Fixed Source Problems Research Articles

Advancing Fixed-Source Reactor Analysis Through OpenNode Implementation

OpenNode, a new open-source Fortran code, enables the simulation of fixed-source reactors. Powered by the nodal expansion method (NEM) and seamless Python integration, OpenNode competes with existing nuclear reactor simulation tools. The fixed-source problem plays a crucial role in radioprotection, providing flux calculations and detailed insights into heating effects and dose rates. Users can define and configure reactor models via a JSON input file, specifying critical parameters like geometry, materials, cross-section data, boundary conditions, and fixed sources. OpenNode further empowers users with a Python interface for preprocessing and postprocessing, streamlining result analysis. Distinguishing itself from conventional NEM codes, OpenNode supports three-dimensional (3D) Cartesian geometries, facilitating intricate reactor design simulations. Customization options, including mesh sizes, polynomial orders, and calculation modes, enhance precision and efficiency. In our comprehensive study, we verified OpenNode’s fixed-source mode using the 3D-IAEA reactor, comparing it with the SANM code KOMODO. The results underscore OpenNode’s exceptional accuracy in computing critical parameters, like the effective multiplication factor, power distribution, and nodal flux, within fixed-source reactors. OpenNode stands as a reliable, user-friendly tool poised to advance nuclear reactor simulations.

Monte Carlo iterative k-source method for fixed-source calculations in nearly critical subcritical systems

In addition to the conventional fixed-source method, the iterative k-source (IKS) method is another Monte Carlo algorithm used for solving fixed-source problems. In contrast to the fixed-source method, the IKS method iterates the fission sources for each generation, which is similar to the k-eigenvalue calculation. The IKS method does not require following long-chain fission reactions until the end, even for a nearly critical subcritical system. The source-multiplication factor, ks, which is supposed to be less than unity, is calculated for each cycle. However, ks occasionally or frequently exceeds unity in a nearly critical system owing to statistical fluctuations, in which case the result of the IKS method is biased. To eliminate this bias, the frequency at which ks exceeds unity should be minimized by increasing the source size per cycle. As demonstrated in this study, the computational efficiency of the IKS method is three to ten times that of the conventional fixed-source method.

A general-purpose code for correlated sampling using batch statistics with MCNP6 for fixed-source problems

Background Correlated sampling can be applied with batch statistics using MCNP6’s tally fluctuation chart (TFC) to reduce the uncertainty of a difference of tallies between a base case and a perturbed case without modifying the source code. The cases are correlated if they are run with the same random number sequence. Equations implemented Equations used to apply correlated sampling to the difference of two tallies are summarized from previous work. Equations used to apply correlated sampling to a ratio of two tallies, to the difference of two tallies divided by a third tally, and to other combinations of ratios are presented. New computer code A general-purpose, command-line-driven computer program, COSUBS (Correlated Sampling Using Batch Statistics), is presented that reads MCNP6 TFCs from output files and applies these equations. The user specifies the input relative perturbation, if desired, and any normalization tally, if desired. Example problems show how to run COSUBS and interpret the output.

An efficient calculation method for particle transport problems based on neural network

Monte Carlo (MC) method is a powerful tool for solving particle transport problems. However, it is extremely time-consuming to obtain results that meet the specified statistical error requirements, especially for large-scale refined models. This paper focuses on improving the computational efficiency of neutron transport simulations. Specifically, this study presents a novel method of efficiently calculating neutron fixed source problems, which has many applications. This type of particle transport problem aims at obtaining a fixed target tally corresponding to different source distributions for fixed geometry and material. First, an efficient simulation is achieved by treating the source distribution as the input to a neural network, with the estimated target tally as the output. This neural network is trained with data from MC simulations of diverse source distributions, ensuring its reusability. Second, since the data acquisition is time consuming, the importance principle of MC method is utilized to efficiently generate training data. This method has been tested on several benchmark models. The relative errors resulting from neural networks are less than 5% and the times needed to obtain these results are negligible compared with those for original Monte Carlo simulations. In conclusion, in this work we propose a method to train neural networks, with MC simulation results containing importance data and we also use this network to accelerate the computation of neutron fixed source problems.

