Greedy method for the model order reduction in neutronics : application of the reduced basis method

We are interested in a methodology that perform a computation in a very short amount of time while preserving the accuracy. A reduced basis approach could meet this requirement.
In the framework of the reduced basis methods [1,3], we devise an approximation space associated to a parameter-dependent partial differential equation. The construction of this approximation space includes a phase of exploration of the space of parameters where it is important to quantify the error between the solution obtained from the approximation space (in construction) and the solution obtained from a standard (fine) discretization.
This crucial step allows to certify the construction of the reduced basis.
Recently, some research work in the laboratory have provided a posteriori error estimator in the context of neutronics [4].
In this context [2], we are interested in generalized non-symmetric eigenvalue problems. Typically, we consider a linear Boltzmann operator of the form:
Find (u, v) such that Lu = Hu + v Fu,
where Lu is the advection operator, Hu is the scattering operator that modelize the collisions of the neutrons, Fu is the fission operator and the unknown u represents the neutron flux. The equation is also called the neutron transport equation. The fact the operator is not symetric comes form the scattering operator.
A first implementation of the reduced basis method based on the Proper Orthogonal Decomposition has been made for the neutron diffusion model in the APOLLO3® code [5]. Reduced basis methods have been studied for the neutron diffusion model [6-8] and the neutron transport model [9-14] with a varying degree of intrusivity.
The objective of this thesis is to contribute to the construction of greedy reduced basis methods for a model of neutronics, especially on assembling the reduced problem and the computation of an a posteriori estimator based on an affine decomposition of the operator. In a second step, many possibilities may be investigated :
- The extension of the reduced basis method to the simplified transport;
- The extension of the reduced basis method to the transport model;
- The application to the loading pattern of a research reactor.

[1] Y. Maday, O. Mula, A generalized empirical interpolation method: application of reduced basis techniques to data assimilation. Analysis and Numerics of Partial Differential Equations, XIII:221-231,2013.
[2] O. Mula, Some contributions towards the parallel simulation of time dependent neutron transport and the integration of observed data in real time, Chapter 1, 2014.
[3] G. Rozza, D. Huynh, and A. Patera, “Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations,” Archives of Computational Methods in Engineering, vol. 15, no. 3, pp. 1–47, 2008.
[4] Y. Conjungo Taumhas, G. Dusson, V. Ehrlacher, T. Lelièvre, F. Madiot. Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations. 2023. ?HAL cea-04156959?
[5] Y. Conjungo Taumhas, F. Madiot, T. Lelièvre, V. Ehrlacher, and G. Dusson. An Application of Reduced Basis Methods to Core Computation in APOLLO3®. M&C 2023
[6] Sartori, A. Cammi, L. Luzzi, M. E. Ricotti, and G. Rozza. Reduced order methods: applications to nuclear reactor core spatial dynamics.15566, in ICAPP 2015 Proceedings, 2015.
[7] S. Lorenzi, An adjoint proper orthogonal decomposition method for a neutronics reduced order model, Annals of Nuclear Energy, 114 (2018), pp. 245–
[8] P. German and J. C. Ragusa, Reduced-order modeling of parameterized multi-group diffusion k-eigenvalue problems, Annals of Nuclear Energy, 134
(2019), pp. 144–157
[9] I Halvic, JC Ragusa. Non-intrusive model order reduction for parametric radiation transport simulations. Journal of Computational Physics 492 (2023), 112385
[10] P Behne, J Vermaak, J Ragusa. Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators. Nuclear Science and Engineering 197 (2), 233-261 (2023)
[11] P Behne, J Vermaak, JC Ragusa. Minimally-invasive parametric model-order reduction for sweep-based radiation transport. Journal of Computational Physics 469, 111525
[12] Z Peng, Y Chen, Y Cheng, F Li. A reduced basis method for radiative transfer equation. Arxiv preprint (2021).
[13] Sun, Y., Yang, J., Wang, Y., Li, Z., & Ma, Y. (2020). A POD reduced-order model for resolving the neutron transport problems of nuclear reactor. Annals of Nuclear Energy, 149, 107799.
[14] Wei, C., Di, Y., Junjie, Z., Chunyu, Z., Helin, G., Bangyang, X., ... & Lianjie, W. (2021). Study of non-intrusive model order reduction of neutron transport problems. Annals of Nuclear Energy, 162, 108495.

