This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the mixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities and the other benchmark imprecisely known parameters, including: (i) the 6 × 180 mixed 2nd-order sensitivities involving the total microscopic cross sections; (ii) the 6 × 21,600 mixed 2nd-order sensitivities involving the scattering microscopic cross sections; (iii) the 6 × 60 mixed 2nd-order sensitivities involving the fission microscopic cross sections; and (iv) the 6 × 60 mixed 2nd-order sensitivities involving the average number of neutrons produced per fission. It is shown that many of these mixed 2nd-order sensitivities involving the isotopic number densities have very large values. Most of the large sensitivities involve the isotopic number density of ²³⁹Pu, and the microscopic total, scattering or fission cross sections for the 12th or 30th energy groups of ²³⁹Pu or ¹H, respectively. The 2nd-order mixed sensitivity of the PERP leakage response with respect to the isotopic number density of ²³⁹Pu and the microscopic total cross section for the 30th energy group of ¹H is the largest of the above mentioned sensitivities, attaining the value −94.91. Full article

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the 1st-order and unmixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity to the isotopic number densities for the two fissionable isotopes have large values, which are comparable to, or larger than, the corresponding sensitivities for the total cross sections. Furthermore, several 2nd-order unmixed sensitivities for the isotopic number densities are significantly larger than the corresponding 1st-order ones. This work also presents results for the first-order sensitivities of the PERP benchmark’s leakage response with respect to the fission spectrum parameters of the two fissionable isotopes, which have very small values. Finally, this work presents the overall summary and conclusions stemming from the research findings for the total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to all model parameters of the PERP benchmark, as presented in the sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”. Full article

By applying the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the polyethylene-reflected plutonium (PERP) benchmark, this work presents results for the first- and second-order sensitivities of this benchmark’s leakage response with respect to the spontaneous fission source parameters. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity of the leakage response to the source parameters for the two fissionable isotopes in the benchmark are all positive, signifying that an increase in the source parameters will cause an increase in the total neutron leakage from the PERP sphere. The 1st- and 2nd-order relative sensitivities with respect to the source parameters for ²³⁹Pu are very small (10⁻⁴ or less). In contradistinction, the 1st-order and several 2nd-order relative sensitivities of the leakage response with respect to the source parameters of ²⁴⁰Pu are large. Large values (e.g., greater than 1.0) are also displayed by several mixed 2nd-order relative sensitivities of the leakage response with respect to parameters involving the source and: (i) the total cross sections; (ii) the average neutrons per fission; and (iii) the isotopic number densities. On the other hand, the values of the mixed 2nd-order relative sensitivities with respect to parameters involving the source and: (iv) the scattering cross sections; and (v) and the fission cross sections are smaller than 1.0. It is also shown that the effects of the 1st- and 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s source parameters on the moments (expected value, variance and skewness) of the PERP benchmark’s leakage response distribution are negligibly smaller than the corresponding effects (on the response distribution) stemming from uncertainties in the total, fission and scattering cross sections. Full article

The stochastic point reactor with random parameters is considered in this work. The hybrid uncertain variations—noise and random parameters—are analyzed with the spectral techniques for the efficiency and high rates of convergence. The proposed hybrid technique enables one to derive an equivalent deterministic system that can be solved to get the mean solution and deviations due to each uncertainty. The contributions of different sources uncertainties can be decomposed and quantified. The deviations in the thermal hydraulics are also computed in the current work. Two model reactors are tested with the proposed technique and the comparisons show the advantages and efficiency compared with the other techniques. Full article

Fuel burnup analysis requires a high computational cost for full core calculations, due to the amount of the information processed for the total reaction rates in many burnup regions. Indeed, they reach the order of millions or more by a subdivision into radial and axial regions in a pin-by-pin description. In addition, if multi-physics approaches are adopted to consider the effects of temperature and density fields on fuel consumption, the computational load grows further. In this way, the need to find a compromise between computational cost and solution accuracy is a crucial issue in burnup analysis. To overcome this problem, the present work aims to develop a methodological approach to implement a Reduced Order Model (ROM), based on Proper Orthogonal Decomposition (POD), in fuel burnup analysis. We verify the approach on 4 years of burnup of the TMI-1 unit cell benchmark, by reconstructing fuel materials and burnup matrices over time with different levels of approximation. The results show that the modeling approach is able to reproduce reactivity and nuclide densities over time, where the accuracy increases with the number of basis functions employed. Full article