4th-Order-SENS: A Software Module for Efficient and Exact 4th-Order Sensitivity Analysis of Neutron Transport

This work presents a software module called 4th-Order-SENS, which enables the efficient computation of exactly obtained expressions for all sensitivities, up to and including the 4th order, of a functional of the particle flux (e.g., the leakage of particles out of a body) with respect to nuclear parameters (total, scattering, and fission cross sections; nu, chi, sources; and number densities) for systems modeled by the neutron transport equation. The 4th-Order-SENS module implements the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Linear Systems (nth-CASAM-L), which is the only practically implementable methodology for obtaining the exact expressions of arbitrarily high-order sensitivities of model responses to model parameters, for response-coupled forward/adjoint large-scale linear systems. In addition to presenting the equations that are solved to obtain the 1st-order through 4th-order sensitivities, this work also describes the components of the module 4th-Order-SENS, including the user interface, input file, output files, and several independent code verification capabilities using symmetries and/or finite-difference formulas. The 4th-Order-SENS module is written in Python and Fortran and runs on Linux platforms. Several illustrative applications involving fixed-source problems in one-dimensional spherical and slab geometries are also presented.

Technique for reducing time in the calculation of eigenvalues and eigenvectors applied in neutron transport fixed-source problems

In this work, a mathematical technique to accelerate the eigenvalue calculations, originally proposed for the Analytical Discrete Ordinates method (ADO), is applied to the coarse-mesh method Modified Spectral Deterministic (MSD), used in neutron shielding problems (fixed-source) in one-dimensional geometry. This technique consists in an arrangement of the neutron transport equation which leads to a reduced eigenvalue calculation on the intranodal analytical solution, where only the positive half of the eigenvalues and its respective eigenvectors are calculated, hence reducing the methods execution time. Precision and performance tests were performed in 2 model-problems, reducing around 80% of the execution time. All the codes were developed using the programming language C++.

Goal-Oriented Mesh Adaptivity of Three-Dimensional Neutron Transport Calculation Using Weighted Difference Scheme and Dual-Weighted Residual Error Indicators

Maintaining a reasonable balance between computational accuracy and overhead is important for neutron transport simulations of engineering problems. This paper presents a goal-oriented mesh adaptive algorithm applied to the multigroup discrete ordinates equation for fixed source and criticality problems. The dual-weighted residual (DWR) approach estimates numerical solution errors and drives local mesh refinement for specific targets, such as detector response, integral flux, and multiplication factor. We employ a reconstruction method to evaluate the spatial residuals of the fluxes obtained by the weighted difference scheme. To improve the performance of adaptive algorithms, new estimation models are proposed for adjoint fluxes needed by the DWR theory, including a regional goal model for fixed source problems and an inconsistent fission source model for k-eigenvalue problems. Additionally, we analyze the impact of the truncation of flux reconstruction and isotropic approximation of adjoint fluxes on grid error indicators and adaptive calculations. Numerical results demonstrate that for the quantities of interest, our adaptive approach saves more than 70% of computational effort and run time when obtaining a level of high accuracy comparable to that of uniform fine grids.

A Consistent Low-Order Acceleration Technique for Coarse Mesh Transport (COMET) Methods

The COarse MEsh Transport (COMET) method, a hybrid continuous energy stochastic and deterministic transport method/tool based on the incident flux response expansion theory, is capable of providing highly accurate and efficient continuous energy whole-core neutron solutions to various heterogeneous reactor cores. In this work, a novel low-order (zeroth-order) acceleration technique is developed to significantly improve COMET’s computational efficiency for core calculations. This new method is based on consistent coupled low-order and high-order calculations to obtain the COMET core solution. In the low-order calculations, COMET is used to converge the total partial current escaping from each coarse mesh and the core eigenvalue. The resulting fixed-source problem in which the off-diagonal terms (equivalent to the scattering and fission neutron sources) are constructed by the zeroth-order solution are efficiently solved by the high-order COMET calculations. The resulting high-order angular flux on each coarse mesh bounding surface is then used to update (collapse) the low-order response coefficients. The coupled low-order and high-order calculations are repeated until both the eigenvalue and the low-order response coefficients are converged. The new acceleration method is implemented into COMET and tested in a set of stylized Advanced High Temperature Reactor (AHTR) benchmark problems. It is found that the core eigenvalues and the local fission density distributions predicted by COMET with the low-order acceleration agree very well with those computed by the original COMET. The eigenvalue discrepancy varies from 0 to 1 pcm, and the average relative differences in the stripewise and assembly-average fission density distributions are in the range of 0.021% to 0.032% and 0.004% to 0.01%, respectively. The comparisons have shown that the new low-order acceleration method can maintain COMET’s accuracy while improving its computational efficiency for core calculations by 12 to 16 times.