Modeling and ALARA optimization of maintenance operations in fusion nuclear power plants with Artificial Intelligence and Virtual Reality techniques

In view to the development of future fusion reactors, the maintenance operations in these nuclear facilities will be a diffculty, as part of them will have to be carried out hands-on. Safety rules govern interventions in a radioactive environment. They take into account the level of effective dose received by the operator, a factor that characterizes the risk to which the operator is exposed (dose depending on ambient dose rate and time).
In the aim of optimizing this dose in line with the ALARA principle and the safety constraints associated with these installations, the prior simulation of operations in Virtual Reality is an asset in terms of design optimization and worker training. Calculating dose during these simulations would be an important contribution to discriminating between different options. The simulation methods currently used to calculate dose rates are in some cases imprecise and in others very costly in terms of simulation time.
The aim of this work is to propose a new method for dynamic dose rate estimation in reduced time (or even real time) as a function of the movements of both the activation sources of a fusion installation, the maintenance operator and the shield protecting the latter. These dynamic configurations are representative of real intervention conditions. This method will implement Artificial Intelligence techniques coupled with Neutronics methods, and should be able to be integrated into a Virtual Reality tool based on existing development environments such as Unity3D.

Development of advanced optimisation methods for nuclear power scenarios

The study of possible nuclear fleet evolution is done through scenario calculations. A scenario models precisely all material flows within the fuel cycle, starting with raw material extraction, following with fuel fabrication, fuel irradiation inside the reactor, spent fuel cooling, fuel reprocessing and waste disposal. The scenario is thus a great tool for decision making. However, a scenario is really dependant on the set of hypotheses considered, that are affected by deep uncertainties. The current way to perform scenario calculation is not well suited to manage such hypotheses changes due to uncertainties.

A new field of research has emerged to deal with these deep uncertainties : the study of scenario robustness and resilience. The objective is no longer to quantify the performances of a precise scenario, but its ability to be modified to answer to the objective or constraint change (such as an installed power variation). To do so, it is necessary to launch several thousands of calculations, among which a large part are not viable.

The goal of this thesis work is to investigate the optimization methods used in logistics in order to build efficient methods to quickly build scenario inputs. The generated inputs should lead to optimal scenarios for a set of given objectives. Then, it would be possible to identify the scenarios that are able to answer to several objectives and assess whether they can be adjusted to answer to new constraints. In other words, this thesis is another step towards the production of resilient scenarios against future uncertainties.

Batch fuel management of molten salt nuclear reactors and consequences on the fuel cycle

Many molten salt reactor concepts rely on so-called continuous salt management, which consists of continually inserting/removing fuel salt into/from the core to compensate for the loss of reactivity due to fuel depletion. In this PhD thesis, we propose to reconsider the molten salt reactor conceptual design in a so-called batch fuel management strategy that involves taking and loading a fraction of the volume of the core after a certain irradiation time and during a reactor maintenance. Contrary to a continuous strategy, the aim of this batch refueling strategy is to take into account technological constraints external to the reactor. The implications of a batch-fueled molten salt reactor with regard to core performances, cycle constraints and salt chemistry constitutes a largely unexplored area of research.

The doctoral student will first focus on evaluating the neutronic impacts of batch management with sensitivity studies (minor actinides burning performances, fuel regeneration performances, cycle time, volume/mass supply). The doctoral student will then focus on the cycle aspects of a molten salt reactor integrated into a given nuclear fleet, including the fresh fuel salt fabrication (actinides solubility) and the used salt reprocessing (process, cooling time, process time) with nuclear scenario calculations. The approach will evaluate the relevance of a batch-fueled molten salt reactor and will be applied to different reactors (burner, breeder) in comparison to continuous salt management.

The thesis will allow the candidate to develop skills in the conceptual design of a fourth generation reactor. He/she will be part of the scientific community working on such complex systems, which opens the door to a job in a R&D lab.

