In UK Energy strategy, nuclear fission is growing rapidly in significance. Government's recent Nuclear Industrial Strategy states clearly that the UK should retain the option to deploy a range of nuclear fission technologies in the decades ahead, and that it should underpin the skill base to do so. The primary aim of Next Generation Nuclear is to provide high quality research training in the science and engineering underpinning nuclear fission technology, focused particularly on developing a multi-scale (from molecular to macroscopic), multi-disciplined understanding of key processes and systems. Nuclear fission research underpins strategic UK priorities, including the safe management of the historic nuclear legacy, securing future low carbon energy resources, and supporting UK defence and security policies. It has become clear that skills are very likely to limit the UK's nuclear capacity, with over half of the civil nuclear workforce and 70% of Subject Matter Experts due to retire by 2025. High level R&D skills are therefore on the critical path for all the UK's nuclear ambitions and, because of the 10-15 year lead time needed to address this shortage, urgent action is needed now. Next Generation Nuclear is a collaborative CDT involving the Universities of Lancaster, Leeds, Liverpool, Manchester and Sheffield, which aims to develop the next generation of nuclear research leaders and deliver underpinning (Technology Readiness Level (TRL) 1-3), long term science and engineering to meet the national priorities identified in Government's Nuclear Industrial Vision. Its scope complements the Nuclear IDC (TRL 4-6), with both Centres aiming to work together and exploit potential synergies. In collaboration with key nuclear industry partners, Next Generation Nuclear will build on the very successful Nuclear First programme to deliver a high quality training programme tailored to student needs; high profile, high impact outreach; and adventurous doctoral research which underpins real industry challenges.

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.

Computational science is a multidisciplinary research endeavour spanning applied mathematics, computer science and engineering together with input from application areas across science, technology and medicine. Advanced simulation methods have the potential to revolutionise not only scientific research but also to transform the industrial economy, offering companies a competitive advantage in their products, better productivity, and an environment for creative exploration and innovation. The huge range of topics that computational science encapsulates means that the field is vast and new methods are constantly being published. These methods relate not only to the core simulation techniques but also to problems which rely on simulation. These problems include quantifying uncertainty (i.e. asking for error bars), blending models with data to make better predictions, solving inverse problems (if the output is Y, what is the input X?), and optimising designs (e.g. finding a vehicle shape that is the most aerodynamic). Unfortunately, the process through which advanced new methods find their way into applications and industrial practice is very slow. One of the reasons for this is that applying mathematical algorithms to complex simulation models is very intrusive; mostly they cannot treat the simulation code as a "black box". They often require rewriting of the software, which is very time consuming and expensive. In our research we address this problem by using automating the generation of computer code for simulation. The key idea is that the simulation algorithm is described in some abstract way (which looks as much like the underlying mathematics as possible, after thinking carefully about what the key aspects are), and specialised software tools are used to automatically build the computer code. When some aspect of the implementation needs to change (for example a new type of computer is being used) then these tools can be used to rebuild the code from the abstract description. This flexibility dramatically accelerates the application of advanced algorithms to real-world problems. Consider the example of optimising the shape of a Formula 1 car to minimise its drag. The optimisation process is highly invasive: it must solve auxiliary problems to learn how to improve the design, and it be able to modify the shape used in the simulation at each iteration. Typically this invasiveness would require extensive modifications to the simulation software. But by storing a symbolic representation of the aerodynamic equations, all operations necessary for the optimisation can be generated in our system, without needing to rewrite or modify the aerodynamics code at all. The research goal of our platform is to investigate and promote this methodology, and to produce publicly available, sustainable open-source software that ensures its uptake. The platform will allow us to make advances in our software approach that enables us to continue to secure industrial and government funding in the broad range of application areas we work in, including aerospace and automotive sectors, renewable energy, medicine and surgery, the environment, and manufacturing.

