Reduction in spatial discretization error in the method of characteristics by using the mobile-chord ray tracing method

The mobile-chord method is applied to the method of characteristics (MOC) to reduce the spatial discretization error in ray traces. In the mobile-chord method, the offset of a ray trace in a strip depends on the azimuthal angle and this variation cancels the spatial discretization error of each ray trace. Although the mobile-chord method has been employed in the evaluation of the collision probability, it has not yet been applied to the MOC. The mobile-chord method is implemented in the AEGIS code, which is a lattice physics code based on the MOC. Verification calculations are carried out for the pin-cell and whole core geometries of the C5G7 benchmark problem by using UO₂ and mixed oxide (MOX) fuels. The calculation results indicate that the spatial discretization error in the mobile-chord method is smaller than that in the equidistant ray tracing method, which is commonly used in conventional MOC codes. Since the mobile-chord method can be used along with the cyclic ray tracing method, it is expected to be an attractive candidate for conventional MOC codes.     

Application of the multigrid amplitude function method for time-dependent MOC based on the linear source approximation

ABSTRACT

An efficient numerical scheme for time-dependent MOC calculations is proposed. In the present scheme, one of the most successful factorization method, the multigrid amplitude function (MAF) method, is employed to achieve faster computation with the minimum degradation for the temporal integration of the scalar flux. In addition, the MAF method is re-derived based on the linear source approximation, which is not applied for time-dependent MOC calculations in the past studies as far as the authors’ knowledge, to reduce the spatial discretization error with the coarser flux region separation. The accuracy and computational time of the present scheme are evaluated through the calculation of the TWIGL and the C5G7-TD 2D benchmark problems. The present calculation results show that the present scheme is 6.2 times faster than the conventional method while achieving the same accuracy in the C5G7-TD benchmark problem.     

Reduction of MOC discretization errors through a minimization of source ratio variances

A new technique to reduce discretization errors for ray tracing in the method of characteristics (MOC) is proposed focusing on depletion calculations of single and multi-assembly geometries. In order to efficiently carry out depletion calculations, a calculation scheme using the superhomogenization (SPH) method can be used. However, the discretization errors are caused by changes of neutron sources and total cross sections according to a depletion. This fact means that improvement of accuracy cannot be expected by the calculation scheme with the SPH method when changes of the above parameters are significant. In order to mitigate this problem, a new approach is developed. In the new approach, the discretization errors are reduced by minimizing a variance of a certain parameter which is composed of a ratio of neutron source to total cross section. The verification results suggest that accuracy is degraded by the SPH method as expected especially in a geometry where neutron sources and total cross sections are drastically changing through a depletion. On the other hand, the new approach gives more accurate results compared to the conventional MOC in all calculation cases. Consequently, improvement of calculation efficiency by the new approach is confirmed.     

一维特征线法中子输运程序开发及验证

为探讨两维/一维综合法堆芯分析方法,本文基于特征线法研制了一维中子输运程序--PEACH-1D.不同于通常的平源近似特征线方法,PEACH-1D可对子区的中子源项作线性近似;程序运用指数函数插值表和渐近源外推技术来加速计算过程.相关数值结果表明,PEACH-1D具有很高的计算精度和效率,线性源近似的特征线法具备处理较粗网格的能力,值得推广.     

Applicability of Constant Flux Approximation in Method of Characteristics with Filtering to Tiny Regions

For the method of characteristic (MOC), a system with large-gradient neutron flux caused by a strong absorber or neutron leakage is reported to entail large errors in spatial mesh discretization and require very fine mesh spacing. To apply the MOC to such fine models, the ray-trace path width has to be fine in order to make many paths cross each region. Our new method intends to obtain good accuracy with a coarse path width. With a coarse path width on the MOC, some tiny regions have less ray-tracing paths. The reliability of flux calculation for the regions can be evaluated with the calculation volumes that are estimated in ray tracing. If the discrepancy between the calculation and true volumes becomes large, the accuracy cannot be expected. In this study, the discrepancies were numerically evaluated, and it is found that the discrepancies occur on a very tiny region. To make the flux calculation of such tiny regions more reliable, an approximation, in which the outgoing flux is equal to the incoming neutron, is applied instead of the usual MOC equation. The criteria switching on the approximation, which is called filtering, was numerically evaluated for PWR assemblies. The method is validated with numerical benchmark calculations.     

