Generalized Perturbation Theory Research Articles

A Novel Adjoint-Based Reduced-Order Model for Depletion Calculations in Nuclear Reactor Physics

The licensing of new reactors implies the use of verified and validated neutronic codes. Numerical validation can rely on sensitivity and uncertainty studies, but they require repeated execution of time-consuming neutron flux and depletion calculations. The computational costs can be reduced by using perturbation theories. However, the uncoupled Depletion Perturbation Theory is restricted to single integral values such as nuclide density. Relying on reduced-basis approaches, which reconstruct all nuclide densities at once, is one way to get around this restriction. Furthermore, the adjoint-based reduced-order model uses the direct and adjoint equations for projection. For diffusion or transport calculations, the Exact-to-Precision Generalized Perturbation Theory was developed. Still, no models for depletion calculations are readily available. Therefore, this paper describes a novel adjoint-based reduced-order model for the Bateman Equation. It uses a range-finding algorithm to create the basis and the uncoupled Depletion Perturbation Theory for the reconstruction of the first order replaced by with a first order formulation. Our paper shows that for several perturbed cases, the depletion reduced-order model successfully reconstructs the nuclide densities. As a result, this serves as a proof of concept for our adjoint-based reduced-order model, which can perform sensitivity and uncertainty burn-up analysis in a shorter time.

Development of a Coupled Depletion Perturbation Theory Methodology in Continuous-Energy Monte Carlo Depletion Simulations

This work develops a depletion perturbation theory methodology coupled with generalized perturbation theory to produce fully coupled nuclide number density sensitivity coefficients for cross sections, initial nuclide number densities, and decay constants. The method is implemented into OpenMC, a high-fidelity Monte Carlo simulation code, utilizing continuous-energy cross sections to produced high resolution sensitivity coefficients over the energy spectrum.

ExROPPP: Fast, accurate, and spin-pure calculation of the electronically excited states of organic hydrocarbon radicals.

Recent years have seen an explosion of interest in organic radicals due to their promise for highly efficient organic light-emitting diodes and molecular qubits. However, accurately and inexpensively computing their electronic structure has been challenging, especially for excited states, due to the spin-contamination problem. Furthermore, while alternacy or "pseudoparity" rules have guided the interpretation and prediction of the excited states of closed-shell hydrocarbons since the 1950s, similar general rules for hydrocarbon radicals have not to our knowledge been found yet. In this article, we present solutions to both of these challenges. First, we combine the extended configuration interaction singles method with Pariser-Parr-Pople (PPP) theory to obtain a method that we call ExROPPP (Extended Restricted Open-shell PPP) theory. We find that ExROPPP computes spin-pure excited states of hydrocarbon radicals with comparable accuracy to experiment as high-level general multi-configurational quasi-degenerate perturbation theory calculations but at a computational cost that is at least two orders of magnitude lower. We then use ExROPPP to derive widely applicable rules for the spectra of alternant hydrocarbon radicals, which are completely consistent with our computed results. These findings pave the way for highly accurate and efficient computation and prediction of the excited states of organic radicals.

Implementation and Testing of Generalized Perturbation Theory Capabilities in TRIPOLI-4®

In this paper, we describe the implementation, verification, and numerical validation of Generalized Perturbation Theory (GPT) capabilities in the TRIPOLI4® Monte Carlo code. The GPT is applied to reaction rate ratios (linear forms) and effective kinetics parameters (bilinear forms). For these latter, we focus in particular on the effective neutron generation time Λ eff , the effective prompt lifetime ℓ eff , and the effective delayed neutron fraction β eff . Verification is achieved comparing the results obtained using TRIPOLI-4® to benchmarks where analytical solutions are available. Numerical validation is performed with respect to independent Monte Carlo calculations using the SCALE 6.2.4 and SERPENT 2.2 codes, to published results, and to direct perturbations. Finally, a comparison with respect to the APOLLO3® deterministic code is performed for the EPICURE experimental configuration of the water-moderated EOLE reactor core, formerly operated at CEA.

