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.     

An Improvement of the Transverse Leakage Treatment for the Nodal SN Transport Calculation Method in Hexagonal-Z Geometry

To overcome the divergent behavior of the NSHEX code, a nodal S_N code for hexagonal geometry, for some transport calculations, an improvement has been made in the calculation of the axial leakage. The axial leakage, previously calculated by using the quadratic transverse leakage approximation (QLA), is calculated by a new method of analytically treating the spatial distribution within a node, based on the axial homogeneity of the ordinary core. The verification tests were performed for the KNK-II model geometry of the NEACRP 3-D Neutron Transport Benchmarks and the large assembly-size KNK-II model. It is found that k_ett values obtained by introducing the new method agree with the reference Monte Carlo calculation results within 0.1% Δk/k for the KNK-II model, although the QLA method did not converge for two cases. Furthemore the new method succeeded in calculations for the large assembly-size model, in which the QLA method failed for all cases. Thus the new method has been found accurate and convergence achieved for the cases in which the QLA method failed.     

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.     

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.     

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.     

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.     

Spherical Harmonics Solutions of Multi-Dimensional Neutron Transport Equation by Finite Fourier Transformation

A method of solution of a monoenergetic neutron transport equation in P_L approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem.     

Evaluation of Critical Bucklings of Light-Water Moderated Low Enriched UO2 Cores by the Variable Loading Method

Abstract

Horizontal extrapolated distances Xh and critical bucklings B² _C for light-water moderated low enriched UO² cores were evaluated using the variable loading method. A set of critical height (water levels) of the cores having various horizontal dimensions was used in this method. Errors of the present method were smaller than the previous ones by the flux shape method in which the horizontal extrapolated distance was determined from the horizontal power distribution. The presently obtained Xh and B² _C reproduced well the change in the critical water level for the whole range of the horizontal core size, while the previous ones could be used only for a limited range. Moreover, the effective multiplication factors obtained through the cell calculation of the SRAC code by using the presently evaluated critical bucklings agreed much better with the rigorous ones by the continuous energy Monte Carlo calculation using a full core model, than those by using the previous values. Therefore, it was concluded that the variable loading method can give more accurate values for Xh and B² _C than the flux shape method for the cores investigated.     

The finite-element method for multigroup neutron transport: Anisotropic scattering in 1-D slab geometry

Proof-tests on 1-D multigroup neutron transport problems are reported for strong anisotropic scattering. These tests have been undertaken as part of the validation of the 3-D multigroup finite-element transport code fel tran for ansisotropic scattering media. To illustrate the treatment of within-group and intergroup anisotropic scattering in the finite-element method the relevant theory is outlined. Ingroup scattering is checked using the backward-forward-isotropic (BFI) scattering law for source and eigenvalue problems. With this law anisotropic scattering problems can be transformed into equivalent isotropic scattering problems. In this way the well-validated isotropic scattering version of fel tran is used to validate the anisotropic version. Intergroup scattering effects are checked by solving few-group source problems for P₁ and P₃ scattering and the BFI scattering law. For P₁ and P₃ scattering checks are made with the discrete-ordinate finite-difference code anisn and the spherical harmonics finite-difference code marc/pn. For the BFI scattering law comparison is made with two-group exact solutions of Williams (1985) for 1-D systems.     

Numerical Solution to Space-Angle Energy-Dependent Neutron Integral Transport Equation

A numerical approach to the steady-state, space-, angle- and energy-dependent neutron transport equation is presented for neutron shielding calculations. The scattering integral, with anisotropic treatment of elastic scattering and isotropic treatment of inelastic scattering, is evaluated by the use of Gaussian and straightforward quadratures. A system of coupled one-group integral equations for all the energy meshes of interest, converted from the energy-dependent integral transport equation, is calculated by performing a line integration along the neutron path in the direction of motion. For this purpose the direction of neutron motion is represented by discrete-ordinate directions Ω_pq on the unit sphere.

