Fixed Source Problems Research Articles

Nuclear Reactor Transient Analysis by Continuous-Energy Monte Carlo Calculation Based on Predictor-Corrector Quasi-Static Method

The continuous-energy Monte Carlo (MC) method is gaining attention not only for nuclear reactor statics but also for transient analysis, as computing power increases with the use of massive parallel computers. This paper presents a practical and accurate MC transient analysis method for heterogeneous, continuous-energy reactor transient problems, based on the predictor-corrector quasi-static (PCQS) method. The transient fixed-source problem of the PCQS method is solved by MC calculation with fission source iteration, where the partial current-based coarse-mesh finite difference (p-CMFD) method is used both to accelerate the convergence of the fission source distributions and to diagnose whether the fission source iteration diverges because of too large a macro-time-step size used for a positive reactivity insertion. To improve the convergence of the fission source iteration, exponential transformation is also applied. In addition, the variances of MC tallies can be reduced by increasing the number of active fission source iterations. For method and code verification, the PCQS method for the MC calculation with fission source iteration is compared with the implicit Euler method for a method-of-characteristics calculation on a two-dimensional TWIGL problem. For both multigroup energy and continuous-energy three-dimensional test problems, the proposed method efficiently reduces computing time with a large macro-time-step size, while the accuracy of the solutions is maintained, compared with those calculated with smaller macro-time-step sizes.

An efficient space-angle subgrid scale discretisation of the neutron transport equation

This article presents a new efficient space-angle subgrid scale formulation (SGS) for the space-angle phase-space discretisation of the Boltzmann transport equation (BTE). It applies both coarse and fine scale discretisations to the space-angle phase-spaces, for which the fine scale solutions yield more accurate but more expensive solutions compared to those of the coarse scale approach. Accuracy and efficiency are maintained by integrating the fine scale components within the coarse scale formulation. This results in a method that generates linear systems the same size as those that would be generated by the coarse scale system. The formulation is based on using finite elements (FEs) to represent the spatial variable of the BTE, whilst spherical harmonics are employed to represent the particles’ directions. In space, the fine and coarse scales are represented through discontinuous and continuous linear finite elements. In angle, the scales are represented by the full spherical harmonics (coarse scale), and a partitioning of these basis functions into subsets that couple only between themselves (fine scale).Whilst the spatial discretisation has been previously documented, it is the angular treatment that is critical for efficient solutions. This is particularly the case for large angular expansion sizes involving angular discretisations that couple moments through the streaming term (i.e. not discrete ordinate schemes), where it is shown here that the computing costs are reduced to O(M) from O(M3), where M denotes the angular expansion size. It is shown how the method increases efficiency of the solving process through the improved efficiency of matrix vector multiplication which forms the most expensive component of a Krylov based solver. This matrix vector product is also computed without the matrix being stored in memory which would be a requirement for large scale radiation transport calculations. The method’s capabilities are demonstrated by solving fixed source problems involving a range of material types – from optically thin to opaque. In all cases the new method yields accurate and stable solutions.

The numerical solution of space-dependent neutron kinetics equations in hexagonal-z geometry using Diagonally Implicit Runge Kutta method

The numerical solution of the space-dependent neutron kinetics equations in hexagonal-z geometry is studied in this paper. The equations are discretized by Diagonally Implicit Runge–Kutta (DIRK) method in the temporal direction. The semi-discretized equations are in the form of the fixed source problems, which can be solved by the function expansion nodal method in the hexagonal-z geometry. The intranodal fluxes in each group are expanded by exponential functions and orthogonal polynomials up to the second order. The two-and three-dimensional benchmark problems are used to verify this method. Compared with the backward Euler method, the numerical results show that the current method is more accurate.

