Neutron Diffusion Problems Research Articles

Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: II. Illustrative application to a paradigm particle diffusion problem

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a paradigm neutron diffusion problem, which is sufficiently simple to admit an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2nd-ASAM indicates that, for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most (2Nα+1) “large-scale” computations using the first-level and, respectively, second-level adjoint sensitivity systems (1st-LASS and 2nd-LASS). Very importantly, however, the illustrative application presented in this work shows that the actual number of adjoint computations needed for computing all of the first- and second-order response sensitivities may be significantly less than (2Nα+1) per response. For this illustrative problem, four “large-scale” adjoint computations sufficed for the complete and exact computations of all 4 first- and 10 distinct second-order derivatives. Furthermore, the construction and solution of the 2nd-LASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the right-sides of the diffusion (differential) operator needed to be modified; the left-side of the differential equations (and hence the “solver” in large-scale practical applications) remained unchanged.All of the first-order relative response sensitivities to the model parameters have significantly large values, of order unity. Also importantly, most of the second-order relative sensitivities are just as large, and some even up to twice as large as the first-order sensitivities. In the illustrative example presented in this work, the second-order sensitivities contribute little to the response variances and covariances. However, they have the following major impacts on the computed moments of the response distribution: (a) they cause the “expected value of the response” to differ from the “computed nominal value of the response”; and (b) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations, and hence would nullify the skewness of the response. Consequently, any events occurring in a response's long and/or short tails, which are characteristic of rare but decisive events (e.g., major accidents, catastrophes), would likely be missed. The 2nd-ASAM is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, e.g., optimization, data assimilation/adjustment, model calibration, and predictive modeling.

Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media

We propose the group-theoretical approach which enables one to generate solutions of equations of mathematical physics in nonhomogeneous media from solutions of the same problem in a homogeneous medium. The efficiency of this method is illustrated with examples of thermal neutron diffusion problems. Such problems appear in neutron physics and nuclear geophysics. The method is also applicable to nonstationary and nonintegrable in quadratures differential equations.

Preconditioned Krylov Solution of Response Matrix Equations

Multigrid-preconditioned Krylov methods are applied to within-group response matrix equations of the type derived from the variational nodal method for neutron transport with interface conditions represented by orthogonal polynomials in space and spherical harmonics in angle. Since response matrix equations result in nonsymmetric coefficient matrices, the generalized minimal residual (GMRES) Krylov method is employed. Two acceleration methods are employed: response matrix aggregation and multigrid preconditioning. Without approximation, response matrix aggregation combines fine-mesh response matrices into coarse-mesh response matrices with piecewise-orthogonal polynomial interface conditions; this may also be viewed as a form of nonoverlapping domain decomposition on the coarse grid. Two-level multigrid preconditioning is also applied to the GMRES method by performing auxiliary iterations with one degree of freedom per interface that conserve neutron balance for three types of interface conditions: (a) p preconditioning is applied to orthogonal polynomial interface conditions (in conjunction with matrix aggregation), (b) h preconditioning to piecewise-constant interface conditions, and (c) h-p preconditioning to piecewise-orthogonal polynomial interface conditions. Alternately, aggregation is employed outside the GMRES algorithm to coarsen the grid, and multigrid preconditioning is then applied to the coarsened equations. The effectiveness of the combined aggregation and preconditioning techniques is demonstrated in two dimensions on a fixed-source, within-group neutron diffusion problem approximating the fast group of a pressurized water reactor configuration containing six fuel assemblies.

Unstructured Grids and the Multigroup Neutron Diffusion Equation

The neutron diffusion equation is often used to perform core-level neutronic calculations. It consists of a set of second-order partial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved by discretizing the neutron leakage term using a structured grid. This work introduces the alternatives that unstructured grids can provide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities they offer. It is by understanding the basic mathematics that lie beneath the equations that model real physical systems; better technical decisions can be made. It is in this spirit that this paper is written, giving a first introduction to the basic concepts which can be incorporated into core-level neutron flux computations. A simple two-dimensional homogeneous circular reactor is solved using a coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements method. Also, the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries, meshing algorithms, basic geometric entities, discretization schemes, and characteristic grid lengths, giving even more insight into the peculiarities that arise when solving the neutron diffusion equation using unstructured grids.

