两维快堆中子扩散方程的有限元逼近

本文讨论了两维快堆中子扩散方程的有限元数值逼近方法,给出了收敛性证明,利用自动剖分技术编制了两维多群中子扩散方程有限元计算程序 FEM2D,对快堆临界问题进行了一系列数值计算,并与国外有限元计算结果及其它方法计算结果进行了分析比较,得到了满意的结论。     

Development of a Method for Evaluation of Pin-wise Power Distribution in Fuel Assemblies of Fast Reactors

In the design of fast reactor core with higher burnup and higher linear power, prediction accuracy of burnup history of fuel pin should be upgraded so as to assure fuel integrity without extra design margin under increased neutron fluence and burnup. A method is studied to predict fuel pin-wise power and its burnup history in fast reactors accurately based on an analytic solution of diffusion theory equation on hexagonal geometry with boundary condition from core calculation by finite-differenced diffusion calculation code. The present method is applied to a fast reactor core model, and its accuracy in predicting fuel pin power is tested. The result is compared with the reference solution by the finite difference calculation with very fine mesh. It is found that the present method predicts the power peaking factors in fuel assemblies accurately. The fuel pin-wise nuclide depletion calculation is also done using neutron fluxes for each fuel pin. The result shows that the fuel pin-wise depletion calculation is very important in predicting the burnup history of the fuel assembly in detail.     

A Krylov–Schur solution of the eigenvalue problem for the neutron diffusion equation discretized with the Raviart–Thomas method

Mixed-dual formulations of the finite element method were successfully applied to the neutron diffusion equation, such as the Raviart–Thomas method in Cartesian geometry and the Raviart–Thomas–Schneider in hexagonal geometry. Both methods obtain system matrices which are suitable for solving the eigenvalue problem with the preconditioned power method. This method is very fast and optimized, but only for the calculation of the fundamental mode. However, the determination of non-fundamental modes is important for modal analysis, instabilities, and fluctuations of nuclear reactors. So, effective and fast methods are required for solving eigenvalue problems. The most effective methods are those based on Krylov subspaces projection combined with restart, such as Krylov–Schur. In this work, a Krylov–Schur method has been applied to the neutron diffusion equation, discretized with the Raviart–Thomas and Raviart–Thomas–Schneider methods.     

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.     

基于任意三角形网格的中子扩散变分节块法

传统的基于矩形和六角形几何的堆芯计算程序已不适用于具有复杂几何的新型反应堆堆芯计算,本文开展了基于任意三角形网格的多群中子扩散变分节块方法研究。首先,采用ANSYS软件对计算区域进行三角形网格剖分,并利用坐标变换将任意三角形变换为正三角形;其次,采用Galerkin变分技术建立包含节块中子平衡方程的泛函,将三角形节块内变量利用正三角形内正交基函数进行展开;最后,利用变分原理,获得中子通量密度与节块边界上分中子流的响应关系,并基于传统的源迭代法对其进行求解。基于上述理论模型开发了程序TriVNM,并采用不同几何基准题进行了验证。结果表明,TriVNM计算的堆芯keff和归一化功率分布与参考解吻合较好,该计算方法适用于复杂几何堆芯扩散计算。     

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.     

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.     

Temperature Fluctuations in a Non-Equilibrium System

Abstract

A modified one-group diffusion scheme and a coarse-mesh diffusion method were newly developed independently, and are combined together to make an effective brief scheme for the solution of few-group diffusion equations in a 3-dimensional BWR core. In the above two methods, the local spatial behavior of the neutron spectrum—the ratio of thermal neutron flux to the fast neutron flux—and of the fast neutron flux within a volume element are approximately taken into account by using analytical procedures, therefore the resultant numerical equation, which is slightly more complicated than those of the conventional brief schemes, gives sufficiently accurate results.

The special behavior of the spectrum in the neighborhood of the core-reflector interface is also treated in the appendix of this paper. An analytical expression for the spectrum on the core-reflector boundary of various shapes is derived and some numerical examples are presented.