Convergence Properties of a Linear Diffusion-Acceleration Method for k-Eigenvalue Transport Problems

Most methods that use low-order operators to accelerate the iterative solution of transport eigenvalue problems employ nonlinear functionals of the transport solution (such as Eddington tensors) in their low-order equations, which are themselves standard eigenvalue problems. Here, we discuss linear diffusion synthetic acceleration (DSA) for -eigenvalue problems, which belongs to a family of methods that has received less attention than its nonlinear counterparts. We review the history of these linear methods as far as we know it and describe theoretical questions that to our knowledge have remained unanswered. With these methods, a low-order problem is solved after each transport step for an updated eigenvalue and an additive correction to the eigenfunction. These low-order problems are not standard eigenvalue problems, for they contain residuals as fixed sources. The low-order problems admit infinitely many solutions (updated and additive correction to the eigenfunction), and the solution that is obtained depends on the initial guess and iterative method chosen for the low-order problems. Experience has shown that when the low-order problems are solved with a powerlike iteration method and certain initial guesses, they yield solutions that cause rapid convergence to the correct high-order solution. We study the convergence properties of this algorithm applied to two model problems: an infinite homogeneous medium and a one-cell problem. For the infinite homogeneous problem, we present a Fourier analysis of the linear DSA method, which demonstrates that when the low-order problems are solved using a powerlike iteration scheme, the linear DSA scheme provides immediate convergence of the -eigenvalue and rapid convergence of the eigenfunction (much like DSA applied to scattering iterations in fixed-source problems). For the one-cell problems, we find that the linear scheme for -eigenvalue problems performs approximately as well as DSA for fixed-source problems. The latter analysis reveals a quantitative bound on the consistency between low- and high-order operators that is necessary and sufficient for convergence of those problems. With some theoretical foundations for the linear methods now established, we turn to numerical testing. We find, as others have before us using different low-order operators, that the method works well in practice. We provide numerical results from reactor problems in which our linear DSA is approximately as effective as the more widely used nonlinear methods. Our theoretical and numerical results add to the body of evidence that the linear methodology offers a simple path to rapid convergence of -eigenvalue problems, especially for codes that already employ linear low-order operators to converge scattering iterations.

Physics-Informed Neural Network Method and Application to Nuclear Reactor Calculations: A Pilot Study

In this paper, the physics-informed neural network (PINN) method is investigated and applied to nuclear reactor physics calculations with neutron diffusion models. The reactor problems were introduced with both fixed-source and eigenvalue modes. For the fixed-source problem, the loosely coupled reactor model was solved with the forward PINN approach, and then, the model was used to optimize the neural network hyperparameters. For the k-eigenvalue problem, which is unique for reactor calculations, the forward PINN approach was modified to expand the capability of solving for both the fundamental eigenvalue and the associated eigenfunction. This was achieved by using a free learnable parameter to approximate the eigenvalue and a novel regularization technique to exclude null solutions from the PINN framework. Both single-energy-group and multiple-energy-group diffusion models were examined in the work to demonstrate the PINN capabilities of solving systems of coupled partial differential equations in reactor problems. A series of numerical examples was tested to demonstrate the viability of the PINN approach. The PINN solution was compared against the finite element method solution for the neutron flux and the power iteration solution for the k-eigenvalue. The error in the predicted flux ranged from 0.63% for simple fixed-source problems up to about 15% for the two-group k-eigenvalue problem. The deviations in the predicted k-eigenvalues from the power iteration solver ranged from 0.13% to 0.92%. These generally acceptable results preliminarily justified the feasibility of PINN applications in reactor problems. The advantageous application potentials as well as the observable computational deficits of the PINN approaches are discussed along with the pilot study.

Optimised Adjoint Sensitivity Analysis Using Adjoint Guided Mesh Adaptivity Applied to Neutron Detector Response Calculations

This article presents a new approach for the efficient calculation of sensitivities in radiation dose estimates, subject to imprecisely known nuclear material cross-section data. The method is a combined application of adjoint-based models to perform, simultaneously, both the sensitivity calculation together with optimal adaptive mesh refinement. Adjoint-based sensitivity methods are known for their efficiency since they enable sensitivities of all parameters to be formed through only two solutions to the problem. However, the efficient solutions can also be obtained by their computation on optimal meshes, here guided by goal-based adjoint approaches. It is shown that both mesh adaptivity and sensitivity can be computed by the same adjoint solution, meaning both can be performed without additional costs. A simple demonstration is presented based on the Maynard fixed-source problem where uncertainties, with respect to material cross-sections, of doses received in local regions are examined. It is shown that the method is able to calculate sensitivities with reduced computational costs in terms of memory and potentially computational time through reduced mesh size when using adaptive resolution in comparison to uniform resolution. In particular, it is the local spatial contributions to the sensitivity that are resolved more effectively due to the adaptive meshes concentrating resolution in those areas contributing the most to its value.

