Transmission probability method for solving neutron transport equation in three-dimensional triangular-z geometry

This paper presents a transmission probability method (TPM) to solve the neutron transport equation in three-dimensional triangular-z geometry. The source within the mesh is assumed to be spatially uniform and isotropic. At the mesh surface, the constant and the simplified P₁ approximation are invoked for the anisotropic angular flux distribution. Based on this model, a code TPMTDT is encoded. It was verified by three 3D Takeda benchmark problems, in which the first two problems are in XYZ geometry and the last one is in hexagonal-z geometry, and an unstructured geometry problem. The results of the present method agree well with those of Monte-Carlo calculation method and Spherical Harmonics (P_N) method.     

Improved nodal expansion method for solving neutron diffusion equation in cylindrical geometry

The challenges encountered in the development of nodal expansion method (NEM) in cylindrical geometry and the method to circumvent these difficulties are introduced and discussed in this paper. Due to the fact that the azimuthal term contains a factor 1/r², the traditional transverse integration fails to produce a 1D transverse integrated equation in θ-direction; a simple but effective approach is employed to obtain the θ-directional transverse integration equation. When the traditional polynomials are used to solve the 1D transverse integral equation in r-direction, some additional approximations, which may undermine the precision of the method, are required in the derivation of the moment equations; in order to preserve the accuracy of calculations, the special polynomial approximation is used to solve the 1D transverse integrated equations in r-direction. Moreover, the Row-Column iterative scheme, which is considered to be the more efficient and convenient schemes in cylindrical geometry, is used to solve the partial currents equations. An improved NEM for solving the multidimensional diffusion equation in cylindrical geometry is implemented and tested. And its accuracy and efficiency are demonstrated through several benchmark problems.     

Solution of the neutron transport problem with anisotropic scattering in cylindrical geometry by the decomposition method

An analytical solution has been obtained for the one-speed stationary neutron transport problem, in an infinitely long cylinder with anisotropic scattering by the decomposition method. Series expansions of the angular flux distribution are proposed in terms of suitably constructed functions, recursively obtainable from the isotropic solution, to take into account anisotropy. As for the isotropic problem, an accurate closed-form solution was chosen for the problem with internal source and constant incident radiation, obtained from an integral transformation technique and the F_N method.     

Spherical harmonics method for neutron transport equation based on unstructured-meshes

Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical harmonics function is used to expand the angular flux. A set of differential equations about the spatial variable, which are coupled with each other, can be obtained. They are solved iteratively by using the finite element method on unstructured-meshes. A two-dimension transport calculation program is coded according to the model. The numerical results of some benchmark problems demonstrate that this method can give high precision results and avoid the ray effect very well.     

Integral transform method for solving multi-dimensional neutron transport problems with linearly anisotropic scattering

A method has been developed to solve the neutron transport equation in multi-dimensional convex and homogenous assemblies with linearly anisotropic scattering. The method consists of solving the Fourier transformed integral transport equations for flux and partial currents and is a generalization of the method developed by Sahni (1972) to treat the one group criticality problem for multi-dimensional geometries. The kernels of the transformed integral equations get factorized into components depending on only one of the dimensions of the assembly. These factorized kernels in each of the dimensions are then decomposed into their respective degenerate forms involving suitable spherical Bessel functions. The transformed flux and partial currents are expanded in a series of products of suitable spherical Bessel functions commensurate with the symmetry and dimensionality of the problem, which can be truncated after very few terms.The one group criticality problem is then converted into the eigenvalue problem of a matrix equation of finite order. The order of this matrix depends upon the truncation order of the transformed flux and partial currents; on the other hand the matrix elements themselves do not depend upon the order of truncation. A striking similarity between one group criticality problem of an infinite rectangular prism of dimensions 2a and 2b along x and y directions and a finite cylinder of diameter 2R and height 2H is brought out, as far as the structure of their matrix equations and calculational procedure of their general matrix elements is concerned. Some results of the criticality problems of the infinite rectangular prism and the finite cylinder are tabulated.     