Evaluation of Improved Contribution Function Method in the Compression of CINDER90 Depletion Library

The improved contribution function (ICF) method based on generalized perturbation theory (GPT) is applied to the compression of CINDER90 depletion library. A series of representative problems for PWR are defined with different fuel materials and operation conditions. According to the characteristics that the reactivity indicator is only meaningful in the neutron radiation calculation while the decay heat indicator is affected by both radiation and decay stage, the ICF method is adopted in the selection of candidate nuclides, then actinides and fission product nuclides of compressed system are obtained. In addition, the absorber nuclides of the compressed system are identified by analyzing the depletion process of the poison with a large absorption cross section separately. Finally, a compressed depletion library consisting of 489 nuclides is generated. The numerical results show that the compressed library maintains high precision in the calculation of k‐inf, nuclide concentration, decay heat power, and pin power. Meanwhile, the use of compressed depletion library leads to a significant reduction of computing resources.

Improving precision and accuracy in cosmology with model-independent spectrum and bispectrum

A new and promising avenue was recently developed for analyzing large-scale structure data with a model-independent approach, in which the linear power spectrum shape is parametrized with a large number of freely varying wavebands rather than by assuming specific cosmological models. We call this method FreePower. Here we show, using a Fisher matrix approach, that precision of this method for the case of the one-loop power spectrum is greatly improved with the inclusion of the tree-level bispectrum. We also show that accuracy can be similarly improved by employing perturbation theory kernels whose structure is entirely determined by symmetries instead of evolution equations valid in particular models (like in the usual Einstein-de Sitter approximation). The main result is that with the Euclid survey one can precisely measure the Hubble function, distance and (k-independent) growth rate f(z) in seven redshift bins in the range z ∈ [0.6, 2.0]. The typical errors for the lower zbins are around 1% (for H), 0.7–1% (for D), and 2–3% (for f). The use of general perturbation theory allows us, for the first time, to study constraints on the nonlinear kernels of cosmological perturbations, that is, beyond the linear growth factor, showing that they can be probed at the 10–20% level. We find that the combination of spectrum and bispectrum is particularly effective in constraining the perturbation parameters, both at linear and quadratic order.

Uncertainty reduction of sodium void reactivity using data from a sodium shielding experiment

ABSTRACT This study investigated the feasibility of reducing the uncertainty associated with fast-reactor-core design by sharing an experimental database between different fields (e.g. reactor physics and radiation shielding) using data assimilation techniques. As the first step in this study, we focused on the ORNL sodium shielding experiment and investigated the possibility of using the experimental data to reduce the uncertainty in sodium void reactivity (SVR), which is the most important safety parameter for sodium-cooled fast reactors. A sensitivity analysis based on the Generalized Perturbation Theory was performed for the sodium shielding experiment. Using the sensitivity coefficients evaluated here and those of the sodium void reactivity previously evaluated by the JAEA, we showed that sodium shielding experimental data can contribute to the uncertainty reduction of SVR by adopting the cross-section adjustment method. Based on this study, the uncertainty reduction effect is expected to be significant, especially for SVR dominated by neutron-leakage phenomena. Although new reactor physics experimental data on SVR may be difficult to obtain, the results of this study suggest that data from sodium shielding experiments can partially substitute for this role. This study demonstrated the value of the mutual use of integral experimental data in fast reactor designs.

Strengths and weaknesses of the modal expansion method for perturbations calculations in nuclear reactor physics

Reactor design and uncertainties propagation request repeated execution of neutron diffusion calculations. To address this, one can rely on the Generalized Perturbation Theory which is restricted to single integral values such as reaction rate ratios. To overpass such limitations, the perturbed flux could be calculated with the modal expansion method. The difficulty to calculate more than one eigenvector has limited its use. Hence, modal expansion lacks a clear explanation of its strengths and weaknesses. Using recent advances in eigenvalues solvers, we have successfully applied modal expansion to the 2D-BIBLIS and IAEA-3D benchmarks. Our results are promising for fission and capture cross sections perturbations. Still, modal expansion is faulty for local perturbations or when it concerns the reflector. In terms of computational cost, modal expansion is more efficient than solving the direct problem with a calculation time divided by 8 for a basis of 20 eigenvectors.

