K-eigenvalue Problems Research Articles

Convergence study of CMFD and lpCMFD acceleration schemes for k-eigenvalue neutron transport problems in 2-D Cartesian geometry with Fourier analysis

The recently developed lpCMFD scheme has been shown to be unconditionally stable and more effective than CMFD scheme for 2D fixed source neutron transport problems by using Fourier analysis. Previous works have derived the Fourier analysis to study the convergence performance of the CMFD acceleration scheme for 1D k-eigenvalue neutron transport problem, but the study has not yet been performed for 2D k-eigenvalue neutron transport problem, which we believe is an important work that complements other studies of neutron transport acceleration methods. In this paper, we extend the Fourier analysis of CMFD and lpCMFD for k-eigenvalue neutron transport problems to 2D space, along with the detailed linearization and formulation derivation. The Fourier analysis results show that for both CMFD and lpCMFD, increasing the number of inner iterations per outer iteration can reduce the spectral radius up to an asymptotic value of the spectral radius. CMFD is found to be unstable at higher coarse mesh optical thickness while lpCMFD is unconditionally stable for scattering ratios c = 0.3, 0.6, 0.9, 0.99 and for a large range of coarse mesh optical thickness up to 100. The results presented here can support the application of lpCMFD for k-eigenvalue problems in 2D space for realistic neutronic simulations.

Iteration Methods with Multigrid in Energy for Eigenvalue Neutron Diffusion Problems

In this paper we present nonlinear multilevel methods with multiple grids in energy for solving the k-eigenvalue problem for multigroup neutron diffusion equations. We develop multigrid-in-energy algorithms based on a nonlinear projection operator and several advanced prolongation operators. The evaluation of the eigenvalue is performed in the space with smallest dimensionality by solving the effective one-group diffusion problem. We consider two-dimensional Cartesian geometry. The multilevel methods are formulated in discrete form for the second-order finite volume discretization of the diffusion equation. The homogenization in energy is based on a spatially consistent discretization of the group diffusion equations on coarse grids in energy. We present numerical results of model reactor-physics problems with 44 energy groups. They demonstrate performance and main properties of the proposed iterative methods with multigrid in energy.

Stability analysis of the CMFD scheme with linear prolongation

The stability analysis was performed on a new CMFD method (lpCMFD) for both k-eigenvalue problem (EVP) and transient fixed source problem (TFSP), and an unconditionally stable variant of lpCMFD method was presented in this paper. The lpCMFD is a new CMFD method which updates the fine cell scalar flux via linear prolongation of local and neighboring coarse mesh fluxes. The stability analysis was performed on the SN-lpCMFD scheme using the Fourier analysis approach. The theoretical results agree well with numerical results. The lpCMFD presented here is shown to be more stable than the conventional CMFD (fpCMFD) method where flat prolongation is used. The unconditionally stable variant of lpCMFD is developed based on the fine cell optical thickness threshold and the insights provided by the optimal diffusive CMFD (odCMFD). The new method is referred to as stabilized CMFD with linear prolongation (SlpCMFD) and is shown to converge faster than all mainstream CMFD methods.

Unstructured coarse mesh finite difference method to accelerate k-eigenvalue and fixed source neutron transport calculations

Unstructured coarse mesh finite difference (CMFD) method was implemented to accelerate the k-eigenvalue and fixed source neutron transport calculations in the nodal SN transport code DNTR, which was based on arbitrary triangular-z meshes. The same triangular-z meshes were adopted in the CMFD calculation to accommodate the arbitrary problem boundaries and simplify the coarse mesh generation and mapping processes. Two-level CMFD formulations were derived for the k-eigenvalue problems, and the few-group CMFD calculation was used to speed up the convergence of the multigroup CMFD calculation. For the fixed source problems in the neutronics analyses of subcritical reactors or transient neutron transport calculations, a special ks iteration method by introducing the source multiplication factor ks was employed in the SN transport calculation to remove the dependence of the convergence rate on the subcriticality. Multigroup CMFD formulations were established to accelerate the convergence of the ks iteration procedure in the fixed source neutron transport calculation. The acceleration effects of CMFD on the k-eigenvalue and fixed source calculations were verified using four representative neutron transport problems, i.e. the small fast breeder reactor with Cartesian boundaries, the BN-600 fast reactor with hexagonal boundaries, the space nuclear power reactor with cylindrical boundaries, and the accelerator driven subcritical reactor with hexagonal boundaries. The numerical results showed that for the first three eigenvalue problems, the multigroup CMFD achieved a speedup of total computation time by a factor of 2–5, and the two-level CMFD reduced the computation time by a factor of 3–7. It was also shown that the multigroup CMFD was able to accelerate the ks iteration procedure in the fixed source problems by a factor of about 6. The implementation of the unstructured CMFD method turns the DNTR code into an efficient neutron transport solver with flexible geometry treatment capability for both k-eigenvalue and fixed source problems.

