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.     

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。     

二维中子输运方程的非结构网格离散纵标数值解法

从一阶中子输运方程出发,对方向变量采用离散纵标法展开,得到一系列关于空间变量的偏微分方程,对这些方程采用最小二乘有限元方法进行离散.编制了二维中子输运方程的非结构网格离散纵标计算程序.对一系列基准问题做了验算,计算结果表明,该方法能用于非结构网格,并具有较高的计算精度.     

The infinite medium Green's function method for multigroup discrete ordinates transport problems in multi-layered slab geometry

Analytic solutions of the multigroup discrete ordinates transport equation with linearly anisotropic scattering and arbitrarily distributed source for multi-layered slab problems are obtained by using the infinite medium Green's function (IMGF) and Placzek's lemma. In this approach, the infinite medium Green's function is derived analytically by using the spectral analysis for the multigroup discrete ordinates transport equation and its transposed equation, and this infinite medium solution is related to the finite medium solution by Placzek's lemma. The resulting equation leads to an exact relation that represents the outgoing angular fluxes in terms of the incoming angular fluxes and the interior inhomogeneous source for each slab. For heterogeneous problems having multi-layered slabs, the slabs are coupled through the interface angular fluxes. Since all derivations are performed analytically, the method gives exact solution with no truncation error. After the interface angular fluxes are calculated by using an iterative method, the continuous spatial distribution of the angular flux (i.e. analytic solution) in each slab is given straightforwardly in terms of the IMGF and the boundary angular fluxes.     

一阶中子输运方程的离散纵标有限元方法研究

利用离散纵标(SN)方法离散SN一阶多群中子输运方程,建立了基于剖分单元的有限元离散与非结构网格扫描方法相结合的求解模型。针对给定的三角形单元应用Galerkin变分,获得线性方程组,通过引入定解条件,求解线性代数方程组,获得该三角形单元所有节点上的角通量,然后对其他三角形单元进行扫描,从而解出所有节点处中子通量密度。根据上述理论模型,编制了相应的计算程序FEGT,对一系列例题进行校验的数值结果表明,该程序的计算精度满足要求。     

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.     

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.     

Finite-moments approaches to the time-dependent boltzmann equation

Two finite-moments techniques to solve the time-dependent discrete ordinates transport equation are discussed. One of the techniques is implicit in time with a finite-moments expansion of the flux in space, and the other uses finite-moments expansions in both time and space. The performance of these algorithms is compared to that of conventional time-dependent discrete ordinates methods. The accuracy of the basic implicit time differencing and the effect of angular discretization are briefly discussed. Finite-moments techniques can offer a significant advantage over conventional methods in certain time-dependent transport problems, especially in multidimensional geometry.     

A linear multiple balance method with high order accuracy for discrete ordinates neutron transport equations

A linear multiple balance method (LMB) is developed to provide more accurate and positive solutions for the discrete ordinates neutron transport equations. In this multiple balance approach, one mesh cell is divided into two subcells with quadratic approximation for the angular flux distribution. Four multiple balance equations are used to relate center angular flux with average angular flux by Simpson's rule. From the analysis of spatial truncation error, the accuracy of the linear multiple balance scheme is O(Δ⁴) whereas that of diamond differencing is O(Δ²). The positivity of the method is also stronger than that of diamond differencing. To accelerate the linear multiple balance method, we also describe an additive angular dependent rebalance factor scheme which combines a modified boundary projection acceleration scheme and the angular dependent rebalance factor acceleration scheme. It is demonstrated, via Fourier analysis of a simple model problem as well as numerical calculations, that the additive angular dependent rebalance factor acceleration scheme is unconditionally stable with spectral radius <0.2069 c (c being the scattering ratio). The numerical results tested so far on slab-geometry discrete ordinates transport problems show that the solution method of linear multiple balance with additive angular rebalance acceleration is effective and sufficiently efficient.     

Three-Dimensional Transport Calculation Method for Eigenvalue Problems Using Diffusion Synthetic Acceleration

A three-dimensional transport code “TRITAC” for solving eigenvalue problems in reactor cores has been developed on the basis of discrete ordinates method with the diffusion synthetic acceleration technique. The Larsen procedure for the diffusion synthetic acceleration method has been extended to three-dimensional geometry. With the procedure a spatially differenced diffusion synthetic equation has been derived and implemented in the TRITAC code. In the X-Y geometry the code yielded the same results as the TWOTRAN-II code. Three-dimensional eigenvalue problems for thermal and fast reactors have been solved and the computational time has been compared with that required for the three-dimensional discrete ordinates calculation with the rebalance acceleration technique.     

Discrete ordinates method-like transport computation with equivalent group condensation and angle-collapsing for local/global iteration

The group condensation of the transport equation is studied in this paper so that the computational burden can be reduced. The group condensation procedure leads to equivalent total cross section that becomes angle dependent. The difficulty of angle dependency has been traditionally treated by consistent P or extended transport approximation in multigroup transport computation. However, in this study, the angle dependency of the total cross section is applied directly to the discrete ordinates equation, and the solution procedure is validated numerically on a test problem. In addition, an angle-collapsing concept is proposed for the purpose of further simplifying the group condensed problem. A local/global iteration framework is also described, in which fine-group discrete ordinates calculation is used in local problems while few-group angle collapsed transport calculation is used in the global problem, with excellent test results in the k_eff and flux estimation.     

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.     

