Fracture propagation in glaciers using discontinuous Galerkin finite element methods

冰山崩解是一个复杂的自然断裂过程，格陵兰岛和南极洲浮冰架和海洋冰川（即边缘冰带）的一半质量损失是冰山崩解造成的。人们主要担心的是，由于与气候动力学有关的冰裂，边缘冰带的快速变化可能会破坏南极和格陵兰冰盖的稳定。据推测，南极冰架的水力压裂和冰崖的破坏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岛和南极洲冰盖的演变，我们必须提高对冰山崩解的断裂力学及其在大尺度数值模型中的表现的理解。基于线性弹性断裂力学（LEFM）的冰山崩解表示已被部署在冰盖模型中。然而，用标准有限元方法模拟裂缝扩展在计算上是低效的。相场（PF）损伤模型的最新进展显示出了模拟断裂扩展的巨大前景。PF模型通过一个类似损伤的标量变量来表示裂缝，该标量变量被建模为单独的空间变化场。完整（未损坏）材料的PF平滑变化为0，域的完全开裂部分为1，该场的值与母材料的刚度耦合。然而，这些模型在模拟混合模式断裂扩展，特别是在多轴应力状态下，还处于初步发展阶段。要建立PF模型的可行性，以捕捉冰川崩解过程的复杂性，还需要取得若干进展。在损伤建模方面，需要新的裂纹驱动力公式来准确描述多轴压应力状态下的流体力学断裂。在数值方法方面，需要新的技术/算法来有效地模拟三维裂纹扩展。该项目的目标是开发一种新的计算方法，使用网格自适应非连续galerkin有限元法（DG-FEM）实现的PF模型来准确有效地模拟冰川冰中混合模式的裂缝扩展。杜伦大学的监督小组Coombs教授和Giani博士是开发和测试新型数值方法（包括DG-FEM）的专家。DG-FEM的主要优点是，它可以有效地模拟PF（和应力场）的演变，通过hp自适应网格细化。该项目还将得到Ravindra Duddu教授（美国范德比尔特大学）的支持，他是计算断裂的专家，包括PF损伤模型和冰盖模型。本博士项目将开发一个三维三维三维三维三维三维数值框架，用于大规模分析冰断裂。现有PF模型的局限性之一是其计算成本，这是由于损坏区域需要足够的元素。由于控制损伤区域宽度的长度尺度减小，从而提供更精确的裂缝表示，因此单元尺寸必须按相同比例减小。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。该项目采用了一种不同的方法，通过误差估计的网格自适应，实现了高精度的裂缝预测，同时保持了计算的可处理性。

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.