Deterministically estimated fission source distributions for Monte Carlo k-eigenvalue problems

The standard Monte Carlo (MC) k-eigenvalue algorithm involves iteratively converging the fission source distribution using a series of potentially time-consuming inactive cycles before quantities of interest can be tallied. One strategy for reducing the computational time requirements of these inactive cycles is the Sourcerer method, in which a deterministic eigenvalue calculation is performed to obtain an improved initial guess for the fission source distribution. This method has been implemented in the Exnihilo software suite within SCALE using the SPN or SN solvers in Denovo and the Shift MC code. The efficacy of this method is assessed with different Denovo solution parameters for a series of typical k-eigenvalue problems including small criticality benchmarks, full-core reactors, and a fuel cask. It is found that, in most cases, when a large number of histories per cycle are required to obtain a detailed flux distribution, the Sourcerer method can be used to reduce the computational time requirements of the inactive cycles.

Jacobian-Free Newton-Krylov Nodal Expansion Methods with Physics-Based Preconditioner and Local Elimination for Three-Dimensional and Multigroup k-Eigenvalue Problems

Motivated by the high accuracy and efficiency of nodal methods on the coarse meshes and the superlinear convergence and high efficiency of Jacobian-free Newton-Krylov (JFNK) methods for large-scale nonlinear problems, a new JFNK nodal expansion method (NEM) with the physics-based preconditioner and local elimination NEM_JFNK is successfully developed to solve three-dimensional (3D) and multigroup k-eigenvalue problems by combining and integrating the NEM discrete systems into the framework of JFNK methods. A local elimination technique of NEM_JFNK is developed to eliminate some intermediate variables, expansion coefficients, and transverse leakage terms through equivalent transformation as much as possible in order to reduce the computational cost and the number of final-solving variables and residual equations constructed in NEM_JFNK. Then efficient physics-based preconditioners are successfully developed by approximating the matrices of the diffusion and removal terms, transverse leakage terms using the three-adjacent-node quadratic fitting methods, and scatter source terms, which make full use of the traditional power iteration. In addition, the Eisenstat-Walker forcing terms are used in the developed NEM_JFNK method to adaptively choose the convergence criterion of linear Krylov iteration within each Newton iteration based on the Newton residuals and to improve computational efficiency further. Finally, the NEM_JFNK code is developed for 3D and multigroup k-eigenvalue problems in neutron diffusion calculations and the detailed study of convergence, computational cost, and efficiency is carried out for several 3D problems. Numerical results show that the developed NEM_JFNK methods have faster convergence speed and are more efficient than the traditional NEM using power iteration, and the speedup ratio is greater for the higher convergence criterion.

An Accurate Globally Conservative Subdomain Discontinuous Least-Squares Scheme for Solving Neutron Transport Problems

In this work, we present a subdomain discontinuous least-squares (SDLS) scheme for neutronics problems. Least-squares (LS) methods are known to be inaccurate for problems with sharp total cross-section interfaces. In addition, the LS scheme is known not to be globally conservative in heterogeneous problems. In problems where global conservation is important, e.g., k-eigenvalue problems, a conservative treatment must be applied. In this study, we propose an SDLS method that retains global conservation and, as a result, gives high accuracy on eigenvalue problems. Such a method resembles the LS formulation in each subdomain without a material interface and differs from LS in that an additional LS interface term appears for each interface. The scalar flux is continuous in each subdomain with the continuous finite element method while discontinuous on interfaces for every pair of contiguous subdomains. The SDLS numerical results are compared with those obtained from other numerical methods with test problems having material interfaces. High accuracy of scalar flux in fixed-source problems and in eigenvalue problems is demonstrated.

Acceleration and Real Variance Reduction in Continuous-Energy Monte Carlo Whole-Core Calculation via p-CMFD Feedback

In the three-dimensional (3-D) continuous-energy whole-core reactor analysis, the partial current–based coarse mesh finite difference (p-CMFD) feedback was applied to the Monte Carlo (MC) k-eigenvalue problem simulation for both inactive and active iterations (cycles). To reduce the stochastic errors in the p-CMFD parameters and their biases due to the ratio-type estimators, the first-in-first-out (FIFO) accumulation scheme was introduced in the MC/p-CMFD procedure. The MC/p-CMFD procedure was tested on a typical pressurized water reactor 3-D continuous-energy whole-core problem while varying the FIFO queue lengths and the results were compared with the conventional power iteration. The Shannon entropy analysis showed that MC/p-CMFD accelerates the convergence of the fission source distributions and mitigates the spatial clustering phenomenon. The real variance analysis also showed that MC/p-CMFD reduces the interiteration correlation, leading to the most real variance reduction in the local MC tallies at the optimum queue length (L = 5). It was also shown that a nontrivial bias was introduced by the p-CMFD feedback, especially for the global tally (keff) with L = 1. However, the bias decreased as the tally bin size became smaller and it was effectively reduced by increasing the queue length (L ≥ 5). In conclusion, the MC/p-CMFD procedure showed promising capability for 3-D continuous-energy whole-core reactor analysis by MC simulation.

