Neutron Transport Research Articles

Negative flux fixups using positivity-preserving limiters for SN-discontinuous finite element scheme of neutron transport equation on unstructured triangular meshes

One significant challenge of spatial discretization for the SN transport equation is the appearance of negative solutions, resulting in numerical algorithms to be unstable or slow iterative convergence in some problems. The discontinuous Galerkin finite element (DGFEM) spatial scheme on unstructured meshes has attracted much attention in recent years, but the issue of negative solutions is still a long-standing problem. This paper studies a negative flux fixup method using positivity-preserving limiters for solving the SN-DGFEM neutron transport equation. This method uses the hierarchical basis functions and triangular reference elements to obtain arbitrary high order DGFEM scheme. In the element where appears negativities, a scaling or a rotation positivity-preserving limiter is used to scale or rotate the polynomial distribution to ensure positive solutions. Five different types of problems are selected to verify the accuracy and convergence of the method. Numerical results demonstrate that the method can produce non-negative solutions and maintain the convergence order of the original scheme, as well as not introduce too much additional computational effort.

Solutions of a Neutron Transport Equation with a Partly Elastic Collision Operators

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups et(T+Ke)t≥0 and et(T+Kc+Ke)t≥0, generated by the operators T+Ke and T+Kc+Ke, respectively, have the same essential type.According to Proposition , it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that R2(t) is compact on L2(Ω×V,dxdv), and, using an interpolation argument (see Proposition ), we establish the compactness of R2(t) on Lp(Ω×V,dxdv)-spaces for 1<p<+∞. To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem , is new for periodic boundary conditions.

Collimated beams of fast neutrons at the NPI CAS

The Nuclear Physics Institute of the Czech Academy of Sciences operates multiple neutron sources that can produce neutrons with energies up to 33 MeV. Recently, a segmented collimator was constructed to facilitate research on collimated beams of fast neutrons. In front of the collimator, a new quasi-monoenergetic neutron source was built using accelerated protons interacting with a 2.5 mm thick beryllium target. The collimated beam provides a neutron flux of approximately 106 n/cm2/s at the standard measurement position.To determine the parameters of the collimated neutron beam, various measurement techniques were employed, including scintillator-based time-of-flight mode, proton recoil telescope, and activation detection through the (n,2-3n) reaction on a yttrium sample. Furthermore, Monte Carlo simulations were conducted to model the neutron transport through the collimator, and the results were subsequently compared to the experimental data obtained.

Development of fuel depletion code for molten salt reactor with very deep burnup

A liquid-fueled molten salt reactor (MSR) can reach a deep burnup based on online reprocessing and continuously refueling, which requires significantly different burnup calculation methods for MSRs compared with those for the traditional reactors. To address the unique burnup features and consider the fidelity of isotopic evolution in an MSR, a fuel depletion code ThorMCB is developed based on the OpenMC coupled with a specific depletion code, MODEC. Furthermore, to lower the computational cost of acquiring the equilibrium state through the time evolution step by step for an MSR, an equilibrium burnup calculation code ThorMCB-eq based on the OpenMC and MODEC is developed, which can obtain the equilibrium burnup efficiently. A single fuel lattice of MSR and an a Molten Salt Fast Reactor (MSFR) benchmark are applied for verifying the correctness of the ThorMCB and ThorMCB-eq codes. Compared with a neutron transport calculation code KENO-VI coupled with MODEC, the maximum deviation of the dominant heavy nuclides (HNs) at equilibrium state by ThorMCB is less than 10%, and that of the total mass of fission products (FPs) is less than 3%. For the MSFR benchmark, the neutronic parameters including temperature reactivity coefficient, the mass evolution of main HNs and FPs and breeding ratio (BR) from ThorMCB agree with the references. The equilibrium behavior can be quickly obtained with ThorMCB-eq, and the relative mass deviations of most nuclides keep around 2% in comparison with the results of step-by-step burnup evolution with ThorMCB. Furthermore, the same fuel contents and micro one-group cross sections at equilibrium are obtained with two different types of start-up fuels and a constant power density and fuel reprocessing scheme. In conclusion, the verified results indicate that ThorMCB and ThorMCB-eq can both provide reliable simulation for depletion evolution and equilibrium burnup for MSR fuel cycle.