Flux expansion nodal method for solving multigroup neutron diffusion equations in hexagonal-z geometry

A flux expansion nodal method (FENM) has been developed to solve multigroup neutron diffusion equations in hexagonal-z geometry. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group. In order to improve the nodal coupling relations, a new type of nodal boundary conditions is proposed, which requires the continuity of both the zero- and first-order moments of partial currents across the nodal surfaces. The response matrix technique is used for the iterative solution of the nodal diffusion equations, which greatly improves the computational efficiency. The numerical results for a series of benchmark problems show that FENM is a very accurate and efficient method for the prediction of criticality and nodal power distributions in the reactors with hexagonal assemblies.     

MCU-FCP program for first-collisions probability calculations of neutron transport

A general scheme, methods, and algorithms, implemented in the MCU-FCP program, making it possible to calculate the neutron-physical characteristics of two-and three-dimensional RBMK cells and polycells taking account of fuel burnup during reactor operation by the first-collisions probability method, are described. Test calculations are preformed and the results are compared with the MCU-REA program. Translated from Atomnaya énergiya, Vol. 105, No. 2, pp. 67–72, August, 2008.     

An approximate method for solving radiation and neutron transport problems in spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. This is done by assuming, a priori, a functional form for the stochastic flux in terms of the members of a random set function. For the case of a two-phase medium, two coupled integro-differential equations are obtained for the deterministic functions that arise and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those of the Levermore–Pomraning (LP) theory, but they offer an opportunity to deal with more general forms of stochastic processes. It is also shown that the coupling coefficient between the phase equations is directly proportional to the gradient of the autocorrelation function evaluated at the origin; a feature which has been noted in other fields in which random media occur. By making plausible assumptions about the functional form of the autocorrelation function, different forms of the transport equations can be obtained, according to the structure of the medium. For the one-dimensional case, we may show an exact correspondence with the LP equations. Discussions are given regarding the application of the method to three-dimensional problems for which we expect it to be a good approximation for the mean. We also note that the equations are applicable to realistic problems, such as grains embedded in a background matrix, and not restricted to slabs. Investigations into the variance have also been made and a simple approximation scheme developed which gives reasonable agreement with the simulation results of Adams et al. [Adams, M.L., Larsen, E.W., Pomraning, G.C., 1989. Benchmark results for particle transport in a binary Markov statistical medium. Journal of Quantitative Spectroscopy & Radiative Transfer, 42, 253].     

用综合核方法求解中子输运临界问题的误差分析

基于中子积分输运方程的综合核近似方法,具有准确、快速的特点,其计算精度和收敛性与求积组的选取密切相关.文章简要介绍了求解中子输运临界问题的综合核方法,采用数值方法分析了综合核近似的计算误差和收敛性,并提出了新的求积组来提高综合核方法的计算精度.应用综合核方法计算了均匀平板介质中各向同性和线性各向异性散射的单群、双群中子临界问题,并与离散纵标法S32结果和文献结果进行了比较.计算结果表明采用合适的求积组,综合核方法在低阶时能够得到较高精度的结果.     

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.     

A high-order discontinuous Galerkin method for the SN transport equations on 2D unstructured triangular meshes

This paper presents high-order numerical solutions to the _SN$S_N$ transport equation on unstructured triangular meshes using a Discontinuous Galerkin Finite Element Method (DGFEM). Hierarchical basis functions, up to order 4, are used for the spatial representation of the solution. Numerical results are provided for source-driven and eigenvalue problems. Convergence rates (as a function of the mesh size and CPU time) are discussed.     

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.     

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.     