Development and verification of a GPU-based MOC transient solver

The rapidly developed computational capability of the Graphic Processing Unit (GPU) provides an alternative platform for transport transient calculation. In this work, a GPU-based 2D method of characteristic (MOC) transient fixed source problem (TFSP) solver is implemented on lattice physics code ALPHA (Advanced Lattice Physics code based on Heterogeneous Architecture), which employs the full implicit method (FIM) and pin-based coarse mesh finite difference (CMFD) acceleration. C5G7-TD benchmark is calculated to verify the transient solver built-in ALPHA. To exploit the potential capability of GPU, sensitivity studies are performed on convergence criteria of MOC, which demonstrate single float-point precision (FP32) can be applied in the transient simulation. The numerical results of the ALPHA code show a good agreement with the results of MPACT and the ALPHA code with FP32 can provide enough accuracy and good performance on executing time.

Monte Carlo sensitivity calculation in fixed source problems with the derivative source method

This paper proposes a new Monte Carlo sensitivity analysis method for fixed source radiation transport problems, which is dubbed the derivative source method (DSM). This method is a variation of the formerly proposed perturbation source method. A transport equation of a derivative particle representing a derivative of a flux with respect to a parameter such as a cross section is derived by differentiating the fixed source radiation transport equation. This paper presents a Monte Carlo algorithm to solve this transport equation. The derivative particle is emitted from a region of interest (ROI) when a radiation particle undergoes a collision in the ROI. The derivative particle is tracked in the same manner as the ordinary radiation particles. The derivative particle that undergoes a collision in the ROI can yield another derivative particle whose derivative order is increased by one from the colliding particle. This higher-order derivative particle can be used for calculating the higher-order derivative of a flux. The differential operator sampling method (DOS), which is commonly used for Monte Carlo sensitivity analyses, suffers from poor performance when an ROI is small. For a small ROI, the efficiency of the DSM can be improved by adding a pseudo scattering cross section in the ROI, thereby increasing the pseudo scatterings and the number of derivative particles emitted from the ROI. The efficiency of the new method is evaluated by comparing the DSM to the DOS. Numerical tests are performed for a two-dimensional geometry containing a small inclusion. The results reveal that the DSM definitely outperforms the DOS for these test problems. The efficiency of the DSM depends strongly on the pseudo scattering cross section added to the inclusion.

On integrating Monte Carlo calculations in and around near-critical configurations – I. Methodology

Monte Carlo with variance reduction (VR) is employed to calculate radiation-induced responses in a variety of fixed source problems. In eigenvalue problems, Monte Carlo is used without VR (apart from implicit capture) for in-core integral responses (keff, reactivity coefficients, burn-up effects, etc.). More differential in- or ex-core responses are either treated in the same way thus limiting the range of solvable problems, or by decoupling. We evaluate an all-Monte Carlo approach developed to calculate differential in- or ex-core responses without decoupling. The ex-core problems involve responses in a GEN III PWR. The in-core problems study the neutron flux and 96Zr(n,γ) rate at the surface of a control rod in the VERA Benchmark. Comparison is made with an empirical approach involving analog Monte Carlo in-core with VR ex-core. The paper consists of two parts: part I tests the methodology; part II compares with PWR GEN II and III ex-core decoupled results.

Relationship between Tone’s method and embedded self-shielding method

The Tone’s and embedded self-shielding methods are based on the equivalence theory. Compared with the traditional equivalence theory, they solve a fixed-source problem for the whole problem domain to consider the shadowing effect in a lattice system rather than use the Dancoff factor. In this paper, we investigate the similarities and differences of these two methods and find the approximation that multi-group absorption cross sections of resonant nuclides are independent of regions is used in Tone’s method and not used in ESSM, which leads to the phenomena that these two methods produce the same numerical results in cases where there is only one resonant region and produce different results in cases where there are several resonant regions. The two methods are applied to the pressurized water reactor analysis and fast spectrum reactor analysis.