Application of the Subgroup Decomposition Method (SDM) for Reactor Simulation

The performance of the TITAN-SDM algorithm for solving a reactor pressure vessel dosimetry problem is evaluated. Douglass and Rahnema recently developed the he subgroup decomposition method (SDM); a methodology which directly couples a consistent coarse-group transport calculation with a set of “decomposition sweeps” to provide a fine-group flux spectrum. The SDM has been implemented into the TITAN three-dimensional transport code and has been shown to accurately solve core criticality problems while significantly reducing computation time. This paper addresses the use of SDM for fixed-source problems. The VENUS-2 dosimetry benchmark problem is selected with an emphasis on fast neutron analysis; therefore, material cross sections are generated from the BUGLE-96 library considering neutron energies greater than 0.1 MeV. The accuracy and efficiency of TITAN-SDM is evaluated by comparison with a standard TITAN multigroup calculation.

Galerkin partial least-square SN (GpLS–SN) method for fixed source problems in the neutron transport theory

The neutron transport equation in the integral form is of difficult solution and therefore, schemes for obtaining an equivalent differential formulation are used. In this paper is used the discrete ordinates method (SN method) in order to obtain the equivalent differential formulation. Inspired in the GLS method (Galerkin least square method) a variant of the GLS (Brooks and Hughes, 1982; Ern and Guermond, 2004; Johnson, 1990) method denoted GpLS (Galerkin partial least-square) is proposed for solving this equivalent differential formulation. The convergence rates as well as the robustness of GLS method are preserved by the GpLS method that possesses computational lower cost. Simulations comparing the convergence rates of GpLS method with other methods are presented, confirming the best performance of GpLS method. The numerical experiments were made using fixed source in multiplying medium, with the objective to test the robustness of GpLS method. All the numerical experiments presented confirm the good properties of approximation of GpLS method.

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.

Improved Monte Carlo Variance Reduction for Space and Energy Self-Shielding

Continued demand for accurate and computationally efficient transport methods to solve optically thick, fixed-source transport problems has inspired research on variance-reduction (VR) techniques for Monte Carlo (MC). Methods that use deterministic results to create VR maps for MC constitute a dominant branch of this research, with Forward Weighted–Consistent Adjoint Driven Importance Sampling (FW-CADIS) being a particularly successful example. However, locations in which energy and spatial self-shielding are combined, such as thin plates embedded in concrete, challenge FW-CADIS. In these cases the deterministic flux cannot appropriately capture transport behavior, and the associated VR parameters result in high variance in and following the plate.This work presents a new method that improves performance in transport calculations that contain regions of combined space and energy self-shielding without significant impact on the solution quality in other parts of the problem. This method is based on FW-CADIS and applies a Resonance Factor correction to the adjoint source. The impact of the Resonance Factor method is investigated in this work through an example problem. It is clear that this new method dramatically improves performance in terms of lowering the maximum 95% confidence interval relative error and reducing the compute time. Based on this work, we recommend that the Resonance Factor method be used when the accuracy of the solution in the presence of combined space and energy self-shielding is important.

General Expressions for Auxiliary Equations of a Nodal Formulation for Two-Dimensional Transport Calculations

A nodal approach is proposed for the solution of discrete ordinates two-dimensional fixed-source problems in Cartesian geometry. For the one-dimensional equations generated by the integration process, explicit solutions in terms of the spatial variables are derived by the analytical discrete ordinates method. The procedure does not require either iterative schemes or domain division into cells, and when associated with the level-symmetric quadrature set it leads to reduced order eigensystems. In particular, the unknown angular fluxes on the boundaries are written in terms of the elementary solutions of the one-dimensional integrated problems and a coupled system for the x and y directions is then obtained. Numerical results generated by the technique are provided and compared to those present in the literature.

