Dmitriy Y. Anistratov

Consistent Spatial Approximation of the Low-Order Quasi-Diffusion Equations on Coarse Grids

Abstract Spatial discretization methods have been developed for the low-order quasi-diffusion equations on coarse grids and corresponding homogenization procedure for full-core reactor calculations. The proposed methods reproduce accurately the complicated large-scale behavior of the transport solution within assemblies. The developed discretization is spatially consistent with a fine-mesh discretization of the transport equation in the sense that it preserves a set of spatial moments of the fine-mesh transport solution over either coarse-mesh cells or its subregions, as well as the surface currents and eigenvalue. To demonstrate accuracy of the proposed methods, numerical results are presented of calculations of test problems that simulate the interaction of mixed-oxide and uranium assemblies.

Multilevel Quasidiffusion Methods for Solving Multigroup Neutron Transport k-Eigenvalue Problems in One-Dimensional Slab Geometry

Abstract The methods for solving k-eigenvalue problems for the multigroup neutron transport equation in one-dimensional slab geometry are presented. They are defined by means of multigroup and effective grey (one-group) low-order quasidiffusion (QD) equations. In this paper we formulate and study different variants of nonlinear QD iteration algorithms. These methods are analyzed on a set of test problems designed using C5G7 benchmark data. We present numerical results that demonstrate the performance of iteration schemes in different types of reactor physics problems. We consider tests that represent single-assembly and color-set calculations as well as a problem with elements of full-core computations involving a reflector zone.

Nonlinear Weighted Flux Methods for Particle Transport Problems

A new parameterized family of iterative methods for the 1‐D slab geometry transport equation is proposed. The new methods are derived by integrating the transport equation over −1≤μ≤0 and 0≤μ≤1 with weight 1+β|μ|α, where α≥0. The asymptotic diffusion analysis enables us to determine a particular method of this family the solution of which satisfies a good approximation of both the diffusion equation and asymptotic boundary condition in the diffusive regions. Note that none of the α‐weighted nonlinear methods possesses this combination of properties. The convergence properties of the proposed method are similar to the properties of the diffusion‐synthetic acceleration (DSA), quasi‐diffusion, and DSA‐like α‐weighted nonlinear methods. Numerical results are presented to demonstrate the performance of the derived method.

Homogenization Method for the Two-Dimensional Low-Order Quasi-Diffusion Equations for Reactor Core Calculations

Abstract In this paper, we develop a homogenization methodology for the two-dimensional low-order quasi-diffusion equations for full-core reactor calculations that is based on a family of spatially consistent coarse-mesh discretization methods. The coarse-mesh solution generated by these methods preserves a number of spatial moments of the fine-mesh transport solution over each assembly. The proposed method reproduces accurately the complicated large-scale behavior of the transport solution within assemblies. To demonstrate the performance of the developed methodology, we present the numerical results of several test problems that simulate mixed-oxide-uranium and assembly-reflector interfacial effects.

Nonlinear Diffusion Acceleration Method with Multigrid in Energy for k-Eigenvalue Neutron Transport Problems

Abstract We present a multilevel method for solving multigroup neutron transport k-eigenvalue problems in two-dimensional Cartesian geometry. It is based on the nonlinear diffusion acceleration (NDA) method. The multigroup low-order NDA (LONDA) equations are formulated on a sequence of energy grids. Various multigrid cycles are applied to solve the hierarchy of multigrid LONDA equations. The algorithms developed accelerate transport iterations and are effective in solving the multigroup NDA low-order equations. We present numerical results for model reactor-physics problems with a large number of groups to demonstrate the performance of different iterative schemes.

Computational transport methodology based on decomposition of a problem domain into transport and diffusive subdomains

A large class of radiative transfer and particle transport problems contain highly diffusive regions. It is possible to reduce computational costs by solving a diffusion problem in diffusive subdomains instead of the transport equation. This enables one to decrease the dimensionality of the transport problem. In this paper we present a methodology for decomposition of a spatial domain of a transport problem into transport and diffusion subregions. We develop methods for solving one-group problems in 1D slab geometry. To identify and locate diffusive regions, we develop metrics for measuring transport effects that are based on the quasidiffusion (Eddington) factor. We present the results of test problems that demonstrate the accuracy of the proposed methodology.

Stability analysis of nonlinear two-grid method for multigroup neutron diffusion problems

Abstract We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.

A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes

We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h^2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.

Multilevel NDA Methods for Solving Multigroup Eigenvalue Neutron Transport Problems

Abstract The nonlinear diffusion acceleration (NDA) method is an efficient and flexible transport iterative scheme for solving reactor-physics problems. This paper presents a fast iterative algorithm for solving multigroup neutron transport eigenvalue problems in one-dimensional slab geometry. The proposed method is defined by a multilevel system of equations that includes multigroup and effective one-group low-order NDA equations. The eigenvalue is evaluated in an exact projected solution space of the smallest dimensionality. Numerical results that illustrate the performance of the new algorithm are demonstrated.

Splitting method for solving the coarse-mesh discretized low-order quasi-diffusion equations

Abstract In this paper, the development is presented of a splitting method that can efficiently solve coarse-mesh discretized low-order quasi-diffusion (LOQD) equations. The LOQD problem can reproduce exactly the transport scalar flux and current. To solve the LOQD equations efficiently, a splitting method is proposed. The presented method splits the LOQD problem into two parts: (a) the D problem that captures a significant part of the transport solution in the central parts of assemblies and can be reduced to a diffusion-type equation and (b) the Q problem that accounts for the complicated behavior of the transport solution near assembly boundaries. Independent coarse-mesh discretizations are applied: the D problem equations are approximated by means of a finite element method, whereas the Q problem equations are discretized using a finite volume method. Numerical results demonstrate the efficiency of the methodology presented. This methodology can be used to modify existing diffusion codes for full-core calculations (which already solve a version of the D problem) to account for transport effects.