A Numerical Method for Solving the Neutron Transport Equation in Finite Cylindrical Geometry

Takeuchi, Kiyoshi

doi:10.1080/18811248.1969.9732925

Cited by 9 publications

(5 citation statements)

References 6 publications

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“…Takeuchi has used the method of characteristics in Cartesian, spherical, and twodimensional (r-θ) cylindrical geometries to solve the transport equation for deep penetration shielding problems. 27,28 In this approach, fluxes are computed at the vertices of a cell. In a two-dimensional problem, it is assumed that the source is known at all four vertices and that the angular flux is known at three vertices.…”

Section: Characteristics Methods In Shielding Applicationsmentioning

confidence: 99%

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

DeHart¹

1992

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A method for applying the discrete ordinates method for solution of the neutron transport equation in arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the traditional shape of a mesh element within a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. Using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons.The method of characteristics, originally developed to eliminate negative fluxes encountered in many finite difference approximations, had been previously applied to regular cell structures in multiple dimensions and various coordinate systems. In such geometries, the geometrical relationships between sides were determined analytically and incorporated directly into the numerical model. However, the present work makes no assumptions about the orientations or the number of sides in a given cell, and computes all geometric relationships between each set of sides in each cell for each discrete direction. A set of nonreentrant polygons can therefore be used to represent any given two-dimensional space.iv Results for a number of test problems have been compared to solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN computer programs. Numerical results were obtained for problems ranging from simple onedimensional geometry to complicated multidimensional configurations. These results have demonstrated the ability of the developed method to closely approximate complex geometrical configurations and to obtain accurate results for problems that are extremely difficult to model using traditional methods. v DEDICATION This work is dedicated to my wife, Leigh. Her patience, understanding and love over the long road leading to this point made the path seem much less rocky; her encouragement and willingness to listen to my problems helped to sustain my efforts; and the daughter she has given us has taught me that miracles are possible. vi ACKNOWLEDGEMENTS I wish to express my sincere and profound gratitude to Ron Pevey and Ted Parish, both of whom provided invaluable support and insight in my endeavors. Their guidance, encouragement, technical assistance and senses of humor were invaluable assets. I also wish to acknowledge the support given to me by the United States Department of Energy, the Westinghouse Savannah River Company, and most importantly, by several members of my direct management and by my colleagues within the Savannah Riv...

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Section: Characteristics Methods In Shielding Applicationsmentioning

confidence: 99%

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

DeHart¹

1992

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“…To solve the Eqs. ( 19) ray tracing techniques (aka the method of long characteristics) are applied [85,86,87,88,89,90,91,92]. In sum, the data-driven VEF model for TRT is constructed with:…”

Section: Variable Eddington Factor Model For Trt With Diffusion-based...mentioning

confidence: 99%

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

Coale

Anistratov

2023

Journal of Quantitative Spectroscopy and Radiative Transfer

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“…Note that if one applies weak condition for the currents and strong conditions for the scalar fluxes, it results in interface relationships that are not valid for the scalar fluxes varying linearly along cell interfaces [49]. For example, in 2D, if the left face of i-th cell is a common face with the m-th and p-th cells, then we define strong continuity conditions for the normal components of the currents 30) and the weak continuity condition for the face-average scalar fluxes 31) where h iω is the cell-edge length.…”

Section: Interface Continuity Conditionsmentioning

confidence: 99%

“…(3.37) and (3.38). We define a small parameter ε, introduce the scaled cross sections and sources 42) and the following ansatz:…”

Section: Asymptotic Diffusion Analysismentioning

confidence: 99%

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Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

Anistratov¹,

Constantinescu²,

Roberts³

et al. 2007

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A Numerical Method for Solving the Neutron Transport Equation in Finite Cylindrical Geometry

Cited by 9 publications

References 6 publications

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

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