AXIAL EXPANSION METHODS FOR SOLUTION OF THE MULTI-DIMENSIONAL NEUTRON DIFFUSION EQUATION. PDF Download
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Author: JOSE FELIPPE BEAKLINI FILHO Publisher: ISBN: Category : Languages : en Pages : 205
Book Description
The feasibility and practical implementation of axial expansion methods for the solution of the multi-dimensional multigroup neutron diffusion (MGD) equations is investigated.
Author: JOSE FELIPPE BEAKLINI FILHO Publisher: ISBN: Category : Languages : en Pages : 205
Book Description
The feasibility and practical implementation of axial expansion methods for the solution of the multi-dimensional multigroup neutron diffusion (MGD) equations is investigated.
Author: Laurent Graziano Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
The purpose of this PhD is the implementation of an axial polynomial approximation in a three-dimensional Method Of Characteristics (MOC) based solver. The context of the work is the solution of the steady state Neutron Transport Equation for critical systems, and the practical implementation has been realized in the Two/three Dimensional Transport (TDT) solver, as a part of the APOLLO3® project. A three-dimensional MOC solver for 3D extruded geometries has been implemented in this code during a previous PhD project, relying on a piecewise constant approximation for the neutrons fluxes and sources. The developments presented in the following represent the natural continuation of this work. Three-dimensional neutron transport MOC solvers are able to produce accurate results for complex geometries. However accurate, the computational cost associated to this kind of solvers is very important. An axial polynomial representation of the neutron angular fluxes has been used to lighten this computational burden.The work realized during this PhD can be considered divided in three major parts: transport, acceleration and others. The first part is constituted by the implementation of the chosen polynomial approximation in the transmission and balance equations typical of the Method Of Characteristics. This part was also characterized by the computation of a set of numerical coefficients which revealed to be necessary in order to obtain a stable algorithm. During the second part, we modified and implemented the solution of the equations of the DPN synthetic acceleration. This method was already used for the acceleration of both inners and outers iteration in TDT for the two and three dimensional solvers at the beginning of this work. The introduction of a polynomial approximation required several equations manipulations and associated numerical developments. In the last part of this work we have looked for the solutions of a mixture of different issues associated to the first two parts. Firstly, we had to deal with some numerical instabilities associated to a poor numerical spatial or angular discretization, both for the transport and for the acceleration methods. Secondly, we tried different methods to reduce the memory footprint of the acceleration coefficients. The approach that we have eventually chosen relies on a non-linear least square fitting of the cross sections dependence of such coefficients. The default approach consists in storing one set of coefficients per each energy group. The fit method allows replacing this information with a set of coefficients computed during the regression procedure that are used to re-construct the acceleration matrices on-the-fly. This procedure of course adds some computational cost to the method, but we believe that the reduction in terms of memory makes it worth it.In conclusion, the work realized has focused on applying a simple polynomial approximation in order to reduce the computational cost and memory footprint associated to a Method Of Characteristics solver used to compute the neutron fluxes in three dimensional extruded geometries. Even if this does not a constitute a radical improvement, the high order approximation that we have introduced allows a reduction in terms of memory and computational times of a factor between 2 and 5, depending on the case. We think that these results will constitute a fertile base for further improvements.
Author: Publisher: ISBN: Category : Languages : en Pages :
Book Description
The usual strategy for solving the neutron diffusion equation in two or three dimensions by nodal methods is to reduce the multidimensional partial differential equation to a set of ordinary differential equations (ODEs) in the separate spatial coordinates. This reduction is accomplished by "transverse integration" of the equation.1 For example, in three-dimensional Cartesian coordinates, the three-dimensional equation is first integrated over x and y to obtain an ODE in z, then over x and z to obtain an ODE in y, and finally over y and z to obtain an ODE in x. Then the ODEs are solved to obtain onedimensional solutions for the neutron fluxes averaged over the other two dimensions. These solutions are found in regions ("nodes") small enough for the material properties and cross sections in them to be adequately represented by average values. Because the solution in each node is an exact analytical solution, the nodes can be much larger than the mesh elements used in finite-difference solutions. Then the solutions in the different nodes are coupled by applying interface conditions, ultimately fixing the solutions to the external boundary conditions.
Author: Jacques Tagoudjeu Publisher: Universal-Publishers ISBN: 1599423960 Category : Mathematics Languages : en Pages : 161
Book Description
This thesis focuses on iterative methods for the treatment of the steady state neutron transport equation in slab geometry, bounded convex domain of Rn (n = 2,3) and in 1-D spherical geometry. We introduce a generic Alternate Direction Implicit (ADI)-like iterative method based on positive definite and m-accretive splitting (PAS) for linear operator equations with operators admitting such splitting. This method converges unconditionally and its SOR acceleration yields convergence results similar to those obtained in presence of finite dimensional systems with matrices possessing the Young property A. The proposed methods are illustrated by a numerical example in which an integro-differential problem of transport theory is considered. In the particular case where the positive definite part of the linear equation operator is self-adjoint, an upper bound for the contraction factor of the iterative method, which depends solely on the spectrum of the self-adjoint part is derived. As such, this method has been successfully applied to the neutron transport equation in slab and 2-D cartesian geometry and in 1-D spherical geometry. The self-adjoint and m-accretive splitting leads to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of minimal residual and preconditioned minimal residual algorithms using Gauss-Seidel, symmetric Gauss-Seidel and polynomial preconditioning are then applied to solve the matrix operator equation. Theoretical analysis shows that the methods converge unconditionally and upper bounds of the rate of residual decreasing which depend solely on the spectrum of the self-adjoint part of the operator are derived. The convergence of theses solvers is illustrated numerically on a sample neutron transport problem in 2-D geometry. Various test cases, including pure scattering and optically thick domains are considered.
Author: H. P. Flatt Publisher: ISBN: Category : Finite differences Languages : en Pages : 40
Book Description
The finite difference approximations used in several one-dimensional neutron diffusion codes are studied from the point of view of conservation of neutrons. A new set of approximation formulae is proposed which conserve neutrons. These formulae differ only slightly from earlier formulae, thus allowing a small effect to be corrected by a small amount of effort."
Author: JOSE FELIPPE BEAKLINI FILHO
Author: JOSE FELIPPE BEAKLINI FILHO
Author: L. A. Hageman
Author: Laurent Graziano
Author: Atomic Energy of Canada Limited
Author: 
Author: 
Author: Jacques Tagoudjeu
Author: R. M. Pearce
Author: H. P. Flatt
Author: