Jaakko Leppänen

Computing the Matrix Exponential in Burnup Calculations

Abstract The topic of this paper is the computation of the matrix exponential in the context of burnup equations. The established matrix exponential methods are introduced briefly. The eigenvalues of the burnup matrix are important in choosing the matrix exponential method, and their characterization is considered. Based on the characteristics of the burnup matrix, the Chebyshev rational approximation method (CRAM) and its interpretation as a numeric contour integral are discussed in detail. The introduced matrix exponential methods are applied to two test cases representing an infinite pressurized water reactor pin-cell lattice, and the numerical results are presented. The results suggest that CRAM is capable of providing a robust and accurate solution to the burnup equations with a very short computation time.

Explicit Treatment of Thermal Motion in Continuous-Energy Monte Carlo Tracking Routines

Abstract This paper introduces a new stochastic method for taking the effect of thermal motion into account on the fly in a Monte Carlo neutron transport calculation. The method is based on explicit treatment of the motion of target nuclei at collision sites and, consequently, requires simply cross sections at a temperature of 0 K regardless of the number of temperatures in the problem geometry. It utilizes rejection sampling techniques to manage the fact that total cross sections become distributed quantities. The method has a novel capability of accurately modeling continuous temperature distributions. The new stochastic method is verified using a simple test program, which compares its results to an analytical reference solution based on NJOY-broadened cross sections. Future implementation to Monte Carlo reactor physics code Serpent is also discussed shortly.

Modeling of Nonuniform Density Distributions in the Serpent 2 Monte Carlo Code

Abstract This paper presents a methodology for applying continuously varying density distributions in Monte Carlo particle transport simulation. The capability is implemented in the Serpent 2 code, as part of an effort for developing a universal multiphysics interface for the coupling of Monte Carlo neutronics to thermal hydraulics and fuel performance codes. The method is based on rejection sampling of particle path lengths, but despite its close resemblance to the Woodcock delta-tracking method, the routine can be used with conventional surface tracking as well. The modified tracking routine is put to the test in a simple boiling water reactor pin-cell calculation with continuously changing void distribution in the coolant channel.

Target Motion Sampling Temperature Treatment Technique with Elevated Basis Cross-Section Temperatures

Abstract The target motion sampling (TMS) temperature treatment technique, previously known as “explicit treatment of target motion,” is a stochastic method for taking the effect of thermal motion on reaction rates into account on-the-fly during Monte Carlo neutron tracking. The method is based on sampling target velocities at each collision site and dealing with the collisions in the target-at-rest frame using cross sections below the actual temperature of the nuclide or, originally, 0 K. Previous results have shown that transport with the original implementation of the TMS method requires about two to four times more CPU time than conventional transport methods, depending on the case. In the present paper, it is observed that the overhead factor may increase even above 10 in cases involving burned fuel. To make the method more practical for everyday use, some optimization is required. This paper discusses a TMS optimization technique in which the temperatures of the basis cross sections are elevated above 0 K. Comparisons show that the TMS method is able to reproduce the NJOY-based reference results within statistical accuracy, both with and without the newly implemented optimization technique. In the specific test cases, the optimization saved 35% to 83% of the calculation time, depending on the case.

Solving Linear Systems with Sparse Gaussian Elimination in the Chebyshev Rational Approximation Method

Abstract The Chebyshev Rational Approximation Method (CRAM) has recently been introduced by the authors to solve burnup equations, and the results have been excellent. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices, and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method requires only a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination, where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on consideration of the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with 1606 nuclides.

Temperature Majorant Cross Sections in Monte Carlo Neutron Tracking

Abstract This paper discusses the generation of temperature majorant cross sections, the type of cross sections required by two separate techniques related to Monte Carlo neutron tracking, namely, the Doppler-broadening rejection correction (DBRC) and target motion sampling (TMS) temperature treatment methods. In the generation of these cross sections, the theoretically infinite range of thermal motion must be artificially limited by applying some sort of a cutoff condition, which affects both the accuracy and the performance of the calculations. In this paper, a revised approach for limiting thermal motion is first introduced, and then, optimal cutoff conditions are determined for both the traditional majorant, commonly used in DBRC implementations and old implementations of the TMS method, and the revised majorant. Using the revised type of temperature majorant cross sections increases the performance of the TMS method slightly, but no practical difference is observed with the DBRC method. It is also discovered that in ordinary reactor physical calculations, the cutoff conditions originally adopted from the SIGMA1 Doppler-broadening code can be significantly relieved without compromising the accuracy of the results. By updating the cutoff conditions for majorant generation, the CPU time requirement of Serpent 2.1.17 is reduced by 8% to 23% in TMS calculations and by 1% to 6% in problems involving DBRC.

Internal Neutronics-Temperature Coupling in Serpent 2

Abstract This paper describes the built-in calculation routines in the reactor physics code Serpent 2 that provide a novel method for solving the coupled problem of the power distribution, temperature distribution, and material property distributions in nuclear fuel elements. All of the coupled distributions are solved during a single simulation with no coupling to external codes. The temperature feedback system consists of three separate built-in parts: an explicit treatment of the thermal motion of target nuclides during the transport calculation, an internal analytic radial temperature profile solver, and internal material property correlations. The internal structure and couplings of the calculation routines are described in detail, after which the results of an assembly-level problem are presented to demonstrate the capabilities and functionality of the system.

Study on computational performance in generation of cross sections for nodal simulators using continuous-energy Monte Carlo calculations

This paper aims to evaluate the practical feasibility of using the continuous-energy Monte Carlo method for producing homogenized group constants for deterministic core simulators. The calculations are carried out using the Serpent 2 Monte Carlo code and ARES nodal diffusion fuel cycle simulator. A test case from a previous validation study is repeated with varying number of neutron histories in group constant generation. The impact of statistical variation in the results of ARES simulations is evaluated, and the corresponding calculation times used to provide an order-of-magnitude estimate for the overall computational cost for generating the full set of group constants covering all state points. It is concluded that, while computationally expensive, Monte Carlo-based spatial homogenization involving burnup and thousands of state points per assembly type is within the range of feasibility using modern computer clusters.