variety of boundary conditions may be specified.The paper includes an example of application to demonstrate the potential usefulness of the method and the code. The case chosen is the calculation of neutron fluxes in a stylized fast reactor.