K-eigenvalue Problems Research Articles

Newton’s Method for the Computation of k-Eigenvalues in SN Transport Applications

Recently, Jacobian-Free Newton-Krylov (JFNK) methods have been used to solve the k-eigenvalue problem in diffusion and transport theories. We propose an improvement to Newton’s method (NM) for solving the k-eigenvalue problem in transport theory that avoids costly within-group iterations or iterations over energy groups. We present a formulation of the k-eigenvalue problem where a nonlinear function, whose roots are solutions of the k-eigenvalue problem, is written in terms of a generic fixed-point iteration (FPI). In this way any FPI that solves the k-eigenvalue problem can be accelerated using the Newton approach, including our improved formulation. Calculations with a one-dimensional multigroup SN transport implementation in MATLAB provide a proof of principle and show that convergence to the fundamental mode is feasible. Results generated using a three-dimensional Fortran implementation of several formulations of NM for the well-known Takeda and C5G7-MOX benchmark problems confirm the efficiency of NM for realistic k-eigenvalue calculations and highlight numerous advantages over traditional FPI.

Acceleration of k-Eigenvalue/Criticality Calculations Using the Jacobian-Free Newton-Krylov Method

We present a new approach for the k-eigenvalue problem using a combination of classical power iteration and the Jacobian-free Newton-Krylov (JFNK) method. The method poses the k-eigenvalue problem as a fully coupled nonlinear system, which is solved by JFNK with an effective block preconditioning consisting of the power iteration and algebraic multigrid. We demonstrate effectiveness and algorithmic scalability of the method on a one-dimensional, one-group problem and two two-dimensional two-group problems and provide comparison to other efforts using similar algorithmic approaches.

Newton’s Method for Solving k-Eigenvalue Problems in Neutron Diffusion Theory

We present an approach to the k-eigenvalue problem in multigroup diffusion theory based on a nonlinear treatment of the generalized eigenvalue problem. A nonlinear function is posed whose roots are equal to solutions of the k-eigenvalue problem; a Newton-Krylov method is used to find these roots. The Jacobian-vector product is found exactly or by using the Jacobian-free Newton-Krylov (JFNK) approximation. Several preconditioners for the Krylov iteration are developed. These preconditioners are based on simple approximations to the Jacobian, with one special instance being the use of power iteration as a preconditioner. Using power iteration as a preconditioner allows for the Newton-Krylov approach to heavily leverage existing power method implementations in production codes. When applied as a left preconditioner, any existing power iteration can be used to form the kernel of a JFNK solution to the k-eigenvalue problem. Numerical results generated for a suite of two-dimensional reactor benchmarks show the feasibility and computational benefits of the Newton formulation as well as examine some of the numerical difficulties potentially encountered with Newton-Krylov methods. The performance of the method is also seen to be relatively insensitive to the dominance ratio for a one-dimensional slab problem.

Acceleration of source convergence in Monte Carlo k-eigenvalue problems via anchoring with a p-CMFD deterministic method

The Monte Carlo method is widely used in neutron transport calculations, especially in complex geometry and continuous-energy problems. However, an extended application of the Monte Carlo method to large realistic eigenvalue problems remains a challenge due to its slow convergence and large fluctuations in the fission source distribution. In this paper, a deterministic partial current-based Coarse-Mesh Finite Difference (p-CMFD) method is proposed that achieves fast convergence in fission source distribution in Monte Carlo k-eigenvalue problems. In this method, the high-order Monte Carlo method provides homogenized and condensed cross section parameters while the low-order deterministic p-CMFD method provides anchoring of the fission source distribution. The proposed method is implemented in the MCNP5 code (version 1.30) and tested on realistic one- and two-dimensional heterogeneous continuous-energy large core problems, with encouraging results.

Use of a New Boundary Condition in Computational Neutron Transport

This paper deals with the numerical evaluation of fundamental- and higher-mode solutions of the well-known K-eigenvalue problem in nuclear reactor physics using neutron transport theory. If the spatial domain has a plane of reflective symmetry, it is customary to find the fundamental K-mode by considering only the half-domain on one side of the plane for the calculation and applying a reflective boundary condition (RBC) on the plane of symmetry. Here, it is shown that the higher antisymmetric K-mode can also be evaluated in a similar way by applying what is called here the anti-RBC (ARBC) on the plane of symmetry. ARBC was implemented in computer codes based on the discrete ordinates method in Cartesian geometry for some sample problems and was found to work well. The implementation of ARBC in existing codes, although very easy, does not seem to be widely used or reported in the literature. For a one-dimensional homogeneous slab with isotropic scattering, the first antisymmetric K-mode found using ARBC is equivalent to the fundamental mode of a sphere, apart from a scaling factor for the total flux. An interesting result is that the fundamental mode of a sphere computed in this way does not contain the unphysical flux dip near the center, commonly obtained by the discrete ordinates codes in spherical geometry. Although not shown here, it appears that ARBC can be implemented in Monte Carlo codes also to find antisymmetric modes.