A computational framework to support probabilistic criticality modelling for the geological disposal of radioactive waste

Nuclear Waste Services is tasked with disposal of the UK’s higher-activity radioactive waste in a Geological Disposal Facility. The disposal of fissile nuclides requires a demonstration that there is no significant concern from criticality, i.e. a fission chain reaction. While waste packages will initially be emplaced in a subcritical configuration, over the long timescales following closure there is potential for waste packages to degrade and for nuclides to be dispersed in the subsurface by groundwater, leading to the potential for a critical system forming. To facilitate modelling, a codebase has been developed which interfaces a probabilistic simulation tool (GoldSim) with a neutron transport code (MONK/MCNP). This allows large ensemble simulations to be run iteratively to determine limiting fissile masses which satisfy a criticality safety criterion. This paper documents the main algorithms and methodologies implemented within this framework, and provides background and example results illustrating the application to post-closure criticality modelling.

Neutronics Analysis on High-Temperature Gas-Cooled Pebble Bed Reactors by Coupling Monte Carlo Method and Discrete Element Method

The High-Temperature Gas-Cooled Pebble Bed Reactor (HTG-PBR) is notable in the advanced reactor realm for its online refueling capabilities and inherent safety features. However, the multiphysics coupling nature of HTG-PBR, involving neutronic analysis, pebble flow movement, and thermo-fluid dynamics, creates significant challenges for its development, optimization, and safety analysis. This study focuses on the high-fidelity neutronic modelling and analysis of HTG-PBR with an emphasis on achieving an equilibrium state of the reactor for long-term operations. Computational approaches are developed to perform high-fidelity neutronics analysis by coupling the superior modelling capacities of the Monte Carlo Method (MCM) and Discrete Element Method (DEM). The MCM-based code OpenMC and the DEM-based code LIGGGHTS are employed to simulate the neutron transport and pebble movement phenomena in the reactor, respectively. To improve the computational efficiency to expedite the equilibrium core search process, the reactor core is discretized by grouping pebbles in axial and radial directions with the incorporation of the pebble position information from DEM simulations. The OpenMC model is modified to integrate fuel circulation and fresh fuel loading. All of these measures ultimately contribute to a successful generation of an equilibrium core for HTG-PBR. For demonstration, X-energy’s Xe-100 reactor—a 165 MW thermal power HTG-PBR—is used as the model reactor in this study. Starting with a reactor core loaded with all fresh pebbles, the equilibrium core search process indicates the continuous loading of fresh fuel is required to sustain the reactor operation after 1000 days of fuel depletion with depleted fuel circulation. Additionally, the model predicts 213 fresh pebbles are needed to add to the top layer of the reactor to ensure the keff does not reduce below the assumed reactivity limit of 1.01.

A boundary-type algorithm based on the discrete ordinates for neutron transport equation

The spatial distribution of neutron flux in the core of a nuclear reactor plays a crucial role in ensuring nuclear safety. It can be determined by solving the neutron transport equation, where computational resource consumption is gradually receiving more attention, especially in problems with multiple reflective boundary conditions. It is common practice to assume initial boundary values in one angular direction to obtain a global solution, and then use reflective boundary conditions to solve for the other directions, iterating until convergence. However, this approach has the drawback of requiring the global solution, and each iteration necessitates storing the neutron flux values, consuming time and storage space. In order to address this issue, this paper proposes a precise and efficient boundary-type method, the Half Boundary Method (HBM). This method establishes relationships between adjacent node values through integration and mathematical deduction, ultimately obtaining the relationship among any node and the boundary values in any direction. As a result, iteration is only required for boundary values, reducing the iteration amount and thus saving time and storage space. Using the discrete ordinate (SN) method for angular discretization and HBM for spatial discretization, this paper solved the steady-state neutron transport equation for two-dimensional systems modeled with Cartesian geometry. The method is validated using several benchmark problems, including the 2D ISSA benchmark, the 2D mono-group k-eigenvalue problem and BWR rod bundle test problem. All of these problems demonstrate good results and effectively reduced computational time.

On the convergence of the fixed point method for solving neutron transport alpha eigenvalue problems

It was shown that the Fixed Point Method (also known as the Rayleigh Quotient Method) is several times faster than the Critical Search Method for solving neutron transport alpha eigenvalue problems. It was also shown that the Fixed Point Method is able to determine the alpha eigenvalues of sub-critical systems that are beyond the reach of the Critical Search Method. Despite these significant advances, the Fixed Point Method remains an unproven algorithm. This report provides a proof.

