High-Fidelity Monte Carlo Simulations of Neutron Noise in Nuclear Power Reactors

Operating nuclear reactors are subject to a variety of perturbations. These can include vibrations of the fuel pins and fuel assemblies due to fluid-structure interactions with the moderator, or even vibrations of the core barrel, baffle, and pressure vessel. All of these perturbations can lead to small periodic fluctuations in the reactor power about the stable average power level. These power fluctuations are referred to as “neutron noise”. Being able to simulate different types of in-core perturbations allows reactor designers and operators to predict how the neutron flux could behave in the presence of such perturbations. In recent years, many different research groups have worked to develop computational models to simulate these sources of neutron noise, and their resulting effects on the neutron flux in the reactor. The primary objective of this PhD thesis will be to bring Monte Carlo neutron noise simulations to the scale of real-world industry calculations of nuclear reactor cores, with a high-fidelity continuous-energy physics representation. As part of this process, the student will add novel neutron noise simulation capabilities to TRIPOLI-5, the next-generation production Monte Carlo particle-transport code jointly developed by CEA and ASNR, with the support of EDF.

Preconditioning of iterative schemes for the mixed finite element solution of an eigenvalue problem applied to neutronics

Neutronics is the study of the behavior of neutrons in matter and the reactions they induce, particularly the generation of power through the fission of heavy nuclei. Modeling the steady-state neutron flux in a reactor core relies on solving a generalized eigenvalue problem of the form:
Find (phi, keff) such that A phi=1/keff B phi and keff is the eigenvalue with the largest magnitude, where A is the disappearance matrix which is assumed invertible, B represents the production matrix, phi denotes the neutron flux, and keff is called the multiplication factor.

The neutronics code APOLLO3® is a joint project of CEA, Framatome, and EDF for the development of a next-generation code for reactor core physics to meet both R&D and industrial application needs [4].
The MINOS solver [2] is developed within the framework of the APOLLO3® project. This solver is based on the mixed finite element discretization of the neutron diffusion model or the simplified transport model. The strategy for solving the aforementioned generalized eigenvalue problem is iterative; it involves applying the inverse power method [6].

The convergence speed of this inverse power method algorithm depends on the spectral gap. In the context of large cores such as the EPR reactor, it is observed that the spectral gap is close to 1, which degrades the convergence of the inverse power method algorithm. It is necessary to apply acceleration techniques to reduce the number of iterations [7]. In neutron transport, the preconditioning called Diffusion Synthetic Acceleration is very popular for the so-called inner iteration [1] but has also recently been applied to the so-called outer iteration [3]. A variant of this method was introduced in [5] for solving a source problem. It is theoretically shown that this variant converges in all physical regimes.

[1] M. L. Adams, E. W. Larsen, Fast iterative methods for discrete-ordinates particle transport calculations, Progress in Nuclear Energy, Volume 40, Issue 1, 2002.

[2] A.-M. Baudron and J.-J. Lautard. MINOS: a simplified PN solver for core calculation. Nuclear Science and Engineering, volume 155(2), pp. 250–263 (2007).

[3] A. Calloo, R. Le Tellier, D. Couyras, Anderson acceleration and linear diffusion for accelerating the k-eigenvalue problem for the transport equation, Annals of Nuclear Energy, Volume 180, 2023.

[4] P. Mosca, L. Bourhrara, A. Calloo, A. Gammicchia, F. Goubioud, L. Mao, F. Madiot, F. Malouch, E. Masiello, F. Moreau, S. Santandrea, D. Sciannandrone, I. Zmijarevic, E. Y. Garcia-Cervantes, G. Valocchi, J. F. Vidal, F. Damian, P. Laurent, A. Willien, A. Brighenti, L. Graziano, and B. Vezzoni. APOLLO3®: Overview of the New Code Capabilities for Reactor Physics Analysis. Nuclear Science and Engineering, 2024.