The subcritical polyethylene-reflected plutonium (PERP) metal fundamental physics benchmark, which is included in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook, has been selected to serve as a paradigm illustrative reactor physics system for the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) that was developed by Cacuci. The 2nd-ASAM enables the exhaustive deterministic computation of the exact values of the 1st-order and 2nd-order sensitivities of a system response to the parameters underlying the respective system. The PERP benchmark is numerically modeled in this work by using the deterministic multigroup neutron transport equation discretized in the spatial and angular independent variables. Thus, the numerical model of the PERP benchmark developed includes the following imprecisely known uncertain parameters: 180 group-averaged total microscopic cross sections, 21,600 group-averaged scattering microscopic cross sections, 120 fission process parameters, 60 fission spectrum parameters, 10 parameters describing the experiment’s nuclear sources, and six isotopic number densities. Thus, the numerical simulation model for the PERP benchmark comprises 21,976 uncertain parameters, which implies that, for any response of interest, there are a total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to the model parameters. Computing these sensitivities exactly represents the largest sensitivity analysis endeavor ever carried out in the field of reactor physics. Only 241,483,276 are distinct from each other, and some of these turned out to be zero due to the symmetry of the 2nd-order sensitivities. The numerical results for all of these sensitivities, together with discussions of their major impacts, will be presented in a sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”. This work is the first in this sequence, presenting formulas of general use for neutron transport problems, along with the numerical results that were produced by these formulas for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic total cross sections. For comparison, this work also presents formulas of general use and numerical results for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic capture cross sections. It has been widely believed hitherto that, for reactor physics systems modeled by the neutron transport or diffusion equations, the second-order sensitivities are all much smaller than the first-order ones. However, contrary to this widely held belief, the numerical results that were obtained in this work prove, for the first time ever, that many of the 2nd-order sensitivities are much larger than the corresponding 1st-order ones, so their effects can become much larger than the corresponding effects stemming from the 1st-order sensitivities. For example, the 2nd-order sensitivities of the PERP leakage response cause the expected value of this response to be significantly larger than the corresponding computed value. The importance of the 2nd-order sensitivities increases as the relative standard deviations for the cross sections increase. For the extreme case of fully correlated cross sections, for example, neglecting the 2nd-order sensitivities would cause an error as large as 2000% in the expected value of the leakage response and up to 6000% in the variance of the leakage response. The significant effects of the mixed 2nd-order sensitivities underscore the need for reliable values for the correlations that might exist among the total cross sections, which are unavailable at this time. The 2nd-order sensitivities with respect to the total cross sections also cause the response distribution to be skewed towards positive values relative to the expected value. Hence, neglecting the 2nd-order sensitivities could potentially cause very large non-conservative errors by under-reporting of the response variance and expected value. Full article

This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size N_r N_r, where N_r denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space. Full article

This work continues the presentation commenced in Part I of the second-order sensitivity analysis of nuclear data of a polyethylene-reflected plutonium (PERP) benchmark using the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM). This work reports the results of the computations of the first- and second-order sensitivities of this benchmark’s computed leakage response with respect to the benchmark’s 21,600 parameters underlying the computed group-averaged isotopic scattering cross sections. The numerical results obtained for the 21,600 first-order relative sensitivities indicate that the majority of these were small, the largest having relative values of O (10⁻²). Furthermore, the vast majority of the (21600)² second-order sensitivities with respect to the scattering cross sections were much smaller than the corresponding first-order ones. Consequently, this work shows that the effects of variances in the scattering cross sections on the expected value, variance, and skewness of the response distribution were negligible in comparison to the corresponding effects stemming from uncertainties in the total cross sections, which were presented in Part I. On the other hand, it was found that 52 of the $21600\times 180$ mixed second-order sensitivities of the leakage response with respect to the scattering and total microscopic cross sections had values that were significantly larger than the unmixed second-order sensitivities of the leakage response with respect to the group-averaged scattering microscopic cross sections. The first- and second-order mixed sensitivities of the PERP benchmark’s leakage response with respect to the scattering cross sections and the other benchmark parameters (fission cross sections, average number of neutrons per fission, fission spectrum, isotopic atomic number densities, and source parameters) have also been computed and will be reported in subsequent works. Full article

The Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) is applied to compute the first-order and second-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental system with respect to the following nuclear data: Group-averaged isotopic microscopic fission cross sections, mixed fission/total, fission/scattering cross sections, average number of neutrons per fission (), mixed /total cross sections, /scattering cross sections, and /fission cross sections. The numerical results obtained indicate that the 1st-order relative sensitivities for these nuclear data are smaller than the 1st-order sensitivities of the PERP leakage response with respect to the total cross sections but are larger than those with respect to the scattering cross sections. The vast majority of the 2nd-order unmixed sensitivities are smaller than the corresponding 1st-order ones, but several 2nd-order mixed relative sensitivities are larger than the 1st-order ones. In particular, several 2nd-order sensitivities for ²³⁹Pu are significantly larger than the corresponding 1st-order ones. It is also shown that the effects of the 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s parameters underlying the average number of neutrons per fission, , on the moments (expected value, variance, and skewness) of the PERP benchmark’s leakage response distribution are negligible by comparison to the corresponding effects (on the response distribution) stemming from uncertainties in the total cross sections, but are larger than the corresponding effects (on the response distribution) stemming from uncertainties in the fission and scattering cross sections. Full article