Yunzhao Li

Block-diagonalization of the variational nodal response matrix using the symmetry group theory

Abstract To further improve the efficiency of the Variational Nodal Method (VNM) for solving the neutron transport equation in hexagonal-z geometry, the nodal response matrix is further block-diagonalized by utilizing the symmetry group theory to decompose the surface basis functions into irreducible components. The block-diagonal property of the nodal response matrix is determined by the symmetry properties of the hexagonal node in geometry, material and basis functions, including both reflection and rotation symmetries. To fully utilize those properties, the symmetry group theory is employed to analyze the symmetry property of the nodal response matrices. It is mathematically proved that the nodal response matrix can be further block-diagonalized into 16 diagonal blocks instead of the current 4 ones by using the symmetry group theory. Numerical comparisons demonstrate that the new approach can reduce the memory storage and computing time by a factor of 2∼3 for P7 angular approximation, compared with the currently employed variables transformation algorithm.

Preconditioned multigroup GMRES algorithms for the variational nodal method

Abstract Combinations of three approaches are examined as options to replace the algorithms presently employed in the variational nodal code VARIANT. They are preconditioned Generalized Minimal Residual (GMRES) algorithms, parallelism in energy, and Wielandt acceleration. Together with partitioned matrix and Gauss-Seidel (GS) preconditioners, two GMRES algorithms are formulated to replace the upscattering iteration and facilitate energy parallelism and Wielandt acceleration. The GMRES algorithms are tested on two-dimensional thermal and fast reactor diffusion problems. The two GMRES algorithms yield higher efficiencies in energy group parallelization and Wielandt acceleration than simple parallelization of the existing GS algorithm. With preconditioning the GMRES algorithms reduce the total computing time by a factor of 2 to 4 and in some cases by a factor of >10. A multilevel iteration optimization scheme is investigated that automatically adjusts the relative error tolerance of the inner iterations according to the estimated convergence rate of the corresponding outer iterations and updates the Wielandt shift magnitude as the calculations progress. Numerical results based on large two-dimensional thermal and fast reactor diffusion problems demonstrate that automated optimization of the multilevel iterative processes reduces iteration numbers by as much as an order of magnitude.

Unstructured Triangular Nodal- SP 3 Method Based on an Exponential Function Expansion

Abstract The isotropic simplified spherical harmonics (SP3) method is employed to cast the neutron transport equation into a coupled set of two equations each of which shares identical mathematical form with the neutron diffusion equation. An exponential function expansion nodal (EFEN) method is presented for an arbitrary triangular grid and implemented to solve the coupled SP3 equations. The EFEN method couples adjacent nodes by defining partial currents on each interface and expanding the detailed flux distribution within each node into a sum of exponential functions to obtain a response matrix between the incoming and outgoing partial currents and a neutron balance condition for each node to obtain the nodal average flux. Numerical results demonstrate that both keff and power distributions agree well with other codes. We find comparable accuracy in most situations, and the new method appears to be faster than the other codes even in cases where EFEN requires a finer unstructured mesh.

Evaluation of Pin-Cell Homogenization Techniques for PWR Pin-by-Pin Calculation

Abstract In general, spatial homogenization, energy group condensation, and angular approximation are all included in the homogenization process. For the traditional pressurized water reactor (PWR) two-step calculation, the assembly homogenization with assembly discontinuity factors plus two-group (2G) neutron diffusion calculation have been proved to be a very efficient combination. However, this changes and becomes unsettled for the pin-by-pin calculation. Thus, this paper evaluates pin-cell homogenization techniques by comparison with the two-dimensional one-step whole-core transport calculation. For the homogenization, both the generalized equivalence theory (GET) and the superhomogenization (SPH) methods are studied. Considering the spectrum interference effect between different types of fuel pin cells, both 2G and 7-group (7G) structures are condensed from the 69-group WIMS-D4 library structure. For practical reactor core applications, the low-order angular approximations, including the diffusion and the SP3 methods, are compared with each other to determine which one is accurate enough for the PWR pin-by-pin calculation. Numerical results have demonstrated that both the GET and the SPH methods work effectively in pin-cell homogenization. In consideration of the spectrum interference effect, the 7G structure is sufficient for the pin-by-pin calculation. Compared with the diffusion method, the SP3 method can decrease the errors dramatically.