[5] O. Palii, M. Schlottbom, On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer, Computers & Mathematics with Applications, Volume 79, Issue 12, 2020.

[6] Y. Saad. Numerical methods for large eigenvalue problems: revised edition. Society for Industrial and Applied Mathematics, 2011.

[7] J. Willert, H. Park, and D. A. Knoll. A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem. Journal of Computational Physics, 2014, vol. 274, p. 681-694.

Fluid–structure interaction in mixtures: theory, numerical simulations and experiments

This PhD project is part of research on fluid–structure interactions (FSI) in complex media, particularly fluid mixtures involving multiple phases (liquid/liquid or liquid/gas) and/or suspended particles. The objective is to develop a thorough, multi-scale understanding of the coupled mechanisms between deformable structures (such as droplets, interfaces, or flexible walls) and the flows of complex mixtures, by combining theoretical modelling, advanced numerical simulations, and comparison with experimental data.

Development and Calibration of an Hyperbolic Phase-Field Model for Explicit Dynamic Fracture Simulation

The numerical simulation of the mechanical behavior of structures subjected to dynamic loads is a major challenge in the design and safety assessment of industrial systems. In the nuclear industry, this issue is particularly critical for the analysis of severe accident scenarios in Pressurized Water Reactors (PWRs) such as the Loss of Coolant Accident (LOCA), during which the rapid depressurization of the primary circuit can lead to pipe rupture. Developing physically representative models and robust, efficient numerical methods to simulate such phenomena with high fidelity remains an active area of research.

Among the existing non-local approaches, phase-field methods have emerged as a interesting framework for simulating crack initiation and propagation. However, most current studies are limited to quasi-static or low-rate dynamic problems, where wave propagation effects can be neglected. In contrast, high-rate dynamic regimes - relevant to accidental loads - require explicit time integration schemes for the mechanical equations, which are sensitive to the stability condition. The classical elliptic formulation of the damage evolution equation is therefore not ideally suited to this context. To address these limitations, recent works have proposed and assessed hyperbolic phase-field formulations, which are naturally more compatible with explicit dynamics and allow better control of crack propagation kinetics.

The objective of this PhD thesis is to advance this emerging modeling strategy through three main research directions:
- Extend the theoretical framework of the hyperbolic phase-field formulation for damage within the context of generalized standard materials, which is suitable for ductile fracture;
- Propose solutions to the negative impact of damage evolution on the critical time step;
- Rely on an dynamic fracture experimental test campaign to calibrate simulations, with a focus on the identification of damage-related parameters

This research is to be conducted in collaboration between CEA Paris-Saclay, ONERA Lille, and Sorbonne Université, with CEA as the main host institution.

Representation of Cross Sections based on the Wavelet Expansion Method, and Development of a Dedicated Solver

The deterministic solution of the neutron transport equation traditionally relies on the use of the multigroup approximation to discretize the energy variable. The energy domain is divided using a one-dimensional mesh, where the volume elements are called "groups" in neutronics. Within each group, all physical quantities (neutron flux, cross sections, reaction rates, etc.) are projected using piecewise constant functions. This homogenization of cross sections, which are the input data of the transport equation, becomes particularly challenging in the presence of resonant nuclei, whose cross sections vary rapidly over several decades. Correcting for this requires computationally expensive on-the-fly treatments to improve the accuracy of the transport solution.

The goal of this thesis is to eliminate the need for the multigroup approximation in the resonant energy range by applying a Galerkin projection of the continuous energy equation onto an orthonormal wavelet basis. The candidate will develop a generic expansion method adapted to mixtures of resonant isotopes, including preprocessing of cross sections, selection of the wavelet basis, and determination of an efficient coefficient truncation strategy. A dedicated neutron transport solver will be developed, with a focus on efficient algorithmic implementation using advanced programming techniques suited to modern architectures (GPU, Kokkos). The results of this thesis research will be valorized through publications in peer-reviewed international journals and presentations at scientific conferences.

