Ricardo C. Barros

Recent advances in spectral nodal methods for X,Y - geometry discrete ordinates deep penetration and eigenvalue problems

Abstract We describe the recent advances in a class of nodal methods applied to multidimensional discrete ordinates (SN) transport problems in Cartesian geometry. This class of coarse-mesh methods is referred to as spectral nodal methods. The basic numerical schemes that we present are the spectral Greens function (SGF) nodal method and the spectral diamond (SD) nodal method. First we describe a spectral nodal method applied to monoenergetic X,Y-geometry deep penetration SN problems with flat approximations for the transverse leakage terms of the transverse integrated SN nodal equations. This method is referred to as the SGF constant nodal (SGF-CN) method. Furthermore, we describe the SGF exponential nodal (SGF-ExpN) method, wherein the transverse leakage terms are approximated by exponential functions. Next, we describe a hybrid spectral nodal method applied to monoenergetic X,Y-geometry SN eigenvalue problems with flat approximations for the transverse leakage terms of the transverse integrated SN nodal equations. For the multiplying regions of the nuclear reactor core, e.g. the fuel regions, we use the SD constant nodal (SD-CN) method, and for the non-multiplying regions, e.g. the reflector regions, we use the SGF-CN method. Numerical results are given to illustrate the accuracy of each method presented.

The application of spectral nodal methods to discrete ordinates and diffusion problems in Cartesian geometry for neutron multiplying systems

We describe in this paper the recent advances in spectral nodal methods applied to discrete ordinates (SN) and diffusion problems in Cartesian geometry for neutron multiplying systems. We divide this paper into three major parts. Part I and II deal with SN and diffusion eingenvalue problems. In Part III we describe the progress of spectral nodal methods applied to time-dependent diffusion model for reactor kinetics calculations. Numerical results to various typical model problems are given, and we close with general concluding remarks and suggestions for future work.

A numerical method for multigroup slab-geometry eigenvalue problems in transport theory with no spatial truncation error

Abstract A new numerical nodal method is developed for multigroup slab-geometry discrete ordinates (SN) eigenvalue problems with no spatial truncation error. The numerical results are exactly those of the dominant analytical solution of the multigroup SN eigenvalue problem on the grid points apart from finite arithmetic considerations and regardless of the spatial grid. We have implemented the option of using the multigroup albedo boundary condition, that is a measure of the reflective power of the neutron reflector, e.g., water or graphite. The Power method used in the outer iterations for convergence of the dominant numerical solution has been accelerated by a scheme based on Tchebycheff extrapolation of the fission source. Numerical results are given to illustrate the methods accuracy.

On the equivalence of discontinuous finite element methods and discrete ordinates methods for the angular discretization of the linearized boltzmann equation in slab geometry

Abstract We describe the equivalence of discontinuous finite element methods and discrete ordinates methods for the angular discretization of the Boltzmann equation in slab geometry. We use the spectral Greens function (SGF) numerical method to solve the discrete ordinates equations on a digital computer. The SGF method is completely free from spatial truncation errors; therefore, we use the ‘exact’ solution generated by the SGF method to reconstruct the flux profile inside each node of the spatial grid. This scheme is referred to as the domain decomposition reconstruction scheme. Numerical results to three typical problems are presented.

A numerical method for monoenergetic slab-geometry fixed-source adjoint transport problems in the discrete ordinates formulation with no spatial truncation error

A numerical method that is free of spatial truncation errors is developed for one-speed slab-geometry constant fixed-source adjoint discrete ordinates (SN) problems. The unknowns in the method are the cell-edge and cell-average adjoint angular fluxes, and the numerical values obtained for these quantities are those of the analytic solution of the adjoint SN equations. The method is based on the use of the standard spatially discretised adjoint SN balance equations, which hold in each spatial cell and for each discrete ordinates direction, and a non-standard adjoint auxiliary equation that contains a Green’s function for the cell-average adjoint angular flux in terms of the exiting cell-edge adjoint angular fluxes and the interior adjoint source. Numerical results are given to illustrate the method’s accuracy.