MC21 v.6.0 – A continuous-energy Monte Carlo particle transport code with integrated reactor feedback capabilities

MC21 is a continuous-energy Monte Carlo radiation transport code for the calculation of the steady-state spatial distributions of reaction rates in three-dimensional models. The code supports neutron and photon transport in fixed source problems, as well as iterated-fission-source (eigenvalue) neutron transport problems. MC21 has been designed and optimized to support large-scale problems in reactor physics, shielding, and criticality analysis applications. The code also supports many in-line reactor feedback effects, including depletion, thermal feedback, xenon feedback, eigenvalue search, and neutron and photon heating. MC21 uses continuous-energy neutron/nucleus interaction physics over the range from 10−5eV to 20MeV. The code treats all common neutron scattering mechanisms, including fast-range elastic and non-elastic scattering, and thermal- and epithermal-range scattering from molecules and crystalline materials. For photon transport, MC21 uses continuous-energy interaction physics over the energy range from 1keV to 100GeV. The code treats all common photon interaction mechanisms, including Compton scattering, pair production, and photoelectric interactions. All of the nuclear data required by MC21 is provided by the NDEX system of codes, which extracts and processes data from EPDL-, ENDF-, and ACE-formatted source files. For geometry representation, MC21 employs a flexible constructive solid geometry system that allows users to create spatial cells from first- and second-order surfaces. The system also allows models to be built up as hierarchical collections of previously defined spatial cells, with interior detail provided by grids and template overlays. Results are collected by a generalized tally capability which allows users to edit integral flux and reaction rate information. Results can be collected over the entire problem or within specific regions of interest through the use of phase filters that control which particles are allowed to score each tally. The tally system has been optimized to maintain a high level of efficiency, even as the number of edit regions becomes very large.

Sub-cell balanced nodal expansion methods using S4 eigenfunctions for multi-group SN transport problems in slab geometry

A highly accurate S4 eigenfunction-based nodal method has been developed to solve multi-group discrete ordinate neutral particle transport problems with a linearly anisotropic scattering in slab geometry. The new method solves the even-parity form of discrete ordinates transport equation with an arbitrary SN order angular quadrature using two sub-cell balance equations and the S4 eigenfunctions of within-group transport equation. The four eigenfunctions from S4 approximation have been chosen as basis functions for the spatial expansion of the angular flux in each mesh. The constant and cubic polynomial approximations are adopted for the scattering source terms from other energy groups and fission source. A nodal method using the conventional polynomial expansion and the sub-cell balances was also developed to be used for demonstrating the high accuracy of the new methods. Using the new methods, a multi-group eigenvalue problem has been solved as well as fixed source problems. The numerical test results of one-group problem show that the new method has third-order accuracy as mesh size is finely refined and it has much higher accuracies for large meshes than the diamond differencing method and the nodal method using sub-cell balances and polynomial expansion of angular flux. For multi-group problems including eigenvalue problem, it was demonstrated that the new method using the cubic polynomial approximation of the sources could produce very accurate solutions even with large mesh sizes.

The Relationship between the Coarse-Mesh Finite Difference and the Coarse-Mesh Diffusion Synthetic Acceleration Methods

The coarse-mesh finite difference (CMFD) and the coarse-mesh diffusion synthetic acceleration (CMDSA) methods are widely used, independently developed methods for accelerating the iterative convergence of deterministic neutron transport calculations. In this paper, we show that these methods have the following theoretical relationship: If the standard notion of diffusion synthetic acceleration as a fine-mesh method is straightforwardly generalized to a coarse-mesh method, then the linearized form of the CMFD method is algebraically equivalent to a CMDSA method. We also show theoretically (via Fourier analysis) and experimentally (via simulations) that for fixed-source problems, the CMDSA and CMFD methods have nearly identical convergence rates. Our numerical results confirm the close theoretically predicted relationship between these methods.

Explicit formulation of a nodal transport method for discrete ordinates calculations in two-dimensional fixed-source problems

Abstract In this work a study of two-dimensional fixed-source neutron transport problems, in Cartesian geometry, is reported. The approach reduces the complexity of the multidimensional problem using a combination of nodal schemes and the Analytical Discrete Ordinates Method (ADO). The unknown leakage terms on the boundaries that appear from the use of the derivation of the nodal scheme are incorporated to the problem source term, such as to couple the one-dimensional integrated solutions, made explicit in terms of the x and y spatial variables. The formulation leads to a considerable reduction of the order of the associated eigenvalue problems when combined with the usual symmetric quadratures, thereby providing solutions that have a higher degree of computational efficiency. Reflective-type boundary conditions are introduced to represent the domain on a simpler form than that previously considered in connection with the ADO method. Numerical results obtained with the technique are provided and compared to those present in the literature.

