A “two-grid” acceleration of binary stochastic mixture deterministic transport iterations in slab geometry

Particle transport in stochastic mixtures has been an area of active research during the last decade. The primary focus has been on the development of simple, efficient models that approximate the behavior of transport solutions which have been ensemble-averaged over a large number of realizations of the mixing statistics. These approximate models are often written as coupled systems of transport equations. The stochastic nature of this problem can be naturally incorporated into Monte Carlo transport codes, but there have also been efforts to solve these coupled equations deterministically. In this paper, we develop an efficient iterative technique for the deterministic solution of the coupled transport equations of the Levermore–Pomraning mix model. The basis for its development is the observation that these equations are similar in structure to traditional multigroup transport equations with upscattering. We provide the results of a Fourier analysis of this technique, and compare these with observed convergence rates from the implemented algorithm.     

Solution of group diffusion equation for X-Y geometry by finite Fourier transformation

A new difference equation to the two dimensional diffusion equation for x-y geometry is derived by using the finite Fourier transformation. This difference equation has a form of a coupled equation of the 3 point difference equations for each coordinate, and can be easily solved by the iterative method of the alternative direction implicit method. Group diffusion equations are solved using this difference equation and sample calculations show that accurate results can be obtained with less mesh points than the usual 5 points difference equation.     

The spherical harmonics method for the multigroup transport equation in x−y geometry

A spherical harmonics equation in the form of a second-order differential equation is derived for the 2-D x−y geometry, including higher-order scattering within a group. Using this equation, a multigroup transport code for the spherical harmonics method of a general order of approximation is developed. Some numerical examples, including typical problems for the ray effect, are presented and compared with those obtained by the discrete-ordinates method. It is shown that the present method gives more accurate results than the discrete-ordinates method, although this spherical harmonics code requires more computer memory than the discrete-ordinates code.     

An exact general solution in spherical harmonics of the Boltzmann equation

A solution of the one-velocity kinetic Boltzmann equation is obtained as a series of spherical harmonics. General expressions are obtained for the terms of the series, derived without any approximately valid assumptions. As particular cases of this solution, we obtain formulas for the known P_N-approximations for the spherical-harmonic method.The exact general solution of the kinetic equation in the form of a series of spherical harmonics contains arbitrary functions which must depend on the formulation of boundary conditions. The general determination of the boundary conditions and the arbitrary functions is not considered. All the results of [4] remain valid for P_N-approximations.Translated from Atomnaya Énergiya, Vol. 18, No. 5, pp. 459–463, May, 1965     

Solution of the neutron diffusion equation in hexagonal geometries

Various ways to solve the neutron diffusion equation in hexagonal geometries are known in the literature. In this paper it is aimed to unify these approaches from different points of view. First conditions are developed for consistency and uniqueness of the various approximations; they are then used to derive estimates of the order of the corresponding rates of convergence. Then several approximations are formulated explicitly and related to the theoretical considerations, and their mutual dependencies are shown. Finally, numerical experiments are reported which support the theoretical findings.     

Bilinear-discontinuous numerical solution of the time dependent transport equation in slab geometry

A bilinear-discontinuous (BLD) discretization in space and time is described for the numerical solution of the discrete ordinates form of the time dependent, one speed neutral particle transport equation in slab geometry. Numerical results using the BLD method are compared with analytical approximations and other space-time discretization schemes. Analysis shows the BLD method is stable and third order accurate in the space and time dimensions independently.     

Solution of diffusion equation in deformable spheroids

The time-dependent diffusion of neutrons in a spheroid as a function of the focal distance has been studied. The solution is based on an orthogonal basis and an extrapolation distanced related boundary condition for the spheroidal geometry. It has been shown that spheres and disks are two limiting cases for the spheroids, for which there is a smooth transition for the systems properties between these two limits. Furthermore, it is demonstrated that a slight deformation from a sphere does not affect the fundamental mode properties, to the first order. The calculations for both multiplying and non-multiplying media have been undertaken, showing good agreement with direct Monte Carlo simulations.     

Neutron flux characterization of the SM1 sub-critical multiplying complex of the Pavia University

SM1 is a thermal Sub-critical Multiplication complex located at the University of Pavia (Italy) and, since its installation in 1962, has been utilized mainly for radiochemistry research. This work focuses on the characterization, by means of the Monte Carlo code MCNP and direct measurements, of the neutron flux distribution inside the complex and on the calculation of the effective multiplicative coefficient (k_eff) in the current SM1 thermal configuration. For two specific irradiation channels, experimental measurements of the neutron fluxes were performed by foils activation technique and neutron spectrum de-convolution based on the SAND II code. Measurements have been compared with the simulation results showing a good agreement. Furthermore, a comparison between the preliminary results of the simulations of the SM1 plant in fast configuration, characterized by a solid lead diffuser, and the actual thermal configuration is also presented. The fast configuration of SM1, if implemented, will give the opportunity to carry out preliminary studies for the analysis of sub-critical fast-neutron installations and their applications.     

Solution of the transport equation by the BIF method

A research code and applications code are described for the numerical solution of the three dimensional linearized and stationary transport equation over a finite domain which is subdivided into a finite number of parallelepipeds. The main program uses only flux densities at the boundaries and interfaces but at no other interior points. This minimizes the order of the associated matrix equations needed for the numerical solution, but requires a good knowledge of the analytic behaviour of the mathematical solution of the problem. Because only Boundary and Interface Functions are used, this approach is called the BIF method.     

A Raviart–Thomas–Schneider solution of the diffusion equation in hexagonal geometry

We present an implementation on the Raviart–Thomas–Schneider finite element method for solving the diffusion equation in hexagonal 3D geometry. This method is dedicated to full-core fuel management and design applications studies of nuclear reactors featuring an hexagonal mesh. The Raviart–Thomas–Schneider method is based on a dual variational formulation defined over lozenges with a Piola transformation of the polynomial basis. An efficient ADI numerical technique was set up to solve the resulting matrix system. Validation results are given for the hexagonal IAEA 2D benchmark and for two additional benchmarks related to the Monju core in 2D and 3D.