Convergence analysis of coarse mesh finite difference method applied to two-group three-dimensional neutron diffusion problem

This article presents the convergence analysis of the coarse mesh finite difference (CMFD) method applied to two-group (2-G) three-dimensional (3D) neutron diffusion problem. Two CMFD algorithms are examined: one-node (1-N) CMFD and two-node (2-N) CMFD. Two test problems are used for the study of the convergence behavior: a model problem of homogeneous 2-G 3D eigenvalue problem and the NEACRP LWR transient benchmark problem. The convergence rates of the 1-N and 2-N CMFD algorithms are numerically measured in terms of the convergence of current correction factors (CCFs). The numerical test results are presented as well as the comparison with the previous analytical study. Overall, 1-N CMFD with the CCF relaxation shows a comparable performance to 2-N CMFD for the realistic 3D rod ejection transients.

A finite element/boundary element hybrid method for 2-D neutron diffusion calculations

A finite element-boundary element hybrid method has been developed for one or two group neutron diffusion calculations. A linear or bilinear finite element formulation for the reactor core and a linear boundary element technique for the reflector which are combined through interface continuity conditions constitute the basis of the developed method. The present formulation is restricted to two-dimensional geometries and has been implemented in the developed computer program. Via comparisons with analytical solutions, the proposed method has been validated. Further comparisons against the pure finite and boundary element formulations show that the proposed method constitutes a viable alternative for the numerical solution of neutron diffusion problems of both the external neutron source and multiplication eigenvalue determination variety.

Interface current method for multi‐region neutron diffusion problems

AbstractThis paper presents an interface current method for solving multi‐region neutron diffusion problems using multi‐dimensional Green's functions. The method is based on evaluation of the net currents at each interface. The unknown interface currents are expanded in terms of eigenfunctions. The orthogonality condition of the eigenfunctions is employed in order to transform these equations to an algebraic system. Numerical examples are provided at the end of the paper to demonstrate the validity of the method, and for comparison with the more established finite difference method. Copyright © 2003 John Wiley & Sons, Ltd.

An Adaptive Approach to Variational Nodal Diffusion Problems

An adaptive grid method is presented for the solution of neutron diffusion problems in two dimensions. The primal hybrid finite elements employed in the variational nodal method are used to reduce the diffusion equation to a coupled set of elemental response matrices. An a posteriori error estimator is developed to indicate the magnitude of local errors stemming from the low-order elemental interface approximations. An iterative procedure is implemented in which p refinement is applied locally by increasing the polynomial order of the interface approximations. The automated algorithm utilizes the a posteriori estimator to achieve local error reductions until an acceptable level of accuracy is reached throughout the problem domain. Application to a series of X-Y benchmark problems indicates the reduction of computational effort achievable by replacing uniform with adaptive refinement of the spatial approximations.

A physical interpretation and benchmarking of nodal diffusion error estimators

An error estimator application was evaluated for the analysis and interpretation of nodal solutions of the neutron diffusion problem extended to reactor core level. This error estimator application relies on the residual of the equation and gives a global and local characterisation of approximate solutions by the determination of unbalances in the phase-space discretisation. It has been provided a physical meaning in terms of spurious or strange neutron reactions per cubic centimetre and second. The error estimator was applied to a wide scenario of benchmark problems, twelve of which are here reported, with the purpose of comparison between classical quality criteria and the new quality criterion based on the error estimator. The results show that the new obervation criterion is able to qualify the approximate solutions in coincidence with the classical criteria but more simply and directly, concluding that approximate solutions producing a global unbalance of about 1 10 , 1 100 , 1 1000 spurious reactions per neutron removed are respectively unacceptable, reasonably acceptable or very precise. This is achieved systematially and rigorously in the global sense because it is not possible to get an absolute local error for the diffusion problem without the help of a complementary solution; however, the observation of local errors provides valuable, practical insight into the nature of the optimal solution obtainable.

Matrix-Type Multiple Reciprocity Boundary Element Method for Solving Three-Dimensional Two-Group Neutron Diffusion Equations

The multiple reciprocity boundary element method has been applied to three-dimensional two-group neutron diffusion problems. A matrix-type boundary integral equation has been derived to solve the first and the second group neutron diffusion equations simultaneously. The matrix-type fundamental solutions used here satisfy the equation which has a point source term and is adjoint to the neutron diffusion equations. A multiple reciprocity method has been employed to transform the matrix-type domain integral related to the fission source into an equivalent boundary one. The higher order fundamental solutions required for this formulation are composed of a series of two types of analytic functions. The eigenvalue itself is also calculated using only boundary integrals. Three-dimensional test calculations indicate that the present method provides stable and accurate solutions for criticality problems.

