Stochastic analysis of the neutron transport equation and applications to nuclear safety

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences


The technical basis of this proposal pertains to the Neutron Transport Equation (NTE), which is used to describe neutron density in a physical environment where nuclear fission is taking place, such as a reactor core. This equation is of prime importance in the nuclear industry as it is used to construct models of reactor cores, nuclear medical equipment (e.g. for proton therapy) and other industrial scenarios where irradiation occurs. Primarily these models are used to assess safety and inform regulatory procedure when handling radioactive materials.

Although the NTE can be derived through physical considerations of mass transport, it can also be derived using entirely probabilistic means. To be more precise, the NTE can be derived from the stochastic analysis of a spatial branching process. The latter models the evolution of neutron particles as they behave in reality, incorporating the features of random scattering and random fission, with increasing numbers of particles as time evolves. The derivation using spatial branching processes has been known since the 1960/70s, however, since then, very little innovation in the literature has emerged through probabilistic analysis. This mirrors a general lull in fundamental mathematical research contributing to modelling of nuclear fission after the 1980s.

In recent years, however, the nuclear power and nuclear regulatory industries have a greater need for a deep understanding the spectral properties of the NTE. Such analytical quantities help e.g. engineers model the criticality and density of nuclear fission activity within a reactor core. In turn this informs optimal reactor design from several different view points (safety, energy production, efficiency etc.) as well as address regulatory constraints. With the decommissioning of old and the construction of new, more efficient and environmentally friendly nuclear power stations the demand for mathematical modelling using the NTE was never greater.

The inhomogeneous nature of the NTE as it is used in practice has seen industry turn to Monte-Carlo techniques based on the underlying probabilistic treatment from 40-50 years ago. Many of the associated algorithms can only be run on supercomputers as they boil down to costly Monte-Carlo cycles of the entire fission processes, in essence replicating a virtual physical reality in a computer. This has the huge drawback that computational parallelization is not possible.

In the decades that new probabilistic developments have been absent from the treatment of the NTE, there has been a significant evolution in the mathematical theory of spatial branching processes and related stochastic processes. The research in this proposal aims to re-align the understanding of the NTE with the modern theory of spatial branching processes. This is principally motivated by the implication that a whole suite of completely new Monte-Carlo techniques can be developed, as desired by industry, which are, fundamentally, of a lower order of complexity than existing algorithms. The overall aim of this project is to develop a `proof of concept' for this completely new approach, providing the theoretical basis and a stochastic numerical analysis that quantifies relative efficiency. In particular, the most important feature of the new algorithms that will emerge is the ability to parallelize computations.

The project will be carried out in close scientific collaboration with industrial partner Amec-Foster-Wheeler, a major UK-based energy consultancies and one of the global leaders in servicing the nuclear energy and nuclear medical industries with simulation software for safety and regulatory purposes.

All research output will be made open source on a webpage dedicated to the project.

Planned Impact

The generation and supply of electricity is at the center of modern life. Almost all imaginable modern endeavours are only possible due to wide availability of electrical power, something that we take for granted. Yet, the UK's energy future is uncertain. As well as an increase in demand, there is a decrease in supply on the horizon. Approximately 20% of the UK's energy is supplied by nuclear power plants, most of which are scheduled to close by about 2024. This means that there is an urgent need within the UK to both extend the life of existing nuclear power plants and build new generating capacity. For example, there has been a recent agreement to construct a reactor at Hinkley Point in Somerset as well as further plans to construct new reactors, Sizewell C in Suffolk and Bradwell in Essex.

Safety and environmental security are key and the research in this proposal feed directly into these aspects of nuclear power by improving the quality of virtual simulation of reactor cores. The techniques proposed in this grant can become significant assets for the UK as a whole with regard to sovereign nuclear capability.

The research agenda in this proposal has emerged from some of the remarkable interactions during a week-long 'Integrative Think Tank' (ITT), hosted by the EPSRC Centre for Doctoral Training in 'Statistical Applied Mathematics at Bath' (SAMBa) with participation from the nuclear industry service provider Amec-Foster-Wheeler (AFW). AFW are a UK-based consultancy to the international energy industry. In particular, they are one of the world's leading service providers of mathematical modelling software for reactor physics, criticality, shielding, medical irradiation, food irradiation, oil well logging and event tree analysis. As indicated by a supporting letter, AFW has also held a long-standing research interaction with the numerical analysis group at the University of Bath.

Thanks to the emergence of a completely new set of probabilistic ideas that form the basis of this proposal, AFW is committed more than ever to continue a strong and lasting partnership with the Department of Mathematical Sciences at Bath (in particular, with the probabilistic research group attached to this proposal) to ensure the maximal impact of our research in the nuclear industry. As an indication of the commitment to this process on both sides, AFW have released GBP70K of funds which will be invested into PhD research aligned with ideas emerging from the ITT, specifically including those described in this research proposal.

We are confident that this level of interaction will deliver strong impact. The Monte Carlo method as currently applied in industry is limited by the prohibitive computational run times of the calculations. The method can be parallelised but its inherent structure, which is based on the virtual simulation of the underlying fission process, makes this hard to do in a way that achieves near optimum parallel scaling. In contrast, the so-called many-to-one algorithms we propose here should allow for significantly more efficient parallelisation by virtue of the fact that there is no need to deal with branching thanks to a mathematical short cut. This offers a step change in the way industry approaches it numerical modelling, as is verified by the willingness of one of the main players in this field to be fully engaged in the research agenda.


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Description We have begun to understand the stochastic analytic interpretation of the neutron transport equation. In particular we have build proofs of how to representation their solutions in terms of paths of single neutron random walks. We have also found a new representation of the NTE for a much more general scenario that incorporates all types of neutron emissions - slow neutrons and fast neutrons - as well as radiation emissions. We have understood how to simulate solutions to the neutron transport equation as well as measure the accuracy.
Exploitation Route The applications of our work suggest new algorithms could be programmed when simulating what happens in nuclear reactor cores.
I am in the beginings of writing a book based on what we produced during this grant and this could influence the nuclear industry
Sectors Construction,Energy,Environment,Other

Description Mathematical Theory of Radiation Transport: Nuclear Technology Frontiers (MaThRad)
Amount £6,001,426 (GBP)
Funding ID EP/W026899/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 08/2022 
End 08/2022
Description Branching structures: Neutron transport days 
Organisation Wood Group
Country United Kingdom 
Sector Private 
PI Contribution The Bath-Paris-Beijing is a series of meetings which have been held over the last decade, largely focused on branching processes, at the three locations in the title of the series. In the next edition, held in Bath on the week of 20th April 2020, we will devote 2 days to neutron transport theory, partly funded by this grant. Note: this was delayed to 2021 because of COVID.
Collaborator Contribution Speakers: Emma Horton (INRIA Bordeaux), Alex Cox (Bath), Andrea Zoia (CEA, Paris), Nick Whitely (Bristol), Pierre Nyquist (Uppsala), Geoff Dobson (Jacobs), Pierre Del Moral (INRIA Bordeaux), Mathias Rousset (INRIA Brittany), Tony Lelievre (Paris), Jere Koskela (Warwick),Eric Dumonteil (CEA Paris), Denis Villemonais (Nancy), Andi Wang (Bristol), Alex Watson (UCL), Minmin Wang (Sussex)
Impact We hope for new collaborations to take place in the neutron transport arena.
Start Year 2021