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.

Stochastic Neutron Noise Estimation Using a Rare-Event Simulation Approach. Application to the Monitoring of Nuclear System Reactivity

This PhD project aims to develop an innovative method to characterize the reactivity of fissile systems by analyzing their stochastic fluctuations, known as zero-power neutron noise. In a subcritical fissile medium, neutrons originating from spontaneous fission can initiate short and random chain reactions, generating a fluctuating signal. This noise carries essential information on the distance of the system to criticality, a key parameter both for the safety of nuclear installations (prevention of criticality accidents) and for the detection of undeclared fissile materials (nuclear security and non-proliferation).

Existing theoretical approaches to infer system reactivity from neutron noise are limited to idealized situations and become unsuitable in realistic configurations, particularly when the system is strongly subcritical or when significant uncertainties exist regarding its geometry or composition (as in the case of the Fukushima Daiichi corium or spent fuel storage). Monte Carlo simulations then appear as a natural alternative, but current simulations rely on variance reduction techniques that fail to correctly preserve stochastic fluctuations.

This thesis proposes to address this scientific challenge by adapting a relatively recent variance reduction method known as Adaptive Multilevel Splitting (AMS), originally developed to efficiently sample rare events while preserving their statistical properties. The goal is to extend this method to neutron transport in multiplying media and to make it a tool capable of faithfully simulating the temporal correlations characteristic of neutron noise. Following the theoretical developments, the algorithm will be implemented in Geant4, compared to analytical benchmark solutions, and experimentally validated through in situ measurements (using neutron sources or research reactors). In the long term, this work may lead to direct applications in nuclear monitoring, safety diagnostics, and detector physics, while also opening perspectives in fundamental physics and medical physics.

Monitoring criticality risk through neutron noise in degraded nuclear environments

Our team at CEA/Irfu is working with ASNR to study the possibility of using neutron noise measurements, i.e., stochastic variations in neutron flux, to estimate the reactivity of subcritical nuclear systems. The aim is to propose this technique for online measurement of the reactivity of the corium at Fukushima Daiichi during future decommissioning operations. The thesis work will focus on evaluating a solution based on Micromegas-type neutron detectors (nBLM detectors) developed by IRFU, which are adapted to the extreme gamma radiation expected in the vicinity of the Fukushima Daiichi corium. The student will participate in experiments at nuclear research facilities in Europe and the United States to test this technical solution and measure neutron noise for a wide range of reactivities. He/she will be responsible for analyzing the data and evaluating the various inversion methods used to estimate reactivity from neutron noise measurements.