Fracture propagation in glaciers using discontinuous Galerkin finite element methods

冰山崩解是一个复杂而自然的断裂过程，造成了浮冰架和海洋终止冰川一半的质量损失（即，格陵兰岛和南极洲的边缘冰区。人们主要担心的是，由于与气候动力学相关的崩解，边缘冰区的快速变化可能会破坏南极和格陵兰冰盖的稳定。据推测，南极洲冰架的水力压裂和冰崖的破裂可能会导致未来几个世纪海平面的快速上升。因此，为了更好地预测格陵兰和南极冰盖的演变，我们有必要提高我们对冰山崩解断裂力学及其在大尺度数值模式中的代表性的理解，基于线弹性断裂力学（LEFM）的冰山崩解代表性已经部署在冰盖模型中。然而，使用具有标准有限元的LEFM来模拟裂缝扩展可能在计算上是低效的。相场（PF）损伤模型的最新进展为模拟裂纹扩展显示了巨大的希望。PF模型通过一个类似损伤的标量变量来表示裂缝，该变量被建模阿萨个单独的空间变化场。PF在0（完整（未损坏）材料）到1（完全开裂的域部分）之间平滑变化，并且该场的值与母体材料的刚度相耦合。然而，这些模型是在发展的初级阶段，用于模拟混合模式的裂缝扩展，特别是在多轴应力状态下。需要几个进步来建立PF模型的可行性，捕捉冰山崩解过程的复杂性从冰川。在损伤建模方面，需要新的裂纹驱动力公式来精确描述多轴压应力状态下的流体力学断裂。为了有效地模拟三维裂纹扩展，需要新的技术/算法。本项目的目标是开发一种新的计算方法，用于精确和有效地模拟混合裂纹扩展。模式断裂在冰川冰中的传播，使用PF模型实现网格自适应间断Galerkin有限元法（DG-FEM）。达勒姆的监督小组，教授库姆斯和博士贾尼，是专家在开发和测试新的数值方法，包括DG-FEM。DG-FEM的主要优点是它可以通过hp自适应网格细化有效地模拟PF（和应力场）的演变。该项目还将得到美国范德比尔特大学的Ravindra Duddu教授的支持，他是计算断裂的专家，包括PF损伤模型和冰盖建模。该博士项目将通过DG-FEM技术在3D中开发大规模冰断裂分析的数值框架。现有PF模型的一个局限性是其计算成本，这是由于在受损区域需要足够的单元。由于控制损伤区宽度的长度尺度减小，这提供了更准确的裂缝表示，单元尺寸必须以相同的比例减小。由于裂缝网络是先验未知的，许多研究人员在整个分析过程中采用精细的计算网格-这是过度浪费的。该项目通过误差估计驱动的网格自适应性遵循不同的方法，允许高度准确的裂缝预测，同时保持计算上的易处理性。

Iceberg calving is a complex, yet natural fracture process and is responsible for half of the mass lost from thefloating ice shelves and marine-terminating glaciers (i.e., marginal ice zones) in Greenland and Antarctica. There ismajor concern that rapid changes in the marginal ice zones due to calving linked to climate dynamics can destabilizethe Antarctic and Greenland ice sheets. It has been hypothesized that hydrofracturing of ice shelves followedby ice cliff failure in Antarctica could contribute to rapid sea level rise over the coming centuries. Therefore, itis essential that we improve our understanding of the fracture mechanics of iceberg calving and its representationin large-scale numerical models, in order to better predict the evolution of Greenland and Antarctica ice sheets.Linear elastic fracture mechanics (LEFM) based iceberg calving representations have been deployed within ice sheetmodels. However, simulating fracture propagation using LEFM with standard finite elements can be computa-tionally inefficient. Recent advances in Phase Field (PF) damage models have shown great promise for simulatingfracture propagation. PF models represent fractures through a damage-like scalar variable which is modelled asa separate spatially varying field. The PF varies smoothly between 0 for intact (undamaged) material to 1 for fullycracked parts of the domain and the value of this field is coupled to the stiffness of the parent material. However,these models are at a preliminary stage of development for simulating mixed-mode fracture propagation, especiallyunder multiaxial stress states. Several advancements are needed to establish the viability of the PF model for captur-ing the complexity of iceberg calving process from glaciers. On the damage modelling front, new crack driving forceformulations are needed to accurately describe hydromechanical fracture under multiaxial compressive stress states.On the numerical method front, new techniques/algorithms are needed to efficiently simulate 3D crack propagation.The goal of this project is to develop a new computational approach for the accurate and efficient simulation ofmixed-mode fracture propagation in glacier ice using a PF model implemented with a mesh-adaptive discontinousGalerkin finite element method (DG-FEM). The Durham supervision team, Prof. Coombs and Dr Giani, areexperts in developing and testing novel numerical methods, including the DG-FEM . The main advantageof the DG-FEM is that it can efficiently simulate the evolution of the PF (and the stress field), via hp adaptivemesh refinement. The project will also be supported by Prof. Ravindra Duddu (Vanderbilt University, USA) whois an expert in computational fracture, including the PF damage models, and ice sheet modelling .This PhD project will develop a numerical framework for large-scale analysis ice fracture via DG-FEM techniquesin 3D. One of the current limitations of existing PF models is their computational cost, driven by the need forsufficient elements in the damaged region. As the length scale controlling the width of the damage zones is reduced,which provides a more accurate fracture representation, the element size must be reduced in the same proportionAs the fracture network is unknown a priori, many researchers resort to using a fine computational meshthroughout the analysis - this is excessively wasteful. This project follow a different approach via error-estimatedriven mesh adaptivity, allowing for highly accurate fracture predictions whilst remaining computationally tractable.