Phase-field computation of brittle fracture: robustness, efficiency, and characterisation of solution non-uniqueness

弹性固体中复杂断裂过程的建模和准确预测，包括裂纹的萌生、扩展、分支和合并，是一个非常具有挑战性的工程课题。我们专注于脆性材料，其中断裂发生在显著的塑性（永久）变形之前。相场方法通过将连续场（裂纹相场）与材料的状态（从完整到完全破裂）联系起来，使这种材料失效过程的逼真模拟成为可能。材料软化模型可能导致数学上的非唯一解。它们出现的概率在理论上是由能量关系决定的，而在实际材料中是由随机的非均质性决定的。在确定性计算中出现以下困难：尖锐裂纹是由相场的急剧变化来模拟的，需要在数值离散中使用非常精细的网格，这可能在计算上非常昂贵。为了通过自适应网格处理这一问题，需要开发高效可靠的误差估计器。由于损伤演化和断裂是不可逆的过程，因此该解随时间演化，计算上采用离散时间步长。时间步长的选择对求解过程和可能出现的非唯一性影响很大。在算法效率和鲁棒性方面，将开发误差指标来控制时间步长选择。在每一个时间步，一个可能的非凸能量泛函必须最小化，显示多个局部最小值，从而给数值算法带来问题。现有的分区和整体方法将在鲁棒性和效率方面得到进一步的改进。将特别注意捕获所有局部极小值。为了捕捉解的非唯一性，它们出现的概率（由它们的能量关系决定）将用随机场来描述。这种非唯一性计算出现在不同的网格细化、时间步进算法和每一步的最小化算法中。所有这些都将通过适当的随机变量在局部参数化，从而通过随机场捕获所有可能解决方案的演变。裂纹扩展过程的描述将空间上每一点和时间上每一刻的位移场和相场作为一个随机变量来建模，这就是一个时空随机场。这描述了材料状态的不同概率，即裂纹发展的概率和相关的位移。这种方法产生了大量描述解及其演化的参数。处理高维参数问题的数值稀疏技术将进一步发展，以适应相场方法。

Modelling and accurate prediction of complicated fracture processes in elastic solids including crack initiation, propagation, branching, and coalescence, is a very challenging topic of engineering interest. We focus on brittle materials where fracture occurs prior to significant plastic (permanent) deformation.The phase-field approach enables realistic simulations of such material failure processes by associating a continuous field (the crack phase-field) to the state of the material, ranging from intact to fully broken. Models of material softening may result in mathematically non-unique solutions. Their probability of appearance is in theory determined by energetic relations, and in the real material by random inhomogeneities.In deterministic computations the following difficulties appear: sharp cracks are modelled by a steep variation of the phase-field, necessitating very fine meshes in numerical discretisations which may be computationally very expensive. To deal with this via adaptive meshing, the development of efficient and reliable error estimators is required. As damage evolution and fracture are irreversible processes, the solution evolves in time, computationally in discrete time-steps. The time-step choice strongly influences the solution process and the possible appearance of non-uniqueness. Error indicators will be developed to control the time-step selection as regards algorithmic efficiency and robustness. In each time-step, a possibly non-convex energy functional has to be minimised displaying multiple local minima and hence causing problems for numerical algorithms. Existing partitioned and monolithic approaches will be further advanced as regards the robustness and efficiency. Particular attention will be paid to the capture of all local minima.To capture the non-unique nature of solutions, the probability of their appearance (determined by their energetic relations) will be described by random fields. This non-uniqueness computationally appears with different mesh-refinements, time-stepping algorithms, and within the minimisation algorithm in each step. All of these will be parameterised locally by appropriate random variables, thus capturing the evolution of all possible solutions via random fields.The description of the crack propagation process will model both the displacement and phase-field in each point in space and each instant in time as a random variable, which then is a spatio-temporal random field. This describes the different probabilities of the state of the material, i.e. probability of crack development and the associated displacement.Such an approach generates a large number of parameters which describe the solutions and their evolution. Numerically sparse techniques for dealing with high-dimensional parametric problems will be further developed to suit the phase-field approach.