Nuclear Data Uncertainty Propagation for the Molten Salt Fast Reactor Design

The development of new reactor technologies requires careful assessments of the various sources of epistemic uncertainties. In this work, nuclear data uncertainties featuring the main isotopes of the U/Th molten salt fast reactor (MSFR) design are propagated through Monte Carlo calculations to quantify the final uncertainty on some relevant integral parameters. In the first part of this paper, some best-estimate calculations are performed by selecting different nuclear data libraries, showing the remarkable impact of this choice on the final responses. Then the Generalized Perturbation Theory routine available in Serpent 2 is adopted for a preliminary sensitivity and uncertainty analyses with respect to keff, highlighting a significant discrepancy between the covariance of the JEFF-3.3 and ENDF/B-VIII.0 libraries. After the selection of a few relevant nuclides, namely, 7Li, 19F, 232Th, and 233U, the Total Monte Carlo method and the unscented transform (UT) are then adopted to estimate the uncertainty of other responses of interest like the conversion ratio and some multigroup constants. Some potential issues of the UT are highlighted, and a mitigation strategy is applied. A relevant result of this analysis concerns the need for better data evaluations for the nuclides constituting the circulating salt for an effective deployment of the MSFR technology.

Index for generalized Fredholm operators and generalized perturbation theory

Abstract In this paper, we develop some generalized Fredholm perturbation results for bounded linear operators acting on non-reflexive Banach spaces satisfying certain conditions. Furthermore, we investigate the stability of the generalized Schechter S-essential spectrum by means of the so-called generalized Fredholm index. This concept is introduced as a semigroup homomorphism satisfying certain properties. Some generalized Fredholm results for block operator matrices acting on a non-reflexive product of two Banach spaces are also developed.

Sensitivity Analysis in Coupled Monte Carlo Radiation Transport Simulations

Sensitivity analysis methods have found extensive use in nuclear criticality safety applications for understanding the impact of uncertain nuclear data on eigenvalue estimates. Significant uncertainty exists in nuclear data and nuclear physics models for photon and electron transport applications, and the goal of this work is to explore whether recently developed adjoint-based, first-order generalized perturbation theory reaction rate sensitivity methods can be extended to coupled Monte Carlo radiation transport simulations. This paper presents a rigorous theoretical derivation for this extended sensitivity analysis method, which is then implemented in a one-dimensional test Monte Carlo code. The adjoint-based sensitivity coefficients are found to agree well with reference direct perturbation and deterministic SENSMG sensitivity coefficients for a simple test problem.

Nuclear data adjustment using a deterministic sampling method with unscented transformation

ABSTRACT A nuclear data adjustment method using a deterministic sampling method based on an unscented transform was developed, and its validity was confirmed through a twin experiment using a critical benchmark problem. Conventional nuclear data adjustment methods require sensitivity analysis using generalized perturbation theory or many forward calculations using stochastically sampled nuclear data. To address this issue, this study focused on unscented transform-based sampling (UTS), which is used in uncertainty quantification. Based on the UTS, perturbed nuclear data can be deterministically sampled to reproduce the population covariance matrix with minimum sample size. Therefore, UTS can significantly reduce the computational cost compared to conventional nuclear data adjustment using random sampling (RS). Furthermore, the UTS was improved to prevent the sampling of negative nuclear data while accurately reproducing the population covariance matrix. The proposed method was applied to the numerical experiment of Godiva, and the adjusted nuclear data were compared with those obtained using conventional methods. Consequently, it was demonstrated that UTS can adjust nuclear data at a lower computational cost than RS.

Sensitivity analysis of the accident rate of a plant equipped with a protection channel under aging by the generalized perturbation theory

We present a sensitivity analysis of a protection system of a nuclear plant equipped with a single aging protection channel. This analysis is related to the plant accident rate and considers the variation of the plant demand rate. Traditionally, the calculation of the values for this analysis has a high computational cost when considering the wear-out period. To overcome this difficulty, we used the Generalized Perturbation Theory (GPT). We developed four case studies for a wide range of demand rates. Due to the use of a first-order GPT approach, the results agree with those of the direct method for the steady-state behavior of the accident rate but not for its transient behavior. Computer times were highly reduced by the application even of the first-order GPT approach to the problem.