Two efficient iteration methods for solving inhomogeneous problems of neutron transport theory

This paper concerns the solution of formally inhomogeneous neutron-physical problems in multiplying media, including in nuclear reactors with neutron illumination, for k_eff close to one. The linear neutron-balance equation with an independent right-hand side is studied in a general operator form. Two unconventional methods of constructing relatively rapidly converging iterations are proposed for calculating the neutron distribution function. A characteristic feature of the iteration algorithms described is that the condition k_eff<1 need not be satisfied directly in order for the iterations to converge. An analytic proof of this assertion is given. For k_eff=1, the computational procedure remains meaningful and gives a result that follows from the well-known Fredholm alternative. The iteration schemes described in the paper can, specifically, substantially reduce the computer time required to solve nonstationary reactor problems, which, as a rule, reduce to a sequence of inhomogeneous stationary problems at the sites of a temporal computational grid. Interest in efficient methods for solving inhomogeneous problems in weakly subcritical reactors has arisen in recent years in connection with work on the design of electronuclear power-generating setups. 5 references. State Science Center of the Russian Federation—A. I. Leipunskii Physics and Power Engineering Institute. Translated from Atomnaya énergiya, Vol. 88, No. 3, pp. 163–169, March, 2000.     

The integral-transform method for solving neutron transport problems with general anisotropic scattering in a cylinder of finite height

In this paper, we present a mathematical technique for solving the integral transport equation for the criticality of a homogeneous cylinder of finite height. The purpose of the present paper is two-fold : firstly, to show that our earlier formalism can be generalized to any order of anisotropy, and secondly to generate the numerical results, which could serve as benchmarks when scattering is linearly anisotropic. We expand the scattering function in spherical harmonics to retain the Lth order of anisotropy. Thereafter, we write the integral transport equations for the Fourier-transformed spherical harmonic moments of the angular flux. In conformity with the integral-transform method for multidimensional geometry, the kernels of these integral equations are represented in their respective factorized form, which consists of a series of products of suitable spherical Bessel functions. The Fourier-transformed spherical harmonic moments are also represented in their separable form by expanding them in a series of products of spherical Bessel functions, commensurate with the symmetry of finite cylindrical geometry. The criticality problem for the cylinder of finite height is then posed as a matrix eigen value problem whose eigen vector is composed of the expansion coefficients mentioned above. The general matrix element is expressed as a product of certain integrals of Bessel functions, which can be evaluated by recursion relations derived in this paper. Finally, a comparison between the present benchmark results and S_N results (twotran) in (r–z) goemetry is presented when scattering is linearly anisotropic.     

Implementation of a second-order upwind method in a semi-implicit two-phase flow code on unstructured meshes

A two-phase flow analysis code, CUPID, has been developed for a realistic simulation of thermal–hydraulic phenomena in nuclear reactor components. In the CUPID code, a two-fluid three-field model is adopted and the governing equations are solved on unstructured meshes. To obtain the numerical solution, the semi-implicit method of the RELAP5 code was used with some modifications for a cell-centered finite volume method. In this work, a second-order upwind method was implemented for the convective terms of the CUPID code. To get the slopes of the convective quantities, we adopted the Frink’s reconstruction method (1994, AIAA Paper 94-0061) and modified it for an application to arbitrary polyhedral cells. To stabilize the numerical solutions, the Barth and Jesperson’s slope limiter was used. In order to evaluate the enhanced accuracy and ensure the robustness of the implemented scheme, numerical tests were performed using conceptual single- and two-phase flow problems, which include a strong phase change and very heterogeneous phase distributions.     

Analysis and improvements of the DPN acceleration technique for the method of characteristics in unstructured meshes

Algebraic preconditioners, renumbering techniques and a two-level algebraic multigrid method have been implemented to speed up the Krylov iterations of the DP_N equations used for the acceleration of the method of characteristics in unstructured meshes. These algorithms were customized to take advantage of the cell-based structure of the DP_N equations. Moreover, two techniques to speed up the solution of the multigroup eigenvalue MOC equations have been implemented. A solution of the multigroup eigenvalue DP_N equation has been developed to provide a first guess for the external transport iterations. Next, a multigroup DP_N acceleration method has been developed to accelerate the thermal iterations. This latter development has been particularly useful because our standard multigroup rebalancing acceleration was counterproductive in the presence of heavy absorbents. All these acceleration techniques have been incorporated in the spectral code APOLLO2. Numerical examples and comparisons are given for the 6-group eigenvalue Atrium benchmark problem. Our best calculation, an initialized ILU0-preconditioned DP₁ scheme with thermal acceleration, was 7.7 times faster that the free iteration calculation, while the total number of transport iterations was divided by 17.