Fracture propagation in glaciers using discontinuous Galerkin finite element methods

冰山崩解是一个复杂而自然的破裂过程，是格陵兰和南极洲漂浮冰架和海洋终止冰川(即边缘冰带)损失的一半质量的原因。人们主要担心，与气候动态有关的崩解导致的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。有人假设，南极洲冰架的水力破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化，我们有必要提高对冰山崩解的断裂力学及其在大尺度数值模式中的表示的理解。基于线弹性断裂力学(LEFM)的冰山崩解表示法已被部署在冰盖模型中。然而，用标准有限元的LEFM模拟裂缝扩展在计算上可能是低效的。相场(PF)损伤模型的最新进展表明，它在模拟裂缝扩展方面具有很大的潜力。PF模型通过一个类似损伤的标量变量来表示裂缝，该变量被建模为单独的空间变化场。对于完整的(未损坏的)材质，PF在0到1之间平稳变化，对于域的完全破裂部分，该字段的值与母材质的刚度相关联。然而，这些模型在模拟复合型断裂扩展方面还处于初步发展阶段，特别是在多轴应力状态下。为了建立从冰川中捕捉冰山崩解过程的复杂性的PF模型的可行性，需要进行几项改进。在损伤建模方面，需要新的裂纹驱动力公式来准确地描述多轴压缩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的扩展。本项目的目标是开发一种新的计算方法，利用网格自适应不连续Galerkin有限元(DG-FEM)实现的PF模型来精确和高效地模拟冰川中的混合型断裂扩展。达勒姆管理小组库姆斯教授和吉亚尼博士是开发和测试新的数值方法的专家，包括DG-有限元方法。DG-有限元的主要优点是通过自适应的hp网格加密，可以有效地模拟PF(和应力场)的演化。该项目还将得到Ravindra Duddu教授(美国范德比尔特大学)的支持，他是计算断裂方面的专家，包括PF损伤模型和冰盖模型。该博士项目将开发一个通过DG-有限元技术在3D中进行大规模冰裂分析的数值框架。现有PF模型目前的局限性之一是它们的计算成本，这是由在受损区域需要足够的单元来驱动的。随着控制损伤区宽度的长度尺度减小，从而提供了更准确的断裂表示，单元尺寸必须以相同的比例减小。由于裂隙网络是先验未知的，许多研究人员在整个分析过程中求助于使用精细的计算单元--这是过度浪费。该项目采用了一种不同的方法，通过误差估计驱动网格自适应，允许高精度的裂缝预测，同时保持计算上的易操纵性。

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.