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.     

Solution of Multi-Group Diffusion Equation in x-y-z Geometry by Finite Fourier Transformation

The multi-group diffusion equation in three-dimensional x-y-z geometry is solved by finite Fourier transformation. Applying the Fourier transformation to a finite region with constant nuclear cross sections, the fluxes and currents at the material boundaries are obtained in terms of the Fourier series. Truncating the series after the first term, and assuming that the source term is piecewise linear within each mesh box, a set of coupled equations is obtained in the form of three-point equations for each coordinate. These equations can be easily solved by the alternative direction implicit method. Thus a practical procedure is established that could be applied to replace the currently used difference equation.

This equation is used to solve the multi-group diffusion equation by means of the source iteration method; and sample calculations for thermal and fast reactors show that the present method yields accurate results with a smaller number of mesh points than the usual finite difference equations.     

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.     

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.     

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.     

三维六角形节块多群中子扩散程序NDHEX

本文介绍用DIF3D (NOD)求解二、三维六角形几何系统下中子扩散方程的理论模型及数值计算方法。六角形节块内的中子通量密度分布采用高次多项式近似表示,最后导出通量矩方程及偏流的响应矩阵方程。应用粗网再平衡和渐近源外推方法加速收敛。参考此方法编制了计算程序NDHEX,并对一些六角形基准问题进行了计算。结果表明:NDHEX的计算结果与DIF3D(NOD)的计算结果符合很好;与差分程序相比,具有更高的精度与计算效率。它可用于快堆计算。     

Solution of group diffusion equation for X-Y geometry by finite Fourier transformation

A new difference equation to the two dimensional diffusion equation for x-y geometry is derived by using the finite Fourier transformation. This difference equation has a form of a coupled equation of the 3 point difference equations for each coordinate, and can be easily solved by the iterative method of the alternative direction implicit method. Group diffusion equations are solved using this difference equation and sample calculations show that accurate results can be obtained with less mesh points than the usual 5 points difference equation.     

Solution of Diffusion Equation in r-z Geometry by Finite Fourier Transformation

It is shown that the monoenergetic diffusion equation in multi-region r-z geometry can be solved by the finite Fourier transformation method which has successfully been applied to x-ygeometry. In this method, a system of linear algebraic equations is derived for Fourier coefficients of fluxes and currents at the material boundaries between regions of constant cross sections, and all the boundary values are determined by solving this equation.

Numerical examples are presented for a problem featuring a fixed source and multiple regions, and the results are compared with those obtained from the current difference method. It is shown that the present method yields a better result with relatively few terms of expansion.     

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.     

Transport consistent diffusion coefficient for CMFD acceleration and comparison of convergence properties

A diffusion coefficient for the coarse mesh finite difference (CMFD) acceleration is derived from the semi-analytic solution of one-group, one-dimensional, even-parity transport equation. The derived diffusion coefficient, i.e., the transport consistent diffusion coefficient (TCD), depends on the optical length of a mesh and shows similar behavior with the artificial grid diffusion (AGD) and the effective diffusion (EffD) coefficients for an optically thick mesh. Convergence properties of typical diffusion coefficients are evaluated using the linearized Fourier analysis. Analyses of the C5G7 3D benchmark problems with and without voided region are carried out to compare the convergence properties. The number of transport sweeps to reach convergence using TCD is smaller than that using EffD or AGD.     

Thermodynamic Isotope Effect of Lithium,Carbon, Nitrogen,Chlorine, Copper and Uranium Compounds

Few studies have so far been reported on isotopic frequency shifts, which present problems in experimentation. In this paper, the G and F matrix method is applied to theoretical calculation of the frequency shifts of linear molecules XYZ and XYZ, as well as of octahedral molecules XY ₂ Using these values, reduced partition function ratios of lithium, carbon, copper, nitrogen, chlorine and uranium compounds have been calculated.     

CFD investigation of vertical rod bundles of supercritical water-cooled nuclear reactor

The commercial CFD code STAR-CD v4.02 is used as the numerical simulation tool for the supercritical water-cooled nuclear reactor (SCWR). The numerical simulation is based on the real full 3D rod bundles’ geometry of the nuclear reactors. For satisfying the near-wall resolution of y⁺ ≤ 1, the structure mesh with the stretched fine mesh near wall is employed. The validation of the numerical simulation for mesh generation strategy and the turbulence model for the heat transfer of supercritical water is carried out to compare with 3D tube experiments. After the validation, the same mesh generation strategy and the turbulence model are employed to study three types of the geometry frame of the real rod bundles. Through the numerical investigations, it is found that the different arrangement of the rod bundles will induce the different temperature distribution at the rods’ walls. The wall temperature distributions are non-uniform along the wall and the values depend on the geometry frame. At the same flow conditions, downward flow gets higher wall temperature than upward flow. The hexagon geometry frame has the smallest wall temperature difference comparing with the others. The heat transfer is controlled by P/D ratio of the bundles.     