Ductile materials, like metals and alloys, are widely used in engineering structures either by themselves or as reinforcement. They usually can sustain a lot of plastic damage before failing. Engineers understand quite well the ways that metals fail and how tolerant they are to damage, so efficient and less massive structures may be designed with well-defined margins of safety or reserve strength to cope with extreme events. By comparison, elastic brittle materials such as glasses and ceramics can fail without prior warning, so much larger safety margins are needed. Quasi-brittle materials are an important class of structural materials. They are brittle materials with some tolerance to damage and include concrete, polygranular graphite, ceramic-matrix composites, geological structures like rocks and bio-medical materials such as bone and bone replacements. Although their damage tolerance is much less than many metals and alloys, it can be quite significant compared to brittle materials such as ceramics and glasses. But this is not accounted for very well when engineers design with, or assess, quasi-brittle materials, as there is not an adequate understanding of the role on their damage tolerance of factors such as the microstructure of the material or the state of stress. Quasi-brittle materials are usually treated as fully brittle, taking little or no account of their damage tolerance, so assessments incorporate very significant safety margins, leading to designs that may be inefficient and unnecessarily bulky. Even when some assessment of damage tolerance is included, the microstructure can change as the material ages over time, and we need ways to measure the effects of this and to predict what it will do to the safety of the structure. This project aims to develop a method to predict the performance and evaluate the integrity of structures and components made from quasi-brittle materials. This will extend opportunities for their use in engineering applications, enabling more efficient design with greater confidence in safety. Quasi-brittleness is a property that emerges from the material's microstructure. A quasi-brittle material can be made from a connected network of very brittle parts (for instance, a porous ceramic). It exhibits a characteristic "graceful" failure as parts break locally when loaded sufficiently, which gives it damage tolerance. The "gracefulness" of the failure is affected by the random variations of strength and stiffness of the network and the form of the connections. Such networks represent a key part of the microstructure of the material, and to understand quasi-brittle fracture we need to construct models that properly describe the microstructure. There is a need to understand and define the mechanisms that control the fracture at the small and the large scale within these quasi-brittle materials. This will allow us to capture sensitivity to microstructure differences and degradation, and to produce general models that are suitable for the wide range of quasi-brittle materials and applications. Three-dimensional models that are faithful to the microstructure can be created using modern 3D microscopy methods, such as X-ray computed tomography. But these models are far too complex to simply scale up to structures very large relative to the microstructure. There is no computer than can do this, yet. We will develop modelling methods that sufficiently represent the complexity of quasi-brittle microstructures over a wide range of length scales, such as cellular automata finite elements. We will use advanced tomography and strain mapping techniques to observe how damage develops and to test and refine our models. We will then use this and the understanding that we gain to design new material tests and characterisation methods so that our methods may be used in a wide range of materials, from concretes to advanced nuclear composites, bone replacement biomaterials and geological materials.

Can green infrastructure form part of a water resources plan? How can we evaluate where the most impact from new wetlands and forests can be made? How robust are these 'natural water retention methods' (NWRM) to climate change? How much will they cost in comparison to conventional methods? These questions are currently being answered in a local context, with case studies and pilot projects well established and their outputs understood. At the national scale there has been far less work done. No model has taken the outputs of individual NWRM and formed a generalised approach which can be used to identify the costs and benefits of a range of options for new NWRM across Great Britain and mapped solutions in financial and carbon terms. This work builds on existing expertise on water resource systems modelling planning within the Oxford research group and uses the existing knowledge of project partners. The main steps in our project are: 1. Explore the knowledge of our project partners on NWRM. We will host discussions on the sites where NWRM has been used and record the accounts of how these schemes have been developed through interviewing expert project partners. We will ask about: -The cost of building, the time taken to develop the sites, and the operational costs. -The impacts that they noticed or monitored on local rivers We will compare the answers we are given with information from reports from elsewhere in the world. 2. Using this information we will make a model of the effects of NWRM on water. This will involve looking at what information on each NWRM can tell us about what the impact on water resources is going to be. 3. Having made a model of individual NWRM, we will apply this to our existing model of water resources for mainland Great Britain. This determines future available supplies and demands for each water resource zone, the management units for water resources. 4. We will investigate how climate change and population will affect the answers from Objective 3, and look at other uncertain factors such as how energy and industry demand for water might change and what happens if people use less water in future. 5. We will work with our partners throughout the project to develop practical and useful reports on our work.