Matrix-type higher order fundamental solutions to three-dimensional two-group neutron diffusion equations

The zero-order and the higher-order fundamental solutions for the 3-D two-group neutron diffusion equations have been derived in such a way that these solutions satisfy the first and the second group equations simultaneously. Each degree of the solutions has a 2 × 2 matrix form based on two types of function, r p exp (−i Br) and r p exp(− kr). Singularities of type (1/ r) are only found at the diagonal components of the zero-order solutions; however, no singularities are found at any components of the higher-order solutions. These solutions can be used for applying the multiple reciprocity boundary element method to 3-D two-group neutron diffusion problems.

Extension of Analytic Function Expansion Nodal Method to Multigroup Problems in Hexagonal-Z Geometry

The analytic function expansion nodal (AFEN) method has been successfully applied to two-group neutron diffusion problems. However, the current AFEN method cannot treat complex eigen-modes, which appear in the general multigroup equations. The AFEN method is extended such that complex eigenmodes are treated within the framework of the original AFEN method for any type of geometry. Also, a suite of new nodal codes based on the extended AFEN theory is developed for hexagonal-z geometry and applied to several benchmark problems. Numerical results obtained attest to their accuracy and applicability to practical problems.

Multiprocessing for neutron diffusion and deterministic transport methods

Multiprocessing is the most exciting development in computer technology in recent years, opening the door for enormous computational power limited only by the imagination and patience of the hardware developer, and the wit and perseverance of the algorithm designer. The potential benefit of this innovation was immediately recognized by researchers in the nuclear field, among others, who explored and reported their experience with concurrent implementation of algorithms, standard and new, to solve neutron diffusion and transport problems. The accumulated experience in, as well as the current status of, this rapidly changing area is reviewed. Collectively these illustrate the strong coupling between parallel algorithms and their target architectures to the extent that standard sequential schemes in nuclear applications are best suited to coarse grained, e.g. shared memory supercomputers, and medium grained, e.g. a few hundred networked powerful CPUs, platforms. The full potential of parallel computing will be better realized by novel, non-traditional, solution algorithms that best utilize all the components of a multiprocessor machine.

Matrix-Type Multiple Reciprocity Boundary Element Method for Solving Three-Dimensional Two-Group Neutron Diffusion Equations.

The multiple reciprocity boundary element method has been applied to three-dimensional two-group neutron diffusion problems. A matrix-type boundary integral equation has been derived to solve the first and the second group neutron diffusion equations simultaneously. The matrix-type fundamental solutions used here satisfy the equation which has a point source term and is adjoint to the neutron diffusion equations. A multiple reciprocity method has been employed to transform the matrix-type domain integral related to the fission source into an equivalent boundary one. The higher order fundamental solutions required for this formulation are composed of a series of two types of analytic functions. The eigenvalue itself is also calculated using only boundary integrals. Three-dimensional test calculations indicate that the present method provides stable and accurate solutions for criticality problems.

Performance Modeling of Parallel Algorithms for Solving Neutron Diffusion Problems

Neutron diffusion calculations are the most common computational methods used in the design, analysis, and operation of nuclear reactors and related activities. Here, mathematical performance models are developed for the parallel algorithm used to solve the neutron diffusion equation on message passing and shared memory multiprocessors represented by the Intel iPSC/860 and the Sequent Balance 8000, respectively. The performance models are validated through several test problems, and these models are used to estimate the performance of each of the two considered architectures in situations typical of practical applications, such as fine meshes and a large number of participating processors. While message passing computers are capable of producing speedup, the parallel efficiency deteriorates rapidly as the number of processors increases. Furthermore, the speedup fails to improve appreciably for massively parallel computers so that only small- to medium-sized message passing multiprocessors offer a reasonable platform for this algorithm. In contrast, the performance model for the shared memory architecture predicts very high efficiency over a wide range of number of processors reasonable for this architecture. Furthermore, the model efficiency of the Sequent remains superior to that of the hypercube if its model parameters are adjusted to make its processors as fast as thosemore » of the iPSC/860. It is concluded that shared memory computers are better suited for this parallel algorithm than message passing computers.« less

A Hennart Nodal Method for the Diffusion Equation

A modification of the Hennart nodal method for neutron diffusion problems is presented. The final system of equations obtained by this method is not positive definite. However, a flux elimination technique leads to a simple positive definite system, which can be solved by the traditional iterative methods. Calculations of a two-dimensional International Atomic Energy Agency benchmark problem are performed and compared with results of the original Hennart nodal method and some finite element methods. The high computational efficiency of this modified nodal method is clearly demonstrated.