Study on Calculation of Regulated Neutron Spectrum Based on Moderator with Multi-dimensional Features

Abstract Neutron transport simulation is one of the key techniques for calculating the regulated neutron spectrum, which is extremely accurate but comes with a significant time loss and is difficult to achieve convergence when there is deep penetration. Accurate regulation of the neutron spectrum is essential for research on experimental neutronics. In this study, a fast way to compute the regulated neutron spectrum using a moderator with multi-dimensional features was developed. The response matrix with different dimensions was generated for each moderator after it first created a moderator database by classifying common shielding materials based on various dimensions (such as shape, size, substance, etc.). Finally, neutron transport modeling enabled the speedy and precise estimation of regulated neutron spectrum via matrix transformation. The results demonstrated that the mean square error (MSE) of the normalized regulated neutron spectra computed using the Monte Carlo approach and the proposed technique were able to reach the order of 10−6. The anticipated method was confirmed by equating the results with those of the Monte Carlo method under a regulation system consisting of moderators of different dimensions. Furthermore, the MSE of the regulated neutron spectrum in the case of deep penetration of the simplified thermal reactor TRIGA of the two methods in the convergent energy domain could also reach the order of 10−6, and in the divergent domain, the suggested method in this study was able to give a heuristic neutron spectrum.

Nonlinear light-output calibration of the oxygenated xylene scintillators used in OMEGA neutron time-of-flight spectrometers.

Neutron time-of-flight (nTOF) spectrometers are essential instruments for measuring and evaluating the performance of inertial confinement fusion implosions. The neutron spectrometers utilized for the OMEGA laser include two liquid-based scintillators, each consisting of a large volume filled with xylene that is coupled to four photomultiplier tubes. Analysis of the signal from these detectors requires detailed knowledge of the scintillator's light output, which is needed to fit the nTOF spectrum, from which the neutron energy spectrum is informed. The light output is nonlinearly proportional to the neutron energy, which, in turn, affects the interpretation of the neutron energy spectrum from a TOF signal. A recent campaign on OMEGA was performed to calibrate the xylene detectors and infer the shape of the light-output curve. The campaign utilized materials with increasing Z placed in the OMEGA target chamber to initiate scattering events with the 14MeV fusion neutrons. This process leads to the production of backscatter neutrons of varying energies that appear as peaks in the nTOF data. Simulations using a neutron transport code were combined with the measured deuterium-tritium neutron yields to calculate the expected backscattered neutron yields from the well-known scattering cross sections of each material. The neutron-energy dependent light output of the scintillator inferred from the experiment is compared to the light-output curve simulated with a neutron transport code for the following neutron energies: 1.5, 2.5, 6, and 14MeV.

KATANA - water activation facility at JSI TRIGA, Part I: Final design and activity calculations

Water as a primary coolant will play an important role in the performance of fusion reactors, as it causes an ionising radiation field throughout the facility after its irradiation and activation and requires improved shielding for instruments and personnel. To support ITER, the KATANA irradiation facility, which utilises a closed-water activation loop, was built, licenced and commissioned in early 2024 at the TRIGA research reactor at the Jožef Stefan Institute in Slovenia.A comprehensive overview of the KATANA irradiation facility, which serves as a well-defined and stable high-energy γ and neutron source, is given, including the final design, experimental set-up, detector systems and overall timeline. The physical characteristics of KATANA were assessed by analyses of neutron transport simulations leading to activity calculations in connection with a conventional analytical approach, without using CFD calculations. The KATANA facility demonstrated the desired operational characteristics in terms of high and stable water flow rates, leading to high activity values of the observed isotopes 16N, 17N and 19O, which is essential for minimising experimental uncertainties.The preliminary calculations provided in-depth operational knowledge and important data for the KATANA facility, forming a basis for more advanced future experiments.

Source identification problems for the neutron transport equations

In this study, the time-dependent source identification problem for the two-dimensional neutron transport equation was studied. For the approximate solution of this problem a first order of accuracy difference scheme was presented. Stability estimates for the solution of these differential and difference problems were established. Numerical results were given.