线性源近似的中子输运方程特征线解法

基于非结构网格的中子输运方程特征线解法已成为堆芯设计程序中组件输运计算的标准方法之一。但现有的大部分特征线法程序均是基于平源近似的步特征线法模型开发的,平源近似是特征线法中除角度变量直接离散外又一基本假定。本工作提出一种基于线性源近似的中子输运方程特征线解法,并提出相关负中子源分布的修正方法。采用自行研制的数值计算软件PEACH,对OECD/NEAC5G7-MOX2D基准问题和自定义沸水堆小堆芯问题进行检验。数值计算结果表明,本工作提出的线性源近似特征线法模型在相同计算精度的前提下,占用更少的系统内存和运行时间。     

Resonance Treatment Based on Ultra-fine-group Spectrum Calculation in the AEGIS Code

The resonance calculation method using the ultra-fine-group spectrum calculations in the AEGIS code is explained in detail. By a simple benchmark problem, it is verified that the effect of anisotropic scattering on effective cross-sections is not very large and the isotropic scattering source approximation is adequate in practical resonance calculations in LWRs. Furthermore, some efficient numerical algorithms in the ultra-fine-group calculations to reduce the computation time without large degeneration of accuracy are presented. In addition, the SPH method for energy collapsing of cross-sections is adopted in the AEGIS code to reduce the error of energy collapsing.

Through the comparison with continuous-energy Monte-Calro calculation in the pin-cell geometry, the validity of the resonance treatment in the AEGIS code is verified.     

Development and verification of an MOC code employing assembly modular ray tracing and efficient acceleration techniques

In this paper, we report the development and verification of a method of characteristics (MOC) code, PEACH, at Shanghai Jiao Tong University. Both the usual flat-source step characteristics (SC) scheme and the linear source (LS) approximation scheme are adopted for the tracking calculation along the neutron trajectory. The assembly-based modular ray tracing (AMRT) technique that possesses a good geometric flexibility and high efficiency is employed, which makes PEACH able to deal with practical LWR assembly and core problems. Moreover, in order to reduce the computational time of the MOC iteration process, both the multi/few-group two-level cell-based coarse mesh finite difference (CMFD) acceleration and the exponential function interpolation technique are used. This results in a significant acceleration. Numerical results for the OECD NEA C5G7 MOX benchmark problem and a 69-group BWR mini-core problem demonstrate that PEACH is accurate and efficient. Numerical results also demonstrate that the LS scheme is more efficient than the SC scheme, taking less time and system memory to generate results of comparable accuracy. In addition, we find that MOC with CMFD acceleration always converges with almost the same number of outer iterations regardless of the physical problem size and the discretization parameters used. This shows an ideal linear relationship between the run time and the geometric size of the problem.     

球床高温气冷堆吸收体价值计算的验证

VSOP程序广泛用于球床高温气冷堆的工程设计。对于被布置在堆芯侧反射层孔道中、用于反应性控制的吸收体,由于物理计算方法的限制,VSOP程序不具备计算其价值的功能,必须借助其他确定论程序进行外部耦合计算,涉及到几何的近似处理、截面的归并和转换,可能引入额外的误差。为此,本文采用蒙特卡罗程序建立了精细的堆芯模型,真实描述了堆芯活性区的球床结构、侧反射层的孔道结构、吸收体的形状和位置,在同样的堆芯状态下,比较了确定论耦合程序和MCNP程序计算得到的吸收体价值。结果表明：确定论耦合程序的计算结果是准确的,从设计角度上是偏保守的。     

Xenon transient simulation of the VVER-1000 nuclear reactor using adiabatic approximation