The results obtained from a series of test calculations indicate that the present method assures an accuracy comparable to the standard few-group fine-mesh finite- difference calculation without requiring much more computing time than those of the conventional brief calculation methods. Thus the present method is particularly suited for improving the computational accuracy of the BWR simulation codes.     

High Order Finite Element Method for the Lambda modes problem on hexagonal geometry

A High Order Finite Element Method to approximate the Lambda modes problem for reactors with hexagonal geometry has been developed. This method is based on the expansion of the neutron flux in terms of the modified Dubiner’s polynomials on a triangular mesh. This mesh is fixed and the accuracy of the method is improved increasing the degree of the polynomial expansions without the necessity of remeshing. The performance of method has been tested obtaining the dominant Lambda modes of different 2D reactor benchmark problems.     

基于先进节块法的六边形三维中子物理程序的开发与验证

六边形燃料组件在液态金属冷却快堆尤其是钠冷快堆中被广泛应用,针对这类堆型的设计与安全分析需要对堆芯中子通量与中子流进行三维全堆芯耦合计算。经过多年发展,目前已有多种解析节块法、积分节块法、节块展开法等先进节块法能在笛卡尔坐标系下较为精确求解多维中子扩散方程。本文通过径向半解析节块法耦合轴向高阶节块展开法的综合节块方法开发了反应堆三维中子物理计算软件SA HNHEX,并对VVER 440二维、三维基准题进行建模与仿真计算。计算结果与参考值符合较好,初步验证了使用该方法进行反应堆堆芯中子扩散计算的正确性。     

A Method of Calculation of Detailed Power Distribution in Fuel Assemblies

Abstract

A method has been developed that effectively estimates the detailed distribution of power generation in the fuel or blanket assemblies in nuclear reactors. A two-dimensional, one-group diffusion model is applied to a region of homogeneous composition enclosed in a contour devoid of concavity viewed from outside. The diffusion equation is reduced to the form of Helmholtz equation, and a non-homogeneous boundary condition of Dirichlet or Neumann type is given on the contour, using neutron fluxes previously obtained in coarse mesh diffusion criticality calculations covering the whole reactor. This boundary value problem in two-dimensional space is made to yield a solution in the form of a potential due to a single or double layer. The method is applied to a hexagonal cell of a fast reactor. The results of calculation are amply accurate in comparison with the corresponding values from the usual fine-mesh diffusion scheme and with much shorter computing time.     

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.     

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.     

Analysis on the neutron kinetics for a molten salt reactor

The neutron kinetics of the molten salt reactor is significantly influenced by the fuel salt flow, which leads to the analysis methods for the conventional reactors using solid fuels not being applicable for the molten salt reactors. In this study, a neutron kinetic model considering the fuel salt flow is established based on the neutron diffusion theory, which consists of two-group neutron diffusion equations for the fast and thermal neutron fluxes and six-group balance equations for delayed neutron precursors. The temperature feedback on the neutron kinetics is considered by introducing a heat transfer model in the core, in which the group constants which are dependent on the temperature are calculated by the code DRAGON. The mathematical equations are discretized and numerically calculated by developing a code, in which the fully implicit scheme is adopted for the time-dependent terms, and the power law scheme is for the convection–diffusion terms. The neutron kinetics is conducted during three transient conditions including the rods drop transient, the pump coastdown transient and the inlet temperature drop transient. The relative power changes and the distributions of the temperature, neutron fluxes and delayed neutron precursors under these three different transient conditions are obtained in the study. The results provide some valuable information for the research and design of this new generation reactor.     

Effective one-group coarse-mesh method for calculating three-dimensional power distribution in fast reactors

An effective one-group coarse-mesh method for calculating three-dimensional power distribution in fast reactors has been developed. This method uses direction dependent one-group diffusion coefficients in each region so as to preserve neutron leakage rates before and after energy condensation. Furthermore, to correct coarse meshes of one point per hexagonal assembly used in the three-dimensional diffusion calculation, one-group cross-sections are modified by applying Askew's method. Based upon this method an effective three-dimensional diffusion code has been made. Results obtained in test cases on a prototype fast reactor indicate that this method is as accurate as six-group fine-mesh diffusion calculations using six mesh points per assembly in each radial plane and the solution of this method is a factor of 40 faster than the six-group fine-mesh calculations.     