The final presentation of the integral transport equation is derived in a difference form convenient for machine computation. A computation program PALLAS has been written in Fortran IV for IBM 360-75 computer to perform neutron transport calculations based on this approach.

Comparisons are given of the numerical solutions with analytical solutions for unscattered fluxes in various geometries such as plane, spherical and two-dimensional cylindrical, for volume sources with self absorption, and with experimental spectra for angular neutron fluxes in graphite-, polyethylene- and water-shield. Excellent agreement is obtained between the present calculations and analytical or experimental results.     

Method for Condensation of the Macroscopic Transport Cross-Sections for Criticality Analyses of FBR MONJU by the Code NSHEX

The problem with the energy-group approximation effect in the criticality analysis of the FBR MONJU by the discrete-ordinate transport code NSHEX has been studied. In order to reduce the existing energy-group collapsing effect in the 18 and 7-group results for the effective multiplication factor, a new algorithm for condensation of the macroscopic transport cross-sections has been proposed and verified. This work presents the definition of the new collapsing algorithm, results from the verification tests and a short discussion from the viewpoint of consistency of the algorithm with the specific finite-difference method of the code NSHEX. According to the presented results, the new collapsing algorithm can be recommended for condensation of the transport cross-sections from 70 into fewer energy-group structures.     

Improvement of Neutron Source Introduction Method for Absolute Measurements of Low Reactor Power

The neutron source introduction method was applied to absolute measurements of low reactor power at the Static Experiment Critical Facility STACY. To obtain the effective neutron source intensity more accurately, which is a key parameter for the source introduction method, the neutron source is newly defined as fission neutrons from the first fission reaction caused by neutrons emitted from the external neutron source. To obtain the newly defined effective neutron source intensity, the probability that a neutron from the external neutron source causes a fission reaction is calculated using the Monte Carlo code MCNP. This calculation took into consideration the three-dimensional complicated core structures. Furthermore, the fission reaction distribution, fundamental mode forward and adjoint flux distribution in a critical state were calculated using the three-dimensional transport code THREEDANT. Following the principle of the neutron source introduction method, an external neutron source was inserted near the STACY core tank and the reactor power was measured. The reactor powers by the neutron source introduction method were in good agreement with the ones from the analyses of the FP activity generated by high power operation.     

Recent activities in the field of nuclear waste management and environmental science

The multiband method has been applied to analyses of critical experiments related to the high-conversion core at the Kyoto University Critical Assembly in order to accurately treat the resonance self-shielding in heterogeneous cells. Three-band parameters were generated using the self-shielding table installed in the SRAC code, and used to calculate the cell-averaged cross sections. The k _∞ values calculated by this method have been compared to those by the VIM Monte-Carlo calculation, the SRAC fine group calculation, Dancoff factor method and/or Tone's method self-shielding calculation. The k∞ values calculated by the present method agree with those by the VIM calculation within 0.3%Δk for all the cases considered.     

The Sandwich Method for Determining Source Convergence in Monte Carlo Calculation

In slow source convergence problems, it is often difficult to ascertain whether the source iteration has converged or not. In order to solve this problem, a new “sandwich method” has been proposed. The essence of this method is that a finally converged eigenvalue k_eff is approached starting from two kinds of initial source guesses which give higher and lower neutron multiplication factors. It is especially important for evaluating nuclear criticality safety to know how to choose a biasing source to obtain an upper limit for k_eff . In this paper, (1) an example is shown to explain the difficulties in ascertaining the source convergence, (2) a method is proposed to obtain the upper and lower limit curves for k_eff by biasing the initial source distribution, (3) the sandwich method is applied to four benchmark problems proposed by the source convergence group of the OECD/NEA Working Party on Nuclear Criticality Safety.

Our calculation results show that the sandwich method is an effective means to confirm source convergence in such slow convergence problems. Appendix is prepared to support the method theoretically.     

