Solution Of Neutron Transport Problems Research Articles

VITAS: A multi-purpose simulation code for the solution of neutron transport problems based on variational nodal methods

Existing neutronic whole-core simulation codes are problem-oriented. With the advancement of computer technology and the development of complicated micro-nuclear reactors, there is a pressing need to create solution methods with broad applicability and the ability to accommodate more complex, high-fidelity, unstructured geometries. This necessity motivates the development of a neutron transport code that can adapt to different types of nuclear reactors. This work describes the code design and theoretical model of VITAS, a multi-purpose simulation code for handling steady-state and time-dependent neutron transport problems. This code incorporates algorithms and models inside a unified variational nodal framework based on the second-order neutron transport equation. The most recent version of VITAS employs the isotropic scattering approximation and permits arbitrary spatial and angular refinement. VITAS is compatible with Cartesian geometry, hexagonal geometry, triangular geometry, and Cartesian geometry with finite element subdivisions in the radial direction (Cartesian-FE). The three-dimensional modeling can be performed by axially extruding the two-dimensional mesh. Users can choose the mesh type and expansion order on-demand for optimal computational efficiency.In this paper, VITAS is validated using benchmark problems, i.e., the TAKEDA3 benchmark and the transient LMW benchmark with Cartesian homogenized geometry, the C5G7 and C5G7-TD exercises with Cartesian-FE geometry, the TAKEDA4 benchmark and the Buckner and Stewart benchmark with hexagonal-z homogenized geometry, and the Dodds benchmark with cylindrical-z geometry described by triangular-z unstructured meshes. The space-angle truncation is extensively examined to demonstrate the asymptotic convergence behavior with p- and h-refinement. The results indicate that VITAS has the potential to become a versatile toolset for solving a variety of nuclear reactor problems.

Spectral analysis of the extended linear discontinuous method for one-dimensional monoenergetic discrete ordinates transport problems in non-multiplying media

Accurate solution of neutron transport problems in the discrete ordinates (SN) formulations is relevant in many areas of engineering and nuclear science. Several researches have led to a variety of numerical differencing schemes in order to generate even more accurate numerical solutions. However, most differencing schemes seem to become unstable for sufficiently large spatial cell width and negative solutions may arise, which are non-physical results. In this work, we present a spectral analysis of the Extended Linear Discontinuous (ELD) and the classical Linear Discontinuous (LD) difference schemes for one-dimensional monoenergetic SN problems with isotropic scattering in non-multiplying media. This analysis can predict analytic threshold values for the discretization cell width, beyond which SN numerical solutions become unstable. Analytical threshold values were calculated by means of the spectrum of spatially discretized SN equations and tested by solving two model problems to compare the practical results with the theoretical predictions offered in this paper.

Quasi-random discrete ordinates method for neutron transport problems

The quasi-random discrete ordinates method (QRDOM) is here proposed for the approximation of the solution of neutron transport problems. Its central idea is to explore a quasi Monte Carlo integration within the classical source iteration technique. It preserves the main characteristics of the discrete ordinates method, but it has the advantage of providing approximate solutions with mitigated ray effect. Application of the QRDOM to neutron transport problems in rectangular domains is discussed in detail. The method is tested against benchmark problems for which DOM solutions are known to suffer from ray effects. Numerical experiments indicate that the QRDOM provides accurate results and that it demands less discrete ordinates per source iteration compared to the classical DOM.

A composite spatial grid spectral Green's function method for one speed discrete ordinates eigenvalue problems in two-dimensional Cartesian geometry

Spectral Green's function nodal methods (SGF) are well established as a class of coarse mesh methods. For this reason, they are widely used in the solution of neutron transport problems in discrete ordinates formulation (SN). When compared with fine mesh methods, SGF are considered efficient, as solutions are as accurate as, using a smaller number of spatial nodes, reducing floating point operations. However, the development of spectral-nodal methods for XY Cartesian geometries, has been limited due to (a) difficulties in implementing efficient computational algorithms and, (b) high algebraic and computational costs. This is because these methods need to use NBI-type (One-Node Block Inversion) sweep schemes. The composite spatial grid methods were developed to overcome these challenges. In this work, we describe a composite spatial grid spectral-nodal method to solve one-speed discrete ordinate eigenvalue problems in XY Cartesian geometry with isotropic scattering. The discretization is developed into two stages and two 1D problems coupled by transverse leakage terms in each domain region are obtained. In order to converge toward the numerical solution, we used an alternating-direction iterative technique and a modified source iteration sweep scheme. Also, we used the conventional power method to estimate the problem's dominant eigenvalue. Numerical results for benchmark problems are presented to illustrate the accuracy and performance of the developed method. This approach offers more accurate and efficient results for integral quantities if compared with others SGF methods.