Transport Corrections and Sensitivities in the Discrete-Ordinates Method

Methods for approximately accounting for the terms neglected in a finite (L’th-order) Legendre expansion of the scattering source in the transport equation are called transport corrections. This paper derives adjoint-based sensitivities of a neutron or gamma-ray transport response for problems that use diagonal, Bell-Hansen-Sandmeier (BHS), or n’th-Cesàro-mean-of-order-2 (Cesàro) transport corrections in the discrete-ordinates method. For diagonal and BHS transport corrections, there is a sensitivity to the L + 1ʹth scattering cross-section moment, and the sensitivity to nuclide and material densities requires this contribution. For the Cesàro transport correction, the sensitivities to the scattering cross section for the l’th moment are multiplied by a simple function of l and the scattering expansion order L. Numerical results for a keff problem and a fixed-source problem verify the derivation and implementation of the sensitivity equations into the SENSMG multigroup sensitivity code. The Cesàro transport correction yields inaccurate responses for both problems.

A Source-expansion-based method for transient Pin-Power reconstruction

In this paper, the transient pin-power reconstruction based on source-expansion method has been proposed. The transient calculation is sufficient to simulate the reactor-physics process in second-time scale, such as the load-dump process and dynamic rod worth measurement. During the transient process, the pin-power distribution is essential for the determination of the power-peak factors, including Fq and FΔH. However, with the widely-applied transverse-integrated nodal methods for transient simulation, the detailed pin-averaged results can’t be provided directly. To address this problem, the transient pin-power reconstruction was proposed. It was derived from the transient fixed-source problem assuming the total-source and the transient fixed-source terms could be respectively expanded by the fourth-order Legendre polynomials and the biquadratic Legendre polynomials constructed from the nine-node problem. Moreover, to improve the accuracy of pin-averaged results, the corner-point condition and corner-discontinuity factors were employed in the reconstruction process. This proposed transient pin-power reconstruction method has been implemented in our self-developed code named SPARK and verified with several benchmarks. Results of the verification indicated that this proposed method for transient pin-power reconstruction can provide a satisfactory accuracy in transient process.

The Advanced Multilevel Predictor-Corrector Quasi-Static Method for Pin-Resolved Neutron Kinetics Simulation

The Advanced Multilevel Predictor-Corrector Quasi-static Method (AML-PCQM) is proposed in this work. The four computational levels, including transport, Multi-Group (MG) Coarse Mesh Finite Difference (CMFD), One-Group (1G) CMFD, and Exact Point-Kinetics Equation (EPKE), are coupled with a new dynamic iteration strategy. In each coupling algorithm, the original Transient Fixed Source Problem (TFSP) is solved in the predictor process using coarse time step, and then the flux distribution is factorized to the functions of amplitude and shape in the next corrector process. Finally, multiple fine time steps are used to adjust the predicted solution. Two heterogeneous single assembly problems with the prompt control rod withdrawal event are used to verify the AML-PCQM scheme’s accuracy and efficiency. The numerical results obtained by different cases are compared and analyzed. The final results indicate that the AML-PCQM performs the remarkable advantages of efficiency and accuracy with the reference cases.

A New Embedded Analysis with Pinwise Discontinuity Factors for Pin Power Reconstruction

Based on embedded analysis, an accurate pin power reconstruction (PPR) method is proposed for conventional nodal analysis. Unlike the common form function (FF) method, the new PPR method, named the embedded pin power reconstruction (EPPR) method, directly solves a two-group fixed-source problem that is defined with pinwise homogenized group constants (HGCs) and coarse-mesh incoming partial currents on the boundary. In the EPPR scheme, the pinwise HGCs including the pinwise discontinuity factor are predetermined from single-assembly lattice calculations, and the boundary partial currents are obtained from two-step nodal analyses. Two EPPR approaches are proposed: One is a 3×3 extended color-set configuration, and the other is a smaller one considering the half-thickness of the surrounding fuel assemblies. The performance of the EPPR methods is evaluated with various benchmark problems including partially mixed oxide–loaded pressurized water reactor cores, and the results are compared with the conventional FF method. Comprehensive results of this work demonstrate that the new EPPR method can provide much better accuracy than the conventional FF-based PPR method.

On the spectral Green’s function-constant nodal method for fixed-source SN problems in X,Y-geometry with arbitrary L’th-order anisotropic scattering

Presented here is an extension of the spectral Green’s function-constant nodal (SGF-CN) method for the numerical solution of energy multigroup, fixed-source, discrete ordinates (SN) problems in X, Y-geometry with arbitrary L’th-order of scattering anisotropy, provided L<N. This analytical coarse-mesh method uses the multigroup SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. The only approximations in the present version of the SGF-CN method occur in these transverse leakage terms, as the energy-group transfer scattering source terms are treated analytically within the offered method. Numerical results to typical model problems are given to illustrate the method’s accuracy and to analyze the efficiency of the offered SGF-CN computer code for neutral particle transport calculations.