Burnup chain compression method preserving neutronics, decay photon source term and decay heat calculation results

A burnup chain compression method, preserving neutronics, decay photon source term and decay heat calculation results, is developed based on the generalized perturbation theory (GPT), to reduce calculation burden for the practical reactor physics simulations. In this method, a certain number of target nuclides which have significant effect on calculation accuracy of the three applications are first selected according to their contribution to the neutron reaction rate, decay heat and decay photon intensity. Then, nuclides important for the accuracy of the nuclide number densities of the target nuclides are selected according to the contribution function calculated based on the GPT. A fine burnup chain containing 1547 nuclides based on ENDF/B-VII.0 is compressed to a burnup chain including 557 nuclides. The compressed burnup chain is adopted to calculate a couple of typical fuel assembly and pin cell problems of PWR. The results are compared with those based on the fine burnup chain. The numerical results show that the compressed burnup chain achieves high accuracy for neutronics, decay photon source term and decay heat calculation, simultaneously.

Nuclear data uncertainty propagation applied to the versatile test reactor conceptual design

The Versatile Test Reactor (VTR) currently under development is a 300 MWth sodium-cooled fast reactor (SFR) fueled with ternary metal alloy fuel, which aims to accelerate the testing of advanced nuclear fuels, materials, instrumentation, and sensors in high flux environments that are necessary to license the next generation of advanced reactor concepts. To support the VTR design process, uncertainties associated with the nuclear data has been propagated through the reactor core neutronics calculation to global parameters of interest, such as the core multiplication factor, kinetic parameters, and various reactivity feedback coefficients, following the sensitivity based uncertainty propagation approach. By folding the sensitivity coefficients, separately computed by the generalized perturbation theory code PERSENT and Monte Carlo code Serpent 2, with the variance–covariance matrices from COMMARA-2.0, we obtain the reaction-wise, isotope-wise, and overall uncertainties for each response of interest due to nuclear data uncertainty. With Serpent 2, the statistical error of the uncertainty is obtained by propagating the statistical error of the sensitivity coefficients through the same process using a newly developed uncertainty propagation method. From both codes, the overall top uncertainty contributors are found to be the cross section of Fe-56 elastic scattering, Na-23 elastic scattering, and U-238 inelastic scattering. The large contributions of the Fe-56 elastic scattering cross sections to global parameters are due to its relatively large relative uncertainty of 5–10% in nuclear data and the large volume of Fe-containing reflector assemblies in the fairly compact VTR core design. Both codes agreed well for the overall uncertainty estimates of all responses of interest, except the delayed neutron fraction, prompt neutron generation time, and the coolant density feedback coefficient, where Serpent 2 yielded a much larger value than PERSENT due to the large statistical error of sensitivity coefficients. The calculated uncertainties are also compared to those associated with other SFR cores. Another outcome of this study is a variance–covariance matrix of reactivity coefficients, which can be used in the subsequent uncertainty propagation to the system level to investigate the impact of identified uncertainties on system responses in the safety analysis.

Uncertainty analysis of infinite multiplication factor and nuclide number density based on the UAM-PWR benchmark with respect to cross sections, fission yields and decay half-life

Uncertainty analysis is performed to qualify the uncertainty of the infinite multiplication factor (kinf) and nuclide number density induced by cross sections, fission yields and decay half-life in ENDF/B-VIII.0. The SUNDEW code, initially developed for burnup sensitivity and uncertainty analysis with respect to cross sections based on the generalized perturbation theory, is extended to consider fission yields and decay half-life. The Bayesian/general least-squares method is adopted to generate the covariance matrix of fission yields. TMI-1 PWR pin cell given in the UAM project is calculated by SUNDEW. In addition to kinf, the uncertainty of the number densities of various nuclides important for different aspects are calculated and analyzed, including nuclides involved in burnup credit, nuclides used for burnup measurement, and 135Xe. Main conclusions are that the cross section covariances in ENDF/B-VIII.0 can induce larger uncertainty to kinf than ENDF/B-VII.1; the fission yields and decay half-life have obvious contributions for the uncertainty of nuclide number density in some cases.

