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

英国承诺到2050年实现净零排放，目标是发展绿色过渡技术，如地热能、工业和直接空气捕集来源的二氧化碳地质储存、氢燃料电池、电池和压缩空气能源，以及通过清洁储存和处置推动核能作为清洁能源。这些技术的成功在很大程度上取决于在变形、流体流动和温度变化的背景下，对地球材料在各种尺度上的行为的了解，这些变化可以影响岩石的破裂方式，以及流体和热量在地下如何通过这些岩石迁移。了解裂缝和其他较小尺度的非均质性如何影响岩石性质也进一步加强了专门用于预测陶瓷、复合材料、生物工程材料、人体骨骼和离子锂电池中的裂缝的数值模型的能力。控制复杂介质中的压裂的过程，以及具有更小和更大尺度的不连续面和材料变化的裂缝的相互作用，通常使用数值模型来研究。这些模型的主要缺点是，它们的性能通常取决于所包含的细节的数量，例如随后裂缝的几何细节，以及倾向于改变材料行为的不同形状的嵌入物的分布。然而，有能力有效和准确地模拟真正的全面、异质、多尺度的问题，对于制定支撑能源转型的强大、低碳和成本效益的战略是必要的。这个项目建议开发一个关键的数学策略来提高计算固体力学方法的性能，同时在材料的描述中加入更多的详细程度。我们建议开发、实现和验证一种有效的三维多尺度数值方法，该方法结合了包含裂缝的物体中的3D体积等距，以及数值误差估计器，以便在计算包含多个裂缝的米到公里尺度的物体的变形时更有效地表示毫米和厘米尺度的非均质性。误差估计器使对整体解决方案精度至关重要的区域能够以更高水平的计算能力为目标，在模拟期间动态调整细节和负载。与其他方法不同的是，在本项目期间开发的特定方法支持材料性质的小变化和大变化，而不会影响解决方案的质量，也不会增加方法的计算成本。计算效率和精度使得该方法能够有效地应用于大型现实世界问题，使得能够在不显著增加计算工作量的情况下考虑更大和更现实的问题。发展模拟这类问题的能力，并通过开放源代码与更广泛的科学界分享这一发展，具有国家重要性。在复杂情况下，量化固体破碎方面的规模之间的关系，直接支持英国负责任的创新，并支持开发低碳和有效的能源发电计划，安全和清洁地沉积废物，延长电子存储设备的寿命和提高其效率。

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.