Space-Dependent Wielandt Shifts for Multigroup Diffusion Eigenvalue Problems

We present a new concept—the space-dependent Wielandt shift (SDWS)—for accelerating the convergence of the power iteration (PI) scheme for multigroup diffusion k-eigenvalue problems. The SDWS improves on standard Wielandt shift (WS) techniques, which are empirical in nature and are typically effective only when the current estimate of the solution is reasonably converged. By accounting for the physics of the problem through SDWS, we are able to improve the acceleration for the initial iterates when the current estimate of the solution is not close to convergence. Numerical results from one-dimensional problems suggest that, compared to standard WS techniques, the new SDWS techniques can provide upward of a 46% reduction in the number of PIs required for convergence and a 40% reduction in the computational time required. This improvement is sensitive to several problem-dependent factors, such as the geometry and energy-dependence of the problem, the spatial discretization, and the initial guess. The reduction in computational time is also dependent on the linear solver in the PI scheme, as it is well known that WSs can significantly worsen the conditioning of the diffusion linear system. In this paper, we provide a detailed study of the impact of WSs on the performance of several iterative linear solvers. Results from our implementation of SDWS in the three-dimensional (3D) code MPACT show that SDWS can provide similar speedups for 3D multigroup diffusion eigenvalue problems. These results also show that moderate speedups can be obtained by applying SDWS to the coarse mesh finite difference (CMFD) solver in a CMFD-accelerated transport scheme. However, the benefit of doing this may be limited because all but the first few CMFD solves are relatively easy to converge, regardless of the WS used.

The multilevel quasidiffusion method with multigrid in energy for eigenvalue transport problems

A multilevel iterative method for solving multigroup neutron transport k-eigenvalue problems in two-dimensional geometry is developed. This method is based on a system of group low-order quasidiffusion (LOQD) equations defined on a sequence of coarsening energy grids. The spatial discretization of the LOQD equations uses compensation terms which make it consistent with a high-order transport scheme on a given spatial grid. Different multigrid algorithms are applied to solve the multilevel system of group LOQD equations on grids in energy. The eigenvalue is evaluated from the LOQD problem on a coarsest grid. To further improve the efficiency of iterative schemes hybrid multigrid algorithms are developed. The numerical results of tests with a large number groups are presented to demonstrate performance of the proposed iterative schemes.

Fourier analysis of iteration schemes for k-eigenvalue transport problems with flux-dependent cross sections

A central problem in nuclear reactor analysis is calculating solutions of steady-state k-eigenvalue problems with thermal hydraulic feedback. In this paper we propose and utilize a model problem that permits the theoretical analysis of iterative schemes for solving such problems. To begin, we discuss a model problem (with nonlinear cross section feedback) and its justification. We proceed with a Fourier analysis for source iteration schemes applied to the model problem. Then we analyze commonly-used iteration schemes involving non-linear diffusion acceleration and feedback. For each scheme we show (1) that they are conditionally stable, (2) the conditions that lead to instability, and (3) that traditional relaxation approaches can improve stability. Lastly, we propose a new iteration scheme that theory predicts is an improvement upon the existing methods.

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

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.

Applying Nonlinear Diffusion Acceleration to the Neutron Transport k-Eigenvalue Problem with Anisotropic Scattering

High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Furthermore, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.

A Hybrid Deterministic/Monte Carlo Method for Solving the k-Eigenvalue Problem with a Comparison to Analog Monte Carlo Solutions

In this article we present a hybrid deterministic/Monte Carlo algorithm for computing the dominant eigenvalue/eigenvector pair for the neutron transport k-eigenvalue problem in multiple space dimensions. We begin by deriving the Nonlinear Diffusion Acceleration method (Knoll, Park, and Newman, 2011; Park, Knoll, and Newman, 2012) for the k-eigenvalue problem. We demonstrate that we can adapt the algorithm to utilize a Monte Carlo simulation in place of a deterministic transport sweep. We then show that the new hybrid method can be used to solve a two-group, two dimensional eigenvalue problem. The hybrid method is competitive with analog Monte Carlo in terms of number of particle flights required to compute the eigenvalue; however it produces a much less noisy eigenvector and fission source distribution. Furthermore, we show that we can reduce the error induced by the discretization of the low-order system by appropriate refinement of the mesh.