Application of the PSI cycle check-up methodology for full-core Monte Carlo simulations verified using pin power estimates and validated for hot zero power conditions

Monte Carlo (MC) based neutron transport codes provide high-fidelity numerical solutions to problems beyond the modelling capabilities of conventional core simulation methods. However, performing core-follow calculations with MC codes remains challenging due to the necessity of incorporating thermal-hydraulic feedback and generating evolved fuel composition for the cycle of interest by taking into account the entire irradiation history of loaded fuel assemblies over previous cycles. Nevertheless, at PSI, the neutronic version of a cycle check-up methodology (CHUP) is under development to address this challenge. This methodology involves extracting operating conditions and nuclide compositions from validated reference deterministic core-follow models (CASMO5/SIMULATE5/SNF), subsequently generating MC neutron transport models for selected operating points. This article presents the verification and validation performed for a hot zero power operating condition of a Swiss pressurised water reactor using the updated CHUP methodology, which automates information extraction from reference core models to minimise human intervention. The MC models were verified by assessing their deviation from criticality, consistently found to be supercritical within the [10,60] pcm range. Additionally, radial relative power distribution verification yield deviations in the [−5,4] % range compared to validated nodal results. Finally, additional validation was performed using start-up measurements from two successive cycles. Eight MC models were found to be subcritical, on average by 51 pcm, with a standard deviation of 25 pcm.

Monte Carlo multiphysics simulation on adaptive unstructured mesh geometry

Monte Carlo simulation based on Constructive Solid Geometry (CSG) brings unique challenges for multiphysics simulation, including establishing field transfers with mesh-based physics codes, the combination of stochastic and deterministic solvers, and high computational expense. In this work, an adaptive, on-the-fly mesh-based Monte Carlo geometry algorithm is implemented in Cardinal to reduce the barrier-to-entry for high-fidelity multiphysics by (i) eliminating ambiguity in defining CSG cells for temperature and density feedback, (ii) enabling simple mesh convergence studies, and (iii) more closely integrating Computer Aided Design (CAD) workflows with Monte Carlo methods. During Picard iterations, an OpenMC mesh geometry is adaptively refined or coarsened by contouring temperature and/or density fields from a thermal-fluid solver. This algorithm is applied to a full-core Molten Salt Fast Reactor (MSFR) geometry with NekRS Large Eddy Simulation (LES) coupled to OpenMC neutron transport. A performance study indicates a net speedup of 2.3× in the OpenMC solver when using an adaptive geometry for cell sizes chosen intermediate to the as-built CAD geometry versus 1:1 element tracking, which points to future algorithmic research in accelerated Monte Carlo mesh tracking.

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

Systematic derivation of [formula omitted] equations, its discretization using GTIN method and development of a switchable SP3 to [formula omitted] neutron transport solver

The Generalized Simplified Spherical Harmonics (GSPN) method has drawn recent interest owing to the fact that theoretically, it is equivalent to the PN and does not rely on any assumption other than piecewise homogeneity. Its level wise approach to PN offers an agreeable balance between the accuracy and computational efficiency making it a valuable mathematical tool for analyzing neutron transport problems pertaining to nuclear engineering and reactor physics. The lowest level approximation for N = 3, expressed as GSP3(0), offers reducing the complexity while still capturing the essential physics. In the current work, GSP3(0) equations are derived along with the interface and boundary conditions for linear anisotropic scattering. These equations are then discretized using the generalized transverse integration nodal method for a two-dimensional system of piecewise homogeneous rectangular regions. The discretization is done with an option kept for reducing the GSP3(0) formalism to the conventional SP3. Subsequently, a computer code has been developed to solve these discretized equations in the multigroup structure for vacuum, reflective or albedo boundaries. At present, this switchable SP3/GSP3(0) code has the capability of estimating k-eigenvalue and position dependent scalar neutron flux within a given two-dimensional rectangular geometry. The code has been verified by solving SP3, GSP3(0) and other benchmark problems from available literature. The results obtained from our code demonstrate that GSP3(0) is superior to the conventional SP3 and comparable to the recently proposed SDPN (N = 1,2,3) [M. Nazari, A. Zolfaghari, M. Abbasi, Prog. Nucl. Energy, 2023; 166, 104,933] in terms of accuracy as well as computation time.

