Modelling hyperbolic and elliptic elasticity with discontinuous coefficients using an error driven adaptive isogeometric basis

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The UK has pledged to achieve net zero emissions by 2050, aiming to develop green transition technologies such as geothermal energy, geological storage of CO2 from industrial and direct air capture sources, hydrogen fuel cells, batteries, and compressed-air energy, as well as advancing nuclear as a clean energy source through clean storage and disposal. The success of these technologies is highly dependent on understanding the behaviour of Earth materials at a range of scales, in the context of deformation, fluid flow, and temperature changes, which can affect how rocks break and how fluids and heat migrate in the subsurface through these rocks. Understanding how fractures and other smaller-scale heterogeneities affect rock properties also furthers the capabilities of numerical models dedicated to predicting fractures in ceramics, composites, bioengineered materials, human bones, and ion lithium batteries, as additional examples.Processes governing fracturing in complex media, and the interaction of fractures with smaller and larger scale discontinuities and material variations, is often investigated using numerical models. The main drawback of these models is that their performance usually depends on the amount of detail included, such as the geometric details of the ensuing fractures, and distributions of differently shaped embedded inclusions that tend to change the material's behaviour. However, having the ability to effectively and accurately model real full-scale heterogeneous multi-scale problems is necessary to the development of robust, low-carbon and cost-effective strategies that underpin the energy transition. This project proposes to develop a key mathematical strategy to enhance the performance of computational solid mechanics methods, while incorporating additional levels of detail in the description of the material. We propose to develop, implement, and validate an efficient three-dimensional multi-scale numerical method, that combines 3D volumetric isogeometry in bodies containing fractures, with numerical error estimators to more efficiently represent mm- and cm-scale heterogeneities when computing the deformation of a meter- to km-scale body containing multiple fractures. Error estimators enable regions critical to overall solution accuracy to be targeted with higher levels of computational power, dynamically adjusting detail and load during the simulation. As opposed to other methods, the specific method to be developed during this project supports both small and large variations in the material properties, without compromising the quality of the solution, and without inflating the computational cost of the method. Computational efficiency and accuracy enable the method to be applied effectively to large real-world problems, enabling the consideration of larger and more realistic problems without significantly increasing computational effort. Developing the ability to model such problems, and sharing the development through open-source code with the wider scientific community, is of national importance. Quantifying the relationship between scales in the context of solid body fracturing, in complex scenarios, directly supports responsible innovation in the UK, and supports the development of low-carbon and effective energy generation schemes, safe and clean deposition of waste materials, and elongating the life and increasing the efficacy of electrical storage devices.