Transmission probability method based on triangle meshes for solving unstructured geometry neutron transport problem

In the advanced reactor, the fuel assembly or core with unstructured geometry is frequently used and for calculating its fuel assembly, the transmission probability method (TPM) has been used widely. However, the rectangle or hexagon meshes are mainly used in the TPM codes for the normal core structure. The triangle meshes are most useful for expressing the complicated unstructured geometry. Even though finite element method and Monte Carlo method is very good at solving unstructured geometry problem, they are very time consuming. So we developed the TPM code based on the triangle meshes. The TPM code based on the triangle meshes was applied to the hybrid fuel geometry, and compared with the results of the MCNP code and other codes. The results of comparison were consistent with each other. The TPM with triangle meshes would thus be expected to be able to apply to the two-dimensional arbitrary fuel assembly.     

Least-squares finite-element SN method for solving three-dimensional transport equation

A discrete ordinates finite-element method for solving three-dimensional first-order neutron transport equation is proposed using a least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation. Different from using the standard Galerkin variation applying to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. The approach allows a continuous finite-element. To eliminate the discontinuity of the angular flux on the fixed flux boundary in the spherical harmonics method, the equation is discretized using the discrete ordinates method for angular dependency. A three-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results demonstrate the accuracy and feasibility of the 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.     

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.     

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.     

An improved three-dimensional wavelet-based method for solving the first-order Boltzmann transport equation

A new angular discretization scheme based on the Daubechies’ wavelets has been developed in recent studies. A decoupled S_N and wavelet expansion method was proposed. This paper discusses the limitations and improvements of this decoupled scheme. The scaling function, instead of the wavelet function, is applied as the basis function. It significantly improved the efficiency and computational stability. A new series of wavelets on the interval are applied instead of the ‘wrapped wavelets’, which eliminate the edge effect in the angular subdomain scheme. Based on the improvements, a wavelet-based neutron transport code package WAVTRAN is developed and the previous work is extended to the three-dimensional calculation and anisotropic scattering calculation. Numerical results demonstrate that the improvements are effective. Further investigations demonstrate that the wavelet-based angular discretization scheme is more powerful than the traditional ones in some highly anisotropic angular flux problems.     

A nodal SN transport method for three-dimensional triangular-z geometry

A new nodal S_N transport method has been developed to perform accurate transport calculation in three-dimensional triangular-z geometry, where arbitrary triangles are transformed into regular triangles via a coordinate transformation. The transverse integration procedure is applied to treat the neutron transport equation in the regular triangle. The neutron angular distributions of intra-node fluxes are represented using the S_N quadrature set, and the spatial distributions of neutron fluxes and sources are approximated by a quadratic polynomial. The nodal-equivalent finite difference algorithm for 3D triangular geometry is applied to establish a stable and efficient iterative scheme. The present method was tested on four 3D Takeda benchmark problems published by the nuclear data agency (NEACRP), in which the first three problems are in XYZ geometry and the last one is in hexagonal-z geometry. The results of the present method agree well with those of the reference Monte-Carlo calculation method, the difference in k_eff being less than 0.1%. This shows that multi-group reactor core/criticality problems can be accurately and effectively solved using the present method.     

On the history of a stochastic ansatz for solving the transport equation

A very useful approximate tool for understanding the role of random material properties on solutions of the transport equation is described and its historical derivation given. The development of this stochastic tool, from its introduction by Randall, to its use in describing current problems involving dichotomic or pseudo-dichotomic Markov processes is discussed.     

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.     

A new three-dimensional method of characteristics for the neutron transport calculation

The method of characteristics (MOC) is a very flexible and effective method for the neutron transport calculation in a complex geometry. It has been well developed in two-dimensional geometries but meets serious difficulty in three-dimensional geometries because of the requirements of large computer memory and long computational time. Due to the demand related to the advanced reactor design for complex geometries, an efficient and flexible three-dimensional MOC is needed. This paper presents a modular ray tracing technique to reduce the amount of the ray tracing data and consequently reduce the memory. In this method, the object geometry is dissected into many cuboid cells by a background mesh. Typical geometric cells are picked out and ray traced, and only the ray tracing data in these typical cells is stored. Furthermore, the Coarse Mesh Finite Difference (CMFD) acceleration method is employed to save computing time. Numerical results demonstrate that the modular ray tracing technique can significantly reduce the amount of ray tracing data, and the CMFD acceleration is effective in shorting the computing time.     

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].     

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.     

The spherical harmonics method for the multigroup transport equation in x−y geometry

A spherical harmonics equation in the form of a second-order differential equation is derived for the 2-D x−y geometry, including higher-order scattering within a group. Using this equation, a multigroup transport code for the spherical harmonics method of a general order of approximation is developed. Some numerical examples, including typical problems for the ray effect, are presented and compared with those obtained by the discrete-ordinates method. It is shown that the present method gives more accurate results than the discrete-ordinates method, although this spherical harmonics code requires more computer memory than the discrete-ordinates code.     

A multigroup finite-element solution of the neutron transport equation—I: X-Y geometry

The result of extending a variational finite element method of solving the neutron transport equation, to energy dependence, is reported. Detailed results are given, in the form of tables and graphs, of P₁ and higher-order transport solutions to a number of benchmark problems in X-Y geometry. The accuracy and flexibility of the method are demonstrated. Some suggestions are made for the future development of the computer implementation of the method.     

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.     

A three-dimensional collision probability method: Criticality and neutron flux in a hexahedron setup

In this work, the collision probability (CP) method is applied to the solution of the one-group criticality problem for a hexahedron. The method is implemented with the spatial domain being partitioned into zones, where the flat-flux approximation is used. A numerical method is employed for computing the required collision probabilities. The method is based on the subdivision of the spatial zones into elements and on the assumption that the interaction between an emission element and a collision element occurs only along the path that connects their centers of mass. The calculation is repeated, increasing the number of elements successively and using Richardson extrapolation to accelerate the convergence of the sequence of results. Some sample cases are studied in detail and a comparison with results from diffusion theory is presented.     

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.