Generalization of New Finite Element Method Using ‘Imaginary’ Nodal Points

A method for improving the accuracy of finite element solutions to diffusion equations has been developed. The author previously suggested a method for improving the accuracy of finite element solutions to neutron diffusion equations, a kind of Helmholtz equations, within a short computing time. The method has been generalized so that it can be applied to problems described by the Laplace equation, too, such as temperature distributions and electric fields. In this generalized method, 3 ‘imaginary’ nodal points are added at the midsides of each data-given triangular element and the element is subdivided into 4 triangular subelements of the same dimension to improve accuracy. Then, approximate expressions, which express solutions at the ‘imaginary’ nodal points using those at ‘real’ nodal points, are derived by Jacobi's iteration method. These approximate expressions are used to reduce the number of unknowns in the final linear equations. The computing time required for the method described here is much shorter than that required for the straightforward method of increasing the number of elements 4 times under the same accuracy.     

Application of the method of implicit non-stationary iteration (MINI) to 3D neutron diffusion problems

This paper extends an earlier one on the method of implicit non-stationary iteration (MINI). The extension is to the solution of symmetric and non-symmetric sets of linear equations arising in a three-dimensional Cartesian finite difference neutron diffusion representation of a reactor. MINI is used to solve the symmetric equations for neutron flux of a particular energy group, as well as the non-symmetric equations for neutron upscatter and region rebalance. Calculations are reported for some real two- and three-dimensional reactors, and the performance of MINI is compared with those of other methods. MINI is found to be particularly beneficial for problems involving extensive upscatter.     

液态金属棒束传热的理论分析

液态金属棒束传热研究在快堆中有重要的应用价值。采用六边形的传热单元，等边三角形布置的棒束传热问题可转化成同心环管的传热问题。引入水力直径修正及温差、热混合两项温度修正后，得到了全新的三角形布置的棒束顺流传热关系式，对实验数据取得了良好的预测效果。同时发现，温度修正的第2项可作为子通道方法中有效混合长度的计算式。     

Nodal calculation of neutron fields in hexagonal lattices

A modified polynomial nodal method of calculating the neutron flux-density distribution in nuclear reactors with a hexagonal fuel-assembly configuration is proposed, with solution of the diffusion equation in rectangular geometry. The method is adapted for hexagonal configurations by conformal mapping of the interior of a hexagon onto a rectangle. The accuracy of solution of the diffusion equation in a two-group diffusional approximation in hexagonal arrays is demonstrated for the example of three-dimensional model problems describing VVéR-1000 and VVéR-440 water-cooled, water-moderated reactors. Analysis of the results indicates high accuracy of the parameter values obtained. Moscow Engineering-Physics Institute. Translated from Atomnaya énergiya, Vol. 87, No. 2, pp. 108–113, August, 1999.     

有限元SN中子输运模拟的区域分解并行

确定论中子输运方法具有计算速度快、可获取物理量的精细场分布、可高效多物理耦合等优点,随着有限元方法在中子输运模拟中的应用,复杂几何结构、大尺度下的屏蔽问题和临界问题都能得到高保真建模和分析。离散纵标（S_N）法是求解中子输运方程的有效数值方法,基于OpenMP并行机制对各独立离散方向进行并行求解,可提高S_N输运模拟的计算速度,但并行规模较有限。对几何空间进行区域分解并采用MPI并行机制,可实现大规模并行扩展,进而实现对大型问题的高精度快速求解。本文采用并行自适应非结构网格应用框架JAUMIN进行区域分解和进程间通信,通过并行S_N扫描实现了自主有限元输运程序ENTER的高效并行,完成正确性检验后在天河Ⅱ号超级计算机上使用1 440个CPU核完成了1.43×10⁷网格单元、2.81×10⁹自由度规模问题的测试,计算时间约7.4 h。表明该程序具备了有效模拟大型复杂结构中子输运问题的能力,具有一定工程应用价值。