Biased Selection of Particle Flight Directions as a Variance Reduction Technique in Monte Carlo Calculations

As a practical variance reduction technique applicable to Monte Carlo shielding calculations, the present article shows a new simple biased sampling technique on particle flight directions. Scattered particles not directed towards the detector positions are killed if they are not so important, that is, if the particle weights are sufficiently small compared to the source weight. In this way, we can reduce the sample size required for obtaining an accurate estimate for the detector response.

The present technique was incorporated into the multigroup neutron and γ-ray transport code MORSE, and sample calculations were performed on spherical fast neutron systems. The results have shown that this biased technique is effective for dealing with neutron multiplication as well as neutron transmission problems. The neutron flux or the effective multiplication factor of a nuclear reactor is estimated more accurately than from the method of path-length stretching with about the same computation time. In addition, it is shown that the flight-direction biasing can further effectively be used by combining it with other variance reduction techniques.     

离散节块输运方法在二维柱坐标系中的应用

讨论粗网离散节块法在二维柱坐标系内中子输运方程数值求解中的应用.给出方法的数学模型,在节块内中子通量采用二次近似,表面泄漏采用常数近似.根据所提出的理论模型,编制了考虑各向异性散射的二维节块输运程序DNSN/2D,并对一系列检验和基准问题作了计算.计算结果表明,与传统S_N方法相比,它具有很高的计算效率,在很粗的节块分割下具有非常高的精度.     

Numerical Solution of Time-Dependent Two-Dimensional P 1 Equation of Neutron Transport

The time-dependent P ₁ equation for two-dimensional neutron transport is numerically solved by a finite difference approximation of the explicit form along the bicharacteristics of the P ₁ equation.

Applying von Neumann's stability condition to this numerical procedure in an infinite space, we can derive the condition necessary for the solution to be stable. This condition is that the mesh widths satisfy the inequality o<λ≦√3/2 with λ=time mesh δt/space mesh δ or δz, where the time t is measured in units of inverse neutron speed l/v. The sufficient stability condition on the ratio λ is to be determined by numerical experiments. It has been found that the upper bound of λ becomes larger for smaller values of space mesh width.

In respect of the stability of numerical solution, the P₁ approximation is more advantageous than the diffusion approximation.

Transient behavior of neutron flux distribution due to a stationary neutron source is numerically determined assuming zero initial values. After the transient state terminates, the steady state distribution is obtained.     

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.     

Reduction of the Spatial Discretization Error in the Method of Characteristics using the Diamond-difference Scheme

In this paper, the diamond-difference (DD) scheme, which is commonly used in discrete-ordinate codes, is applied to the method of characteristics (MOC) to reduce the spatial discretization error of the flat flux approximation. Smaller spatial discretization error allows coarser background mesh division, which leads to smaller computational burden. Some theoretical considerations on the DD scheme are discussed to clarify the strength of this method. An absorption cross section weighted DD scheme (AWDD), which utilizes macroscopic absorption cross section to set the weight, is also discussed. The DD and AWDD schemes are implemented to AEGIS, which is a lattice physics code based on MOC. Then the AEGIS code is applied to two different benchmark problems whose spatial discretization errors are large. The calculation results indicate that from the viewpoint of spatial discretization error, the AWDD scheme is superior to the conventional MOC in which the step characteristics approximation is commonly used. Since incorporation of the AWDD scheme to current MOC codes is very simple, it will be a good candidate of spatial discretization method for MOC codes.     

基于OpenMP的中子输运方程特征线法并行计算研究

特征线法是目前求解反应堆中子输运方程的主要计算方法之一。本文开发了基于OpenMP的中子输运方程特征线法并行计算程序,以提高特征线法的计算效率。OpenMP是共享存储体系结构上的一个并行编程模型,采用Fork-Join并行执行方式,适合于SMP共享内存多处理系统和多核处理器体系结构。通过相关基准题测试验证,表明所开发的程序在有效增殖因数以及相对中子通量(归一化栅元功率)分布等参数上都能取得良好的精度,且使用OpenMP能取得良好的加速效果,使计算时间显著减少。