Spatial Homogenization of Transport Discretization Schemes

Neutron transport problems for whole reactor core calculations result in a very large amount of unknowns. Some difficulties related to dimensionality of this kind of problem can be resolved by solving a well-posed homogenized transport problem on a coarse grid. We apply homogenization methodology to develop discretization schemes for solving k-eigenvalue problems on coarse meshes in 1D slab geometry. The step characteristic (SC) method is used for fine-mesh transport calculations. We develop spatially homogenized transport schemes on a basis of two different transport discretization methods: SC and linear discontinuous schemes. The basic approach is to formulate a discretization that is spatially consistent with the given fine-mesh transport discretization. The presented numerical results demonstrate features of the developed methods in preserving grid functions of the angular flux computed with the SC method on fine spatial meshes. We study sensitivity of the proposed schemes to various types of perturbations in spatially averaged cross sections and other homogenization parameters.

A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton–Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.

Prompt Fission Neutron Spectrum Uncertainty Propagation Using Polynomial Chaos Expansion

The polynomial chaos expansion-stochastic collocation method (PCE-SCM) is demonstrated to be a computationally efficient approach for propagating nuclear data uncertainties evaluated for the prompt fission neutron spectra (PFNS) of n + 235U and n + 239Pu fission reactions through two fast neutron critical benchmark experiments. A principal component decomposition of the PFNS covariance matrices yields an efficient representation of the uncertainty in terms of two to four random variables. Both normal and uniform distributions are considered for these random variables, and the random output variables (angular flux and k-eigenvalue) are expressed in terms of Hermite and Legendre chaos expansions, respectively. Tensor product Hermite and Legendre Gauss quadrature sets, respectively, are used to relate the deterministic chaos expansion coefficients to solutions of independent transport k-eigenvalue problems, and the resulting polynomial chaos expansion provides a complete statistical characterization of the uncertainty in the output variables. Direct random sampling of the PFNS followed by repeated solution of the transport problem to create an ensemble of solutions is used to benchmark results obtained from the PCE-SCM implementation. Both direct random sampling and the PCE-SCM implementation yield comparable results where, for the Jezebel and Lady Godiva critical assemblies, the calculated uncertainties in keff resulting from the PFNS propagated uncertainties are found to be of the same order or larger than reported experimental measurement uncertainties, respectively. The PCE-SCM implementation results obtained require orders of magnitude less computational resources compared with the direct random sampling approach.

Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem

We compare a variant of Anderson Mixing with the Jacobian-Free Newton–Krylov and Broyden methods applied to an instance of the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.

Hybrid and Parallel Domain-Decomposition Methods Development to Enable Monte Carlo for Reactor Analyses

** This paper describes code and methods development at the Oak Ridge National Laboratory focused on enabling high-fidelity, large-scale reactor analyses with Monte Carlo (MC). Current state-of-the-art tools and methods used to perform “real” commercial reactor analyses have several undesirable features, the most significant of which is the non-rigorous spatial decomposition scheme. Monte Carlo methods, which allow detailed and accurate modeling of the full geometry and are considered the “gold standard” for radiation transport solutions, are playing an ever-increasing role in correcting and/or verifying the deterministic, multi-level spatial decomposition methodology in current practice. However, the prohibitive computational requirements associated with obtaining fully converged, system-wide solutions restrict the role of MC to benchmarking deterministic results at a limited number of state-points for a limited number of relevant quantities. The goal of this research is to change this paradigm by enabling direct use of MC for full-core reactor analyses. The most significant of the many technical challenges that must be overcome are the slow, non-uniform convergence of system-wide MC estimates and the memory requirements associated with detailed solutions throughout a reactor (problems involving hundreds of millions of different material and tally regions due to fuel irradiation, temperature distributions, and the needs associated with multi-physics code coupling). To address these challenges, our research has focused on the development and implementation of (1) a novel hybrid deterministic/MC method for determining high-precision fluxes throughout the problem space in k-eigenvalue problems and (2) an efficient MC domain-decomposition (DD) algorithm that partitions the problem phase space onto multiple processors for massively parallel systems, with statistical uncertainty estimation. The hybrid method development is based on an extension of the FW-CADIS method, which attempts to achieve uniform statistical uncertainty throughout a designated problem space. The MC DD development is being implemented in conjunction with the Denovo deterministic radiation transport package to have direct access to the 3-D, massively parallel discrete-ordinates solver (to support the hybrid method) and the associated parallel routines and structure. This paper describes the hybrid method, its implementation, and initial testing results for a realistic 2-D quarter core pressurized-water reactor model and also describes the MC DD algorithm and its implementation.

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

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.