Preconditioned krylov solution of response matrix equations

Abstract Multigrid-preconditioned Krylov methods are applied to within-group response matrix equations of the type derived from the variational nodal method for neutron transport with interface conditions represented by orthogonal polynomials in space and spherical harmonics in angle. Since response matrix equations result in nonsymmetric coefficient matrices, the generalized minimal residual (GMRES) Krylov method is employed. Two acceleration methods are employed: response matrix aggregation and multigrid preconditioning. Without approximation, response matrix aggregation combines fine-mesh response matrices into coarse-mesh response matrices with piecewise-orthogonal polynomial interface conditions; this may also be viewed as a form of nonoverlapping domain decomposition on the coarse grid. Two-level multigrid preconditioning is also applied to the GMRES method by performing auxiliary iterations with one degree of freedom per interface that conserve neutron balance for three types of interface conditions: (a) p preconditioning is applied to orthogonal polynomial interface conditions (in conjunction with matrix aggregation), (b) h preconditioning to piecewise-constant interface conditions, and (c) h-p preconditioning to piecewise-orthogonal polynomial interface conditions. Alternately, aggregation is employed outside the GMRES algorithm to coarsen the grid, and multigrid preconditioning is then applied to the coarsened equations. The effectiveness of the combined aggregation and preconditioning techniques is demonstrated in two dimensions on a fixed-source, within-group neutron diffusion problem approximating the fast group of a pressurized water reactor configuration containing six fuel assemblies.

Exponential Function Expansion Nodal Diffusion Method

An exponential function expansion nodal diffusion method is proposed to take care of diffusion calculation in unstructured geometry. Transverse integral technique is widely used in nodal method in regular geometry, such as rectangular and hexagonal, while improper in arbitrary triangular geometry because of the mathematical singularity. In this paper, nodal response matrix is derived by expanding detailed nodal flux distribution into a sum of exponential functions, and nodal balance equation can be obtained by strict integral in the polygonal node. Numerical results illustrate that the exponential function expansion nodal method in rectangular and triangular block can solve neutron diffusion equation in regular and irregular geometry.Copyright

Improved Variational Nodal Method Based on Symmetry Group Theory

Abstract To reduce the calculation effort and memory requirement for high-order PN expansion calculation in the Variational Nodal Method (VNM), the surficial irreducible basis functions based on the symmetry group theory have been employed to block-diagonalize one of the four nodal response matrices. Its effectiveness encourages our further investigation on the application of the symmetry group theory to volumetric expansion to block-diagonalize the remaining three of the nodal response matrices in this paper. By using the symmetry group theory, the neutron transport problem for each node can be decoupled into several independent subproblems as long as both the geometry and the material distribution of the node are symmetric. Each of these subproblems can be solved by using variational principles as in the traditional VNM, providing their nodal response matrices as the diagonal blocks of the corresponding entire ones. For hexagonal-z node, each nodal response matrix can be reduced into 16 diagonal blocks, among which only 12 have to be calculated due to the properly selected irreducible basis functions. In addition, it is also proved that the response matrices with anisotropic scattering can also be block-diagonalized as the same. Calculation results based on typical problems demonstrate that the new method reduces the time cost for the response matrice calculation by one order of magnitude compared with our previous work. For the total computing time, the speedup ratio is about 2 for P3 calculation and 4 for P5 calculation. Furthermore, almost 40% of the memory requirement can be saved.

A Depletion System Compression Method Based on Quantitative Significance Analysis

Abstract The depletion systems defined by the general purpose evaluated nuclear data libraries are unnecessarily complex for most applications in nuclear reactor physics analysis. However, the corresponding compression methods are confronted with two difficulties. On one hand, the number of possible compressed depletion systems is excessively large. On the other hand, the complicated neutronic-depletion coupling effects should be properly considered. In spite of the legacy empirical-based or semi-empirical-based methods, a generalized depletion system compression method based on quantitative significance analysis is proposed in this paper. First, a quantitative significance pair was defined for each basic unit compression operation (BUCO) with respect to the neutron production density, neutron absorption density, and number densities of selected important nuclides. Second, a series of representative problems was composed according to the problem definition domain and simulated by using the original depletion system. Third, the significance pairs were evaluated based on the simulation results of the representative problems, and then employed as the quantitative guidance for accepting or rejecting each BUCO. The commpressed depletion systems have been obtained based on the newly proposed method, and typical pressurized water reactor problems were employed to verify the compresssed depletion systems. Numerical results demonstrated that by adopting the compressed depletion systems generated by the proposed method, significant computing time and storage savings can be achieved while maintaining demanded accuracy.

Neutron Up-Scattering Effect in Refined Energy Group Structure

Aiming at generating a 361-group library, this paper investigated neutron up-scattering effect in the 361-group Santamarina-Hfaiedh Energy Mesh (SHEM). Firstly, the Doppler Broadening Rejection Correction (DBRC) method is implemented to consider the neutron up-scattering effect in Monte Carlo (MC) method. Then the MC method is employed to prepare resonance integral table and scattering matrix for afterward calculation. Numerical results show that the neutron up-scattering affects kinf by ~200 pcm at most for UO2 pin cell problems in the 361-group SHEM, while the fuel temperature coefficient (FTC) is also influenced by 12~13%. It has also been found that both of the above two influences acts through scattering matrix rather than self-shielded absorption cross sections. In addition, the self-shielding effect of cladding is studied and it’s been found that it affects kinf by 30~70 pcm.