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.     

The analytic nodal method in cylindrical geometry

Nodal diffusion methods have been used extensively in nuclear reactor calculations, specifically for their performance advantage, but also for their superior accuracy. More specifically, the Analytic Nodal Method (ANM), utilising the transverse integration principle, has been applied to numerous reactor problems with much success. In this work, a nodal diffusion method is developed for cylindrical geometry. Application of this method to three-dimensional (3D) cylindrical geometry has never been satisfactorily addressed and we propose a solution which entails the use of conformal mapping. A set of 1D-equations with an adjusted, geometrically dependent, inhomogeneous source, is obtained. This work describes the development of the method and associated test code, as well as its application to realistic reactor problems. Numerical results are given for the PBMR-400 MW benchmark problem, as well as for a “cylindrisized” version of the well-known 3D LWR IAEA benchmark. Results highlight the improved accuracy and performance over finite-difference core solutions and investigate the applicability of nodal methods to 3D PBMR type problems. Results indicate that cylindrical nodal methods definitely have a place within PBMR applications, yielding performance advantage factors of 10 and 20 for 2D and 3D calculations, respectively, and advantage factors of the order of 1000 in the case of the LWR problem.     

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.     

Analytical reconstruction scheme for the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates transport model in slab geometry

Coarse-mesh numerical methods are very efficient in the sense that they generate accurate results in short computational time, as the number of floating point operations generally decrease, as a result of the reduced number of mesh points. On the other hand, they generate numerical solutions that do not give detailed information on the problem solution profile, as the grid points can be located considerably away from each other. In this paper we describe two steps for the analytical reconstruction of the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates (S_N) transport model in slab geometry. The first step of the algorithm is based on the analytical reconstruction of the coarse-mesh solution within each discretization cell of the grid set up on the spatial domain. The second step is based on the angular reconstruction of the discrete ordinates solution between two contiguous ordinates of the angular quadrature set used in the S_N model. Numerical results are given so we can illustrate the accuracy of the two reconstruction techniques, as described in this paper.     

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.     

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.     

Scalar product formulas for nodal methods in hexagonal-z geometry

A polynomial expansion technique is proposed to reconstruct the neutron flux inside the nodes of a coarse-mesh nodal method in hexagonal-z geometry. It is shown to be a valid approach to derive the scalar product formulas needed when coupling a nodal method with the quasistatic formalism to solve reactor dynamics problems.     

An efficient parallel algorithm of variational nodal method for heterogeneous neutron transport problems

The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-fidelity models has been challenging due to the prohibitive computational costs.This paper presents an efficient parallel algorithm tailored for HVNM based on the Message Passing Interface standard.The algorithm evenly distributes the response matrix sets among processors during the matrix formation process,thus enabling independent construction without communication.Once the formation tasks are completed,a collective operation merges and shares the matrix sets among the processors.For the solution process,the problem domain is decomposed into subdomains assigned to specific processors,and the red-black Gauss-Seidel iteration is employed within each subdomain to solve the response matrix equation.Point-to-point communica-tion is conducted between adjacent subdomains to exchange data along the boundaries.The accuracy and efficiency of the parallel algorithm are verified using the KAIST and JRR-3 test cases.Numerical results obtained with multiple processors agree well with those obtained from Monte Carlo calculations.The parallelization of HVNM results in eigenvalue errors of 31 pcm/-90 pcm and fission rate RMS errors of 1.22％/0.66％,respectively,for the 3D KAIST problem and the 3D JRR-3 problem.In addition,the parallel algorithm significantly reduces computation time,with an efficiency of 68.51％using 36 processors in the KAIST problem and 77.14％using 144 processors in the JRR-3 problem.     

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.     

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.     

求解中子输运SP3方程的半解析节块方法

中子输运简化P3 (SP3)方法是对中子输运方程PN的一种近似,可以转换为与中子扩散方法相似的形式.采用节块方法中有效的半解析方法求解中子输运SP3方程,同时也基于粗网有限差分(CMFD)方法采用细网有限差分(FMFD)形式同样求解该方程.通过对NEACRP-L-336基准题(修改)的数值计算,验证了通过Pin-By-Pin的节块计算能够获得与FMFD几乎相同的结果,而Pin-By-Pin的CMFD计算结果与FMFD计算结果有一定的偏差.     