Generalized Integral Method for Solving the Neutron Transport Equation with the Hybridized Discontinuous Galerkin Method

Accurate modeling of the neutron transport equation (NTE) with anisotropic scattering is crucial to the understanding of neutron interactions within various mediums. The primary challenges in this domain are (1) the considerable computational resources demanded by anisotropic calculations and (2) the numerical instabilities that arise due to the transport correction approximation. This study introduces a novel, generalized integral method based on the hybridized discontinuous Galerkin framework for solving the second-order NTE with anisotropic scattering. This method employs a spherical harmonics expansion to define the partial current at the mesh interface and applies an angular integral approach to the flux treatment within the mesh. This dual approach facilitates an efficient computational process while preserving accuracy. The integral method has been validated through comparisons with the standard discrete ordinates method (SN) using two eigenvalue problems. The integral method showcases several significant improvements over the traditional SN method. First, it repositions the P0 scattering sources during the formulation process, effectively circumventing the convergence issues associated with transport correction. Second, this strategic repositioning substantially enhances the convergence rates of iterative calculations. Last, a standout feature of the integral method is its capability to perform angular integrals during assembling matrices, successfully reducing the floating-point operations for local flux retrieval and eliminating the ray effect.

Monte Carlo Simulations of Spent Nuclear Fuel Dry Storage Cask Measurements with a New External Remote Monitoring System for Developing Safeguards Approaches

Until a long-term solution for the disposal of spent nuclear fuel (SNF) is available, interim dry casks will be increasingly used for the storage of SNF discharged from civilian nuclear power reactors. Dry casks containing commercial SNF may hold several significant quantities of plutonium, so appropriate nuclear material safeguards monitoring is needed. An external remote monitoring system (RMS) has been developed to advance dry cask safeguards monitoring beyond the current method of containment and surveillance used to maintain continuity of knowledge. In this study, neutron transport simulations of SNF assemblies in a dry cask were performed for several special nuclear material diversion scenarios. The simulations considered various loading patterns and fuel storage durations as long as 100 years. For each fuel loading pattern and storage time investigated, the simulation results were used to calculate the required measurement time to achieve a nondetection probability β ≤ 10% for the diversion of any single fuel assembly in the cask. The calculations were performed for false alarm probabilities α as low as 0.0001% (or 10−6). A Monte Carlo postprocessing approach was developed to consider the impact on the required measurement time of uncertainty in the burnup of fuel assemblies. The study found that the external RMS is well suited for the surveillance of SNF in dry cask storage for nuclear safeguards or other purposes and is able to detect the diversion of a single SNF assembly even after decades of storage and with a very low false alarm probability.

Exploring run-and-tumble movement in confined settings through simulation.

Motion in bounded domains is a fundamental concept in various fields, including billiard dynamics and random walks on finite lattices, and has important applications in physics, ecology, and biology. An important universal property related to the average return time to the boundary, the Mean Path Length Theorem (MPLT), has been proposed theoretically and experimentally confirmed in various contexts. We investigated a wide range of mechanisms that lead to deviations from this universal behavior, such as boundary effects, reorientation, and memory processes. This study investigates the dynamics of run-and-tumble particles within a confined two-dimensional circular domain. Through a combination of theoretical approaches and numerical simulations, we validate the MPLT under uniform and isotropic particle inflow conditions. This research demonstrates that although the MPLT is generally applicable for different step length distributions, deviations occur for non-uniform angular distributions, non-elastic boundary conditions, or memory processes. These results underline the crucial influence of boundary interactions and angular dynamics on the behavior of particles in confined spaces. Our results provide new insights into the geometry and dynamics of motion in confined spaces and contribute to a better understanding of a broad spectrum of phenomena ranging from the motion of bacteria to neutron transport. This type of analysis is crucial in situations where inhomogeneity occurs, such as multiple real-world scenarios within a limited domain.

Point containment algorithms for constructive solid geometry with unbounded primitives

We present several algorithms for evaluating point containment in constructive solid geometry (CSG) trees with unbounded primitives. Three algorithms are presented based on postfix, prefix, and infix notations of the CSG binary expression tree. We show that prefix and infix notations enable short-circuiting logic, which reduces the number of primitives that must be checked during point containment. To evaluate the performance of the algorithms, each algorithm was implemented in the OpenMC Monte Carlo particle transport code, which relies on CSG to represent solid bodies through which subatomic particles travel. Two sets of tests were carried out. First, the execution time to generate a rasterized image of a 2D slice of three CSG models of varying complexity was measured. Use of both prefix and infix notations offered significant speedup over the postfix notation that has traditionally been used in particle transport codes, with infix resulting in a 6× reduction in execution time relative to postfix for a model of a tokamak fusion device. We then measured the execution time of neutron transport simulations of the same three models using each of the algorithms. The results and performance improvements reveal the same trends as for the rasterization test, with a 5.52× overall speedup using the infix notation relative to the original postfix notation in OpenMC for the tokamak model.