Applications of the multidimensional equations to complex fuel assembly problems

For the analysis of reactors with complex fuel assemblies or fine mesh applications as pin by pin neutron flux reconstruction, the usual approximation of the neutron transport equation by the multigroup diffusion equation does not provide good results. A classical approach to solve the neutron transport equation is to apply the spherical harmonics method obtaining a finite approximation known as the P_L equations. In this line, a nodal collocation method for the discretization of these equations on a rectangular mesh is used in this paper to analyse reactors with MOX fuel assemblies. Although the 3D P_L nodal collocation method becomes feasible due to the improvements in computer hardware, a complete treatment of the detailed structure of the fuel assemblies in actual three-dimensional geometry is still prohibitive, thus, an assembly homogenization method is necessary. A homogenization method compatible with our multidimensional P_L code is proposed and tested performing heterogeneous and homogenized calculations. In this work, we apply the method to 2D complex fuel assembly configurations.     

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.     

On the Donnell's equations in the creep range

This paper is concerned with assessing the accuracy of Donnell's approximation when employed in the creep analysis of a class of circular cylindrical shells. Basic formulation of a general method describing the creep behaviour of two-dimensional cylindrical shells is first presented. The terms affected by Donnell's approximation are then pointed out. The solution of governing equations is obtained through coupling the ‘extended Newton's method’ and finite difference technique in an iterative procedure. A number of examples having geometries falling within the shallow shell definition, around the limit, and beyond the range of applicability, are solved using both theories. It has been noted that the parameter α, representing the shell geometry, has a pronounced effect on the accuracy of Donnell's simplification. As α increases the deviation between the two theories decreases. It is concluded that for the class of circular cylindrical shells considered herein Donnell's approximations yield accurate results for creep analysis, particularly for higher values of creep exponent n. Of course, employing Donnell's approximations results in simpler formulation and a reduction in computational time.     

Application of Hexagonal Element Scheme in Finite Element Method to Three-Dimensional Diffusion Problem of Fast Reactors

The finite element method is applied in Galerkin-type approximation to three-dimensional neutron diffusion equations of fast reactors. A hexagonal element scheme is adopted for treating the hexagonal lattice which is typical for fast reactors. The validity of the scheme is verified by applying the scheme as well as alternative schemes to the neutron diffusion calculation of a gas-cooled fast reactor of actual scale. The computed results are compared with corresponding values obtained using the currently applied triangular-element and also with conventional finite difference schemes.

The hexagonal finite element scheme is found to yield a reasonable solution to the problem taken up here, with some merit in terms of saving in computing time, but the resulting multiplication factor differs by 1% and the flux by 9% compared with the triangular mesh finite difference scheme. The finite element method, even in triangular element scheme, would appear to incur error in inadmissible amount and which could not be easily eliminated by refining the nodes.     

Discretizing the diffusion equation on unstructured polygonal meshes in two dimensions

We derive a discretization of the two-dimensional diffusion equation for use with unstructured meshes of polygons. The scheme is presented in r–z geometry, but can easily be applied to x–y geometry. The method is “node”- or “point”-based and is constructed using a finite volume approach. The scheme is designed to have several important properties, including second-order accuracy, convergence to the exact result as the mesh is refined (regardless of the smoothness of the grid), and preservation of the homogeneous linear solution. Its principle disadvantage is that, in general, it generates an asymmetric coefficient matrix, and therefore requires more storage and the use of non-traditional, and sometimes slowly-converging, iterative linear solvers. On an unstructured triangular grid in x–y geometry, the scheme is equivalent to the linear continuous finite element method with “mass-matrix lumping”. We give computational examples that demonstrate the accuracy and convergence properties of the new scheme relative to other schemes.     

Influence of the external neutron sources in the criticality prediction using 1/M curve

The influence of external neutron sources in the process to obtain the criticality condition is estimated. To reach this objective, the three-dimensional neutron diffusion equation in two groups of energy is solved, for a subcritical PWR reactor core with external neutron sources. The results are compared with the solution of the corresponding problem without external neutron sources, that is an eigenvalue problem. The method developed for this purposes it makes use of both the nodal method (for calculation of the neutron flux) and the finite differences method (for calculation of the adjoint flux). A coarse mesh finite difference method was developed for the adjoint flux calculation, which uses the output of the nodal expansion method. The results regarding the influence of the external neutron source presence for attaining criticality have shown that far from criticality it is necessary to calculate the reactivity values of the system.     

Simulating control rod and fuel assembly motion using moving meshes

A prerequisite for designing a transient simulation experiment which includes the motion of control and fuel assemblies is the careful verification of a steady state model which computes k_eff versus assembly insertion distance. Previous studies in nuclear engineering have usually approached the problem of the motion of control rods with the use of nonlinear nodal models. Nodal methods employ special approximations for the leading and trailing cells of the moving assemblies to avoid the rod cusping problem which results from the naive volume weighted cell cross-section approximation. A prototype framework called the MOOSE has been developed for modeling moving components in the presence of diffusion phenomena. A linear finite difference model is constructed, solutions for which are computed by SLEPc, a high performance parallel eigenvalue solver. Design techniques for the implementation of a patched non-conformal mesh which links groups of sub-meshes that can move relative to one another are presented. The generation of matrices which represent moving meshes which conserve neutron current at their boundaries, and the performance of the framework when applied to model reactivity insertion experiments is also discussed.