Numerical study of scattering Legendre moments and effect of anisotropic scattering on S_N shielding calculation

In neutron and photon transport problems, anisotropic scattering is of great importance for the particle flux, especially when the angular flux has a strong forward peak in shielding analyses. The conventional Legendre expansion is widely used in discrete ordinates transport codes because of algebraic simplifications with spherical harmonics for the scattering source. However, negative cross sections caused by the finitely truncated expansion may give rise to a negative source and flux. A simple method is adopted, based on integrating functions of scattering moments, to evaluate anisotropy and convergence of expanded functions. A series of problems were designed with angular fluxes of different anisotropy, and numerical simulations were performed using the ARES transport code to study different treatments and algorithms for scattering. Results show that the diagonal transport approximation is more stable and obtains a similar accuracy with the extended approximation. A conservative fixup for the negative source could ensure particle balance and improve computational accuracy significantly for photon transport. The effect of anisotropic scattering is problem-dependent, and no distinct differences among various methods are observed for volume source problems with a continuous energy source. For beam source problems, flux results are sensitive to negative scattering functions, and strictly nonnegative cross sections need to be implemented.     

基于二十面体间断有限元离散求积组的IRI-TUB基准题验证

精确可靠的屏蔽设计是保证核装置安全性的重要组成部分,离散纵标法是应用最广泛的确定论屏蔽计算方法。对于角通量密度各向异性较强的屏蔽问题,求积组精度不足会导致离散误差较大,严重影响屏蔽计算的准确性与可靠性。本文结合间断有限元思想,构造正二十面体线性及二次间断有限元离散求积组,并优化求积组权重及方向保证权重严格非负。采用球谐函数数值积分及IRI-TUB基准题验证求积组的计算精度与适应性。数值结果表明,二十面体线性间断有限元离散求积组在1/20球面内能准确积分对应0阶和1阶球谐函数,且具有4阶收敛性;对于IRI-TUB基准题,反应率计算值与实验测量值的相对偏差小于25%。二十面体间断有限元离散求积组能适用于角通量密度各向异性较强的屏蔽问题,从而提高屏蔽计算的可靠性。     

角度自适应离散纵标输运计算方法

精确的屏蔽计算方法是核装置辐射屏蔽设计的重要基础,离散纵标法(SN)是主要的屏蔽计算方法之一。本文基于价值理论的目标导向与角度自适应相结合的方法,有效地减弱了角度的离散误差。求解输运共轭方程获得目标函数的重要性分布,采用局部角度离散误差与目标函数的重要性加权,产生后验误差估计,为角度自适应过程提供判断依据。角通量密度的映射采用多项式权重法和球谐函数拟合法。数值结果表明,对于具有直孔道或曲折孔道的屏蔽问题,在相同精度下离散角度数减少了1～2个数量级,极大地减少了计算量。角度自适应方法以较少的离散方向获得了准确的计算结果,有效地减弱了角度离散误差对屏蔽计算精度的影响。     

基于多次碰撞源的屏蔽计算方法研究

离散纵标（SN）方法在求解过程中将空间变量和角度变量进行离散,空间变量和角度变量的离散误差控制对保证计算精度至关重要。本文基于射线追踪研究了多次碰撞源方法,通过计算在选定区域内粒子发生多次碰撞的通量密度,将孤立源等效为计算模型内的分布源进行离散纵标输运计算。选取自设屏蔽问题及Kobayashi基准题进行测试验证并对结果进行分析。数值结果表明,自设屏蔽问题中多次碰撞源方法较首次碰撞源方法能有效缓解二次射线效应问题;Kobayashi基准题计算结果与基准值相对误差的均方根小于3%。多次碰撞源方法有效地减弱了离散误差,提高了屏蔽计算的准确性与可靠性。     

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.     

Discrete Ordinates Numerical Integration Method for Neutron Transport Equation in Slab Geometry

A method is presented for solving neutron transport problems in slab geometry with the application of discrete ordinates numerical integration to the Boltzmann transport equation.

Neutron spectra for water, iron and carbon layers are calculated and compared with experimental data. Good agreement is obtained between computed and measured values for both angular distributions and the energy spectra of neutrons. The calculated neutron spectra in an iron-water layer are also presented as an example of solutions derivable for neutron transport problems in multi-layered shield.     

基于求积组局部细化技术的SN屏蔽计算方法

辐射屏蔽设计是保证核装置安全性的重要组成部分,离散纵标法是屏蔽计算的主要方法之一。在具有狭长孔道的屏蔽问题中,由于中子角通量密度呈强各向异性分布,特别在孔道内其分布存在极大峰值,传统求积组难以实现计算精度与效率之间的平衡。为此,本文基于勒让德-切比雪夫求积组的离散特点,研究局部范围内多层极角细化技术,提高求积组积分角通量密度的精度。在极角细化的基础上,进一步研究偏倚求积组以提高计算效率,并开展相关收敛分析。对国际权威基准题Kobayashi的测试分析表明,极角细化技术可有效提高带有孔道屏蔽问题的计算精度。     

A parallel algorithm with interface prediction and correction for spherical geometric transport equation

When solving transport equation by the discrete ordinates method, the solution procedure must accord with the moving direction of particle due to the stability restriction. Thus it results in a strong correlation among these unknown variables, and the intrinsic serial nature of the corresponding algorithmic procedure becomes the bottleneck for transport equation calculations on large-scale parallel computers. The conventional parallel sweep method has restrictive but in no means high degree of parallelism for two and three spatial dimensional problems, and it has no spatial parallel degree for one-dimensional problems. In this paper, a spatial domain decomposition method is adopted in the computational domain, and different interface prediction and correction methods are introduced to solve the 1-D spherical geometric transport equation. It is natural to generalize to multidimension problems. In order to avoid producing negative flux, we employ the exponential method for variable discretization. The numerical experiments by using MPI show that our parallel method with explicit prediction and implicit correction has good precision, parallelism and simplicity.