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[en] Fourier methods are applied to the convergence analysis of the neutron diffusion eigenvalue problem. Of particular interest was the application of the methods to a 2-D/1-D coupling method for solving the 3-dimensional neutron diffusion eigenvalue problem. The analysis shows that the method diverges with a small mesh size as it does in the fixed source problem. Analysis also reveals that the spectral radius of the eigenvalue problem approaches unity as the mesh size increases while it approaches zero in the fixed source problem. The techniques presented in this paper should be useful for the convergence analysis of other algorithms used to solve neutronics eigenvalue problems. (authors)
[en] It is shown in this paper that traditional formulas for reactivity effects calculation based on diffusion approximation small-perturbation theory sometimes need updating. The corresponding specifications are provided. These are caused by the unbalance of neutron leakages between subdomains (zones, cells, fuel subassemblies, etc.) of the reactor as a result of perturbation. The unbalance occurs because of the jumps of relative increments of diffusion coefficients on the subdomain interfaces. (author)
[en] A new moving-mesh Finite Volume Method (FVM) for the efficient solution of the two-dimensional neutron diffusion equation is introduced. Many other moving-mesh methods developed to solve the neutron diffusion problems use a relatively large number of sophisticated mathematical equations, and so suffer from a significant complexity of mathematical calculations. In this study, the proposed method is formulated based on simple mathematical algebraic equations that enable an efficient mesh movement and CV deformation for using in practical nuclear reactor applications. Accordingly, a computational framework relying on a new moving-mesh FVM is introduced to efficiently distribute the meshes and deform the CVs in regions with high gradient variations of reactor power. These regions of interest are very important in the neutronic assessment of the nuclear reactors and accordingly, a higher accuracy of the power densities is required to be obtained. The accuracy, execution time and finally visual comparison of the proposed method comprehensively investigated and discussed for three different benchmark problems. The results all indicated a higher accuracy of the proposed method in comparison with the conventional fixed-mesh FVM
[en] We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.
[en] A convergence analysis was performed for three methods used to couple the 2-D radial and 1-D axial solutions of the 3-D neutron diffusion equation. In the first and second methods, the axial net currents and partial currents at plane interfaces were used for the inter-plane couplings, respectively. In a newly proposed third method, the current correction factors from the axial two-node kernel are used to couple the planes similar to the conventional CMFD formulation with a 2-node kernel. The new method has at least two advantages compared to the other methods. First, it is always stable whereas the net current method diverges for small mesh sizes. Second, the new method uses a Gauss-Seidel planar sweeping, and in the range of practical mesh sizes provides the best performance in terms of convergence rate. (authors)
[en] We develop a fully analytical study of the spectrum of the neutron diffusion operator both for spatially homogeneous and reflected reactors in a multi-group energy model. We illustrate and discuss the results of the analysis of the time spectrum of the diffusion operator, to highlight some general properties of the neutronic evolution in a multiplying system. Various new results are presented, particularly regarding the possible existence of complex time eigenvalues, the appearance of a continuum part of the spectrum and the orthogonality properties of the eigenfunctions in the case of an infinite reflector.
[en] We study convergence of the quasi-diffusion (QD) method on two-dimensional spatially periodic problems with strong heterogeneities. A Fourier analysis of the linearized QD equations in the vicinity of the solution is performed. The analysis shows that in Periodic Horizontal Interface (PHI) problems the QD iteration method loses its effectiveness and even diverges in some cases. Numerical results of finite-medium PHI problems are presented to demonstrate the behavior of the QD method that was theoretically predicted. (authors)
[en] This paper presents the modeling of a 3-D Supercritical Water-Cooled Reactor unit cell. The pre-homogenization of the geometries allows us to reduce the size of the problem without affecting significantly the results of lattice cell calculations. The differences observed between the results of 2-D and 3-D calculations are explained by the isotropic reflective boundary conditions associated with the Z surfaces in the 3-D case. The lattice code DRAGON, which has the ability to solve the 3-D neutron transport equation, has been used in this study. (author)
[en] This paper presents an error analysis of the Nodal Integral Method (NIM) applied to the two-dimensional neutron diffusion equation. The geometry of the problem under consideration consists of a homogeneous material unit square. This geometry is transformed by scaling out the diffusion length. The NIM formalism is presented then used to solve the neutron diffusion equation in this specific problem. The Maximum Principle is proved to be valid for the NIM formalism and an error analysis is performed by applying the Maximum Principle to the truncation error and a comparison function. Results show that the convergence order for the NIM solution to the exact solution is O(a2), where a is half the scaled length of a computational cell, and a bound on the error is derived. Numerical results are also presented in verification of this result. (authors)
[en] Authors investigate the impact of the spectral gap, also known as dominant ratio, on the spatial stability of the field of neutrons in nuclear reactor. It is noted that the magnitude of the spectral gap, which characterizes the amplitude of perturbation of neutrons in the reactor, is the universal criterion for the spatial instability of neutron field
[ru]В работе рассматриваются вопросы влияния спектрального зазора, называемого также доминантным отношением, на пространственную устойчивость (или, наоборот, неустойчивость) поля нейтронов в ядерном реакторе и на эффекты реактивности при возмущении свойств реактора. Отмечается, что величина спектрального зазора, характеризующая амплитуду возмущения поля нейтронов в реакторе, является универсальным критерием пространственной неустойчивости поля нейтронов в реакторе