Monte Carlo simulation of the reactor transfer function to improve neutron noise measurement analyses

The neutron population in a reactor fluctuates due to the random nature of neutron emission and various sources of mechanical vibrations, which can impact macroscopic neutron cross sections. The reactor can be seen as a system with a transfer function that connects an excitation (such as a vibration or the random nature of neutron emissions from fission) to the neutron population. The study and measurement of this transfer function allow us to deduce essential neutron parameters related to the kinetics of delayed neutron emission or even the source of thge vibrations. However, the theoretical expression of this transfer function is often based on the kinetics of the point reactor, which in some cases does not reliably exploit the measurements.
In this thesis work, we propose to study various extensions of the neutron transfer function formalism using Monte Carlo simulations. First, we will simulate fluctuations using a simplified C++ model to confirm the assumptions of theoretical equations for "neutron noise" that can be used to "measure" the effective fraction of delayed neutrons. We will then seek to optimize the positioning of detectors in a reactor and interpret certain effects related to positioning already observed in past experiments conducted by CEA.

Innovative modeling for multiphysics simulations with uncertainty estimates applied to sodium-cooled fast reactors

Multiphysics modeling is a powerful tool for analyzing nuclear reactors, but the uncertainty propagation between disciplines is often disregarded. This PhD thesis proposes innovative approaches to improve the accuracy of multiphysics modeling by accounting for these uncertainties. The primary goal is to propose optimal modeling approaches tailored to diverse accuracy requirements. This information is of prime interest to researchers and industry professionals involved in the development and utilization of multiphysics models. Specifically, the thesis will assess various uncertainty propagation techniques applicable to multiphysics simulations. This involves exploring surrogate modeling through avenues like reduced-order modeling and polynomial chaos expansion. The goal is to identify and categorize input parameters with the most significant impact on system outputs, irrespective of their physical domain. Subsequently, uncertainty propagation will be executed using two core modeling types: a ‘high-fidelity’ model based on the CEA's reference simulation tools and a ‘best-estimate’ model accounting for the "industrial" objective of the calculations). The similarities and differences between these approaches will be analyzed to assess model biases. These uncertainty evaluations employing the above methods will be tested on an extensive set of experiments performed in SEFOR, a sodium-cooled fast reactor, representing a diverse range of experimental data for various reactor conditions.

Development of similarity estimators for simulation/experiment validation: application to Lead Cooled nuclear reactors

The validation of simulation tools involves comparison with experience for all the quantities of interest related to reactor physics. In the frame of neutronics, this process makes use of national and international databases (IRPHE, ICSBEP) that offer a set of configurations on which these comparisons are possible. In order to ensure a high level of confidence of simulation tools for a specific target core design (industrial design, prototype, demonstrator,...) to be built, it is necessary to select the relevant experiments for validation. Up to now, this selection is made according to a criterion of “similarity” or “representativeness”. In practice, neutronic sensitivity coefficients can be used to build integral estimators or indicators that can help quantifying the degree of “similiarity” between two concepts or experiments.
The reduced number of estimators built with these sensitivities does not allow an “efficient” selection between several configurations. The present work does propose to characterize experiments using alternate estimators (to be defined) related, as close as possible, to the physical phenomena (slowing down of neutrons, neutron spectra, mean free paths, etc.) to improve the selection tools (such as NDAsT) and make it possible to better discriminate between the available experiments.
Once implemented, this new methodology should make it easier to identify phenomena that are not covered by existing databases. New experiments can then be considered to fill this gap and broaden the validation domain for the target concepts.
A specific application will be made on lead-cooled fast neutron reactors.

Development of a holistic approach of neutronics code validation based on Bayesian inference and deep learning techniques

The evaluation of nuclear data with the production of international files integrated into an international library such as the European library (JEFF) is of major importance for calculating current and future nuclear systems and reactors. Understanding and mastering the uncertainties related to these nuclear data is a particularly delicate task, which requires the use of advanced Bayesian inference techniques. The objective of this PhD thesis is to develop a BEPU (Best Estimate Plus Uncertainty) approach of the CEA neutronics codes which is holistic in the sense that all the known sources of uncertainty are taken into account (nuclear data, geometry and material description data, model approximations,...) when solving the neutron transport equation. In oder to account for these aleatory and epistemic uncertainties, we will use both a standard Bayesian framework and recent machine-learning methods (Deep Learning). In particular, this PhD thesis will focus on the difficult task of assimilating data from integral (critical and post-irradiation) measurements such as those available in the IRPhE international database. This work is essential for the validation of the new JEFF4 nuclear data library.