Variational nodal transport solutions of multigroup criticality problems

Variational nodal methods are extended to treat multigroup and criticality problems and are implemented as a module for the production code DIF3D at Argonne National Laboratory. New within-group solution algorithms based on the partitioning of the response matrix equations result in computing times that are comparable to those obtained with a widely employed nodal transport method. Accuracy and timing comparisons are made for three few-group criticality problems in X-Y geometry.     

NODHEX: A 2-D diffusion code based on the nodal expansion method in hexagonal geometry and its validation

A 2-D neutron diffusion theory computer code NODHEX for hexagonal geometry has been developed. The nodal algorithm is based on the nodal expansion method proposed by Lawrence. The nodal equation formulation is accomplished by using a second-order polynomial approximation for the flux. The equations include additional terms of discontinuity which occur in the expression of transverse leakage for the hexagonal geometry, unlike the nodal equations (using a second-order polynomial approximation) formulated by Lawrence. The code has been validated by comparing its predictions for the SNR-300 and VVER-1000 benchmarks with the results of other standard computer codes like DIF3D and SNAP. The inclusion of the additional terms of discontinuity is found to improve the predictions relative to Lawrence's predictions, though the same second-order polynomial approximation was used for solving the nodal equations.     

A posteriori error estimator and AMR for discrete ordinates nodal transport methods

In the development of high fidelity transport solvers, optimization of the use of available computational resources and access to a tool for assessing quality of the solution are key to the success of large-scale nuclear systems’ simulation. In this regard, error control provides the analyst with a confidence level in the numerical solution and enables for optimization of resources through Adaptive Mesh Refinement (AMR).     

An analytical approach for a nodal scheme of two-dimensional neutron transport problems

In this work, a solution for a two-dimensional neutron transport problem, in cartesian geometry, is proposed, on the basis of nodal schemes. In this context, one-dimensional equations are generated by an integration process of the multidimensional problem. Here, the integration is performed for the whole domain such that no iterative procedure between nodes is needed. The ADO method is used to develop analytical discrete ordinates solution for the one-dimensional integrated equations, such that final solutions are analytical in terms of the spatial variables. The ADO approach along with a level symmetric quadrature scheme, lead to a significant order reduction of the associated eigenvalues problems. Relations between the averaged fluxes and the unknown fluxes at the boundary are introduced as the usually needed, in nodal schemes, auxiliary equations. Numerical results are presented and compared with test problems.     

Improved quasi—static nodal Green‘s function method

To solve the multi-dimensional transient neutron diffusion equations,improved quasi-static Green‘s function method(IQS/NEFM) is adopted to deal with the temporal problesm,which will increase the time step as long as possible so as to decrease the number of times of spatial calculation.The time step of IQS/NGFM can be increased to 5-10 times longer than that of full implicit differential method.In spatial calculation.The theory of NGFM is used to get the distribution of shape function,with coarse meshes which can be nearly 20 times larger than that of traditional finite differential method.So the IQS/NGFM is considered as an efficinet kinetic method.     

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 spatial rehomogenization method in nodal calculations

Infinite medium flux weighted cross-sections used in nodal calculations enable equivalence with the corresponding fine configuration if the following condition is satisfied: the flux shape inside the assembly in the core is close to the infinite medium flux shape (computed in lattice calculations). In presence of big flux gradients this condition is not satisfied and the absence of information about cross-sections distributions inside a node does not permit to predict the reaction rates with the same accuracy attained in ordinary situations. This tendency is amplified in case of high heterogeneous regions where tilting the flux causes big changes in reaction rates. The method presented here uses information coming from the lattice calculations that produced the homogenized cross-sections, in order to predict the right reaction rate even in presence of high tilted flux shapes. This is done in evaluating a variation of the cross-sections equivalent to the variation in reaction rate, but no variation is applied to the discontinuity factors. The accuracy of the method and its limitations are shown in several significant configurations. Its implementation in the Areva NP reactor core simulation system SCIENCE has shown better evaluation of control rod worth in comparisons with experimental results.