Scaling Up Dislocation Dynamics Simulations for the Study of Nuclear Material Aging

Materials used in nuclear energy production systems are subjected to mechanical, thermal, and irradiation condition, leading to a progressive evolution of their mechanical properties. Understanding and modeling the underlying physical mechanisms involved is a significant challenge.

Dislocation Dynamics simulation aims to understand the behavior of the material at the crystal scale by explicitly simulating the interactions between dislocations, microstructure, and crystal defects induced by irradiation. The CEA, CNRS, and INRIA have been developing the NUMODIS calculation code for this purpose since 2007 (Etcheverry 2015, Blanchard 2017, Durocher 2018).

More specific work on zirconium alloys (Drouet 2014, Gaumé 2017, Noirot 2025) has allowed the validation and enhancement of NUMODIS's ability to handle these individual physical mechanisms by directly comparing them with experiments, through in situ tensile tests under a transmission electron microscope. However, these studies are limited by the current inability of the NUMODIS code to handle a sufficiently high and representative number of defects, and thus to obtain the mechanical behavior of the grain (~10 microns).

The objective of the proposed work is to implement new algorithms to extend the functionalities of the code, propose and test new numerical algorithms, parallelize certain parts still processed sequentially, and ultimately demonstrate the code's ability to simulate the deformation channeling mechanism in an irradiated zirconium grain.

The work will focus primarily on algorithms for calculating velocities, junction formation, and time integration, requiring both mastery of dislocation physics and the corresponding numerical methods. Algorithms for integration recently proposed by Stanford University and LLNL will be implemented and tested for this purpose.

Significant work will also be devoted to adapting the current code (hybrid MPI-OpenMP parallelism) to new computing machines that favor GPU processors, through the adoption of the Kokkos programming model.

Building on both previous experimental and numerical work, this study will conclude with the demonstration of NUMODIS's ability to simulate the channeling mechanism in an irradiated zirconium grain and to identify or even model the main physical and mechanical parameters involved.

At the interface between several fields, the candidate must have a good foundation in physics and/or mechanics, while being comfortable with programming and numerical analysis.

References:
1. Etcheverry Arnaud, Simulation de la dynamique des dislocations à très grande échelle, Université Bordeaux I (2015).
2. Blanchard, Pierre, Algorithmes hiérarchiques rapides pour l’approximation de rang faible des matrices, applications à la physique des matériaux, la géostatistique et l’analyse de données, Université Bordeaux I (2017).
3. Durocher, Arnaud, Simulations massives de dynamique des dislocations : fiabilité et performances sur architectures parallèles et distribuées (2018).
4. Drouet, Julie, Étude expérimentale et modélisation numérique du comportement plastique
des alliages de zirconium sous et après irradiation (2014).
5. Gaumé, Marine, Étude des mécanismes de déformation des alliages de zirconium
après et sous irradiation (2017).
6. Noirot, Pascal, Etude expérimentale et simulation numérique, à l'échelle nanométrique et en temps réel, des mécanismes de déformation des alliages de zirconium après irradiation (2025).

Detailed Numerical investigations on highly-concentrated bubbly flows

To assess the safety of industrial facilities, the CEA develops, validates, and uses thermohydraulic simulation tools. Its research focuses on modelling two-phase flows using various approaches, from the most detailed to the largest system-scale. In order to better understand two-phase flows, Service of Thermal-hydraulic and Fluid Mechanics (STMF) is working on implementing a multi-scale approach in which high-fidelity simulations (DNS, Direct Numerical Simulation of two-phase flows) are used as “numerical experiments” to produce reference data. This data is then averaged to be compared with models used on a larger scale. This approach is applied to high-pressure flows where the bubbly flow regime is maintained even at very high void fractions. The Laboratory of Development at Local Scales (LDEL) belonging to STMF has developed a DNS method (Front-Tracking) implemented in its open-source thermo-hydraulics code: TRUST/TrioCFD [1] (object-oriented code, C++). In several PhDs, it has been used to perform massively parallel simulations to describe interfaces in detail without resorting to models, for example in groups of bubbles (called swarms) [2][3][4].
Currently applied to low-concentration two-phase bubbly flows (volume fraction less than 12%), the objective of this thesis will be to evaluate and use the method at higher void fractions. Reference HPC simulations of bubble swarms will be conducted on national supercomputers up to gas fractions of 40%. The quality of the results will be evaluated before extracting physical models of bubble interactions under these conditions. The objective of these models is to recover the overall dynamics of the bubble swarm at much lower resolutions, thereby enabling the study of larger systems in disequilibrium (external forcing of imposed turbulence generation, imposed average velocity gradient, etc.).