Calculation of the power factor using the neutron diffusion hybrid equation

In this paper, we used a neutron diffusion hybrid equation with an external neutron source to calculate nuclear power factors in each fuel element in the reactor core. We used the nodal expansion method to obtain the neutron flux for a given control rods bank position. The results were compared with results obtained for eigenvalue problem near criticality condition and fixed source problem during the start-up of the reactor, where external neutron sources are extremely important for the stabilization of external neutron detectors.

ANALOG COMPUTING FOR A NEW NUCLEAR REACTOR DYNAMIC MODEL BASED ON A TIME-DEPENDENT SECOND ORDER FORM OF THE NEUTRON TRANSPORT EQUATION

This paper considers the concept of analog computing based on a cellular neural network (CNN) paradigm to simulate nuclear reactor dynamics using a time-dependent second order form of the neutron transport equation. Instead of solving nuclear reactor dynamic equations numerically, which is time-consuming and suffers from such weaknesses as vulnerability to transient phenomena, accumulation of round-off errors and floating-point overflows, use is made of a new method based on a cellular neural network. The state-of-the-art shows the CNN as being an alternative solution to the conventional numerical computation method. Indeed CNN is an analog computing paradigm that performs ultra-fast calculations and provides accurate results. In this study use is made of the CNN model to simulate the space-time response of scalar flux distribution in steady state and transient conditions. The CNN model also is used to simulate step perturbation in the core. The accuracy and capability of the CNN model are examined in 2D Cartesian geometry for two fixed source problems, a mini-BWR assembly, and a TWIGL Seed/Blanket problem. We also use the CNN model concurrently for a typical small PWR assembly to simulate the effect of temperature feedback, poisons, and control rods on the scalar flux distribution.

Comparison between correlated sampling and the perturbation technique of MCNP5 for fixed-source problems

Correlated sampling and the differential operator perturbation technique are two methods that enable MCNP (Monte Carlo N-Particle) to simulate small response change between an original system and a perturbed system. In this work the performance of the MCNP differential operator perturbation technique is compared with that of the MCNP correlated sampling method for three types of fixed-source problems. In terms of precision of predicted response changes, the MCNP perturbation technique outperforms correlated sampling for the problem involving variation of nuclide concentrations in the same direction but performs comparably with or even underperforms correlated sampling for the other two types of problems that involve void or variation of nuclide concentrations in opposite directions. In terms of accuracy, the MCNP differential operator perturbation calculations may predict inaccurate results that deviate from the benchmarks well beyond their uncertainty ranges for some of the test problems. However, the accuracy of the MCNP differential operator perturbation can be improved if the midpoint correction technique is used.

Solution of moving-load-induced soil vibrations based on the Betti–Rayleigh Dynamic Reciprocal Theorem

Based on the Betti–Rayleigh Dynamic Reciprocal Theorem, the reciprocal property of the Green's functions is demonstrated, on the basis of which the analytical solution of soil vibration subjected to moving loads is derived. By application of the Betti–Rayleigh Dynamic Reciprocal Theorem, the moving source problem is converted to the fixed source problem with receiver point moving in opposite direction, which greatly simplifies the complex analytical solution of soil/ground vibration induced by a moving load. A computer code for solving the ground responses subjected to the moving constant load and harmonic load is developed via MATLAB and is employed to perform a case study. The analyzed results show that the ground vibration induced by moving constant load is of typical low-frequency feature; the frequency range of ground vibration is controlled by the Rayleigh's wave velocity of the top layer soil; when the receiver nears the source path, the R-wave contributes to the ground vibration more than the P-wave and the S-wave; while when the receiver is far from the source path, the contribution of the P-wave is more obvious.