A Geometric Buckling Expression for Regular Polygons: II. Analyses Based on the Multiple Reciprocity Boundary Element Method

A procedure is presented for the determination of geometric buckling for regular polygons. A new computation technique, the multiple reciprocity boundary element method (MRBEM), has been applied to solve the one-group neutron diffusion equation. The main difficulty in applying the ordinary boundary element method (BEM) to neutron diffusion problems has been the need to compute a domain integral, resulting from the fission source. The MRBEM has been developed for transforming this type of domain integral into an equivalent boundary integral. The basic idea of the MRBEM is to apply repeatedly the reciprocity theorem (Green’s second formula) using a sequence of higher order fundamental solutions. The MRBEM requires discretization of the boundary only rather than of the domain. This advantage is useful for extensive survey analyses of buckling for complex geometries.The results of survey analyses have indicated that the general form of geometric buckling is B2g = (aN/Rc)2, where Rc represents the radius of the circumscribed circle of the regular polygon under consideration. The geometric constant aN depends on the type of regular polygon and takes the value of π for a square and 2.405 for a circle, an extreme case that has an infinite number of sides. Values of aN for a triangle, pentagon, hexagon, and octagon have been calculated as 4.190, 2.821, 2.675, and 2.547, respectively.Although the discussion is restricted to simple regular polygons, the proposed solution technique based on the MRBEM can easily be applied to many other complex geometries.

Space-Dependent Core/Reflector Boundary Conditions Generated by the Boundary Element Method for Pressurized Water Reactors

The boundary element method is used to generate energy-dependent matrix-type boundary conditions along core/reflector interfaces and along baffle-plate surfaces of pressurized water reactors. This method enables one to deal with all types of boundary geometries including convex and concave corners. The method is applicable to neutron diffusion problems with more than two energy groups and also can be used to model a reflector with or without a baffle plate. Excellent eigenvalue and flux shape results can be obtained when the boundary conditions generated by this technique are coupled with core-only finite difference calculations.

２次元増倍系における空間高次モードの計算

The improved power method in the higher-harmonic eigenvalue calculation of nuclear plant thermal-hydraulic dynamics was applied to the λ- and ωp-eigenvalue problems of thermal and fast reactor systems to obtain the two-dimensional, few-energy-group higher-harmonic eigenvalues and eigenfunctions. As a result, the efficiency of the method was confirmed in the large-scale neutron diffusion problems consisting of the variables amounting to tens of thousands, and the interesting spatial patterns of the eigenfunction were obtained.Even though a harmonic order may be lower, the discrepancy between λ- and ωp-eigen-functions, which are usually assumed to be approximately identical, is significant in a thermal system with low-absorbing region (e.g. reflector). In the higher ωp-harmonics of the system, zero loci of the eigenfunctions depend on neutron energy groups. Therefore, the particular attention should be paid to placements and energy-response of detectors in the kinetic experiments.

The Cell Analytical‐Numerical Method for Solution of the Advection‐Dispersion Equation: Two‐Dimensional Problems

In this paper we present the development and evaluation of a new numerical scheme for efficient solution of groundwater solute transport problems. The scheme, which we have named the cell analytical‐numerical (CAN) method, is an extension of the so‐called nodal methods which have been developed in the nuclear engineering area for solving neutron diffusion problems. The CAN method is based upon decomposition of the solution domain into a number of rectangular cells (volume subdomains); each cell is homogeneous so that a local analytic solution to the solute transport equation can be obtained. A first‐order accurate finite difference approximation of the time derivative followed by a transverse averaging procedure is used to transform the governing partial differential equation into a set of coupled one‐dimensional ordinary differential equations for the transverse moments of the concentration. The local analytical solution is therefore found in terms of these moments. Solute mass flux continuity across cell surfaces, along with the local analytical solution, is used to construct an algebraic relationship between concentration moment values at adjacent cell surfaces. Assembling all the cells together results in a set of coupled tridiagonal matrix equations, one set for each spatial direction, which can be solved very efficiently. The CAN method is applied to several simple test problems involving uniform flow in homogeneous aquifers. While the method lacks geometrical flexibility, it offers a spatial approximation that is demonstrated to have high accuracy and minimal grid orientation error, even when applied to coarse meshes.