A Laplace transform exponential method for monoenergetic three-dimensional fixed source discrete ordinates problems in Cartesian geometry

We describe a Laplace transform exponential method applied to x-y-z geometry heterogeneous neutron transport problems in the discrete ordinates (SN) formulation that we refer to as the LTSN – Exp method. This numerical method uses the space Laplace transform technique to solve the one-dimensional transverse-integrated SN exponential equations within one of the homogeneous regions of the domain of solution. Based on the physics of shielding problems where the neutron flux attenuates exponentially with increasing distance from the source, we approximate the transverse leakage terms by exponential functions. We show in two numerical experiments that the LTSN – Exp method generates very accurate results in highly absorbing media.

On the decomposition method applied to linear and non-linear discrete ordinates problems in slab geometry

Abstract We describe the applications of the decomposition method to discrete ordinates (S N ) problems in slab geometry, which model neutral particle transport in shields, and coupled conductive-radiative heat transfer phenomena. Numerical results to typical model problems are shown to illustrate the accuracy of each application.

Shifting Strategy in the Spectral Analysis for the Spectral Green’s Function Nodal Method for Slab-Geometry Adjoint Transport Problems in the Discrete Ordinates Formulation

We present the positive features in the shifting strategy that we use in the homogeneous component of the general solution of the monoenergetic, slab-geometry, adjoint discrete ordinates (S N ) equations inside each discretization node for neutral particle source-detector transport problems. The adjoint spectral Green’s function (SGF) method uses the standard spatially discretized adjoint S N balance equations and nonstandard SGF adjoint auxiliary equations, which have parameters that need to be determined to preserve this local solution. We remark that the shifting strategy scales the N exponential functions of the local solution in the interval (0, 1). One advantage is to avoid the overflow in computational finite arithmetic calculations in high-order angular quadrature and/or coarse-mesh calculations.

Analytic Representation of the Solution of Neutron Kinetic Transport Equation in Slab-Geometry Discrete Ordinates Formulation

Described here is a semi-analytical numerical method for the solution of simplified monoenergetic SN kinetics equations in a homogeneous slab, assuming one group of delayed neutron precursors. The basic idea relies on the solution of the neutron SN kinetics equation following the idea of the Decomposition method. To this end, the neutron angular flux and the concentration of the delayed neutron precursors are expanding in a truncation series of unknown functions, \(\sum _{k=0}^{M}\psi _{m}^{(k)}(x,t)\) and \(\sum _{k=0}^{M}C^{(k)}(x,t)\). Replacing these expansions in the SN transport equation we come up with one equation, in which, a simple count indicates that the kinetic problem has been reduced to a set of two equations in (2N + 2) unknowns, namely the expansions modes ψ(k) and C(k) for k = 1: M. In order to determine these unknown functions, we construct a recursive scheme of SN kinetic equations with the property that all equations satisfy the resulting equation, with the error being governed by the order M of the expansion. We remark that the first equation of the recursive system satisfy the boundary conditions of the SN kinetic equation whereas the remaining equations satisfy the homogeneous boundary conditions. Further we note that the construction of the recursive system is not unique. We support our choice with the argument that the analytical representation for each equation attained is known and given by the Time Laplace Transform Sn approach, i.e., the TLTSN method. This method has been implemented for the solution of a homogeneous slab and the numerical results have been compared with the ones available in the literature. Also, a convergence analysis for the recursive system are to be discussed.

The Error Bounds for Three-Dimensional Nodal LTSN Method

In this paper we present a proof about the convergence of the 3D Nodal-LTSN Method in order to solve the transport problem in a parallelepiped domain. For that, we define functions associated to the errors, one in the approximated flux, another in the quadrature formula and establish a relation between them. We present a Nodal-LTSN method to generate an analytical solution for discrete ordinates problems in three-dimensional cartesian geometry. We first transverse integrate the SN equations and then we apply the Laplace transform. The essence of this method is the diagonalization of the LTSN transport matrices and the spectral analysis garantees this. The transverse leakage terms that appear in the transverse integrated SN equations are represented by exponential functions with decay constants that depend on the characteristics of the material of the medium the particles leave behind. We present numerical results generated by the offered method applied to typical shielding model problems.Copyright