Fourier Convergence Analysis of the Rebalance Methods for Discrete Ordinates Transport Equations in Eigenvalue Problems

This paper analyzes the convergence of the rebalance iteration methods for accelerating the power iteration method of the discrete ordinates transport equation in the eigenvalue problem. The rebalance iteration methods include the coarse mesh rebalance (CMR), the coarse mesh finite difference (CMFD), and the partial current-based CMFD methods. The convergence analysis is performed with the well-known Fourier analysis through linearization. In the linearized form, these rebalance methods are formulated in a unified way where the rebalance methods are different only in a parameter. The analyses are applied for both one- and two-group problems in a homogeneous infinite medium and a finite medium having periodic boundary conditions. The theoretical analysis shows that the convergences of the rebalance methods for the eigenvalue problems are closely related with the ones for the fixed source problems and that the convergences for the eigenvalue problems can be analyzed with the formula for the fixed source problem after transforming the scattering cross sections into a different cross-section set. The numerical tests show that the Fourier convergence analysis provides a reasonable estimate for the numerical spectral radii for the model problems.

An Improved Inverse Analysis Model for Fuel Loading Pattern Optimization

An improved inverse analysis model for fuel loading pattern (LP) optimization is proposed to reproduce a desired in-core power distribution from available fuel assemblies based on the inverse analysis of diffusion equations. In the present approach, the inverse analysis is simplified as a fixed-source problem rather than an eigenvalue problem in a previous work. By using the one-to-one correspondence between the power distribution and reactivity distribution, a target reactivity distribution is solved from a desired power distribution through the inverse analysis. The reactivity of available assemblies is evaluated using a modified effective fast-group model with a higher order correction of leakage effect. A candidate LP is determined by individually matching the reactivity of available assemblies to the target in-core reactivity distribution using a stable matching algorithm. The process of predicting the reactivity of available assemblies and matching them to the target reactivity distribution is repeated until the convergence condition is satisfied. Validation of the model has been confirmed with a reference PWR core. The model is expected to be applicable to a problem of multicycle loading pattern optimization.

Subgroup weight generation based on shielded pin-cell cross section conservation

A method for generating subgroup weights is presented employing the idea of conserving resonance shielded cross sections for representative pin cells. Consistency in the generation and usage steps of the subgroup weights is imposed in a subgroup formulation involving the subgroup fixed source problem to resolve the subgroup level and space dependent self shielding. Consequently, the subgroup level dependent background cross sections are determined and used in the generation step. The requirement of reconstructed cross section error minimization forms a constrained least square problem which is solved by the Lagrange multiplier technique augmented with a regularization term introduced to enhance solution stability. The generated subgroup weights are verified by analyzing various pin cell benchmark problems and comparing with corresponding Monte Carlo solutions. The results for the Rowlands and Mosteller Doppler-effect benchmark problems show small errors of 0.1% in the resonance cross sections and about 70pcm in multiplication factor, and also good agreement in the reactivity vs. fuel temperature behavior. The problem of the slight over-prediction of the fuel temperature coefficients is noted and discussed as well.

An Improved Inverse Analysis Model for Fuel Loading Pattern Optimization

An improved inverse analysis model for fuel loading pattern (LP) optimization is proposed to reproduce a desired in-core power distribution from available fuel assemblies based on the inverse analysis of diffusion equations. In the present approach, the inverse analysis is simplified as a fixed-source problem rather than an eigenvalue problem in a previous work. By using the one-to-one correspondence between the power distribution and reactivity distribution, a target reactivity distribution is solved from a desired power distribution through the inverse analysis. The reactivity of available assemblies is evaluated using a modified effective fast-group model with a higher order correction of leakage effect. A candidate LP is determined by individually matching the reactivity of available assemblies to the target in-core reactivity distribution using a stable matching algorithm. The process of predicting the reactivity of available assemblies and matching them to the target reactivity distribution is repeated until the convergence condition is satisfied. Validation of the model has been confirmed with a reference PWR core. The model is expected to be applicable to a problem of multicycle loading pattern optimization.