The neutronic behavior of very slow transients like fuel burn up and xenon studies can be performed with the sequence of instantaneous criticality calculations. Such a scheme is known as the adiabatic approximation. ¹³⁵Xe as a fission product has an enormous thermal absorption cross section, on the order of a million barns, therefore the study of xenon poisoning and its effect on flux and power distribution is very important in thermal reactors. In this work xenon transient analysis of the VVER-1000 nuclear reactor and its effect on the flux and power distribution from reactor start up to xenon saturation and the change of power from nominal to 25% of nominal is carried out using WIMS and CITATION codes. We used the WIMS cell calculation code and found some relations between xenon concentration and group constants of different FA (Fuel Assemblies); in this way we bypassed the WIMS running at each time step. Also, the CITATION code is used as a core calculation code to find the effective multiplication factor as well as flux and power distributions. In order to link WIMS and CITATION codes and facilitate numerous executions, a VISUAL BASIC program has been developed. The results have a good agreement with the safety analysis report of the reference plant such that the relative differences in most cases are less than 10%.     

An Accurate Method to Interpolate First-Flight Collision Probabilities

A method is proposed to approximately calculate first-flight collision probabilities for use in resonance integral calculations with good accuracy and within a short computing time. In the proposed method, collision probabilities at three different energy points are rigorously calculated and stored beforehand. They are used to obtain approximate collision probabilities at an arbitrary energy point by Takahashi's zero-th approximation. Then, these approximate values are used to obtain more accurate values by quadratic interpolation with the neutron total cross section change as an interpolation parameter. The proposed method was applied to the calculation of collision probabilities in a two-region LWR fuel pin cell, and it was found that the collision probabilities can be calculated with an error of less than 1%, a great improvement in accuracy compared to the zero-th approximation, when the total cross section of the fuel region changes 30 times. The proposed method shortens computing time of lattice cell neutronic calculation codes based on the first-flight collision probability method.     

数字化反应堆高保真中子学程序SHARK研发

SHARK程序是由中国核动力研究设计院新近研发的基于全堆芯确定论非均匀输运理论体系的数字化反应堆软件。该软件从多群数据库的截面与共振数据出发,采用改进子群方法刻画有效共振截面的复杂非均匀效应,采用二维/一维或准三维特征线方法开展堆芯层面非均匀输运计算。目前该程序的定态微观问题计算能力已建立完毕。数值结果显示,SHARK程序对于商用压水堆相关基准问题具有良好的计算精度和效率。     

非均匀变分节块法

传统的节块方法均属于均匀节块法,要求节块内的材料必须是均匀分布的。对于传统的固体燃料核反应堆,该近似处理带来的误差是可接受的;但对于液体燃料的熔盐堆,流动的燃料由于空间上连续变化的温度和核密度分布而具有连续变化的宏观截面,使传统的节块方法无法在保证计算精度的同时取得较高的效率。为尝试解决该问题,本文在一维扩散近似的情况下,基于变分节块法推导了能处理空间连续变化截面的非均匀变分节块法,并开发了相应的计算程序Violet-Het1D。数值结果表明,在相同的节块划分和展开阶数条件下,非均匀变分节块法和均匀变分节块法计算时间相当,但前者的计算精度高于后者。     

适用于任意几何的特征线边界条件处理方法

边界条件处理是特征线方法（MOC）向任意三维几何拓展时遇到的难点之一。本文提出一种边界条件处理方法,既保留循环特征线中首尾相连的特性,又能像插值方法一样适用于任意几何。首先推导了平源近似下的特征线方程,提出了一种将源项和边界角通量分离处理的内迭代解法。然后证明了该解法具有唯一解,并类似于循环特征线方法给出解的构造方法。最后借助数值积分和权重插值给出迭代计算流程。采用Takeda算例、单铀球水腔模型和C5G7算例进行验证计算,k_eff的最大计算误差分别为21、319和138.8 pcm,表明方法可靠。该方法可应用于任意几何,且不需存储边界通量和进行边界迭代。     

Validation of a Computational Simulation Method for Evaluating Thermal Stratification in the Reactor Vessel Upper Plenum of Fast Reactors