A Functional Monte Carlo Method for k-Eigenvalue Problems

In Monte Carlo simulations of k-eigenvalue problems for optically thick fissile systems with a high dominance ratio, the eigenfunction is often poorly estimated because of the undersampling of the fission source. Although undersampling can be addressed by sufficiently increasing the number of particles per cycle, this can be impractical in difficult problems. Here, we present a new functional Monte Carlo (FMC) method that minimizes this difficulty for many problems and yields a more accurate estimate of the k-eigenvalue. In the FMC method, standard Monte Carlo techniques do not directly estimate the eigenfunction; instead, they directly estimate certain nonlinear functionals that depend only weakly on the eigenfunction. The functionals are then used to more accurately estimate the k-eigenfunction and the eigenvalue. Like standard Monte Carlo methods, the FMC method has only statistical errors that limit to zero as the number of particles per cycle and the number of cycles become large. We provide numerical results that illustrate the advantages and limitations of the new method.

A scheme for the evaluation of dominant time-eigenvalues of a nuclear reactor

This paper presents a scheme to obtain the fundamental and few dominant solutions of the prompt time eigenvalue problem (referred to as α-eigenvalue problem) for a nuclear reactor using multi-group neutron diffusion theory. The scheme is based on the use of an algorithm called Orthomin(1). This algorithm was originally proposed by Suetomi and Sekimoto [Suetomi, E., Sekimoto, H., 1991. Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations. Ann. Nucl. Energy 18 (4), 205–227] to obtain the fundamental K-eigenvalue ( K-effective) of nuclear reactors. Recently, it has been shown that the algorithm can be used to obtain the further dominant K-modes also. Since α-eigenvalue problem is usually more difficult to solve than the K-eigenvalue problem, an attempt has been made here to use Orthomin(1) for its solution. Numerical results are given for realistic 3-D test case.

New applications of Orthomin(1) algorithm for K-eigenvalue problem in Reactor Physics

This paper presents some new investigations on the use of Orthomin(1) algorithm for the solution of K-eigenvalue problem in Nuclear Reactor Physics. The algorithm was first used by Suetomi and Sekimoto [Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations, Annals of Nuclear Energy, 18(4) (1991) 205–227.] to find fundamental mode solution of K-eigenvalue problem based on multi-group neutron diffusion theory. It was shown to be an efficient alternative to the conventional methods based on power iterations. The use of this algorithm needs software development to evaluate product of coefficient matrices with vectors. Here, it is shown that the algorithm can also be applied to the K-eigenvalue equation cast in terms of fission matrix. This opens up the possibility of using conventional well-established diffusion codes to implement Orthomin(1) so that new software is not needed. Apart from this, a remarkable feature of Orthomin(1) brought out in this paper is its utility to find higher mode solutions of the K-eigenvalue problem. This procedure is radically different from the methods commonly used for finding higher K-modes. It may also be of interest for finding higher mode eigensolutions in other scientific fields.

05/02341 Krylov sub-space methods for K-eigenvalue problem in 3-D neutron transport

05/02341 Krylov sub-space methods for K-eigenvalue problem in 3-D neutron transport

Krylov sub-space methods for K-eigenvalue problem in 3-D neutron transport

The K-eigenvalue problem in nuclear reactor physics is often formulated in the framework of Neutron Transport Theory. The fundamental mode solution of this problem is usually obtained by the power iteration method. Here, we are concerned with the use of a Krylov sub-space method, called ORTHOMIN(1), to obtain a more efficient solution of the K-eigenvalue problem. A Matrix-free approach is proposed which can be easily implemented by using a transport code which can perform fixed source calculations. The power iteration and ORTHOMIN(1) schemes are compared for two realistic 3-D multi-group cases with isotropic scattering: an LWR benchmark and a heavy water reactor problem. In both the schemes, within-group iterations over self-scattering source are required as intermediate procedures. These iterations are also accelerated using another Krylov method called conjugate gradient method. The overall work is based on the use of Sn-method and finite-differencing for discretisation of transport equation.