Generalized Perturbation Theory Based Total Sensitivity and Uncertainty Analysis for High-Fidelity Neutronics Calculation

Neutronics calculation for nuclear reactor with high-fidelity technology can significantly reduce the uncertainties propagated from numerical approximation error and model error. However, the uncertainty of input parameters inevitably exists, especially for nuclear data. On the other hand, resonance self-shielding calculation is essential for multi-group assumption based high-fidelity neutronics calculation, which introduce the implicit effect for calculation responses. In order to fully consider the implicit effects in the process of uncertainty quantification, a generalized perturbation theory (GPT) based implicit sensitivity calculation method is proposed in this paper. Combining the explicit sensitivity coefficient, which can be quantified using classic perturbation theory, the total sensitivity coefficient of calculation responses is obtained. Then the total sensitivity and uncertainty module is established in self-developed neutron transport code with high-fidelity technology-HNET. To verify the accuracy of the sensitivity calculation methods proposed in this paper, a two-dimensional fuel pin problem is chosen to verify the sensitivity results, and the numerical results show good agreement with results calculated by a direct perturbation method. Finally, uncertainty analysis for two-dimensional fuel pin problem is performed and some general conclusions are obtained from the numerical results.

On the statistical evolution of the large-scale structure and the effective dynamics

In this paper, we study the statistical evolution of the large-scale structure (LSS), focusing on the joint probability distribution function (PDF) of the coarse-grained cosmic field and its role in constructing effective dynamics.As the most comprehensive statistics, this PDF encodes all cosmological information of large-scale modes, therefore, could serve as the basis in the LSS modelling. Following the so-called PDF-based method from turbulence, we write down this PDF's evolution equation, which describes the probability conservation.We show that this conservation equation's characteristic curves follow the same PDF history and could be considered as an effective dynamics of the coarse-grained field.Unlike the EFT of LSS, which conceptually would work at both realization and statistics level, this effective dynamics is valid only statistically. However, this `statistical equivalence' also provides valuable insight into scale interactions at the statistical level. It also enables predicting a wide variety of statistics beyond the typical N-point polyspectra, including, e.g. topologies, density PDF and non-linear covariance matrices etc. Our formula expresses the small-scale effect as the ensemble average of their interactions conditional on the large-scale modes. This suggests an interesting way to measure effective terms directly from simulation. By applying the Gram-Charlier expansion, we demonstrate a different structure of these effective terms.This formalism is a natural framework for discussing the evolution of statistical properties of large-scale modes, and provides an alternative view for understanding the relationship between general effective dynamics and standard perturbation theory.

Many-body perturbation theory for the superconducting quantum dot: Fundamental role of the magnetic field

We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated $0$-phase. It controls the critical behavior of the $0-\pi$ quantum transition, lifts the degeneracy in the $\pi$-phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in $0$- and $\pi$-phases. While the magnetic susceptibility vanishes in the $0$-phase it becomes of the Curie type and diverges in the $\pi$-phase at zero temperature.

Improved generalized perturbation theory method for sensitivity analysis of generalized response function

Sensitivity and uncertainty analyses of generalized response functions are essential for nuclear designs because they provide useful information about how changes in the input parameters influence the neutronics response. Several methods have been developed to compute generalized sensitivity coefficients for nuclear data. One method is the GEneralized Adjoint Responses in Monte Carlo (GEAR-MC) method which can be used to analyze the sensitivity coefficients of reaction rate ratios. However, this method requires the generalized adjoint function to estimate the sensitivity coefficients. The generalized adjoint function can be calculated using a method like the iterated fission probability (IFP) method which has a huge memory requirement. In this paper, the superhistory-based GEAR-MC method is developed to reduce the huge memory requirement induced by calculating the generalized adjoint function. This newly developed superhistory-based GEAR-MC method is found to be accurately predict the sensitivities of the reaction rate ratios for the Godiva, Flattop and the UAM TMI PWR pin cell benchmark problems. The calculations also show that the memory usage of the superhistory-based GEAR-MC method is 98% less than for the traditional power iteration-based GEAR-MC method. In addition, it is shown that the superhistory-based GEAR-MC method will increase the runtime and this increase is not totally linearly proportional to the number of inner generations in each superhistory for all test cases. Moreover, the equivalence of GEAR-MC method and first-order differential operator method are also proved in this paper.