The quasi-static method revisited

The quasi-static method is a standard tool for the space–time solution of neutron transport problems in multiplying media. Its basic principle lies in a factorization of the angular flux into the product of two functions, ‘amplitude’ and ‘shape’, where the amplitude depends only on time (and contains the major part of the time-dependence) while the shape function depends on all variables, time included. The shape equation is solved on a long time-scale, while the amplitude is determined on a short time-scale. The factorization is made unique by proper normalization conditions for the shape function. Most implementations replace the basic equation (transport or diffusion) by the set of coupled amplitude and shape equations derived from the factorization, the so-called ‘Improved Quasi-static Method’ (IQM). In this paper we describe an alternate approach already known for some time we have called ‘Predictor–Corrector Quasi-static Method’ (PCQM). We discuss its efficiency for both solid- and liquid-fuel systems dynamics.

Exact Controllability for the Time Dependent Transport Equation

Exact Controllability for the Time Dependent Transport Equation

A Boundary Functional for the Least-Squares Finite- Element Solution of Neutron Transport Problems

The least-squares finite-element framework for the neutron transport equation is based on the minimization of a least-squares functional applied to the properly scaled neutron transport equation. This approach is extended by incorporating the boundary conditions into the least-squares functional. The proof of the V-ellipticity and continuity of the new functional leads to bounds of the discretization error for different regimes. For a P1 approximation of the angular dependence the resulting system of partial differential equations for the moments is explicitly derived. In the diffusion limit this system is essentially a Poisson equation for the zeroth moment and has a divergence structure for the set of moments of order 1. One of the key features of the least-squares approach is that it produces a posteriori error bounds. The use of these bounds is demonstrated in numerical examples for a spatial discretization using trilinear finite elements on a uniform tessellation into cubes.

An efficiency study of the r-function method applied as solid modeler for Monte Carlo calculations

Unlike conventional methods the R-function theory can be efficiently applied both to deterministic and stochastic solution of neutron transport problems. This paper presents two procedures for Monte Carlo particle tracking based on R-function solid modeling. An extensive series of calculations is performed on three families of geometric models and timing comparison with the MCNP code is given. The presented results clearly confirm the potentials of the R-function approach.

A Practical Adjoint Monte Carlo Technique for Fixed-Source and Eigenfunction Neutron Transport Problems

An adjoint Monte Carlo technique is described for the solution of neutron transport problems. The optimum biasing function for a zero-variance collision estimator is derived. A simple approximation to this optimum biasing function has been chosen to arrive at a problem-independent sampling scheme. The transport kernel for the adjoint particles is almost the same as for neutrons. The sampling of the collision kernel needs the introduction of so-called adjoint cross sections. The optimum treatment of an analogon of a one-velocity thermal group has also been derived. The method is extended to multiplying systems, especially for eigenfunction problems to enable the estimate of averages over the unknown fundamental neutron flux distribution. A versatile computer code, FOCUS, has been written, based on the described theory. Numerical examples are given for a shielding problem and a critical assembly, illustrating the performance of the FOCUS code.

Generation of neutron transport benchmark problems using BEAPAC-3T

A method is described how a quantitative measure can be obtained for the accuracy of numerical methods for the solution of neutron transport problems in an arbitrary homogeneous multidimensional medium. An expansion for the combining coefficients of singular eigenfunctions is developed that yields exact elementary plane-wave solutions that can be evaluated very accurately in terms of exponential integral functions. Linear combinations of these solutions are automatically generated which approximate, in the least squares sense, a user specified incident flux distribution on the boundary of a multidimensional benchmark cell domain. The generated solution is exact for a new benchmark problem which has incident flux given precisely by the generated solution. The quantitative error analysis method has been implemented in the code BEAPAC-3T and is illustrated by application to the TWOTRAN-II code for a square domain problem.

Linear analysis and solution of neutron transport problems

Abstract Modern operator techniques are used to give a rigorous analysis and solution of the time independent isotropic linear transport equation for infinite and semi-infinite media. The theory utilizes von Neumnan spectral theory and is presented both in L2 and in spaces of Holder continuous functions.

Solution of neutron transport problems in L1

Communications on Pure and Applied MathematicsVolume 28, Issue 6 p. 729-746 Article Solution of neutron transport problems in L1 Edward W. Larsen, Edward W. LarsenSearch for more papers by this author Edward W. Larsen, Edward W. LarsenSearch for more papers by this author First published: November 1975 https://doi.org/10.1002/cpa.3160280604Citations: 16AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume28, Issue6November 1975Pages 729-746 RelatedInformation

Asymptotic solution of neutron transport problems for small mean free paths

A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter ε, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in ε, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to ε−1. Similarly ψ0 decays at a rate proportional to ε−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half-space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from the initial instant and from the boundary, respectively. The results are illustrated by specializing them to the one speed case. The method may make it possible to treat more realistic and more complex problems than can be handled by other methods.