This work is funded by the French ANR, in collaboration with IMFT and LMFL, in parallel with two other theses with which there will be strong interactions. It will be performed at CEA-Saclay, in the STMF/LDEL laboratory. It includes numerical aspects (validation), computer developments (C++), and a physical analysis of the flows obtained.

GPU-ACCELERATED CHARACTERISTICS METHOD FOR 3D NEUTRON TRANSPORT COMBINING THE LINEAR-SURFACE METHOD AND THE AXIAL POLYNOMIAL EXPANSION

This thesis falls within the framework of advancing numerical computation techniques for reactor physics. Specifically, it focuses on the implementation of methods that incorporate higher-order spatial expansions for neutron flux and cross-sections. The primary objective is to accelerate both existing algorithms and those that will be developed through GPU programming. By harnessing the computational power of GPUs, this research aims to enhance the efficiency and accuracy of simulations in reactor physics, thereby contributing to the broader field of nuclear engineering and safety.

Simulation of nuclear glass gels at the mesoscopic scale using a quaternary system.

This research work is part of studies conducted on the long-term behavior of nuclear glass used to immobilize radioactive waste and potentially intended for geological disposal. The challenge lies in understanding the mechanisms of alteration and gel formation (a passivating layer that can slow down the rate of glass alteration) by water and in predicting the kinetics of radionuclide release over the long term.

The proposed simulation approach aims to predict, at a mesoscopic scale, the maturation process of the gel formed during the alteration of glass by water using a ternary “phase field model” composed of silicon, boron, and water (leachate), to which aluminum will be added.

The underlying quaternary mathematical model will consists of a set of coupled nonlinear partial differential equations. These are based on Allen-Cahn and transport equations. The numerical solution of the associated equations is performed using the Lattice Boltzmann Method (LBM) programmed in C++ in the massively parallel LBM_saclay calculation code, which runs on several HPC architectures, both multi-CPUs and multi-GPUs.

The proposed research requires a solid foundation in applied mathematics and programming in order to develop the algorithms necessary for the correct resolution of the new system of strongly coupled equations.

HPC two-phase simulations with lattice Boltzmann methods and adaptative mesh refinement

CEA/STMF develops computational fluid dynamics (CFD) codes in thermohydraulics that aim to quantify mass and energy transfers in nuclear cycle systems such as reactors and management devices of radioactive wastes. This thesis focuses on Lattice Boltzmann Methods (LBM) adapted to Adaptive Mesh Refinement (AMR) inside a generic computing environment based on Kokkos and executable on multi-GPU supercomputers. The proposed work consists in developing LB methods in the Kalypsso-lbm code to simulate coupled partial differential equations (PDEs) modelling incompressible two-phase and multi-component flows such as those encountered in downstream cycle devices. Once the developments have been completed, they will be validated with reference solutions. They will allow a comparison of various interpolation methods between blocks of different sizes in the AMR mesh. A discussion will be held on the refinement and de-refinement criteria that will be generalized for these new PDEs. Finally, benchamrks of performance will quantify the contribution of AMR for 3D simulations when the reference simulation is performed on a static and uniform mesh. This work will use supercomputers which are already operational (e.g., Topaze-A100 from CEA-CCRT), as well as the future exascale supercomputer Alice Recoque depending on the progress of its installation.