Validation of a numerical simulation method is carried out for thermal stratification phenomena in the reactor vessel upper plenum of advanced sodium-cooled fast reactors. The study mainly focuses on the fundamental applicability of commercial computational fluid dynamics (CFD) codes as well as an inhouse code to the evaluation of thermal stratification behavior including the simulation methods such as spatial mesh distribution and RANS-type turbulence models in the analyses. Two kinds of thermal stratification tests are used in the validation, which is done for relatively simple- and conventional-type upper plenum geometries with water and sodium as working fluids. Quantitative comparison between the simulation and test results clarifies that when used with a high-order discretization scheme of the convection term, the investigated CFD codes are applicable to evaluations of the basic behaviors of thermal stratification and especially the vertical temperature gradient of the stratification interface, which is important from the viewpoint of structural integrity. No remarkable difference is seen in the simulation results obtained using different RANS turbulence models, namely, the standard kε model, the RNG k-ε model, and the Reynolds stress model. It is further confirmed in a numerical experiment that the distribution of two or more meshes within the stratification interface will lead to accurate simulation of the interface temperature gradient with less than 10% error.     

Boundary Condition Processing Method for MOC Calculation in Arbitrary Geometry

Boundary condition processing is one of the difficulties encountered in the application of method of characteristics (MOC) to arbitrary three-dimensional geometry. In this paper, a boundary condition processing method was proposed, which not only preserved the track continuity as cyclic track method, but also could be applied to arbitrary geometry as the interpolation method. The MOC equation was derived under the flat source approximation and an internal iterative method was proposed in which the source term and the boundary angular flux were processed separately. It was proved that the equation had a unique solution which could be constructed similarly to the cyclic track method. The iterative calculation flow was given by numerical integration and weight interpolation. Takeda benchmark, single uranium sphere model with water cavity and C5G7 benchmark were calculated to test the accuracy. The maximum error of k_eff is 21, 319 and 138.8 pcm respectively, which shows that the method is reliable. This method can be applied to arbitrary geometry without storing boundary fluxes and performing boundary iteration.     

基于OpenMC的多群截面库制作及有效性验证

OpenMC是麻省理工大学计算反应堆物理组开发的开源蒙特卡罗程序,能够方便地制作适用于特定堆芯中子能谱分布的多群反应截面及高阶勒让德散射截面以用于离散坐标输运程序ANISN的计算。本文基于ENDF/B-Ⅶ.1和CENDL-3.1评价数据库,利用OpenMC计算制作了ANSIN格式的多群截面并通过基准题的计算验证计算结果的准确性。通过截面转换程序的编写,将OpenMC给出的堆芯各阶勒让德散射分量,堆芯中子能谱分布,散射、吸收反应率以及裂变中子产生速率等信息转换为ANISN程序可读取的截面库格式。采用制作的截面库利用ANINS计算有效中子增殖因子及堆芯中子通量分布。结果表明,ANISN确定论的计算结果与OpenMC给出的蒙特卡罗计算结果相吻合,验证了这种方法可有效地为ANISN提供截面数据,将来可推广应用于二维、三维确定论中子输运计算。     

GPU加速MOC输运计算性能分析研究

特征线方法(MOC)在求解堆芯规模中子输运方程时面临计算时间长的问题,加速和并行算法是目前研究的热点。基于MOC在特征线和能群层面的并行特性,采用统一计算设备构架(CUDA)编程规范,实现了基于图形处理器(GPU)的并行二维MOC算法。测试了菱形差分和步特征线法分别在双精度、混合精度及单精度浮点运算下的计算精度、效率及GPU加速效果。采用性能分析工具对GPU程序性能进行了分析,识别了程序性能瓶颈。结果表明:菱形差分和步特征线法在不同浮点运算精度下均表现出良好的计算精度;相比于CPU单线程计算,GPU加速效果在双精度和单精度情况下分别达到35倍和100倍以上。     

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.     

以栅元为模块进行特征线跟踪的中子输运方程解法

为解决复杂几何条件下中子输运方程的求解问题,分析了特征线法理论模型,探讨了以栅元为模块的高效特征线产生方法,以及与之相关的空间角度离散和边界条件处理问题.采用自行研制的特征线法数值计算软件--PEACH,对经济合作组织核能机构(OECD/NEA)UO2和MOX燃料混合装载的7群(C5G7MOX)基准问题的数值进行了检验.结果表明,无论是计算Keff还是棒功率分布该方法都具有很高的精度.